Download Effective Operators for Dark Matter Detection

Effective Operators for
Dark Matter Detection
Dissertation submitted for the degree of
Philosophiae Doctor
November 2012
Particle Physics & Cosmology
University of Southern Denmark
Institut for Fysik, Kemi og Farmaci
Centre for Cosmology and
Particle Physics Phenomenology CP3 -Origins
and the Danish Institute for Advanced Study DIAS
Candidate
Eugenio Del Nobile
Academic Advisor
Prof. Francesco Sannino
Esperti degli spazi
dalla terra alle stelle
ci perdiamo nello spazio
dalla terra alla testa.
Wislawa Szymborska, Agli amici
traduzione di Pietro Marchesani
Effective Operators for
Dark Matter Detection
Abstract
In this Thesis we investigate possible generalizations of the usual assumptions
behind Dark Matter (DM) modeling in relation with detection experiments. In
fact, while model independent analyses should in principle allow to investigate the
DM problem in its full generality, strong assumptions are usually made in order
to reduce it to a form that can be tackled with our limited tools and knowledge.
Investigation of only a handful of possible interactions, or assumptions regarding
the DM interaction with ordinary particles, might reveal themselves on the other
hand as oversimplifications of the problem, preventing us to understand the true
nature of Dark Matter. For instance, the present situation of DM direct searches is
unclear, with potential DM signals detected by some experiments being excluded
by others. Given the difficulties in accommodating the results of the various direct
detection experiments within the standard scenario, a generalization of the usual
assumptions is therefore in order.
After presenting the complete list (up to dimension six in mass) of interaction
operators between a scalar DM and the Standard Model particles, we show that the
phenomenon of quantum interference between some of these operators drastically
changes the DM phenomenology in direct detection searches, respect to the one
usually assumed. In fact, relaxing the customary assumption of equal DM-proton
and DM-neutron interactions, it can be shown that several experiments that are
found to disagree one with the other in the standard interpretation can find better
agreement allowing for a certain degree of isospin violation. We propose quantum
interference between exchange processes of two different interaction mediators as
a concrete mechanism for obtaining isospin violation, and investigate which pairs
of mediators could lead to viable interference.
In the last part of this Thesis we consider a DM particle featuring a magnetic
dipole moment, thus interacting with the photon. This can be seen again as an
application of the effective operators framework, as the magnetic moment operator arises at loop order or in composite DM models. The interesting feature of
this interaction is that it is of long-range type, in contrast with the contact type
that is usually assumed. The phenomenology of these two kinds of interaction is
very different and in fact we show that magnetic moment DM can accommodate
all present direct detection experiments, assuming a conservative estimate of the
XENON100 low energy threshold.
Effective Operators for
Dark Matter Detection
Resumé
I denne afhandling undersøger vi mulige generaliseringer af de sædvanlige antagelser bag mørkt stof modellering i forbindelse med påvisningseksperimenter. Mens
modeluafhængige analyser i princippet gør det muligt at undersøge mørkt stof i
al almindelighed, laves der som regel kraftige antagelser for at reducere problemet
til en form, der kan løses med vores begrænsede værktøjer og viden. Analyser
af kun en håndfuld af mulige vekselvirkninger eller formodninger om mørkt stofs
vekselvirkning med almindelige partikler, kan vise sig at være en overforenkling
af problemet, som forhindrer os i at forstå mørkt stofs sande natur. For eksempel er der uklarhed omkring de nuværende resultater fra eksperimenter designet til
direkte at påvise mørkt stof, da potentielle mørkt stof signaler målt i nogle eksperimenter samtidig udelukkes gennem andre forsøg. I betragtning af vanskelighederne
med at imødekomme resultaterne af de forskellige påvisningseksperimenter, er en
generalisering af de sædvanlige antagelser derfor nødvendig.
Efter at have fremlagt den komplette liste (til sjette orden) af vekselvirkningsoperatorer mellem en skalar mørkt stof partikel og standardmodelpartikler, viser
vi at kvanteinterferens mellem nogle af disse operatorer drastisk ændrer mørkt stof
fænomenologien i påvisningseksperimenter i forhold til hvad der normalt antages.
Faktisk viser det sig at man ved at fjerne den sædvanlige antagelse om lige mørkt
stof-proton og mørkt stof-neutron vekselvirkninger, kan vise at adskillige eksperimenter, man tidligere har ment at være modstridende, i virkeligheden er forenelige,
hvis man antager en vis grad af brud på isospin-symmetri. Vi foreslår kvanteinterferens for udvekslingsprocesser mellem to forskellige vekselvirkningsfrembringere
som en konkret mekanisme til at opnå isospin asymmetri, og undersøger hvilke par
af frembringere der fører til den rette vekselvirkning.
I den sidste del af denne afhandling betragter vi en mørkt stof partikel med
et magnetisk dipolmoment, som altså vekselvirker med fotoner. Dette kan igen
ses som en anvendelse af listen over de effektive operatorer, idet den magnetiske
dipolmomentoperator enten opstår i højere ordens perturbationsteori eller i modeller hvor mørkt stof er sammensatte partikler. Det interessante ved denne vekselvirkning er at den er langtrækkende i modsætning til kontaktvekselvirkninger,
som der normalt antages for mørkt stof. Fænomenologien af disse to former for
vekselvirkninger er meget forskellige, og vi viser at mørkt stof med magnetisk
moment er forenelig med alle nuværende resultater fra påvisningseksperimenter,
under forudsætning af et konservativt estimat for lavenergitærsklen af XENON100
eksperimentet. Danske sprøg såks.
Contents
Preface
ix
1 Introduction
1
2 Evidence for Dark Matter
2.1 Evidence on galactic scales . . . .
2.2 Evidence from clusters of galaxies
2.3 Evidence on large scales . . . . .
2.4 Evidence from Cosmology . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 Scalar Dark Matter Effective Field Theory
3.1 Construction of the effective operators . . . . . . . . . . . .
3.2 Singlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Interaction with SM gauge bosons . . . . . . . . . . .
3.2.2 Interaction with SM fermions . . . . . . . . . . . . .
3.2.3 φ as a pseudo-Goldstone boson . . . . . . . . . . . .
3.2.4 DM-Higgs interaction . . . . . . . . . . . . . . . . . .
3.2.5 DM self-interaction . . . . . . . . . . . . . . . . . . .
3.2.6 Interaction with more than one type of SM fields . .
3.3 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Interaction of the doublet with SM gauge bosons . .
3.3.2 Interaction of the doublet with SM fermions . . . . .
3.3.3 Doublet-Higgs interaction . . . . . . . . . . . . . . .
3.3.4 Doublet self-interaction . . . . . . . . . . . . . . . . .
3.3.5 Interaction of the doublet with more than one type of
SM fields . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Triplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Interaction of the triplet with SM gauge bosons . . .
3.4.2 Interaction of the triplet with SM fermions . . . . . .
3.4.3 Triplet-Higgs interaction . . . . . . . . . . . . . . . .
3.4.4 Triplet self-interaction . . . . . . . . . . . . . . . . .
iv
8
. 9
. 11
. 17
. 19
.
.
.
.
.
.
.
.
.
.
.
.
.
26
28
29
29
30
31
31
32
32
33
33
34
35
36
.
.
.
.
.
.
36
38
38
39
39
39
Contents
3.4.5
v
Interaction of the triplet with more than one type of
SM fields . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Interference Patterns for Isospin Violating Dark Matter
4.1 Isospin violating Dark Matter . . . . . . . . . . . . . . . . .
4.2 An application of the effective field theory: direct detection of
a DM scalar singlet . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Low energy dark sector . . . . . . . . . . . . . . . . .
4.2.2 High energy dark sector . . . . . . . . . . . . . . . .
4.3 Examples of interfering Dark Matter . . . . . . . . . . . . .
4.3.1 Interference between photon and Higgs . . . . . . . .
4.3.2 Interference between Z and Z 0 . . . . . . . . . . . . .
4.3.3 Interference between Z 0 and Higgs . . . . . . . . . . .
4.3.4 Interference within the two Higgs doublet model . . .
5 Magnetic Moment DM in Direct Detection Searches
5.1 The event rate . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Kinematics . . . . . . . . . . . . . . . . . . . .
5.1.2 Model and differential cross section . . . . . . .
5.1.3 Nuclear recoil rate . . . . . . . . . . . . . . . .
5.2 Theoretical predictions . . . . . . . . . . . . . . . . . .
5.2.1 Light Dark Matter . . . . . . . . . . . . . . . .
5.2.2 Heavy Dark Matter . . . . . . . . . . . . . . . .
5.3 Data sets and analysis technique . . . . . . . . . . . . .
5.4 Fit to the direct detection experiments . . . . . . . . .
5.5 Relic Abundance . . . . . . . . . . . . . . . . . . . . .
5.6 Constraints . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Epoch of reionization and CMB . . . . . . . . .
5.6.2 Present epoch γ-rays . . . . . . . . . . . . . . .
5.6.3 Collider and other astrophysical constraints . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
42
. 44
.
.
.
.
.
.
.
.
46
47
48
50
50
51
53
54
.
.
.
.
.
.
.
.
.
.
.
.
.
.
56
57
57
57
59
62
66
67
69
73
75
77
78
79
80
6 Conclusions
82
A DM Effective Interaction Terms with SM Gauge Bosons
A.1 Singlet’s interaction terms with SM gauge bosons . . . . . .
A.1.1 Interaction with gluons . . . . . . . . . . . . . . . . .
A.1.2 Interaction with photons only . . . . . . . . . . . . .
A.1.3 Interaction with Electroweak gauge bosons W , Z and
A.1.4 Terms with four φ’s . . . . . . . . . . . . . . . . . . .
A.2 Doublet’s interaction terms with SM gauge bosons . . . . . .
A.2.1 Interaction with gluons only . . . . . . . . . . . . . .
.
.
.
A
.
.
.
85
86
86
86
86
94
95
95
Effective Operators for Dark Matter Detection
Contents
vi
A.2.2 Interaction with photons only . . . . . . .
A.2.3 Interaction with Electroweak gauge bosons
A.2.4 Terms with four D’s . . . . . . . . . . . .
A.3 Triplet’s interaction terms with SM gauge bosons
A.3.1 Interaction with Electroweak gauge bosons
A.3.2 Terms with four T ’s . . . . . . . . . . . .
. .
W,
. .
. .
W,
. .
. . . .
Z and
. . . .
. . . .
Z and
. . . .
. 95
A 96
. 106
. 108
A 108
. 111
B Nucleus Interaction with Higgs and Gauge Bosons
114
B.1 Nucleons coupling to gauge bosons . . . . . . . . . . . . . . . 114
B.2 Nucleons coupling to the Higgs boson . . . . . . . . . . . . . . 115
B.3 Interaction with the nucleus . . . . . . . . . . . . . . . . . . . 116
C Corrections to the Articles Related to This Thesis
118
Effective Operators for Dark Matter Detection
Acknowledgements
First, I am really grateful to my supervisor Francesco Sannino for the possibility he gave me to undertake the PhD studies. Without his (wise) decision,
nothing of this would have happened in precisely the same way. Besides, I
also owe him my interest in classical music.
I thank mamma, papà and Umberto for their l♥ve.
Many thanks to everybody in CP3 for the nice time spent together, the
useful discussions, and much more: Marco and Paolo, Tuomas, Claudio and
Francesca, Matin, Toto, Ulrik, Jakob and Esben, Jussi, Chris, and Stefano. I
thank the Danes for having taught me Danish English; and in general I wish
to thank everybody for the nice atmosphere. In particular, I thank Lone for
her smile.
Some friends helped me in surviving during these years in Denmark. In
particular I wish to express my gratitude to my Polish friends and almostcountrymen: Magda Makowska and her family (Artur and the sweet Kaja,
not to forget Makowska’s mother and the cat Pyla), Karolina and Andrzej.
I thank all the people in Blomsterkassen for the nice time spent together (put
aside the huge amount of time not spent together), in particular Henrik. He
is the only Dane with which I managed to establish something that I would
dare to call a friendship. He is different from the other Danes, that’s probably
the reason for it. Then there are also the foreigners, Luis, Guillem, and Dess,
that like me had hard time living in a house with seven other people and
never seeing anybody around.
Sandra was a very important part of my stay here. Grazie Sandra.
There are of course other people that helped alleviating the Danish suffering
and took good care of me, and deserve therefore many thanks. Some of these
are Alessia (one or two), Betta, Daniele aka il Milano, Diana, Eleonora, Elia,
Gerda, Ietta, Isabel, Jan, Moira and her family, Oriella, Peiwen, Stefania,
vii
Acknowledgements
viii
the whole Tiberi family (including Stefano), the International Club and the
International Staff Office at SDU, and Søren and Mette of the Odense Boulder
Klub.
Please forgive me if your name is not in the list and you think it should be.
You can buy a slot for 500 bucks. Well yeah OK, not to unplease anybody,
I also acknowledge everybody who is not in this list. Fine?!?
Last, and therefore first - when reading upside down, I thank Lord Marcus.
Effective Operators for Dark Matter Detection
Preface
The material presented in this Thesis was originally elaborated in the articles
[I, II, III, IV], which have all been published. Lucky me!!!
Ref. [I] contains the complete list of interaction operators of a scalar
DM with the SM particles, up to dimension six in mass, which is presented
in Chapter 3 and in Appendix A. It also contains a phenomenological application in the context of isospin violating DM, which has been here inserted in Chapter 4. This contains as well the work originally presented
in Ref. [II, III], where quantum interference between exchange processes of
different DM-nuclei interaction mediators was proposed and investigated as
viable mechanism for isospin violation.
After those articles appeared, though, newer XENON100 results became available [1] which now exclude most of the DAMA/CoGeNT overlapping region;
moreover, the isospin violating DM paradigm has been found to be in tension
also with other DM searches, see e.g. Ref. [2, 3].
Finally, Chapter 5 contains the work presented in Ref. [IV] about the
magnetic dipole moment DM. Also in this article, the XENON100 2011 data
were used, while a new analysis was published by the experimental collaboration with more data later in 2012 [1], as commented above.
While writing this Thesis, few typos have been found in papers [I,III] and
have been corrected here. To favor the comparison with the original articles,
we have kept track of the relevant changes in Appendix C. We stress that
none of these modifications change in any case the conclusions drawn in the
articles, being due in many cases only to typos or misprints which do not
affect the subsequent results.
ix
Chapter 1
Introduction
On July 4th , 2012, the discovery of a particle with properties compatible
with the Standard Model Higgs boson was announced by the CMS [4] and
ATLAS [5] collaborations at CERN. The mass of this new particle has been
measured with high accuracy as 125 GeV. The CDF and D0 collaborations
at Tevatron [6] also found an excess in the data around this value.
This historical event projected us in a new era of measurements with the
Large Hadron Collider (LHC) experiment. The next step is to establish the
exact nature of this particle, studying to which extent it might be connected
with physics beyond the Standard Model (SM). Indeed, it might turn out that
this is not the actual Higgs boson of the Standard Model, providing us with
a direct link to new physics. Many are the extensions of the Standard Model
featuring a Higgs particle with different properties respect to the SM one; the
most famous are Supersymmetry, Extra Dimensions and Technicolor (see [7],
[8] and [9] for updated reviews). A careful determination of the properties of
this newly discovered particle could therefore lead us to a vast new landscape
to explore. One of the main points in this program is to finally establish the
mechanism which underlies the breaking of the Electroweak symmetry, giving
mass to the Weak gauge bosons. Not less important, understanding the origin
of the masses of the SM fermions could provide a window on new physics,
giving us the opportunity to study their complicated flavor structure.
Before the discovery of this new particle, we already knew that something
needed to be there around the TeV scale in order to unitarize the W W scattering; in the Standard Model this task is accomplished by the Higgs boson,
even though this is not the only possibility. In other words, the Standard
Model is not renormalizable without the Higgs boson, so its non existence
would automatically imply the presence of some new physics. Now that we
found a new particle, were this the SM Higgs boson or not, strictly speaking
1
Chapter 1. Introduction
2
there are no theoretical reasons to expect that we will find something else
at the LHC, as long as this new particle is proven to accommodate the high
energy scattering of the longitudinal Weak gauge bosons.
Even without theoretical grounds, though, expectations of new physics
around the Electroweak scale vEW ' 246 GeV come from a variety of other
reasons. Following the naturalness principle, for instance, one would expect
some new physics around the TeV scale in order to solve the Higgs hierarchy
problem, i.e. to avoid fine-tuning of its couplings and provide a ‘natural’ explanation for its light mass. It is important to notice that this is an aesthetic
criterion rather than a physical one, and that the Standard Model is perfectly consistent as it is now, even though the values of its couplings might
not seem ‘natural’.
The particle observed at LHC has properties remarkably similar to those of
the SM Higgs boson. Still, the initial measurements of its branching fraction
to two photons show a small departure from the Standard Model value (see
e.g. [10, 11]). If this feature will be confirmed by the next data, this most
likely means that new particles are behind the corner of discovery by LHC
(see e.g. [12]).
The ATLAS and CMS detectors have been designed to detect and discover
almost anything that could be directly produced at LHC. Were these two
experiments not to find any new physics beyond the Standard Model, though,
LHC could still have a say in new physics searches. The study of flavorchanging and CP-violating interactions with, for instance, the LHCb detector
[13], is in fact indirectly sensitive to much higher energy scales respect to the
two main experiments.
Even in the worst-case scenario of no physics beyond the Standard Model
showing up at LHC, anyway, there are several other places where to look for
it. Various other experiments are in fact underway to investigate phenomena
that are not accounted for in the Standard Model. The observation of neutrino flavor oscillations requires for instance the introduction of a mechanism
to generate neutrino masses, the most famous being the so called seesaw ; in
its simplest incarnation, type-I seesaw, it requires to extend the Standard
Model with right-handed neutrinos (see e.g. [14] for a viable model), but also
more exotic extension are possible like for instance with SU (2)L triplets in
type-II and type-III.
Another notorious physical phenomenon not accounted for in the Standard Model is the presence of Dark Matter (DM) in the Universe. This is a
new kind of matter, whose existence has been put forward as an explanation
for several astrophysical and cosmological observations. These observations
sum up to say, in general, that the observed matter is by far not enough
Effective Operators for Dark Matter Detection
Chapter 1. Introduction
3
to account for the inferred gravitational potential on a variety of different
length scales. Since the visible matter is observed through emission or scattering of light, we conclude that this new matter must be dark. While also
some modifications of Gravity were proposed to accommodate the astrophysical observations (see e.g. [15, 16]), the Dark Matter paradigm is currently
favored.
A curious coincidence gained a lot of interest to a certain class of hypothetical DM particles. In the usual assumption of a Dark Matter with
thermal origins, one can compute the DM relic abundance ΩDM by numerically solving the Boltzmann equation, which describes the time evolution
of the particle number density. In the simplest case of s-wave annihilations,
i.e. when the annihilation cross section times the DM velocity σann v does
not depend on v, one has this simple formula for the relic abundance (see
e.g. [17]):
ΩDM h2 ' 0.11 ·
1 pb
3 × 10−26 cm3 /s
' 0.11 ·
,
hσann vi
hσann vi
(1.1)
where h ' 0.7 is the present Hubble parameter, and h i denotes thermal
average. From the value ΩDM ' 0.23 measured by WMAP [18–20], it follows
that hσann vi ' 1 pb, i.e. a tipical Weak-scale cross section. This striking
coincidence suggests that the Dark Matter could be connected with new
physics at the Electroweak scale. DM particles with Weak-scale interactions,
and usually with mass in the GeV–TeV range or above, are referred to as
WIMPs (Weakly Interacting Massive Particles).
Several DM detection experiments are now underway to establish the nature of this new kind of particles, in the assumption they interact with the SM
particles also through forces other than Gravity. Different kinds of detection
strategies have been devised in order to fulfill this program. While one hope
is to be able to produce and study these new particles at colliders, where
the environmental conditions (i.e. the initial state) are human-controlled,
the study of Dark Matter in non-controlled conditions is carried out with so
called direct and indirect searches.
Indirect detection of Dark Matter consists of detecting cosmic photons
and charged particles (cosmic rays) in order to assess whether or not part of
these could be produced by DM annihilation or decay. Underlying this kind of
searches is the assumption that the Dark Matter consists of both particles and
anti-particles in sufficient amount to leave an observable imprint in the cosmic
rays spectrum, or that otherwise the Dark Matter particles are unstable but
their abundance is not spoiled by decays. A delicate point here is that, in
Effective Operators for Dark Matter Detection
Chapter 1. Introduction
4
order to establish an excess of cosmic rays due to DM annihilation/decay, the
experimental data must be compared with an a priori unknown background.
In recent times, an intriguing excess in the positron fraction has been observed by the satellite borne experiment PAMELA [21], and later confirmed
by the Large Area Telescope (LAT) onboard the FERMI satellite [22]. Interpreting this signal as due to Dark Matter is made difficult by the absence of
an expected corresponding raise in the anti-proton fraction [23]; attempted
explanations point to a DM particle with mass from hundreds of GeV to
tenths of TeV (see e.g. [24, 25]).
More recently, a gamma-line has been found in the FERMI−LAT data [26,
27]. Since such a signal would be very difficult to explain with astrophysical
sources, it is considered as the ‘smoking gun’ signal for Dark Matter. The
evidence, at the level of at least 3σ, is now under scrutiny and would lead, if
confirmed, to a DM particle of mass 130 GeV or more.
The approach described so far implies detection of Dark Matter annihilation or decay products, hence called indirect. Direct searches aim instead
at measuring recoil energies of nuclei scattered by DM particles; experiments
employing this technique consist of detectors placed in caves deep underground to be screened by cosmic rays, and further shielded against environmental radioactivity.
DM direct detection experiments are providing exciting results. For example the DAMA/NaI collaboration [28] has claimed to have observed the
expected annual modulation of the DM induced nuclear recoil rate [29, 30],
due to the rotation of the Earth around the Sun. The rotation, in fact, causes
a different value of the flux of DM particles hitting the Earth depending on
the different periods of the year. The upgraded DAMA/LIBRA detector has
further confirmed [31] the earlier result adding much more statistics, and it
has reached a significance of 8.9σ C.L. for the cumulative exposure [32].
Interestingly the DAMA annual modulation effect has been shown to be
compatible with a DM interpretation which, for the case of coherent spinindependent scattering, leads to a range of DM masses spanning from a few
GeV up to a few hundred GeV, and cross sections between 10−42 cm2 to
10−39 cm2 [28, 31, 32], with some noticeable differences due to the galactic
halo modeling [33, 34].
More recently, the CoGeNT experiment first reported an irreducible excess
in the counting rate [35], which could also be in principle ascribed to a DM
signal. Later on, the same experiment reported an additional analysis which
shows that the time-series of their rate is actually compatible with an annual
modulation effect [36]. The evidence of such a modulation for CoGeNT is at
the level of 2.8σ C.L.
Effective Operators for Dark Matter Detection
Chapter 1. Introduction
5
Also the CRESST collaboration observed an excess [37]. In particular,
67 counts were found in the DM acceptance region, where the estimated
background is not sufficient to account for all the observed events. The
analysis made by the collaboration rejects the background-only hypothesis
at more than 4σ [37].
The interesting feature is that the DAMA and CoGeNT results appear
to be compatible for relatively light DM particles, in the few GeV to tens
of GeV mass range and coherent scattering cross section around 10−40 cm2 .
CRESST points somehow to larger DM masses, but it is still compatible with
the range determined by the other two experiments. The actual relevant
range of masses and cross sections depends on assumptions of the galactic
DM properties, namely the velocity distribution function and the local DM
density [34].
The CDMS II and XENON100 experiments have recently reported a small
number of events which pass all the selection cuts. Specifically, they have 2
events for CDMS [38] and 6 events for XENON100, reduced to 3 events after
post-selection analysis [39]1 , which are still too few to be interpreted as potential DM signal. They can therefore provide upper bounds on the DM scattering cross section. These constraints seem to set severe bounds on the DM
parameter space, and therefore to rule out much of the allowed regions of the
other experiments (relevant analyses can be found e.g. in [40, 41]). Last, the
PICASSO experiment [42] seems to provide even more stringent constraints
for the low mass DM interpretation of DAMA, CoGeNT and CRESST.
However, there are at least two caveats when interpreting the results from
the experiments mentioned above. The first is that one has to pay attention
to the fine details associated to the results quoted by each experiment, since
a number of factors can affect the outcome. For example the actual response
of the XENON and CDMS detectors can be uncertain and model dependent
for a low energy signal [43–45]. Moreover, for crystal detectors like DAMA, an
additional source of uncertainty is provided by the presence of an unknown
fraction of nuclear recoils undergoing channeling. This effect is currently
being investigated [46–48]. If confirmed, it would lead to a significant shift
of the DAMA allowed regions in the DM parameter space. Another source of
uncertainty is associated with our still poor knowledge of the matrix elements
at the nucleon level and the nuclear form factors, for each interaction and
for each experiment (see e.g. [49]).
The second caveat is related to the interpretation of the actual data within a
very simple-minded model of the DM interaction with nuclei. In compliance
1
After the work illustrated in this Thesis was done, newer XENON100 results became
available, see Ref. [1].
Effective Operators for Dark Matter Detection
Chapter 1. Introduction
6
with the standard references on DM direct detection, Ref. [17, 50, 51], the
following assumptions are usually made on DM-nucleon interactions:
• The interaction amplitude does not depend on the DM velocity nor
on the exchanged energy. Here one has in mind a contact interaction,
i.e. due to the exchange of a heavy mediator, as opposed to a long-range
interaction that is mediated by a massless or nearly massless mediator.
• For the spin-independent case, when the interaction is not explicitly
known, the DM is assumed to couple with equal strength to proton
and neutron. The framework in which this hypothesis is relaxed is
called ‘isospin violating Dark Matter’ (see e.g. [52]), and allows for a
less stringent interpretation of the constraints.
This Thesis explores both directions, first by proposing viable models for
isospin violating Dark Matter, and then inspecting the effect of a long-range
interaction via a magnetic dipole moment for the DM particle.
Several models exist, which attempt to describe the Dark Matter from
a particle physics perspective. The most famous Dark Matter candidate is
certainly the supersymmetric neutralino, but many others have also been
envisaged. Some Technicolor models, for instance, feature DM candidates
that are composite states (see e.g. Ref. [53–55]); while the existence of these
states is regulated by intricate strong dynamics and is often assumed without
analytical proof, lattice simulations were used to show that they can actually
appear in the spectrum of these models [56].
Anyway, given the inconclusive results of the experimental searches so
far, a broader view could be worth taking, to go beyond the limitations of
the single models. In this respect, one could opt for a less model-dependent
approach. A possibility could be to only pay attention to the physics at low
energy, disregarding the different possible UV completions that one could
imagine for the same low energy description. This can be done in the context of an ‘effective field theory’, in which the high energy degrees of freedom
are integrated out from the theory; as a result one obtains an effective Lagrangian with an infinite number of terms, which can be organized according
to a derivative expansion in powers of ∂/Λ, where Λ is the cutoff (usually
identified with the energy scale of the lightest UV degrees of freedom). Different UV theories will dictate specific relations among the coefficients of the
terms in the expansion; nevertheless, all these models can be tested simultaneously by using the appropriate low energy effective theory description.
This Thesis is organized as follows. In Chapter 2 we present a concise list
of evidences for Dark Matter from Astrophysics and Cosmology. In Chapter
Effective Operators for Dark Matter Detection
Chapter 1. Introduction
7
3 we introduce the effective field theory for a scalar DM φ up to dimension
six operators. The full list of Lagrangian terms is completed by the tables
given in Appendix A. In Chapter 4 we introduce the isospin violating DM
paradigm, aimed at accommodating the disagreement between the different
direct detection experiments. In this context we show how the effective theory of Chapter 3 could be used as a template for scalar isospin violating DM
models, via quantum mechanical interference. Moreover we investigate the
possibility of having an isospin violating DM χ of spin-1/2. In Chapter 5
we focus on the effective theory of fermionic DM interactions with photons,
singling out the magnetic and electric dipole moment operators as the most
relevant for direct detection searches. We show that a 10 GeV DM with
magnetic dipole about 1.5 × 10−18 e cm can accommodate the various direct
detection experiments, assuming a conservative bound from XENON100. Finally, in Chapter 6 we summarize and conclude.
This is just an introduction and there is no will to be complete.
Effective Operators for Dark Matter Detection
Chapter 2
Evidence for Dark Matter
Cosmology and Astrophysics rely today on the standard paradigm of an homogeneous, isotropic and flat Universe (on cosmological scales) governed by
the laws of Einstein’s General Relativity. While there is no theoretical motivation for it, flatness has been pointed at by a huge deal of experimental
measurements; defining ρc as the critical energy density for making up a flat
Universe and with ρ the actual energy density of the Universe, Ω ≡ ρ/ρc = 1.
Furthermore, the same experiments (plus others) show compelling evidence
for the presence of two other ingredients, beside the known matter content
constituted mainly by baryons: Dark Energy and Dark Matter. While being
both dubbed ‘dark’, they are two very different entities and we are still far
from understanding whether they could have a common origin or be somehow connected. Dark Energy is a sort of distributed pressure that is now
leading Universe’s accelerated expansion, it has the equation of state of an
energy and, unlike matter, it doesn’t cluster; for our purposes, its effect can
be expressed by a cosmological constant Λ in Einstein’s equations. Dark
Matter (DM) behaves instead like matter but, unlike baryons, it is dissipationless, i.e. it doesn’t emit/absorb nor diffuse light; while it is thought to
‘feel’ gravity in the same fashion as the known matter, its reduced or absent
interaction with photons makes it very difficult to detect it with the usual
astronomical tools, for which reason it was dubbed ‘dark’.
With the inclusion of these dark ingredients, which have not yet been verified
by laboratory experiments, Cosmology works remarkably well. The data consistently point to a large Dark Energy contribution ΩΛ to the energy budget
of the Universe, and to a Dark Matter component ΩDM that dominates over
the other known forms of matter. The ΛCDM model, named after a nonzero
cosmological constant and a substantial portion of cold (i.e. non-relativistic)
Dark Matter, is considered today the standard cosmological paradigm and
is almost universally accepted as the best description of the present data.
8
Chapter 2. Evidence for Dark Matter
9
The latest Wilkinson Microwave Anisotropy Probe WMAP 7-year fit to the
ΛCDM model yields the figures [18–20]
ΩΛ ' 0.72 ,
ΩDM ' 0.23 ,
Ωb ' 0.05 ,
(2.1)
where Ωb is the baryon contribution to the total energy density, that is the
only relevant contribution among the known particles.
In the rest of this chapter we focus on the experimental motivations and
evidence for Dark Matter. We also provide motivations for its key properties: dark, dissipationless, cold, and non-baryonic1 . Instead of taking an
historical and chronological perspective, we list the evidences according to
the astrophysical scale at which they show up. We start from the smaller
scale (galaxies) to arrive to the largest cosmological ones, probing the early
times of our Universe.
2.1
Evidence on galactic scales
One of the first and most convincing evidence for the presence of non visible
matter comes from the rotation curves of galaxies. Assuming that the orbits
of stars within a galaxy closely mimic the rotations of the planets within
our solar system, their centripetal acceleration is expected to be (in the
approximation of circular orbits)
GN M (r)
v(r)2
=
,
r
r2
(2.2)
thus yielding a circular velocity
r
v(r) =
GN M (r)
.
r
Here GN is Newton’s gravitational constant and
Z
M (r) = 4π ρ(r) r2 dr
(2.3)
(2.4)
is the mass contained inside a sphere of radius r, ρ(r) being the mass density
profile. Luminous material in these galaxies is concentrated in the central
regions, so the angular rotation of stars ought to slow
√ at large radii, reproducing the classical Keplerian behavior v(r) ∝ 1/ r for an object in the
1
For a more comprehensive list of Dark Matter properties, see [57].
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
10
Figure 2.1: Two examples of galactic rotation curves. The flat behavior of
v(r) is apparent outside the galactic center, implying the existence of a dark
halo with M (r) ∝ r i.e. ρ(r) ∝ 1/r2 . Left: rotation curve for the spiral
galaxy NGC6503 from Ref. [59] (adapted from Ref. [60]). The points are
the measured circular rotation velocities as a function of distance from the
center of the galaxy. The dashed and dotted curves are the contribution to
the rotational velocity due to the observed disk and gas, respectively, and the
dot-dash curve is the estimated contribution from the dark halo needed to fit
the data. Right: Milky Way’s rotation curve from Ref. [61].
optical disk. On the other hand observations show that stars in the outskirts rotate at the same rate as those near the centre, displaying the flat
behavior for the rotational curve depicted in Fig. 2.1. The fact that v(r)
is approximately constant implies therefore the existence of an halo with
M (r) ∝ r i.e. ρ(r) ∝ 1/r2 , if we believe Newton’s gravitational law to be
correct. Moreover, given such high velocities of their constituents even in the
outskirts, galaxies should pull themselves apart; preserving their existence
requires a deeper gravitational potential and therefore more mass respect to
the observed luminous one [58].
Perhaps the most notable type of galaxies exhibiting this mass discrepancy are the dwarf spheroidal galaxies that are satellites of the Milky Way
and of Andromeda. These satellites are tiny by galaxy standards, possessing
a small number of stars (millions, or in the case of the so-called ultrafaint
dwarfs, only thousands). These are close enough to allow for a precise measurement of the velocity dispersion of their components. The mass inferred
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
11
from their motions greatly exceeds the mass visible in luminous stars2 : indeed, these dim satellite galaxies exhibit some of the largest mass discrepancies observed (on the contrary, bright giant elliptical galaxies exhibit remarkably modest mass discrepancies).
It is worth noting that, while we readily assumed that the galactic rotation
curves can be explained by some form of Dark Matter, another explanation
exists: indeed the data can be fitted also without the Dark Matter assumption, but modifying Newtonian gravity in the limit of low accelerations. This
paradigm is known as MOdified Newtonian Dynamics or MOND (see [15,16]
for recent reviews), and is not definitely ruled out despite the Dark Matter
assumption is by far the most favored.
Another very strong evidence of the missing mass problem comes from
gravitational lensing studies (see [58] for a recent review). These exploit the
fact that light propagates along geodesics, which deviate from straight lines
when passing near gravitational fields. When the bending of light is caused
by a large gravitational mass (the ‘lens’) between a background source and
the observer, this effect is very apparent and is called ‘strong lensing’ (see
Fig. 2.2); an example is the image of a quasar getting distorted by the presence of a galaxy on the line of sight connecting the quasar to the observer.
If the source is located exactly behind a circular massive object in the foreground, a complete ‘Einstein ring’ appears; in more complicated cases, like a
background source that is slightly offset or a lens with a complex shape, one
can still observe arcs or multiple images of the same source. The mass distribution of the lens can then be inferred by the measurement of the ‘Einstein
radius’ or more in general by the positions and shapes of the source objects.
Comparing the measured mass of the lens with its luminosity, that is supposed to map the amount of baryonic matter in it, a mismatch is found again.
This leads once more to think that a large portion of the mass of the galaxies
is actually in the form of Dark Matter.
2.2
Evidence from clusters of galaxies
The methods outlined above, namely the study of the rotation curves and the
strong lensing effect, can also be employed with larger astronomical objects,
like clusters of galaxies, to measure their total mass. In the first case one maps
2
Contrarily to clusters of galaxies, in which the majority of the visible mass consists
of diffuse intra-cluster gas, almost all the bright matter forming galaxies is in the form of
stars [58].
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
12
Figure 2.2: Examples of strong lensing. Top: Many of the brightest blue
images are of a single ring-like galaxy which happens to line-up behind the
giant cluster of galaxies CL0024+1654. Cluster galaxies here typically appear
yellow and, together with the cluster’s Dark Matter, act as a gravitational
lens. Credit: NASA, ESA, H. Lee & H. Ford (Johns Hopkins U.). Bottom
left: the Luminous Red Galaxy LRG 3-757 acting as lens on a much more
distant blue galaxy. Here the alignment is so precise that the background
galaxy is distorted into a horseshoe (a nearly complete ring), so that it is
now called ‘the Cosmic Horseshoe’. Credit: ESA/Hubble & NASA. Bottom
right: the cluster MACS J1206.2-0847 lensing the image of a yellow-red
background galaxy into the huge arc on the right. Credit: NASA, ESA, M.
Postman (STScI) & the CLASH Team.
the rotation velocities of the outer galaxies as a function of their distance from
the center of the cluster, while in the second case one considers galaxy clusters
as lenses. The baryonic contribution to the cluster’s mass can be inferred
by studying the profile of X–ray emission, that traces the distribution of
hot emitting gas in the cluster; other probes of the baryonic components of
the cluster are the temperature of the intra-cluster gas and the SunyaevZel’dovich effect, by which the Cosmic Microwave Background radiation gets
spectrally distorted through inverse-Compton scattering from hot electrons.
The observations on this scale reproduce the results obtained for smaller
objects.
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
13
Figure 2.3: The various regimes of gravitational lensing image distortion.
Along typical lines of sight through the Universe, an intrinsically circular
source is distorted into an ellipse by weak lensing shear (the distortion has
been exaggerated here for illustration). Nearer concentrations of mass, the
distortion begins to introduce flexion curvature. Along lines of sight passing
near the most massive galaxies of clusters of galaxies, and through the most
curved space-time, strong gravitational lensing produces multiple imaging and
giant arcs. From Ref. [58].
The mass of clusters can also be measured in few other ways. One method
involves for instance the observation of virialized objects inside the cluster:
knowing the observed distribution of radial velocities of the individual galaxies, one can infer the gravitational potential responsible for their motion by
means of the virial theorem:
hU i = −2 hT i ,
(2.5)
where U and T are the potential and kinetic energies, respectively, and h i
denotes time average. The comparison of the total cluster’s mass to the mass
inferred by its luminosity provided one of the first hints that the luminous
matter is not enough to sustain the internal cluster dynamics, and that some
dark mass is needed.
Another powerful way of measuring the mass present in a certain part of
the sky is provided by the ‘weak lensing’ (see Fig. 2.3). While the methods
outlined above take into account single gravitationally bound objects, like
galaxies and cluster of galaxies, this technique analyzes a large number of
independent galaxies in a statistical fashion. It can be applied when the line
of sight to the observed patch of sky does not pass near a strong gravitational
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
14
lens, and light deflection is very slight. In these conditions, the distortion of
individual sources cannot be seen because too small compared to the range
of intrinsic shape variation in galaxies. The power of this methods comes
from its statistical nature. When considering a large number of galaxies lying in the same small patch of sky, one expects their intrinsic shapes to be
uncorrelated and therefore, in absence of lensing, if there is no preferred direction in the Universe, they must average out as circular. The presence of
some mass along the line of sight pointing to that patch of sky introduces
instead a coherent ‘shear’ distortion, that can therefore be measured averaging over a large number of galaxies (averaging over ∼ 100 galaxies yield
a signal-to-noise ratio of unity in shear [58]). The observed shear field can
then be converted into a map of the projected mass distribution.
Bridging the gap between strong and weak lensing is the more subtle effect
known as ‘flexion’ (see Fig. 2.3). This can be observed as a tiny curvature
in the shape of a galaxy, when the foreground gravitational potential is not
deep enough to produce an arc or a ring, as in strong lensing. The amplitude
of the flexion signal is lower than the shear signal, but so is also the noise
due to the intrinsic curvature of typical galaxy shapes. Statistical techniques
similar to those used in weak lensing can therefore be applied, when the light
deflection is too small for strong lensing, but the number of lensed sources is
too low for a significant weak lensing analysis.
A further way to measure the mass of a cluster is to study the X–ray
emission of its electron component. This allows to infer the temperature of
the gas, which in turn gives information about its mass through the equation of hydrostatic equilibrium (see e.g. [62]). For a system with spherical
symmetry,
1 dP
= −a(r) ,
(2.6)
ρ dr
where P , ρ, and a are, respectively, the pressure, density, and gravitational
acceleration of the gas, at radius r. For an ideal gas, we can use the equation
of state P V = N kB T to rewrite this in a more suitable form. First we express
the total number of particles in the gas, i.e. electrons and ionized nuclei, as
N = M/mp µ, where M is the total mass of the gas, mp is the proton mass
and µ ' 0.6 is the so called ‘average molecular weight’. Since M/V = ρ, the
equation of hydrostatic equilibrium for an ideal gas reads now
d log ρ d log T
µmp
+
=−
ra(r) .
d log r
d log r
kB T (r)
(2.7)
The temperature of clusters is roughly constant outside their cores and the
density profile of the observed gas at large radii roughly follows a power–law
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
15
with an index between −2 and −1.5. We then find that the temperature
should obey the relation
M (r)
1 Mpc
kB T ≈ (1.3 ÷ 1.8) keV
(2.8)
1014 M
r
for the baryonic mass of a typical cluster, where M (r) is the mass enclosed
within the radius r. The disparity between the temperature obtained using
this calculation and the corresponding observed temperature, kB T ≈ 10 keV,
when M (r) is identified with the baryonic mass, suggests the existence of a
substantial amount of Dark Matter in clusters.
The most striking example of mass discrepancy in clusters of galaxies
is undoubtedly the so-called Bullet Cluster 1E 0657-56 [63–65], shown in
Fig. 2.4. The Bullet Cluster is strictly two clusters that collided about 150
million years ago. Since individual galaxies within the clusters (and stars
within those galaxies) are well-spaced, they have a very low collisional cross
section: most continued moving during the collision, and today lie far from
the point of impact. On the other hand, the intra-cluster gas which forms the
bulk mass of the incident clusters was uniformly spread. This had a large
interaction cross section and was slowed dramatically during the collision.
The two concentrations of hot gas, seen in X–ray emission, have now passed
through each other, but have not travelled far from the point of impact. The
collision speed and gas density were sufficient for a shock front to be observed
in the gas from the smaller of the two clusters, allowing the determination
of the collision speed.
Exploiting the weak lensing effect, the gravitational potential of the system
was mapped in [65], showing that it does not trace the plasma distribution
(the dominant baryonic mass component), but rather approximately traces
the distribution of galaxies. An 8σ significance spatial offset of the center
of the total mass from the center of the baryonic mass peaks was found,
requiring a great deal of extra dark mass to be located near the galaxies. To
have travelled so far, this mass must have a self-interaction collisional cross
section σ/mDM < 1.25 cm2 g−1 at 68% confidence (or σ/mDM < 0.7 cm2 g−1
under the assumption that the mass-to-light ratio of the initial clusters was
the same), where mDM is the DM particle mass [66].
The Bullet Cluster has provided the most direct empirical evidence for Dark
Matter, also due to the common belief to be extremely hard to explain within
the MOND paradigm (see however [16]).
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
16
Figure 2.4: The Bullet Cluster 1E 0657-56, shown here in a composite image. The background image shows the location of galaxies, with most of the
larger yellow galaxies associated with one of the colliding clusters. The overlaid red features show X–ray emission from hot, intra-cluster gas; the gas
cloud at the right emerged distorted into the distinctive bullet-shape from the
collision. The overlaid blue features show a reconstruction of the total mass
from measurements of gravitational lensing. This appears coincident with the
locations of the galaxies, implying it has a similarly small interaction crosssection. However, there is more mass than that present in the optical galaxies
and X–ray gas combined; this, plus the clear separation of the center of the
potential well from the gas, otherwise considered to be the bulk gravitational
component of the cluster, is considered strong evidence for the existence of
Dark Matter. Credit: X–ray: NASA/CXC/CfA/ M.Markevitch et al.; Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/ D.Clowe et al.;
Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al.
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
2.3
17
Evidence on large scales
On large scales, the Universe shows a wealth of structure: galaxies are gathered into clusters, clusters are part of superclusters, and superclusters are
arranged into large-scale sheets, filaments and voids (see Fig. 2.5). This cosmic scaffolding has been revealed by large-scale surveys such as the 2-degree
Field Galaxy Redshift Survey (2dFGRS, [67]) and the Sloan Digital Sky Survey (SDSS, [68]). Since looking far in space means also looking back in time,
we can reconstruct the distribution of mass in distant structures as it was
when the light passed near and was lensed by them many billions of years
ago. Therefore, we can assume that the patterns of structures at decreasing
distances reflect the history of gravitational clustering of matter. The possible presence of Dark Matter or a modification of gravitational dynamics
respect to the predictions of General Relativity, should have influenced the
pattern of large scale structures (LSS) we see today and the history of their
formation.
Large-scale cosmological ‘N-body’ simulations like e.g. the Millennium simulation [69] demonstrate that the observed large scale structure of luminous
matter could only have been formed in the presence of a substantial amount
of Dark Matter. Furthermore, the bulk of Dark Matter must be both nondissipative3 and non-relativistic (‘cold’) for the correct structures to be produced. This is because a particle species that is relativistic (‘hot’) at the
time structures start forming, will free-stream out of galaxy-sized overdense
regions, so that only very large structures can form early. This leads to
a top-down hierarchy in the formation of structures, with small structures
like galaxies forming by the fragmentation of larger ones. This behavior is
nowadays strongly disfavored in view of observations of galaxies at very high
redshift, showing that galaxies are older than superclusters. Cold species
move instead very non-relativistically at the time they decoupled and have
therefore a short free-streaming length, making it possible to clump also on
smaller scales and seed galaxy formation. This favors a hierarchical structure
formation, with small clumps merging in larger ones, forming galaxy halos
and successively larger structures, in accordance with N-body simulations.
Another possibility, dubbed ‘warm’ DM because with velocity dispersion between that of hot and cold DM, was proposed in order to solve some problems
of the cold DM paradigm (see e.g. [70]). Its larger free-streaming length, with
respect to the cold DM case, suppresses the formation of small structures: for
instance, a warm DM particle with a mass of 1 keV and an abundance that
3
DM must be non-dissipative in order to prevent it from cooling and collapsing with
the luminous matter, which would produce larger and more abundant galactic disks than
observed.
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
18
Figure 2.5: Large scale structures of the Universe. Left: particular of a
slice through the SDSS 3-dimensional map of the distribution of galaxies.
Earth is at the center, and each point represents a galaxy, typically containing
about 100 billion stars. Galaxies are colored according to the ages of their
stars, with the redder, more strongly clustered points showing galaxies that
are made of older stars. The outer circle is at a distance of two billion light
years. The region between the wedges was not mapped because dust in our own
Galaxy obscures the view of the distant Universe in these directions. Credit:
M. Blanton and the Sloan Digital Sky Survey. Right: the large-scale Dark
Matter distribution in the Universe as generated by the Millennium simulation.
From the Millennium Simulation Project webpage.
matches the correct Dark Matter density, has a free-streaming length of order
of galaxy scales [71]. Measurements of the Cosmic Microwave Background
power spectrum, the growth of structures in galaxy clusters and Lyman-α
forest can then be used to set a lower bound on the mass of the warm DM
particles around the keV scale [72] (the same is also done for hot particles,
such as neutrinos [19]).
Cold Dark Matter (CDM) remains however the standard paradigm. Small
amounts of hot and warm Dark Matter can still be tolerated, provided they
are compatible with large scale structures and Cosmic Microwave Background
data.
Large-scale galaxy surveys not only provide information as to the amount
and pattern of structure present in the Universe, but also a handle on the total
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
19
mass contained within it [68,73,74]. Dark Matter plays a key role in structure
formation because it feels only the force of gravity. As a result, it begins to
collapse into a complex network of Dark Matter halos well before ordinary
matter; the latter, indeed, dominated by baryons, is tightly bound to photons
in the primordial plasma, which prevent it to clump due to the pressure
force they exert. At the time baryonic matter decouples from photons, their
pressure push fades and baryons promptly fall in the potential wells created
by the Dark Matter halos. Without Dark Matter, baryons would clump much
slower and galaxy formation would occur substantially later in the Universe
than is observed.
The complex process of structure formation depends therefore critically on
the presence of Dark Matter, and we can expect that the pattern of structures
encodes quantitative information about its distribution and its properties.
Such a kind of information can be inferred e.g. from the structures power
spectrum, i.e. the galaxy correlation function (in other words, the statistical
probability distribution that any two galaxies are separated by a certain
distance) [68, 73, 74]. These studies give an independent assessment of the
ΛCDM model, that is in remarkable agreement with the other very different
experimental tests.
2.4
Evidence from Cosmology
Cosmology, in its study of the history of the Universe, offers some invaluable
tools to probe its matter content and distribution. During the evolution of
the hot particle plasma after the Big Bang, in fact, some phenomena occurred
which produced relics; these relics remained in the cosmos almost unaltered,
giving us the possibility to directly probe the state of the Universe at the time
they were produced. This also brings us informations about the quantity and
distribution of baryons and overall matter present at the time.
One of the most important relics of the early Universe, if not the most
important, is the Cosmic Microwave Background radiation (CMB). This consists of photons emitted during the short period of recombination, when protons bound to electrons to form electromagnetically neutral atoms thus allowing photons to free-stream across the Universe. The CMB can be now
detected and is seen to feature a perfect black body spectrum of temperature 2.725 K [75,76]. Moreover it embodies a huge deal of information on the
state of the Universe at recombination, allowing to fit several cosmological
parameters (see e.g. [19]).
Anisotropies in the angular distribution of temperatures of the CMB sky,
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
20
Figure 2.6: The detailed, all-sky picture of the infant Universe created from
the 7-year WMAP data. The image reveals temperature fluctuations in a
range of ±200 µK, shown as color differences, that correspond to the seeds
that grew to become galaxies. The signal from our own galaxy was subtracted
using the multi-frequency data. At the center-left one can spot Stephen
Hawking’s initials [77]. From the WMAP website, courtesy of the WMAP
Science Team.
shown in Fig. 2.6, map the presence of overdensities and underdensities in the
primordial plasma before recombination. For this reason, information on the
baryon and matter distribution of the Universe can be read in the spectrum
of CMB anisotropies. These are defined as a function of the angular position
in the sky θ,
+∞ X
+`
X
δT
(θ) =
a`m Y`m (θ) ,
(2.9)
T
`=2 m=−`
where Y`m (θ) are spherical harmonics. Notice that one skips in the definition
the monopole ` = 0 and the dipole ` = 1 since these are just the CMB mean
temperature and the anisotropy due to the motion of the Earth relative to
the CMB rest frame, respectively. Given the high degree of uniformity of the
sky on large scales, the anisotropies are extremely small: δT /T ∼ 10−5 . One
then defines the variance C` of a`m as
`
X
1
C` ≡ h|a`m | i ≡
|a`m |2 .
2` + 1 m=−`
2
(2.10)
On small sections of the sky where its curvature can be neglected, the spherEffective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
21
ical harmonic analysis becomes ordinary Fourier analysis in two dimensions.
In this limit ` becomes the Fourier wavenumber. Since the angular wavelength is θ = 2π/`, large multipole moments correspond to small angular
scales.
If the temperature fluctuations are assumed to be Gaussian (i.e. with the
a`m distributed according to a Gaussian with mean 0 and variance 1), as
appears to be the case at a high degree of accuracy, all of the information
contained in CMB map is expressed by the power spectrum, usually displayed
as `(` + 1)C` T 2 /2π (see Fig. 2.7). This features a number of peaks due to
the acoustic oscillations of the plasma before photon decoupling. In fact,
the tendency of baryons to clump under the effect of gravitational attraction
was contrasted in the primordial plasma by the pressure due to photons; this
caused the plasma to ‘oscillate’, forming pressure/density waves traveling at
constant speed. The imprint of these oscillations remained in the CMB after
the baryons decoupled from photons at recombination: in fact, the CMB
temperature fluctuations map the heating and cooling of the primordial fluid
that is compressed and rarefied by acoustic waves. Modes that are caught
at maxima or minima of their oscillation at recombination correspond to
peaks in the power spectrum. These are actually all higher harmonics of
the first (fundamental) peak, occurring at ` ' 200; the first peak represents
the mode that compressed once inside potential wells before recombination,
the second the mode that compressed and then rarefied, the third the mode
that compressed then rarefied then compressed again, and so on. Since the
odd numbered acoustic peaks are associated with how much the plasma compresses, they are enhanced by an increase in the amount of baryons in the
Universe. The even numbered peaks are associated instead with how much
the plasma rarefies, and do not depend on the baryon density. Thus with
the addition of baryons the odd peaks are enhanced over the even peaks.
Fits to the CMB power spectrum allow then to determine the baryon and
matter content of the Universe in the context of the ΛCDM model, yielding
the values reported in Eq. (2.1) at the beginning of this chapter. The CMB
measurements therefore independently confirm the large mismatch between
the baryon and matter density, pointing to a prevailing (cold) Dark Matter
component.
Another crucial phenomenon occurred in the early Universe is the synthesis of light nuclei, dubbed Big Bang Nucleosynthesis or BBN, which took
place within the first three minutes. The Big Bang Nucleosynthesis offers
one of the deepest reliable probes of the early Universe, being based on wellunderstood Standard Model physics. Predictions of the abundances of the
light elements, D, 3 He, 4 He, and 7 Li are in good overall agreement with the
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
10
22
25. Cosmic microwave background
1
o
10’
Angular scale
5’
6000
WMAP
ACBAR
ACT
QUAD
SPT
( + 1)C /2π [µK2]
5000
4000
3000
2000
1000
0
0
500
1000
1500
2000
2500
Multipole
Figure 25.2: Band-power estimates from the WMAP, ACBAR, ACT, QUAD, and
SPT experiments (omitting some band-powers which have larger error bars). Note
TT experiments and have not been plotted.
that the widths of the #-bands vary between
`
` available experimental
This figure represents only a selection of
results, with some
other data-sets being of similar quality. The multipole axis here is linear, so the
Sachs-Wolfe plateau is hard to see. However, the acoustic peaks and damping region
are very clearly observed, with no need for a theoretical curve to guide the eye; the
curve plotted is a best-fit model from WMAP plus other CMB data. At high # there
is some departure from the model due to secondary anisotropies.
Figure 2.7: The CMB temperature power spectrum. C = C here corresponds to C` T 2 in the text. Left: the 7-year WMAP data. The curve is
the ΛCDM model best fit to the 7-year WMAP data. Figure courtesy of the
WMAP Science Team, from Ref. [18]. Right: power spectrum data from a
The
of this
allowsalone.
for six different
crosscurve
power spectra to
selection of experiments, extending
theexistence
reach
inlinear
` ofpolarization
WMAP
The
be determined from data that measure the full temperature and polarization anisotropy
information.
Parity
considerations
makeFrom
two of these
zero,[20].
and we are left with four
is a best fit model from WMAP plus
other
CMB
data.
Ref.
potential observables: C TT , C TE , C EE , and C BB . Because scalar perturbations have
!
!
!
!
no handedness, the B-mode power spectrum can only be sourced by vectors or tensors.
Moreover, since inflationary scalar perturbations give only E-modes, while tensors
generate roughly equal amounts of E- and B-modes, then the determination of a non-zero
B-mode signal is a way to measure the gravitational wave contribution (and thus
potentially derive the energy scale of inflation), even if it is rather weak. However, one
must first eliminate the foreground contributions and other systematic effects down to
very low levels.
The oscillating photon-baryon fluid also results in a series of acoustic peaks in the
polarization C! ’s. The main ‘EE’ power spectrum has peaks that are out of phase with
those in the ‘TT’ spectrum, because the polarization anisotropies are sourced by the
primordial abundances inferred from observational data. Abundances are,
however, usually observed at much later epochs, after stellar nucleosynthesis has commenced. The ejected remains of this stellar processing can alter
the light element abundances from their primordial values, and also produce
heavy elements (‘metals’). Thus, one seeks astrophysical sites with low metal
abundances, in order to measure light element abundances which are closer
to primordial.
The reaction rates for the formation of light elements depend on the
density of baryons (strictly speaking, nucleons) nb , which is usually expressed
normalized to the relic black body photon density as η ≡ nb /nγ . As shown
in Fig. 2.8, all the light-element abundances can be explained with η × 1010
in the range 5.1 ÷ 6.5 (95% CL). With nγ fixed by CMB measurements, this
can be stated as the allowed range for the baryonic fraction of the critical
density, Ωb ' (0.019 ÷ 0.024)h−2 , where h ' 0.7 is the present Hubble
parameter. Therefore BBN measurements independently confirm the value
of Ωb as measured by WMAP, Eq. (2.1). Comparing then with the total
amount of matter in the Universe, as inferred for instance by large scale
structures studies, we are forced once more to assume the existence of a
predominant Dark Matter component. Furthermore, BBN provides us with
yet another invaluable piece of information about the nature of Dark Matter,
namely that it must be predominantly non-baryonic, in order not to alter
considerably the element abundances today.
June 18, 2012
16:19
Effective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
22. Big-Bang nucleosynthesis 3
Baryon density Ωbh2
0.005
0.27
0.01
0.02
23
0.03
4He
0.26
0.25
Yp
D 0.24
___
H
0.23
10 − 4
3He/H
CMB
D/H p
BBN
10 −3
5
6
p
10 − 5
10 − 9
7Li/H
5
p
2
10 − 10
1
2
3
4
Baryon-to-photon ratio η × 1010
7
8 9 10
Figure 22.1: The abundances of 4 He, D, 3 He, and 7 Li as predicted by the standard
model of Big-Bang nucleosynthesis [14]
− the bands
show the 95%
Boxes
Figureindicate
2.8: the
Theobserved
abundances
of 4 He,
D, 3 He,
and 7 Li CL
as range.
predicted
by the
light element abundances (smaller boxes: ±2σ statistical
4
standard
model
of
Big
Bang
nucleosynthesis
(Y
customary
indicates
the
He
errors; larger boxes: ±2σ statistical and systematic errors).
The narrow vertical
p
band
indicates
the
CMB
measure
of
the
cosmic
baryon
density,
while
the
wider
mass fraction). The bands show the 95% CL range on the theoretical result.
band indicates the BBN concordance range (both at 95% CL).
Boxes indicate the observed light element abundances (smaller boxes: ±2σ
statistical errors; larger boxes: ±2σ statistical and systematic errors). The
narrow vertical band indicates the CMB measure of the cosmic baryon denJune 18, 2012 16:19
sity, while the wider band indicates
the BBN concordance range (both at 95%
CL). From Ref. [20].
To summarize, the CMB, BBN, and LSS, all consistently point to a significant dark mass fraction that is four to five times larger than the baryonic
mass contained in the Universe. The CMB, together with observations of
Type Ia supernovae, imply the further presence of a Dark Energy compoEffective Operators for Dark Matter Detection
Chapter 2. Evidence for Dark Matter
24
nent, responsible for the Universe’s accelerated expansion today, that contributes the 72% of the total energy density. In fact, while the position of
the CMB peaks depend on ΩΛ and can be used to fit its value [18–20], distant supernovae are apparently dimming, indicating that the expansion of
the Universe is accelerating [78,79]. All these observations are in remarkable
mutual agreement and conspire to paint an entirely self-consistent picture
of the Universe, dubbed ‘Concordance Cosmology’ (see Fig. 2.9). Nevertheless, we still ignore the fundamental nature of both Dark Energy and Dark
Matter.
Effective Operators for Dark Matter Detection
23
ΩM
0.287+0.029+0.039
−0.027−0.036
0.285+0.020+0.011
−0.020−0.009
0.265+0.022+0.018
−0.021−0.016
0.274+0.016+0.013
−0.016−0.012
+0.020+0.011
0.285−0.019−0.011
0.285+0.020+0.010
−0.020−0.010
B
B
B
Ωk
0 (fixed)
0 (fixed)
0 (fixed)
0 (fixed)
−0.009+0.009+0.002
−0.010−0.003
−0.010+0.010+0.006
−0.011−0.004
w
-1 (fixed)
−1.011+0.076+0.083
−0.082−0.087
−0.955+0.060+0.059
−0.066−0.060
−0.969+0.059+0.063
−0.063−0.066
-1 (fixed)
−1.001+0.069+0.080
−0.073−0.082
Chapter 2. Evidence for Dark Matter
25
TABLE 6
s ΩM , Ωk and w. The parameter values are followed by their statistical (σstat ) and
ameter values and their statistical errors were obtained from minimizing the χ2 of
esults in a χ2 of 310.8 for 303 degrees of freedom with a ∆χ2 of less than one for
were obtained from fitting with extra nuisance parameters according Eq. 5 and
2
2 )1/2 .
sulting error, σw/sys , the statistical error: σsys = (σw/sys
− σstat
2.0
-0.7 w/ sys
w
No Big Bang
-1.0
-1.3
0.2
0.3
0.4
1.5
-0.7 w/o NB99
w
-1.0
5
-1.3
0.2
0.3
0.4
m
1.0
ence level contours on
shows the individual
SN set, as well as the
tatistical errors only).
ding systematic errors.
the SCP Nearby 1999
d BAO are consisen in Table 6). Fig.
ts in the ΩM − ΩΛ
SNe
0.5
CM
B
ate constraints on
(EOS) parameter
ms of
,
(10)
cellent approximaand other dark ener aspects of time
suming a flat Unih constraints from
the present value
e of its time varimological constant
neracies within the
matter density ΩM ,
ppreciable leverage
adding other meaoken and currently
d.
on of the SN data
e BAO constraints.
Fl
BAO
0.0
0.0
0.5
at
1.0
Fig. 15.— 68.3 %, 95.4 % and 99.7% confidence level contours
on ΩΛ and ΩM obtained from CMB, BAO and the Union SN set,
as well as their combination (assuming w = −1).
Figure 2.9: The current Concordance model of Cosmology, indicating that
the Universe is dominated by Dark Energy and Dark Matter, and essentially
Theflat.
results
are similar; note
that
including either
one reIllustrated
is the
remarkable
agreement
between cosmological fits to BBN,
sults in a sharp cut-off at w0 +wa = 0, from the physics as
observations
of
the
CMB,
baryon
acoustic
oscillations
(BAO) measurements
mentioned in regards to Eq. 9. Since w(z " 1) = w0 +wa
in this
parameterization,
model
with more
from
the 2dFGRSany
LSS
survey,
andpositive
Type Ia supernovae (SNe). Contours in
high-redshift w will not yield a matter-dominated early
the
figure
on
the
left
indicate
1σ,
2σ
Universe, altering the sound horizon in conflict withand
ob- 3σ CL, whilst areas plotted on the
servations.
right are 1σ ranges. Left figure from [80], right figure from [81], adapted
Note that BAO do not provide a purely “low” redshift
from [82]
with
additional
2dFGRS
results
constraint,
because
implicit
within the
BAO data
anal- from [73].
ysis, and hence the constraint, is that the high redshift
Universe was matter dominated (so the sound horizon
Effective Operators for Dark Matter Detection
Chapter 3
Scalar Dark Matter Effective
Field Theory
In this chapter we introduce a model independent and organized study of
the interaction terms of a generic scalar DM candidate. The purpose is to
illustrate the wealth of terms that one would have in a general setting. For
instance, in effective operator studies such as [83–105], the DM field is usually
considered a singlet under the SM, thus reducing the number of interaction
terms to few; moreover, these operators are not even considered together, as
a truly effective field theoretical approach would require, but they are usually
taken into account separately one from the other. While this is an almost
necessary assumption to reduce the otherwise overwhelming complexity of
the problem, it could nevertheless be an oversimplification, thus preventing
us to discover the true nature of Dark Matter.
Here we show for example how the number of interaction terms drastically
increases as one drops the assumption that the DM is neutral under the Weak
interactions. We retain color neutrality based on the strong limits imposed on
strongly interacting DM (see e.g. [57]). Electrical charges for DM particles are
bound to be only a small fraction of the electron charge [57]; models in which
this possibility is realized (for example via kinetic mixing of the ordinary
photon with a dark photon, see e.g. [106]) are dubbed millicharged DM.
We do not consider this possibility here and take the DM as an electrically
neutral field. Nevertheless we allow the low energy dark sector to feature
also charged particles.
As customary, we also assume the DM being charged under a non-SM global
U (1) symmetry to protect it against decays.
In the framework of the effective theory, we have no control over the underlying dynamics generating the interactions of the DM, and therefore we
26
Chapter 3. Scalar Dark Matter Effective Field Theory
27
must be as general as possible. Integrating out some weakly charged states,
for instance, explicitly breaks the Electroweak symmetry. If the DM is a
bound state, as it can occur in Technicolor theories, the lack of understanding of its complicate internal dynamics makes necessary again to recur to an
effective field theory approach with all the possible interactions. Giving up
the symmetry constraints, one has therefore a huge deal of new interaction operators that are not gauge-invariant under the Electroweak symmetry. Some
of these operators could be absent or feature functional dependence among
their coefficients depending on the underlying theory, but in the generic approach in which one avoids any specific model for DM interaction, all of them
are generally present and unrelated. In this setting, one can also encounter
interesting types of interaction that are not often discussed in literature, as
for instance flavor violating interactions.
In this initial classification of the possible terms we consider the low
energy dark sector as composed by an electrically neutral scalar and at most
two charged particles with charges ±1. We take a modular approach. We
first elaborate the scenario with a single neutral scalar, writing down all
its interactions with the SM fields as well as its self-interactions. We then
complete the list of interactions, in the case a singly positively or negatively
charged state is added to the theory as part of the IR spectrum. Finally,
we also introduce interactions in which the positively and the negatively
charged states appear together, as well as the neutral one. In this last case
the states are denoted T + , T 0 and T − ; since the simplest UV completion
featuring these states consists of the SM being augmented with an SU (2)L
triplet, we will collectively call these states a ‘triplet’. For the same reason,
the previous case of a single charged particle in addiction to the neutral one
will be dubbed ‘doublet’, and the states denoted D+ , D0 . The equivalent
choice D0 , D− could have also been possible. The case in which only the
neutral scalar belongs to the IR spectrum will be accordingly denoted as the
‘singlet’, φ. It is understood that, when increasing the number of states from
the singlet to the doublet and from the doublet to the triplet, the following
dictionary applies: φ = D0 = T 0 , D± = T ± . These states arise for example
in Technicolor models, as objects made of elementary matter mimicking the
bright side of the Universe (see e.g. [53–55]).
We consider all the states as complex scalars, charged under a non-SM
U (1) symmetry. The case of real scalars subject to a Z2 symmetry is simply
deduced from our list of terms by imposing reality of the fields; in this case,
some of the terms are redundant and vanish identically.
This chapter is organized as follows. After a brief comment on the effective operators approach, we start with classifying the interaction terms of the
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
28
complex neutral scalar state with ordinary matter. All the listed operators
are hermitian and are intended to be multiplied by real coefficients, unless
otherwise noted. The complete classification, up to (in mass) dimension six
operators, of the independent interaction terms with the SM gauge bosons is
reported in Appendix A.1; in the main text we show only a few terms, some
of which already appeared in model dependent phenomenological applications like e.g. [53–55]. We also classify the interactions of the complex scalar
with the Higgs particle in the unitary gauge up to dimension six operators,
as well as the DM self-interactions. Moreover, we discuss the possibility that
the DM particle is a pseudo-Goldstone boson, as in the case of the model
introduced in [54]. Still in the main text we provide a complete classification
of the interaction terms with SM fermions including potentially interesting
DM induced flavor-changing operators.
We perform a similar analysis for the doublet and triplet complex scalar
DM fields. In the corresponding sections we only report the terms that have
not been listed yet. This means that the full set of operators for D also
includes the interactions listed for φ, and in turn it is part of the set of
interactions of T . The summary of the interactions of the doublet and the
triplet with the SM gauge bosons can be found respectively in Appendices
A.2 and A.3.
In Section 4.2 we provide an application of the terms introduced in the
case of the DM singlet to direct detection searches.
3.1
Construction of the effective operators
When constructing an effective field theory, the following standard strategy
applies:
1. Identify the correct degrees of freedom, here the SM and DM fields.
2. Use the intact global and gauge symmetries to classify the operators.
3. Provide a counting scheme.
Using the rules above we introduce all possible operators, to the best of
our knowledge, ordered in the inverse powers of the cutoff scale Λ. This scale
is assumed to be the one above which a more fundamental theory of DM
emerges.
At low energies the SU (2)L ⊗U (1)Y symmetry is broken to the electromagnetic one and therefore we classify the operators requiring invariance under
color and electromagnetic interactions. Of course the Weak interaction is not
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
29
required to be broken also at high energies. One could in principle construct
an effective Lagrangian invariant under the full Electroweak symmetry, but
in this case information about the high energy regime of the theory might
be needed, thus making the effective theory more model dependent. In any
case, this alternative procedure would lead to the same set of operators at
low energy given that the relevant intact gauge symmetries are respected.
3.2
Singlet
A scalar SM singlet is one of the most used templates for models of DM. It
emerges in a plethora of more or less natural models. It is for this reason
that we start our analysis from this DM prototype. We are interested in providing a complete model independent classification of the interaction terms
of a generic complex scalar1 with the SM fields. The natural framework for
this analysis are effective theories. We, therefore, first identify a new energy
scale Λ representing the cutoff of our effective theory above which a more
fundamental description arises. Since we are interested in low energy phenomena typically associated to DM detection and cosmic ray production, the
low energy effective theory we construct respects explicitly only the SU (3)c
and U (1)EM symmetries.
3.2.1
Interaction with SM gauge bosons
In this section we provide the complete list of operators up to dimension four
describing the interaction with the SM gauge bosons. For illustration we also
provide a partial list of the operators up to dimension six. The complete list
of allowed interaction terms up to and including dimension six operators is
provided in Appendix A.1. The dimension four operators are:
φ∗ φ Z µ Zµ ,
φ∗ φ W +µ Wµ− ,
(3.1)
(3.2)
Jµ Z µ ,
φ∗ φ (∂µ Z µ ) ,
(3.3)
(3.4)
←
→
where Jµ ≡ i φ∗ ∂µ φ ≡ i [φ∗ (∂µ φ) − (∂µ φ∗ )φ].
Gauge invariance forbids operators with mass dimension less than six
involving the ordinary photons and gluons. Therefore the first operators
1
Real scalar DM candidates with Z2 symmetry are easily recovered in our framework
by breaking the U (1) global symmetry down to Z2 .
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
30
emerge at dimension six and are:
φ∗ φ µν
F Fµν ,
Λ2
∂ν F µν
Jµ
,
Λ2
φ∗ φ
Fµν F̃ µν ,
2
Λ
(3.5)
for the interactions with the photon and with Λ the energy scale at which
these interactions are generated. Here F̃ µν ≡ εµνρσ Fρσ , with Fρσ the photon field strength. The operators ∂µ ∂ν (φ∗ φ)F µν , ∂µ ∂ν (φ∗ φ)F̃ µν and Jµ ∂ν F̃ µν
vanish identically.
µν
For the interactions with gluons, with G̃µν
a defined analogously to F̃ ,
φ∗ φ a µν
G G ,
Λ2 µν a
φ∗ φ a µν
G G̃ .
Λ2 µν a
(3.6)
The interactions with the gluons are expected to dominate with respect to the
ones with the photon. Depending on the model, however, the interaction with
the gluons can be further suppressed. We do not allow the U (1) symmetry
acting on φ to break spontaneously here, to prevent generating mass terms
for the massless gauge bosons.
For each gauge boson field one must multiply by one power of the associated coupling constant the interaction term in which the field appears. For
example the operator φ∗ φ Z µ Zµ should be understood as multiplied by g 2
with g the Weak coupling constant. Besides the SM coupling constants one
has also to multiply each term by an independent dimensionless coefficient
whose specific value is fixed once the underlying model of DM is specified.
3.2.2
Interaction with SM fermions
The possible interaction terms between φ and Dirac spinors ψ up to dimension six are
φ∗ φ ψ̄ψ ,
∂µ (φ∗ φ) ψ̄γ µ ψ ,
Jµ ψ̄γ µ ψ ,
←
→
/ψ,
φ∗ φ ψ̄ i D
φ∗ φ ψ̄γ 5 ψ
∂µ (φ∗ φ) ψ̄γ µ γ 5 ψ
Jµ ψ̄γ µ γ 5 ψ
←
→ 5
/γ ψ
φ∗ φ ψ̄ i D
,
,
,
(3.7)
(3.8)
(3.9)
,
(3.10)
where ψ̄ and ψ are any two SM fermions such that their combination is
←
→
←
→
colorless and electrically neutral. Dµ ≡ 21 ∂µ − ieQAµ is the ‘hermitianized’
form of the usual covariant derivative Dµ ≡ ∂µ −ieQAµ , which introduces also
the minimal coupling to the photon; one has to add the color term −igs T a Gaµ
when the covariant derivative is applied to quark fields. Each operator should
be divided by the appropriate power of the cutoff scale Λ.
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
31
Given the operators above, it is possible to have for instance
φ∗ φ µ̄e
and
←
→
/c,
φ∗ φ ū D
(3.11)
where the sum over colors in the second operator is understood. It is clear
that the dark sector can break flavor universality and these operators should
be included for a true model independent analysis of the experimental constraints.
3.2.3
φ as a pseudo-Goldstone boson
An intriguing possibility is that φ itself is a Goldstone boson emerging from
the spontaneous breaking of non-abelian global symmetries, see e.g. Ref. [54].
In this case all its non-derivative interactions vanish unless a mass-term
m2φ φ∗ φ, with mφ small with respect to the cutoff scale Λ, is explicitly introduced. We introduce this mass term and the DM becomes a pseudoGoldstone boson with the non-derivative interactions suppressed by a factor
mφ /Λ per field φ. For example we would have:
φ∗ φ Wµ± W ∓µ →
m2φ ∗
φ φ Wµ± W ∓µ .
Λ2
(3.12)
According to the new counting of the non-derivative operators the leading
dimension four operators are now
Jµ Z µ
3.2.4
and φ∗ φ (∂µ Z µ ) .
(3.13)
DM-Higgs interaction
In the unitary gauge the interaction terms involving the physical Higgs h,
were it composite or elementary, up to dimension six are:
2
2
X
X
hn
hn
hn
∗
2
µ ∗
bn n + (∂ φ )(∂µ φ)
cn n +
φφ
an n−2 + (φ φ)
Λ
Λ
Λ
n=1
n=1
n=1
∗
4
X
1
X
1
X
hn
hn
(∂ µ h)(∂µ h)
µ
∂ (φ φ)(∂µ h)
dn n+1 + J (∂µ h)
en n+1 + f φ∗ φ
,
Λ
Λ
Λ2
n=0
n=0
(3.14)
µ
∗
where the coefficients are dimensionless and real.
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
3.2.5
32
DM self-interaction
The Dark Matter can in principle self-interact. The Lagrangian terms involving only the DM field up to dimension six are:
(∂ µ φ∗ )(∂µ φ) − m2φ φ∗ φ +
3
X
n=2
gn
(φ∗ φ)n
k
+
(∂ µ ∂µ φ∗ )(∂ ν ∂ν φ)+
2n−4
2
Λ
Λ
1
(l1 ∂ µ (φ∗ φ)∂µ (φ∗ φ) + l2 ∂ µ (φ∗ φ)Jµ + l3 J µ Jµ ) , (3.15)
Λ2
where the coefficients are dimensionless and real.
3.2.6
Interaction with more than one type of SM fields
We list here the independent interaction terms with more than one type of
SM fields up to dimension six.
The terms involving both DM, the SM gauge bosons and the Higgs are:
φ∗ φ Z µ Zµ h ,
φ∗ φ Z µ Zµ h2 ,
φ∗ φ W +µ Wµ− h ,
(3.16)
(3.17)
(3.18)
φ∗ φ W +µ Wµ− h2 ,
(3.19)
∂µ (φ∗ φ) Z µ h ,
∂µ (φ∗ φ) Z µ h2 ,
Jµ Z µ h ,
Jµ Z µ h2 ,
φ∗ φ (∂µ Z µ )h ,
φ∗ φ (∂µ Z µ )h2 .
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
The terms involving both DM and SM fermions plus either a gauge boson or
the Higgs are:
φ∗ φ ψ̄γ µ ψ Zµ ,
φ∗ φ ψ̄γ µ γ 5 ψ Zµ ,
φ∗ φ ψ̄ψ h ,
φ∗ φ ψ̄γ 5 ψ h ,
(3.26)
(3.27)
(3.28)
(3.29)
where ψ̄ and ψ are any two SM Dirac fermions such that their combination
is colorless and electrically neutral. Each operator should be divided by the
appropriate power of the cutoff scale Λ.
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
33
The full list of Lagrangian terms introduced here constitutes the first
and most comprehensive model independent study of the relevant operators
allowing a complex scalar DM candidate to interact with ordinary matter.
From the large number of terms it is clear that any recent analysis of DM has
explored an incredibly tiny portion of the parameter space in the couplings.
3.3
Doublet
Here we generalize the analysis above to the case of the doublet. Its states
are here denoted as D+ , D0 and feature the same nonzero charge under the
new global U (1) symmetry. We could equally have chosen D0 , D− , that is
∗
recovered from the former case by substituting D+ → D− (in the Appendix
A.2, both these possibilities are explicited). Since at very low energy weak
isospin is broken, D0 and D+ split and the operators involving only D0 are
identical to the ones obtained in the previous section by replacing φ with D0 .
Therefore we write here only the new operators.
3.3.1
Interaction of the doublet with SM gauge bosons
The dimension 4 operators can be divided in two types:
i) The terms involving both the neutral and charged components of D:
∗
D0 D+ Wµ− Z µ ,
0∗
(3.30)
(∂ D ) D Wµ− ,
∗
D0 (D µ D+ ) Wµ− ,
µ
+
(3.31)
(3.32)
which are not written here in hermitian form (for each operator, the
real and the imaginary part are therefore independent and hermitian).
These terms are relevant for the production of D0 from scattering of
D+ with nuclei or from decays of D+ if the mass difference between
the charged and the neutral component is larger than the W mass.
ii) The interaction terms between the charged elements of the multiplet:
∗
(D µ D+ )(Dµ D+ ) ,
D
D
+∗
+∗
+
µ
D Z Zµ ,
D W Wµ−
+∗ +
µ
+
+µ
(3.33)
(3.34)
,
(3.35)
D )Z ,
←
→
∗
i (D+ Dµ D+ ) Z µ .
(3.36)
∂µ (D
(3.37)
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
34
→
∗←
i ∂ µ (D+ Dµ D+ ) could also be present, but being a total divergence we
will ignore it.
The complete list up to and including dimension six operators is reported
in Appendix A.2.
3.3.2
Interaction of the doublet with SM fermions
The possible interaction terms between D+ and Dirac spinors ψ up to dimension six are
∗
∗
D+ D+ ψ̄γ 5 ψ ,
D+ D+ ψ̄ψ ,
∂µ (D
+∗
+
+∗
µ
D ) ψ̄γ ψ ,
→
∗←
i (D+ Dµ D+ ) ψ̄γ µ ψ ,
←
→
∗
/ψ,
D+ D+ ψ̄ i D
+
µ 5
∂µ (D D ) ψ̄γ γ ψ ,
→
∗←
i (D+ Dµ D+ ) ψ̄γ µ γ 5 ψ ,
←
→ 5
∗
/γ ψ,
D+ D+ ψ̄ i D
(3.38)
(3.39)
(3.40)
(3.41)
where ψ̄ and ψ are any two SM fermions such that their combination is
colorless and electrically neutral. Each operator should be divided by the
appropriate power of the cutoff scale Λ. For instance it is possible to have,
as in the φ case,
∗
D+ D+ µ̄e
and
←
→
∗
/c,
D+ D+ ū D
(3.42)
where the sum over colors in the second operator is understood. The dark
sector can break flavor universality and these operators should be included
for a true model independent analysis of the experimental constraints.
Type
Leptons
Quarks
SU (2)L
Y
SU (3)c
(νL , eL )
ecR
(uL , dL )
ucR
dcR
−1/2
1
1
3
3̄
3̄
1
1/6
−2/3
1/3
Table 3.1: SM fermion multiplets (one generation only) and their representations under the SM gauge groups. Fermions are denoted as left-handed Weyl
spinors.
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
35
The possible interaction terms between D0 , D+ and the SM Weyl fermions
up to dimension six are
∗
∗
D0 D+ eLi νj
∂µ (D0 D+ ) eLi σ µ ν̄j
→
∗←
i (D0 Dµ D+ ) eLi σ µ ν̄j ,
∗
D0 D+ eLi iσ µ Dµ ν̄j
∗
∗
D0 D+ ν̄j ēcRi
∂µ (D0 D+ ) νj σ µ ēcRi
→
∗←
i (D0 Dµ D+ ) νj σ µ ēcRi ,
∗
D0 D+ νj iσ µ Dµ ēcRi
∗
∗
∂µ (D0 D+ ) ucRi σ µ d¯cRj
→
∗←
i (D0 Dµ D+ ) ucRi σ µ d¯cRj ,
D0 D+ ucRi dLj
0∗
D D+ ucRi iσ µ Dµ d¯cRj
∗
∗
D0 D+ d¯cRj ūLi
∂µ (D0 D+ ) dLj σ µ ūLi
→
∗←
i (D0 Dµ D+ ) dLj σ µ ūLi ,
∗
D0 D+ dLj iσ µ Dµ ūLi
(3.43)
(3.44)
(3.45)
(3.46)
(3.47)
(3.48)
(3.49)
(3.50)
where i, j are generation indices and σ µ = (I, σ i ). Here ν denotes both
the SM left-handed neutrino νL and the charge conjugate of an hypothetical
right-handed neutrino, νRc . See Table 3.1 for the notation regarding the SM
fermions. Each operator should be divided by the appropriate power of the
cutoff scale Λ. The couplings of these operators can be complex.
3.3.3
Doublet-Higgs interaction
In the unitary gauge the interaction terms involving the Higgs field as well
as the charged component of the DM doublet, up to dimension six, are:
4
X
2
2
X
X
hn
hn
hn
∗
∗
∗
∗
D+ D+
an n−2 +(D+ D+ ) D0 D0
bn n + D + D +
cn n
Λ
Λ
Λ
n=1
n=1
n=1
µ
(D D
µ
∂ (D
+∗
+∗
!
+
2
X
µ
hn
+ ∗ + (∂ h)(∂µ h)
+
)(Dµ D )
dn n + e D D
Λ
Λ2
n=1
+
1
X
→ +
hn
hn
+∗←
D )(∂µ h)
fn n+1 + i (D Dµ D ) (∂µ h)
gn n+1 , (3.51)
Λ
Λ
n=0
n=0
+
1
X
where the coefficients are dimensionless and real.
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
3.3.4
36
Doublet self-interaction
The doublet has an electrically charged component, therefore in its derivative interactions both the partial as well as the covariant derivative appear.
This results in an unavoidable entanglement of the doublet’s self-interaction
terms and electromagnetic interaction terms. Here we show both the interactions with the purpose to list the entire set of self-interaction operators;
see Appendix A.2.2 for the list of all the interaction terms with the photon
field.
The Lagrangian terms including the doublet’s charged component D+ are,
up to dimension six:
∗
∗
∗
∗
∗
(D µ D+ )(Dµ D+ ) − m2+ D+ D+ + D+ D+ k1 D+ D+ + k2 D0 D0 +
1 +∗ + +∗ +
0∗ 0
0∗ 0 2
+∗ + 2
D D l1 (D D ) + l2 D D D D + l3 (D D ) +
Λ2
1
∗
∗
∗
∗
r1 (D µ D+ )(Dµ D+ ) D0 D0 + r2 D+ D+ (∂ µ D0 )(∂µ D0 ) +
2
Λ
→
∗
∗
∗
∗←
r3 ∂ µ (D+ D+ ) ∂µ (D0 D0 ) + i r4 (D+ Dµ D+ ) ∂ µ (D0 D0 ) +
→
s
∗←
∗
∗
r5 ∂ µ (D+ D+ ) Jµ + i r6 (D+ Dµ D+ ) J µ + 2 (D µ Dµ D+ )(D ν Dν D+ ) ,
Λ
(3.52)
→
∗←
where Jµ ≡ i D0 ∂µ D0 and the coefficients are dimensionless and real.
3.3.5
Interaction of the doublet with more than one
type of SM fields
We list here the independent interaction terms introduced by the presence
of an electrically positive DM charged state, beside the neutral one, up to
dimension six.
The terms involving both the doublet, the SM gauge bosons and the Higgs
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
37
are:
∗
D+ D+ Z µ Zµ h ,
D
+∗
+∗
(3.53)
D+ Z µ Zµ h2 ,
(3.54)
D D W Wµ− h ,
∗
D+ D+ W +µ Wµ− h2 ,
∗
λ1 D0 D+ Wµ− Z µ h + h.c. ,
∗
λ2 D0 D+ Wµ− Z µ h2 + h.c. ,
∗
∂µ (D+ D+ ) Z µ h ,
∗
∂µ (D+ D+ ) Z µ h2 ,
+
+µ
→
∗←
i (D+ Dµ D+ )Z µ h ,
→
∗←
i (D+ Dµ D+ )Z µ h2 ,
D
+∗
+
(3.55)
(3.56)
(3.57)
(3.58)
(3.59)
(3.60)
(3.61)
(3.62)
µ
D (∂µ Z ) h ,
(3.63)
D+ D+ (∂µ Z µ ) h2 ,
(3.64)
∗
0∗
λ3 (∂ µ D )D+ Wµ− h + h.c. ,
0∗
λ4 (∂ D )D Wµ− h2 +
∗
λ5 D0 (D µ D+ )Wµ− h +
∗
λ6 D0 (D µ D+ )Wµ− h2 +
∗
λ7 D0 D+ (D µ Wµ− ) h +
∗
λ8 D0 D+ (D µ Wµ− ) h2 +
µ
+
(3.65)
h.c. ,
(3.66)
h.c. ,
(3.67)
h.c. ,
(3.68)
h.c. ,
(3.69)
h.c. .
(3.70)
The terms involving both the doublet, the SM fermions and the gauge bosons
are:
∗
D+ D+ ψ̄γ µ ψ Zµ ,
D
+∗
+
(3.71)
µ 5
D ψ̄γ γ ψ Zµ ,
(3.72)
0∗
λ9 D D+ ψ̄γ µ ψ Wµ− + h.c. ,
0∗
λ10 D D+ ψ̄γ µ γ 5 ψ Wµ− + h.c. ,
0∗
+
0∗
+
µ
λ11 D D eLi σ ν̄j Zµ + h.c. ,
λ12 D D νj σ µ ēcRi Zµ +
∗
λ13 D0 D+ ucRi σ µ d¯cRj Zµ +
∗
λ14 D0 D+ dLj σ µ ūLi Zµ +
h.c. ,
(3.73)
(3.74)
(3.75)
(3.76)
h.c. ,
(3.77)
h.c. ,
(3.78)
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
38
while the terms involving both the doublet, the SM fermions and the Higgs
are:
∗
D+ D+ ψ̄ψ h ,
D
+∗
0∗
+
5
(3.79)
D ψ̄γ ψ h ,
(3.80)
+
(3.81)
λ15 D D eLi νj h + h.c. ,
0∗
λ16 D D+ ν̄j ēcRi h + h.c. ,
0∗
λ17 D D+ ucRi dLj h + h.c. ,
∗
λ18 D0 D+ d¯cRj ūLi h + h.c. .
(3.82)
(3.83)
(3.84)
Here all the λ’s are complex coefficients, and ψ̄ and ψ are any two SM Dirac
fermions such that their combination is colorless and electrically neutral.
Each operator should be divided by the appropriate power of the cutoff scale
Λ.
3.4
Triplet
The complex scalar triplet allows for new terms beyond the ones deduced
by replacing D0 with T 0 , D+ with T + and, of course, keeping in mind that
the terms involving only T 0 are the ones for φ. The terms involving T −
with T 0 and/or T + are obtained from the ones of the doublet by the formal
∗
replacement of D+ with T − . What we are still missing are therefore only
the interactions featuring oppositely charged DM fields, i.e. in the language
of the doublet both D+ and D− .
3.4.1
Interaction of the triplet with SM gauge bosons
As we have done for the other multiplets, in order to keep the main text
easier to read, we list here only the new dimension four operators linking the
plus with the minus components of T :
∗
∗
λ T + T − W +µ Wµ+ + λ∗ T − T + W −µ Wµ− ,
(3.85)
where λ can be a complex number. The complete list up to and including
dimension six operators is reported in Appendix A.3.
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
3.4.2
39
Interaction of the triplet with SM fermions
The triplet introduces the following possible interaction terms (up to dimension six) between the DM and the SM fermions:
∗
(3.86)
+∗
(3.87)
T + T − ēLi ēLj ,
T T − ecRi ecRj ,
∗
(Dµ T + )T − ecRi σ µ ēLj
∗
T + (Dµ T − ) ecRi σ µ ēLj
,
(3.88)
,
(3.89)
where i, j are generation indices and σ µ = (I, σ i ). See Table 3.1 for the
notation regarding the SM fermions, written here as Weyl spinors. Each
operator should be divided by the appropriate power of the cutoff scale Λ.
The couplings for these operators can be complex.
3.4.3
Triplet-Higgs interaction
In the unitary gauge the interaction terms involving the physical Higgs field
h as well as both the charged component of the DM triplet, up to dimension
six, are:
2
X
hn
+∗ + −∗ −
(3.90)
T T T T
kn n ,
Λ
n=1
where the coefficients kn are dimensionless and real.
3.4.4
Triplet self-interaction
As for the doublet, the presence of electrically charged components in the
triplet causes the electromagnetic covariant derivative to appear, when
needed in the interactions terms. Here we summarize the whole list of
self-interaction terms up to dimension six which do not include the covariant
derivative as well as the terms including up to two covariant electromagnetic
derivatives. In the appendices the reader will find the complete list of the
interactions with the photon to the same order in the fields.
The Lagrangian terms including both the triplet’s charged components T +
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
40
and T − are, up to dimension six:
∗
∗
∗
∗
a T + T + T − T − + ã (T + T 0 T − T 0 + h.c.)+
1 +∗ + −∗ −
∗
∗
∗
T T T T b1 T 0 T 0 + b2 T + T + + b3 T − T − +
2
Λ
1 +∗ −
∗
∗
∗
∗
T T b̃1 T 0 T + T 0 T + + b̃2 T + T + T − T + +
2
Λ
∗
∗
∗
∗
c̃1 T − T 0 T − T 0 + c̃2 T − T + T − T − + h.c. +
1 +∗ 0 −∗ 0 ˜ 0∗ 0 ˜ +∗ + ˜ −∗ −
T
T
T
T
d
T
T
+
d
T
T
+
d
T
T
+
h.c.
+
1
2
3
Λ2
→ + µ −∗ −
1
µ
+∗ +
−∗ −
+∗←
c
∂
(T
T
)
∂
(T
T
)
+
i
c
(T
D
T )+
2
µ T ) ∂ (T
1
µ
Λ2
→
→
→
∗
∗←
∗←
∗←
i c3 ∂ µ (T + T + ) (T − Dµ T − ) + c4 (T + D µ T + ) (T − Dµ T − )+
∗
∗
∗
∗
c5 (D µ T + )(Dµ T + ) T − T − + c6 T + T + (D µ T − )(Dµ T − ) +
1
∗
∗
∗
∗
ẽ1 (∂ µ T 0 )(∂µ T 0 )T + T − + ẽ2 ∂ µ (T 0 T 0 )(Dµ T + )T − +
2
Λ
∗
∗
∗
∗
+ ẽ3 ∂ µ (T 0 T 0 )T + (Dµ T − ) + ẽ4 T 0 T 0 (D µ T + )(Dµ T − ) + h.c. . (3.91)
The coefficients are dimensionless and real, apart from the ones denoted by
a tilde that are complex.
3.4.5
Interaction of the triplet with more than one
type of SM fields
We list here the independent interaction terms introduced by the simultaneous presence of electrically positive and negative charged DM states, up to
dimension six:
∗
(3.92)
λ2 T
+∗
T − W +µ Wµ+ h2 + h.c. ,
(3.93)
λ3 T
+∗
T − ecRi σ µ ēLj Zµ + h.c. ,
(3.94)
λ1 T + T − W +µ Wµ+ h + h.c. ,
+∗
−
µ
λ4 T T νj σ ēLi Wµ+ + h.c. ,
∗
λ5 T + T − ecRi σ µ ν̄j Wµ+ + h.c. ,
∗
λ6 T + T − dcRj σ µ ūcRi Wµ+ + h.c. ,
∗
λ7 T + T − uLi σ µ d¯Lj Wµ+ + h.c. ,
∗
λ8 T + T − ēLi ēLj h + h.c. ,
∗
λ9 T + T − ecRi ecRj h + h.c. .
(3.95)
(3.96)
(3.97)
(3.98)
(3.99)
(3.100)
Effective Operators for Dark Matter Detection
Chapter 3. Scalar Dark Matter Effective Field Theory
41
All the λ’s are complex coefficients. Each operator should be divided by the
appropriate power of the cutoff scale Λ.
Effective Operators for Dark Matter Detection
Chapter 4
Interference Patterns for
Isospin Violating Dark Matter
From an experimental perspective, the situation of direct detection searches
is unclear and still far from being settled. The DAMA/LIBRA [31,32], CoGeNT
[35,36] and CRESST−II [37] experiments detect events that can be attributed
to WIMP-nuclei collisions, while all the other searches find null evidence for
Dark Matter. In particular, the CDMS II [38, 107], XENON10/100 [39, 108]
and PICASSO [42] experiments impose severe constraints on the WIMPnucleons cross sections, excluding much of the parameter space allowed by the
experiments featuring a signal. Moreover, the DAMA, CoGeNT and CRESST
allowed regions do not coincide, thus making difficult a simultaneous interpretation of their signal in terms of Dark Matter. This situation is depicted in
the left panel of Fig. 4.1, assuming spin-independent DM-nucleus interaction
(for the rest of the chapter we will stick to this case).
Underlying the experimental analysis’ which determine the picture outlined above, there are few decisive assumptions on the particle physics properties of the interaction:
• The interaction amplitude does not depend on the DM velocity nor on
the exchanged energy between the DM and the nucleus. Actually, the
WIMP-nucleon interaction is always assumed to be of contact type,
which leads to a differential cross section that is constant or decreasing
with the exchanged momentum; in this context results are usually expressed in the zero transferred momentum limit, that means that one
approximates the full cross section with the leading order in a smallmomentum expansion.
• The DM couples equally to protons and neutrons.
42
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
43
10-39
fn  f p = 1
PICASSO
DAMA
-40
Σ p in cm2
10
CoGeNT
10-41
CRESS
T
XE
NO
CDMS
N1
00
XENON
10
10-42
8
9
10
11
12
DM mass in GeV
Figure 4.1: Favored regions and exclusion contours for spin-independent interactions in the standard case fn /fp = 1. The green contour is the 3σ favored
region by DAMA [109] assuming no channeling [47] and that the signal arises
entirely from Na scattering; the blue region is the 90% CL favored region by
CoGeNT1 [35]; the cyan contour is the 2σ favored region by CRESST−II, assuming only scattering off O and Ca, in equal amount [37]; the dashed line is
the exclusion plot from the CDMS II low energy analysis2 [107]; the red line is
the exclusion by PICASSO [42]; and the black and blue lines are respectively
the exclusion plots drawn from the XENON10 [108] and XENON100 [39] 2011
data.
With these assumptions, the WIMP-nucleus cross section has a simple mathematical expression which only depends on two parameters, usually taken to
be the WIMP mass and the WIMP-proton cross section σp (that is equal
to the WIMP-neutron cross section; we here neglect the small mass difference between proton and neutron). For a nucleus with Z protons and A − Z
neutrons, the WIMP-nucleus cross section is
σA =
µ2A 2
A σp ,
µ2p
(4.1)
1
After the work illustrated in this section was done, newer CoGeNT and XENON100
results became available, respectively Ref. [36] and [1].
2
The CDMS II low energy analysis is more constraining respect to the ‘standard’ CDMS
bound [38] for DM masses below 9 GeV.
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
44
with µA and µp the DM-nucleus and DM-nucleon reduced mass, respectively.
Notice the typical scaling of the cross section with A2 , which motivates our
neglecting the possible spin-dependent part of the interaction as it doesn’t
receive this important enhancement (see Appendix B). The WIMP-proton
cross section is defined as
µ2p
σp = |fp |2 ,
(4.2)
π
where fp is the WIMP-proton coupling.
Generally speaking although such interactions where protons and neutrons are indistinguishable do exist, e.g. in the case of a Higgs exchange,
other interactions can potentially distinguish protons from neutrons. Examples of the latter case are the photon exchange (that obviously couples only to
the protons, and therefore the cross section will scale as Z2 ), and the Z boson
exchange that couples protons and neutrons differently, having a scaling for
the cross section as (A−Z+ Z)2 where = 1−4 sin2 θW ' 0.08 (θW being the
Weak angle). However, due to the fact that in most stable nuclei the number
of protons is close to the one of neutrons, the discrepancy between DAMA,
CoGeNT, CRESST and CDMS, XENON remains when separately considering
WIMP-proton and WIMP-neutron interactions.
4.1
Isospin violating Dark Matter
Giving up the assumption of equal interaction between the WIMP and proton/neutron, the DM-nucleus cross section generalizes to
σA =
µ2
µ2A
|Zfp + (A − Z)fn |2 = σp 2A |Z + (A − Z)fn /fp |2 .
π
µp
(4.3)
One has now an extra parameter, which is the ratio between the WIMPneutron and the WIMP-proton couplings fn /fp . As it is apparent from
Eq. (4.3), a positive value for this parameter will increase the cross section, while a negative value can decrease it and even make it vanish with the
choice fn /fp = −Z/(A − Z). This parameter plays therefore an important
role as already pointed out by e.g. [52, 110], and can in principle drastically
change the interpretation of the experimental results in terms of bounds and
favored regions. Varying the value of fn /fp will in principle change these
constraints; we need therefore to translate the experimental results from the
case fn = fp to the more general case fn /fp 6= 1.
While experiments measure the event rate R, the collaborations conventionally set constraints on the WIMP-nucleon cross section σp , assuming
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
45
fn = fp , as shown in Fig. 4.1. The event rate for generic couplings fn and fp
is
X
µ2
R = σp
ηi A2i IAi |Z + (Ai − Z)fn /fp |2 ,
(4.4)
µ
p
isotopes
whereas the experimentally constrained rate (fn = fp ) can be cast in the
form
X µ2A
(4.5)
R = σpexp
ηi 2i IAi A2i .
µp
i
σpexp is defined by Eq. (4.5), and can be read directly from Fig. 4.1. Here ηi
is the abundance of the specific isotope Ai in the detector material, and IAi
contains all the astrophysical factors as well as the nucleon spin-independent
form factor FAi (ER ). For a given isotope we have
Z
Z vesc
mAi 2
F (ER ) .
(4.6)
IAi = NT nφ dER
d3 v f (v)
2vµ2Ai Ai
vmin
Here mAi is the mass of the target nucleus, NT is the number of target
nuclei, nφ is the local number density of DM particles, and f (v) is their
local velocity distribution. The velocity integration is limited between the
minimum velocity requiredpin order to transfer a recoil energy ER to the
scattered nucleus, vmin = mA ER /2µ2A , and the escape velocity from the
galaxy vesc .
Finally, equating Eqs. (4.4) and (4.5) yields the experimental constraints
on the generic WIMP-proton cross section σp (with arbitrary couplings fp
and fn ):
P
2
2
i ηi µAi IAi Ai
exp
(4.7)
σp = σp P
2 .
2
η
µ
I
|Z
+
(A
−
Z)f
/f
|
i
A
i
n
p
A
i
i
i
Provided that the factors IAi do not change significantly from one isotope to
another (as we checked), they cancel out from numerator and denominator.
Recently, it was observed in [52] that a relative strength of the couplings of
protons and neutrons fn /fp ' −0.71 can cause an overlap of the DAMA and
CoGeNT regions, leaving even a small region of phase space that evades the
tightest bounds coming from CDMS II, XENON10 and from XENON100 2011
data. This is visible in Fig. 4.2, where the region of the parameter space which
is allowed by all the experiments is shown in light red. The favored value for
the WIMP mass mφ ranges between 7.5 and 8.5 GeV, and the WIMP-proton
cross section is σp ' 2×10−38 cm2 = 2×10−2 pb; assuming mφ = 8 GeV, this
corresponds to |fp | ' 1.51×10−5 GeV−2 and |fn | ' 1.07×10−5 GeV−2 . There
is not much freedom to change fn /fp , since even small changes in the ratio
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
46
10-37
fn  f p = -0.71
Σ p in cm2
CoGeNT
10-38
DAMA
PICASSO
XENON
CDMS
10
CRESS
T
XEN
ON1
10-39
8
9
10
00
11
12
DM mass in GeV
Figure 4.2: Same as Fig. 4.1, but for the case fn /fp = −0.71. The CoGeNT
and DAMA overlapping region passing the constraints by CDMS and XENON
is shown in light red.
drive the DAMA/CoGeNT overlapping region within the excluded area by
either CDMS or XENON. For example, for fn /fp = −0.70 CDMS II excludes
the whole DAMA region, while for fn /fp = −0.72 the XENON10 line excludes
both DAMA/LIBRA and CoGeNT.
4.2
An application of the effective field theory: direct detection of a DM scalar singlet
We now provide an example in which we show how to use the operators listed
in Chapter 3 and in Appendix A in the framework of isospin violating Dark
Matter.
In preparing for a phenomenological study of DM direct detection, one
has to determine the possible interactions with nucleons. In principle we can
have different types of interactions which range from a photon exchange, via
for example a tiny dipole (or higher) moment for the WIMP, to an exchange
of an Higgs or Z boson, or still another particle, heavier than the DM, for the
effective description to be valid. Within the framework developed in Chapter
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
47
3 one can now provide a general interaction amplitude including all of the
relevant operators, ordered by power counting. We can therefore study in
full generality what are the interactions which can yield naturally the desired
value fn /fp ' −0.71, and exclude the cases that cannot reproduce it or that
require an unnatural amount of fine-tuning.
We find useful to distinguish between two different regimes: one in which
the dark sector, responsible for the non-renormalizable interactions described
in this work, emerges at an energy scale Λ lower than the Electroweak one,
and the other in which this hierarchy is inverted. The difference is that, in the
first case, the Electroweak gauge bosons and the Higgs field are integrated
out together with the unknown dark sector, while in the second case one
takes specifically into account the contribution of the W and Z bosons (and
of the Higgs, if light enough), and integrates out only the dark sector. Notice
that the possible interaction with massless degrees of freedom, such as the
photon, is always present regardless of the value of the UV cut-off.
4.2.1
Low energy dark sector
In the case the dark sector emerges below the Electroweak scale, MZ > Λ &
mφ , we can describe the DM-nucleon interactions with contact operators.
These can be deduced from the ones with quarks listed in Section 3.2.2; for
our purposes the relevant operators are
i
X h sN
vN 1
vN 2
∗
∗
µ
µ
φ φ N̄ N + 2 ∂µ (φ φ) N̄ γ N + 2 Jµ N̄ γ N , (4.8)
Lcontact =
Λ
Λ
Λ
N =n,p
←
→
where Jµ ≡ i (φ∗ ∂µ φ) and with s denoting the scalar couplings to nucleons (N̄ N ) and v the vector ones (N̄ γ µ N ). These interactions are all spinindependent. We did not consider pseudo-scalar (N̄ iγ 5 N ) and axial-vector
couplings (N̄ γ µ γ 5 N ), although they are possible, because they describe a
mainly spin-dependent interaction and therefore their contribution to the
cross section is negligible respect to the one given by the operators considered here (see Appendix B).
We also consider interaction with the photon; being φ overall electromagnetically neutral, this interaction can only arise as an effective operator. The
most relevant operator for this analysis is the first one in (3.5), leading to
the Lagrangian term
cγ
Lγ = e 2 Jµ (∂ν F µν ) .
(4.9)
Λ
This dimension 6, dipole-type term, appears naturally in any model of composite DM similar to Technicolor Interacting Massive Particles [53–55]: in
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
48
fact, although electrically neutral, the DM can carry an electric dipole moment if its component are electrically charged.
Using the effective Lagrangian Lcontact + Lγ , we obtain the following DM
couplings to proton and neutron:
sp 1
vp2
cγ
+ 2 + 4παEM 2 ,
Λ 2mφ
Λ
Λ
vn2
sn 1
+ 2 ,
fn =
Λ 2mφ
Λ
fp =
(4.10a)
(4.10b)
αEM = e2 /4π being the electromagnetic coupling. The vN 1 couplings do not
contribute by virtue of the conservation of the nuclear current N̄ γ µ N .
Since Λ & mφ , in order to obtain the fitting values for fp and fn we need the
dimensionless couplings in equations (4.10) to be of order O(10−2 ) ÷ O(10−1 ).
Given the large number of parameters involved, it is not an issue to obtain
the right value for fn /fp .
4.2.2
High energy dark sector
In case the scale of the dark sector, responsible for coupling φ to the SM,
is heavier than or of the order of the Electroweak scale, we must consider
explicitly the DM couplings to the heavy SM fields. In particular we will
consider the case Λ > mh , mh being the physical Higgs mass, that we take
here of the order of 100 GeV. The Lagrangian we must consider is therefore
Lcontact + Lγ + LWeak , with
LWeak = cZ1 Jµ Z µ + cZ2 φ∗ φ (∂µ Z µ ) + ch vEW φ∗ φ h ,
(4.11)
where vEW is the Higgs vacuum expectation value. These terms have been
classified in Section 3.2.1. Lcontact takes now into account only the physics
directly related to the dark sector, without the contribution of the SM fields.
Once again we did not consider pseudo-scalar and axial-vector nucleon couplings, because they give a smaller contribution with respect to the ones
considered here. For the same reason we neglect the axial-vector couplings
of the Z boson with the quarks in our calculation, and therefore we write the
Z-nucleon couplings as cZ,N N̄ γ µ N Zµ (see Appendix B). This effective interaction arises from the coupling of the vector current of the quarks with the Z
boson. Since the vector current is conserved, the coupling is only sensitive to
those quarks in the sea which give the nucleon its quantum numbers, i.e. the
valence quarks; therefore we have
cZ,p = 2vZ,u + vZ,d ,
cZ,n = vZ,u + 2vZ,d ,
(4.12)
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
49
where again v stands for only the vector couplings:
g
1 4 2
g
1 2 2
vZ,u =
− sin θW ,
vZ,d = −
− sin θW .
2 cos θW 2 3
2 cos θW 2 3
(4.13)
mp
The Higgs-nucleon Lagrangian interaction is vEW f N̄ N h (see Appendix B),
with mp the nucleon mass (we neglect the small mass difference between
proton and neutron) and f ' 0.3 [111–113].
We find then
ch f mp
sp 1
vp2
cZ1 cZ,p
cγ
+
+
+ 2 + 4παEM 2 ,
2
2
MZ
2mφ mh
Λ 2mφ
Λ
Λ
ch f mn
sn 1
vn2
cZ1 cZ,n
+
+
+ 2 .
fn = −
2
2
MZ
2mφ mh
Λ 2mφ
Λ
fp = −
(4.14a)
(4.14b)
As for the couplings vN 1 , also cZ2 doesn’t contribute to the cross section due
to the conservation of the nuclear current.
For a not too heavy dark sector, 1 TeV > Λ > mh , supposing no big difference
between the numerical values of the dimensionless couplings,
cZ1 cZ,p sp 1
+
,
MZ2
Λ 2mφ
cZ1 cZ,n sn 1
fn ' −
+
,
MZ2
Λ 2mφ
fp ' −
(4.15a)
(4.15b)
and the couplings need to be of order O(10−1 ) in order to fit the experimental
data. Also in this case it is possible to obtain the fitting value for fn /fp
(notice that cZ,p and cZ,n have opposite sign, and that also sp , sn can differ
by the sign).
For a heavier dark sector Λ > 1 TeV, instead,
fp ' −
cZ1 cZ,p
,
MZ2
fn ' −
cZ1 cZ,n
,
MZ2
(4.16)
so that if cZ1 6= 0 it is not possible to fit the data, since fn /fp ' cZ,n /cZ,p '
−10. The only possibility is for the DM not to be coupled to the Z boson.
In this case, for Λ < 10 TeV the scattering amplitude due to the contact
operator is comparable to the one due to Higgs exchange (still in the assumption that mh ∼ 100 GeV), and therefore there is still a possibility to
get fn /fp ' −0.71. For larger Λ, instead, getting this value is only possible
if also the DM coupling to the Higgs is 0, i.e. if the DM couples to quarks
only via dark interactions.
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
4.3
50
Examples of interfering Dark Matter
While the effective theory applied in the previous section is very general,
and spanning such a large parameter space, it only rarely allows to draw
firm conclusions and set significative constraints. It is therefore instructive
to focus on some concrete examples to understand whether models of isospin
violating Dark Matter can actually be realized in simple setups and still be
viable.
In this section we discuss therefore some examples of quantum interference between two possible channels of the DM-nucleus interaction, and check
whether such models are able to fit the experimental data. We first discuss a
model of scalar DM, featuring interference of two SM mediators: the photon
and the Higgs boson. In the remaining examples we focus on fermionic DM
and determine under which conditions it is possible to achieve interference
introducing one or two beyond the SM mediators, namely Z 0 and extra Higgs
bosons.
4.3.1
Interference between photon and Higgs
We here take a special case of the effective theory of a scalar DM developed
in the previous section, namely we consider a DM scalar singlet φ interacting
with nuclei via exchange of a photon or a Higgs boson. Such a DM candidate
can naturally arise, for instance, in models of composite DM (see discussion
after Eq. (4.9); a coupling to the Higgs boson can be natural in Technicolor
theories in which also this particle is composite).
The Lagrangian of the model, already appeared in Ref. [53], is
2
dB
vEW
)φ∗ φ − dH H † H φ∗ φ + 2 e Jµ ∂ν F µν , (4.17)
2
Λ
√EW
in the unitary gauge with
where H is the Higgs doublet, H = 0, h+v
2
vEW = 246 GeV. Notice that the DM-photon interaction, although in principle of long-range type, results here in an interaction amplitude that does
not depend on the exchanged energy, as in the case of short-range contact
interactions. This fact entitles us to use the zero momentum transfer limit.
With the DM-nucleus interaction cross section specified by Eq. (4.3), the
effective DM-proton and DM-neutron couplings are therefore
L = ∂µ φ∗ ∂ µ φ − (m2φ − dH
fn = dH f
mp
,
2m2h mφ
f p = fn −
4παEM dB
,
Λ2
(4.18)
exactly as in Eq. (4.14) with ch = −dH , cγ = dB and setting to zero all the
remaining couplings. Fixing fn /fp ' −0.71 and σp ' 2 × 10−38 cm2 allows
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
51
us to determine both dB and dH , provided we also fix the Higgs mass and
the scale Λ:
dB
' 2.8 × 10−4 GeV−2 ,
2
Λ
dH
' 6.1 × 10−4 GeV−2 .
2
mh
(4.19)
Assuming a Higgs mass of the order of O(100 GeV) and Λ ' vEW we find
dB ∼ dH ∼ O(1) − O(10). Therefore interfering DM emerging from a new
strong dynamics at the Electroweak scale can resolve the experimental puzzle
via the isospin violation mechanism.
4.3.2
Interference between Z and Z 0
Here and for the rest of this section we consider a spin-1/2 DM field χ. The
Z-χ and Z-nucleons interaction Lagrangian, including only renormalizable terms,
reads
LZ =
g
Zµ χ̄(vχ − aχ γ 5 )γ µ χ +
2 cos θW
g
Zµ p̄ γ µ (vp − ap γ 5 )p + n̄ γ µ (vn − an γ 5 )n ,
2 cos θW
(4.20)
where the Z-DM couplings vχ (vector) and aχ (axial-vector) are normalized to the
usual Weak coupling strength. p and n refer respectively to protons and neutrons
and the Z-nucleon vector and axial-vector couplings are
vp =
1
− 2 sin2 θW ,
2
vn = −
1
,
2
ap = 0.68 ,
an = −0.59 ,
(4.21)
where we have made use of Eq. (4.13) to determine vp , vn and we used the numerical
values from [49, 113] to estimate ap and an , as explained in Appendix B. We are
however not concerned with the axial-vector couplings, since their contribution
to the cross section is mainly spin-dependent and it is therefore suppressed with
respect to the spin-independent one given by the vector couplings (see Appendix
B).
Similarly the Z 0 -χ and Z 0 -nucleons interaction Lagrangian can be written as
LZ 0 =
g
Z 0 χ̄(vχ0 − a0χ γ 5 )γ µ χ +
2 cos θW µ
g
Zµ0 p̄ γ µ (vp0 − a0p γ 5 )p + n̄ γ µ (vn0 − a0n γ 5 )n .
2 cos θW
(4.22)
As for the Z, also in this case the axial-vector couplings contribution to the cross
section is negligible. Possible constraints from colliders on a light Z 0 can be safely
avoided assuming a leptophobic Z 0 . In this case in fact the LEP bounds do not
apply, and at present LHC searches in hadronic channels only set bounds for Z 0
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
52
masses above 1 TeV, see e.g. Ref. [114]. Since as we will see we require the Z 0 to
be lighter than 1 TeV, we can evade all present collider constraints.
Using Eqs. (4.20) and (4.22), we can write the DM-nucleus spin-independent
cross section at zero momentum transfer as we already did in Eq. (4.3),
σA =
2G2F µ2A
|Zfp + (A − Z)fn |2 .
π
(4.23)
Here GF is the Fermi constant and the dimensionless couplings to protons and
neutrons are
fp = vχ vp + vχ0 vp0
m2Z
,
m2Z 0
fn = vχ vn + vχ0 vn0
m2Z
.
m2Z 0
With σp recovered from Eq. (4.23) with the choice A = Z = 1 we get
r
σp π
|fp | =
,
2G2F µ2p
(4.24)
(4.25)
which determines fp as a function of σp ' 2 × 10−38 cm2 , while the condition
fn /fp = −0.71 fixes fn . Dividing by the known parameters vp = 0.055 and vn =
−0.5 we get the two constraints
m 0 −2
vp0 m2Z
Z
0 0
=
v
+
15
v
v
= ±17 ,
χ
χ p
vp m2Z 0
100 GeV
m 0 −2
v 0 m2
Z
vχ + vχ0 n 2Z = vχ − 1.7 vχ0 vn0
= ±1.3 .
vn m Z 0
100 GeV
vχ + vχ0
(4.26a)
(4.26b)
The Z-χ coupling vχ can be constrained using the measurements of the Z decay
width into invisible channels. The LEP experiment set strict limits on the number
of SM neutrinos, i.e. Nν = 2.984 ± 0.008 [20]. The error in the measurement
can be used to constrain non-SM contributions to the Z decay width. Using the
uncertainty in the LEP result δLEP = 0.008, this yields
vχ2 β(3 − β 2 ) + 2a2χ β 3 < δLEP ,
(4.27)
q
where β = 1 − 4m2χ /m2Z is the velocity factor. Assuming a DM mass of ' 8
GeV, with no axial-vector coupling (aχ = 0), the vector coupling vχ can assume
its maximal allowed value |vχ | < 0.063, while for aχ = vχ this constraint gives
|vχ | < 0.045. Taking into account this strong bound in Eq. (4.26), it is evident that
the bulk contribution to isospin violation is due to the Z 0 alone. Therefore, while
a Z 0 could produce the right amount of isospin violation on its own, interference
with the SM Z boson is not relevant to explain the data.
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
4.3.3
53
Interference between Z 0 and Higgs
As we have seen, LEP data set a severe constraint on the couplings of a light DM
particle to the SM Z boson. For this reason, it is not possible to obtain a DMproton cross section of the desired size when only the Z mediates the interaction.
For the same reason, in the possibility the DM can exchange different mediators
with the nucleons, the Z contribution will be subdominant. Therefore, a Z-Higgs
interference would not explain the experimental data. However, as we will show
here, interference between a Z 0 and the SM Higgs is a viable possibility.
The relevant Lagrangian describing the interactions of the physical Higgs h
with the DM χ and with the nucleons (see Appendix B) is
Lh = mχ χ̄χ − h χ̄(dh + ah γ 5 )χ −
mp
f h (p̄p + n̄n) ,
vEW
(4.28)
where dh and ah are the dimensionless scalar and pseudo-scalar h-χ couplings
respectively. We have explicited a mass term for χ to point out that it does not
need to be generated by the vacuum expectation value of the Higgs field (we assume
χ to be a vector-like state). The use of the effective Lagrangian is practical and
natural as long as the origin of the Electroweak symmetry breaking and the actual
nature of the Higgs field are still unknown, and we are mostly interested in the
DM phenomenology (see discussion in Chapter 3).
Combining the scalar interaction from this Lagrangian, with the vector one
for the Z 0 as it appears in (4.22), we get the DM-nucleus spin-independent cross
section as in Eq. (4.23), where now the dimensionless couplings to protons and
neutrons read
fp = vχ0 vp0
m2Z
f mp vEW
− dh
,
2
mZ 0
m2h
fn = vχ0 vn0
m2Z
f mp vEW
− dh
.
2
mZ 0
m2h
(4.29)
As for the Z-Z 0 case, the pseudo-scalar and pseudo-vector couplings of the DM
with the Higgs and the Z 0 respectively lead to negligible contributions to the cross
section compared to the scalar and vector ones investigated here. The constraints
are now (see Eq. (4.25))
|fp | = 0.92 ,
fn = −0.71 fp = ±0.65 ,
(4.30)
corresponding to
m
−2
mZ 0 −2
h
− 8.3 × 10−3 dh
= ±1.1 ,
100 GeV
100 GeV
m 0 −2
m
−2
h
Z
vχ0 vn0
− 8.3 × 10−3 dh
= ∓0.78 .
100 GeV
100 GeV
vχ0 vp0
(4.31)
(4.32)
If all the couplings are of order unity and mZ 0 , mh ∼ 100 GeV, the Higgs contribution to the interference is negligible, and the Z 0 has to directly account for
the isospin violation needed to get the desired value of fn /fp . A substantially
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
54
lighter Higgs, around 50 GeV with a coupling dh in the range 5 − 10, can lead to
a phenomenologically viable interference. Note that such a light Higgs-like state
is not immediately ruled out by collider experiments since this state has new decay modes, e.g. to two DM particles which are not accounted for in the SM (see
e.g. [115]).
4.3.4
Interference within the two Higgs doublet model
We now discuss the possibility of having interference between two Higgses in a
two Higgs doublet model (see Ref. [116] for a recent review). We take one of the
Higgs fields to couple to up type quarks, while the other couples to down type
quarks. Two Higgs doublets count a total of 8 degrees of freedom, 3 of which are
the Goldstone bosons eaten up by the Weak gauge bosons; the remaining degrees
of freedom divide into two neutral and one charged scalars, plus a pseudoscalar.
As usual we find convenient to employ the effective theory at low energy, rather
than a renormalizable model. The reason is that many models that are different
at high energies might behave identically at low energies, so that our description
encompassess a large class of theories. Therefore we only consider here the lightest
states, assuming these are the two neutral scalars that we dub h1 and h2 . We also
make the customary assumption that the CP symmetry is preserved by the Higgs
sector. Our fermionic DM candidate χ has Yukawa-type interactions with the two
Higgs fields, but being a vector-like state its mass is not required to arise from the
Higgs vevs. The interaction Lagrangian is
L2h =λDM
h1 χ̄χ + λp1 h1 p̄p + λn1 h1 n̄n+
1
λDM
h2 χ̄χ + λp2 h2 p̄p + λn2 h2 n̄n ,
2
(4.33)
where we also wrote the Higgses effective interactions with the proton p and neutron n (see Appendix B). The nucleon couplings are
cos θ X
sin θ X
hp|mqu q̄u qu |pi −
hp|mqd q̄d qd |pi ,
v1 q
v2 q
u
d
cos θ X
sin θ X
λn1 =
hn|mqu q̄u qu |ni −
hn|mqd q̄d qd |ni ,
v1 q
v2 q
u
d
cos θ X
sin θ X
p
λ2 =
hp|mqu q̄u qu |pi +
hp|mqd q̄d qd |pi ,
v1 q
v2 q
u
d
sin θ X
cos θ X
λn2 =
hn|mqu q̄u qu |ni +
hn|mqd q̄d qd |ni ,
v1 q
v2 q
λp1 =
u
(4.34a)
(4.34b)
(4.34c)
(4.34d)
d
where the sums over up type (qu ) and down type (qd ) quarks account for the scalar
quark currents within the nucleons. v1 and v2 are the vacuum expectation values
2 /2 = (174 GeV)2 in
of the two Higgs fields, which obey the relation v12 + v22 = vEW
Effective Operators for Dark Matter Detection
Chapter 4. Interference Patterns for Isospin Violating Dark Matter
55
order to provide the right masses to the Weak gauge bosons. θ is the mixing angle
which diagonalizes the Higgs system, and here is a free parameter. The matrix
elements hp, n|mq q̄q|p, ni in (4.34) are obtained in chiral perturbation theory, when
dealing with light quarks, using the measurements of the pion-nucleon sigma term
[117], and in the case of heavy quarks, from the mass of the nucleon via the trace
anomaly [111]. The experimental uncertainties, especially in the pion-nucleon
sigma term, cause large uncertainties in the value of these matrix elements. This
concerns us because, as long as λpi and λni are not identical, isospin violation can
always be guaranteed. To evaluate the matrix elements we follow Ref. [49] which
makes use of the results found in [111, 117]:
X
X
hp|mqd q̄d qd |pi ' 417 MeV , (4.35a)
hp|mqu q̄u qu |pi ' 105 MeV ,
qd
qu
X
qu
hn|mqu q̄u qu |ni ' 100 MeV ,
X
hn|mqd q̄d qd |ni ' 426 MeV . (4.35b)
qd
The spin-independent DM-nucleus cross section reads then
σA =
where
fp =
µ2A
|Zfp + (A − Z)fn |2 ,
π
p
p
λDM
λDM
2 λ2
1 λ1
+
,
m2h1
m2h2
fn =
n
n
λDM
λDM
1 λ1
2 λ2
+
.
m2h1
m2h2
(4.36)
(4.37)
Substituting the couplings from (4.34), (4.35) into (4.37), and imposing the fitting
values for mχ , fn /fp and σp , we get the following constraint equations for the
unknown parameters:
v1 −1
v1 mh1 −2
DM
λ1
cos θ − 4.0 sin θ
+
vEW
v2
100 GeV
v2 mh2 −2
v2 −1
DM
cos θ + 0.25 sin θ
= ±3.5 × 102 , (4.38)
4.0 λ2
vEW
v1
100 GeV
λDM
1
−1 v1 mh1 −2
cos θ − 4.3 sin θ
+
v2
100 GeV
v2 −1
v2 mh2 −2
cos θ + 0.23 sin θ
= ∓2.6 × 102 .
vEW
v1
100 GeV
v1
vEW
4.3 λDM
2
(4.39)
For natural values of v1 and v2 , i.e. of the order of vEW , and for mh1 and mh2
of the order of 100 − 1000 GeV, the DM couplings to the Higgs fields need to be
of O(103 ) to fit the data. This large couplings are of course unnatural. Thus we
must conclude that the simple, yet quite generic template considered here within
the two Higgs doublet model, can not account for the desired amount of isospin
violation in DM direct detection experiments.
Effective Operators for Dark Matter Detection
Chapter 5
Magnetic Moment DM in
Direct Detection Searches
In this chapter we perform a thorough investigation of the effects of a magnetic
dipole moment of the DM on the direct detection experiments. This is a very
peculiar type of Dark Matter, for at least two reasons: first because it interacts with
photons through its dipole momentum, thus being not completely dark. Second,
such an interaction is of long-range type and therefore is very different from the
contact interactions usually considered in direct detection analyses (see discussion
in Chapter 1). Long-range interactions modify substantially the recoil spectrum
with respect to a contact interaction, because the interaction probability is higher
at lower energies and therefore low-energy threshold detectors like CoGeNT and
DAMA are especially sensitive. For this reason, the hope exists in this framework
to get a better agreement among the various direct detection experiments. Given
the difficulties in accommodating the experimental outcomes within the contact
interaction paradigm, an investigation of the properties of this DM candidate is
therefore in order. Previous works on the same subject are for instance [118–131].
In this chapter we will show how the DM magnetic moment interaction affects
the interpretation of the different direct detection experiments. In Sec. 5.1 we
present the DM interaction Lagrangian and provide some considerations concerning the scattering of DM off nuclei. This will allow us to determine the nuclear
recoil rate for a given experiment. In Sec. 5.2 we show how the experimental
favored regions and constraints are modified with respect to the naı̈ve contact interaction case. In Sec. 5.3 we describe the experimental data set we use and our
statistical analysis performed to determine the favored regions and constraints for
the DM candidate envisioned here. We summarize these results in Sec. 5.4. We
determine the associated thermal relic density in Sec. 5.5, and in Sec. 5.6 we report
the constraints imposed by indirect searches, colliders and by the observations of
compact stars.
56
Chapter 5. Magnetic Moment DM in Direct Detection Searches
5.1
5.1.1
57
The event rate
Kinematics
When a DM particle scatters off a nucleus, depending on the DM properties, one
can envision at least two distinct kinematics, the elastic and the inelastic. The
elastic scattering is represented by
χ + N(A, Z)at rest → χ + N(A, Z)recoil ,
(5.1)
χ + N(A, Z)at rest → χ0 + N(A, Z)recoil .
(5.2)
while the inelastic is
In (5.1) and (5.2), χ and χ0 are two DM particle states, and A, Z are respectively
the mass and atomic numbers of the nucleus N. In the detector rest frame, a DM
particle with velocity v and mass mχ can scatter off a nucleus of mass mA , causing
it to recoil. The minimal velocity providing a recoil energy ER is:
s
mA ER
µA δ
vmin (ER ) '
1+
,
(5.3)
mA ER
2µ2A
where µA is the DM-nucleus reduced mass and δ = m0χ − mχ is the mass splitting
between χ and χ0 , and the equation above holds for δ m0χ , mχ . Elastic scattering
occurs for δ = 0, while δ 6= 0 implies inelastic scattering. Here we will only consider
the case of elastic scattering.
5.1.2
Model and differential cross section
We concentrate on the possibility of having a massless mediator of the DM-nuclei
interaction able to yield long-range interactions absent in contact interactions usually assumed. The obvious candidate mediator in the SM is the photon while other
exotic possibilities can be also envisioned such as a dark photon, see e.g. [106].
Assuming that the Dark Matter is either a neutral spin-0 or spin-1/2 one can
then use the language of the effective theories to select the relevant operators.
Stability of the specific candidate can be easily ensured by charging it under an
unbroken global symmetry. The most popular choices are either a new U (1) or a
Z2 depending on the reality of the specific candidate DM field.
According to the classification of the possible interaction operators made in
Chapter 3, there is only one gauge-invariant operator coupling a DM complex
boson φ to a photon up to dimension six, namely
←
→
i(φ† ∂µ φ)∂ν F µν .
(5.4)
DM-nucleus scattering mediated by this operator has been already studied in
Sec. 4.3.1, where it was shown that it actually leads to a contact interaction and
will henceforth not be used here.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
58
On the other hand, considering a fermionic DM χ one can show that the only
gauge-invariant couplings to the photon, up to dimension five, are the magnetic
and electric moment-mediated interactions:
1
int
LM
= − λχ χ̄σµν χF µν ,
2
1
LEint = − i dχ χ̄σµν γ 5 χF µν .
2
(5.5)
int and L int vanish identically, therefore χ has
For a Majorana fermion both LM
E
to be a Dirac fermion. λχ and dχ are the magnetic and electric dipole moment,
respectively, and are usually expressed in units of e × cm. The scales ΛM ≡ e/λχ
and ΛE ≡ e/dχ can be interpreted as the energy scales of the underlying interaction
responsible for the associated operators to arise. For instance, in models in which
the interactions (5.5) are explained by the DM being a bound state of charged
particles, the Λ’s could represent the compositeness scale.
The differential cross sections for elastic scattering are (see e.g. [122, 123])
SI (v, E )
dσM
dσ SD (v, ER )
dσM (v, ER )
R
=
+ M
=
dER
dER
dER
)
( 2
αEM λ2χ
2m
+
m
m
E
λ̄
E
A
χ
A
nuc
R
R
2
F 2 (ER ) ,
Z2 1 − 2
FSI
(ER ) +
ER
v
2mA mχ
λp
v 2 3m2p SD
(5.6)
dσE (v, ER )
αEM Z2 2 2
=
d F (ER ) ,
dER
ER v 2 χ SI
(5.7)
where v is the speed of the DM particle in the Earth frame, αEM = e2 /4π ' 1/137
is the fine structure constant, λp = e/2mp is the nuclear magneton and FSI (FSD )
denotes the spin-independent (spin-dependent) nuclear form factor which takes
into account the finite dimension of the nucleus. Here

1/2
X
Si + 1 
λ̄nuc = 
fi λ2i
,
(5.8)
Si
isotopes
is the weighted dipole moment of the target [123], where fi , λi and Si are respectively the abundance, nuclear magnetic moment and spin of the i-th isotope; the
values for these quantities are taken from [132, 133] and agree exactly with the
values provided in Fig. 1 of [123].
The differential cross section in (5.6) features both a spin-independent (SI) and
a spin-dependent (SD) part. The SI part arises from the DM magnetic moment
interaction with the protons electric charge. Neglecting for a moment the form
−1
factor, this contains two terms: an energy dependent term with a ER
drop-off of
the cross section, and an energy independent one. In contrast the common contact
interactions only feature the constant term, typically with the 1/v 2 dependence
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
59
on the DM velocity; the SD part, arising from the dipole-dipole DM-nucleon interactions, is of this kind. Notice that, for low enough energies (that might be
also below threshold for some experiments, in principle), the interaction is always
−1
SI due to the ER
divergence. As the recoil energy rises, the interaction becomes
mostly SD for target nuclei with large magnetic moment, namely 19 F, 23 Na and
127 I (λ̄
nuc /λp = 4.55, 2.86, 3.33 respectively); the SD term is instead negligible for
all the other nuclei.
The differential cross section in (5.7), instead, has only an energy dependent
term. We included here only the electric dipole-charge interaction, that is expected
to be dominant. We observe that the cross section in the electric case is enhanced
with respect to the magnetic one by a factor 1/v 2 ∼ 106 translating in a value for
dχ circa 103 times lower than the one for λχ , when one tries to fit the experiments.
This is confirmed for instance by Ref. [122], where the authors find that, in order to
fit the CoGeNT data alone, a ΛM of the order of the TeV is needed for a magnetic
moment interaction, while for electric moment interaction ΛE is around the PeV.
Given that such a high scale is hardly reconcilable with other attempts to study
DM and in general new physics, we will treat from now on only the magnetic
int .
dipole moment interaction LM
As for FSI and FSD , we use the nuclear form factors provided in Ref. [134]. For
the SI interaction we have checked that FSI matches with the standard Helm form
factor [135]. We recall that all the parameters used in the parameterization of the
nuclear form factors may be affected by sizable uncertainties.
5.1.3
Nuclear recoil rate
The differential recoil rate of a detector can be defined as:
Z
dR
dσM (v, ER )
= NT
v dnχ ,
dER
dER
(5.9)
where NT = NA /A is the total number of targets in the detector (NA is the
Avogadro’s number) and dnχ is the local number density of DM particles with
velocities in the elemental volume d3 v around ~v . This last factor can be expressed
as a function of the DM velocity distribution fE (~v ) in the Earth frame, which
is related to the DM velocity distribution in the galactic frame fG (w)
~ by the
galilean velocity transformation fE (~v ) = fG (~v + ~vE (t)); here ~vE (t) is the timedependent Earth (or detector) velocity with respect to the galactic frame. The
prominent time-dependence (on the time-scale of an experiment) comes from the
annual rotation of the Earth around the Sun, which is the origin of the annual
modulation effect of the direct detection rate [29, 30]. More specifically:
~vE (t) = ~vG + ~vS + ~v⊕ (t) .
(5.10)
The galactic rotational velocity of our local system ~vG and the Sun’s proper motion
~vS are basically aligned and their absolute values are vG ≡ v0 = 220 ± 50 km/s and
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
60
vS = 12 km/s, while the Earth rotational velocity ~v⊕ (t) has a size v⊕ = 30 km/s,
period of 1 year and phase such that it is aligned to ~vG around June 2nd and it is
inclined of an angle γ ' 60◦ with respect to the galactic plane. More details can
be found, for instance, in Ref. [136]. Summarizing:
dnχ = nχ fE (~v ) d3 v ,
(5.11)
where nχ = ξχ ρ0 /mχ is the local DM number density in the Galaxy and is determined by the local Dark Matter density ρ0 and, in general, by a scaling factor ξχ
which accounts for the possibility that the specific DM candidate under consideration does not represent the whole amount of DM. Here we assume ξχ = 1. In
Eq. (5.11) the velocity distribution function needs to be properly normalized: this
can be achieved by requiring that in the galactic frame
Z
d3 v fG (~v ) = 1 ,
(5.12)
v6vesc
where vesc denotes the escape velocity of DM particles in the Milky Way. For
definiteness, we will adopt here vesc = 650 km/s.
When considering the differential cross section given in equation (5.6), the rate
of nuclear recoils reduces to
2
ξχ ρ0 αEM λ2χ 2
dR
2
2
(t) = NA
Z GSI (vmin , t)FSI
(ER ) + λ̄nuc /λp GSD (vmin , t)FSD
(ER ) ,
dER
A mχ ER
(5.13)
where
2mA + mχ
I1 (vmin (ER ), t)
GSI (vmin (ER ), t) = I(vmin (ER ), t)
− ER
, (5.14)
I(vmin (ER ), t)
2mA mχ
GSD (vmin (ER ), t) = I(vmin (ER ), t)
mA ER
,
3m2p
(5.15)
and
fE (~v )
I(vmin , t) =
d v
,
v>vmin (ER ) v
Z
3
I1 (vmin , t) =
Z
d3 v v fE (~v ) ,
(5.16)
v>vmin (ER )
with vmin (ER ) given by Eq. (5.3). The detection rate is function of time through
the velocity integrals I(vmin , t) and I1 (vmin , t) as a consequence of the annual motion of the Earth around the Sun. Their actual form depends on the velocity
distribution function of the DM particles in the halo. In this paper we will consider an isothermal sphere density profile for the DM, whose velocity distribution
function in the galactic frame is a truncated Maxwell-Boltzmann:
fG (~v ) =
(v0
√
exp(−v 2 /v02 )
.
2 /v 2 )
π)3 erf(vesc /v0 ) − 2v03 π(vesc /v0 ) exp(−vesc
0
(5.17)
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
61
Under this assumption, and defining the normalized velocities η ≡ (vG +vS )/v0 ≡
v /v0 , ηE (t) ≡ vE (t)/v0 , ηmin (ER ) ≡ vmin (ER )/v0 and ηesc ≡ vesc /v0 , the velocity
integrals can be written analytically as [122]
I(ηmin , t) =
1
1
2
[erf(η+ ) − erf(η− )] − √
(η+ − η− ) e−ηesc
2 v0 ηE (t)
π v0 ηE (t)
(5.18)
and
I1 (ηmin , t) = v0
η
1
√ −
+√
2 π ηE (t)
π
2
−η−
e
−
η
1
√ +
−√
2 π ηE (t)
π
e
2
−η+
v0
1 + 2ηE2 (t) [erf(η+ ) − erf(η− )]
(5.19)
4 ηE (t)
v0
1 2
3
3
− √ 2+
e−ηesc ,
(ηmin + ηesc − η− ) − (ηmin + ηesc − η+ )
3ηE (t)
π
+
where η± (ER ) = min(ηmin (ER ) ± ηE , ηesc ).
Since in Eq. (5.10) the rotational velocity of the Earth around the Sun v⊕ , is
relatively small compared to the main contribution represented by vG + vS , we can
approximate ~vE (t) with its component directed toward the galactic center. We can
then write [136]
ηE (t) ' η + ∆η cos [2π(t − φ)/τ ] ,
(5.20)
where ∆η = v⊕ cos γ/v0 , with ∆η η , and where φ = 152.5 days (June 2nd ) is
the phase and τ = 365 days is the period of the Earth motion around the Sun. By
means of Eq. (5.20) we can then expand the recoil rate, assuming that the velocity
distribution is not strongly anisotropic:
dR
∂ dR dR +
∆η cos [2π(t − φ)/τ ] .
(5.21)
(t) '
dER
dER ηE =η ∂ηE dER ηE =η
To properly reproduce the recoil rate measured by the experiments, we should
take into account the effect of partial recollection of the released energy (quenching), and the energy resolution of the detector:
Z
X dRi dR
E0
0
0
(Edet ) = dE K(Edet , E )
ER =
.
(5.22)
dEdet
dER
qi
i
Here the index i denotes different nuclear species in the detector, Edet is the
detected energy and qi are the quenching factors for each of the nuclear species.
The function K(Edet , E 0 ) reproduces the effect of the energy resolution of the
detector; as is generally done, we assume for it a Gaussian behavior.
Finally, the recoil rate of Eq. (5.22) must be averaged over the energy bins
of the detector. For each energy bin k of width ∆Ek we therefore define the
unmodulated components of the rate S0k and the modulation amplitudes Smk as:
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
Z
1
dR S0k =
dEdet
,
∆Ek ∆Ek
dEdet ηE =η
Z
∂ dR 1
dEdet
Smk =
∆η .
∆Ek ∆Ek
∂ηE dEdet ηE =η
62
(5.23)
(5.24)
S0k and Smk are the relevant quantities that we use for the analysis of the experiments which address the annual modulation effect, namely DAMA and CoGeNT.
For the other experiments, only the S0k are relevant.
5.2
Theoretical predictions
We attempt here an analytical study of the effect of the magnetic moment interaction on the observed differential rate. More precisely we provide a simple
comparison between the results arising from the interaction studied here and the
standard picture, i.e. the spin-independent coherent contact interaction, shown in
Fig. 5.1. To this aim, Eq. (5.13) can be rewritten as
ξχ ρ0 mA 2
dR
2
(t) = NA
A αEM λ2χ Θ(ER )I(ηmin (ER ), t)FSI
(ER ) ,
dER
A mχ 2µ2p
(5.25)
where µp is the DM-nucleon reduced mass, αEM λ2χ plays the role of the spinindependent DM-proton cross section σp and we defined
2 (E ) ΘSD (ER ) FSD
R
≡ ΘSI (ER ) (1 + r(ER )) , (5.26)
Θ(ER ) = ΘSI (ER ) 1 +
2 (E )
ΘSI (ER ) FSI
R
where
ΘSI (ER )
ΘSD (ER )
!
2
2µ2p
Z
GSI (ηmin (ER ), t)
=
,
(5.27)
A
mA ER
I(ηmin (ER ), t)
!
2 2
2µ2p
1
λ̄nuc
GSD (ηmin (ER ), t)
=
A
λp
mA ER
I(ηmin (ER ), t)
2
=
3
2 2 µp 2
1
λ̄nuc
.
A
λp
mp
(5.28)
The function Θ measures the deviation of the allowed regions and constraints with
respect to the standard spin-independent picture; since GSI /I depends itself on
mχ , we expect an asymmetric shift of the favored regions and constraint lines
in the (mχ , σp ) plane due exclusively to the modified dynamics of the long-range
interaction considered here. The ratio r parametrizes the relevance of the SD
interaction respect to the SI one: if r > 1 the interaction is largely SD, while it is
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
ΣL
PICASSO H5
10-40
10-41
10-42
Σ
H5
L
DAMA H8Σ,7ΣL
CoGeNT H1ΣL
CRESST H4Σ,3ΣL
ON
10-43
N
XE
Contact SI Cross Section Σ p @cm2 D
MB Halo Hv0 =220 kmsL
H5ΣL
10-39
S
CDM
10-38
63
10-44
10
102
DM Mass m Χ @GeVD
Figure 5.1: DM-proton spin-independent interaction cross section σp as a
function of the Dark Matter mass mχ , in the “standard” case of coherent
contact interaction. The galactic halo has been assumed in the form of an
isothermal sphere with velocity dispersion v0 = 220 km/s and local density
ρ0 = 0.3 GeV/cm3 . In this figure we show the allowed regions compatible
with the annual modulation effects in DAMA and CoGeNT, as well as the
region compatible with the CRESST excess, when interpreted as a DM signal.
Specifically, the solid green contours denote the regions compatible with the
DAMA annual modulation effect [31, 32], in absence of channeling [46]. The
short-dashed blue contour refers to the region derived from the CoGeNT annual modulation effect [36], when the bound from the unmodulated CoGeNT
data is included. The dashed brown contours denote the regions compatible
with the CRESST excess [37]. For all the data sets, the contours refer to
regions where the absence of excess can be excluded with a C.L. of 7σ (outer
region), 8σ (inner region) for DAMA, 1σ for CoGeNT and 3σ (outer region),
4σ (inner region) for CRESST. For XENON, the constraints refer to a threshold of 4 photoelectrons (published value [39], lower line) and 8 photoelectrons
(our conservative estimate, upper line), as discussed in Sec. 5.3. The two
lines for PICASSO enclose the uncertainty in the energy resolution [42].
mostly SI for r < 1. One can see from Fig. 5.2 that the dipole-dipole term plays a
major role for DAMA (Na, I) and PICASSO (F), while being negligible for all the
other experiments considered, as already commented above.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
10
m Χ = 10 GeV
F
10
Na
1
64
m Χ = 50 GeV
F
Na
I
1
I
10-1
0
5
10
15
20
Recoil Energy ER @keVD
10-6
25
0
5
10
15
DAMA-I
PICASSO
10-5
DAMA-I
XENON 8PE
10-6
W
O
10-4
W
DAMA-Na
PICASSO
10-5
CoGeNT 2nd bin
10-4
Ca
10-3
Ca
XENON 8PE
O
XENON 4PE
CDMS
CRESST
10-3
10-2
DAMA-Na
rHER L
Ge
Xe
Ge
CoGeNT 2nd bin
Xe
10-2
XENON 4PE
CDMS
CRESST
rHER L
10-1
20
25
Recoil Energy ER @keVD
Figure 5.2: The SD to SI rates ratio r defined by Eq. (5.26) for the two cases
of mχ = 10 GeV (left) and mχ = 50 GeV (right). For illustrative purposes
we approximated here FSI (ER ) ' FSD (ER ), and we considered a Maxwellian
halo with local dispersion velocity 220 km/s. Supposing the major part of the
signal to come from the lower energy threshold (or from the second bin in the
case of CoGeNT), indicated by a vertical solid line, r can be approximately
determined in the figure by the point where the dashed line meets the vertical
line with the same color, for each nuclear element.
We remark that this model, despite being so different from the standard scenario, is very predictive, and it actually features the same number of free parameters as the usually assumed non isospin violating models with contact interaction,
mχ and σp (or alternatively λχ ). In contrast, the isospin violating scenario introduced in Chapter 4 counts one more parameter to be fitted: in fact in such models
the role of Θ is played by the expression ([Z + fn /fp (A − Z)]/A)2 , and therefore one
needs to specify the ratio of the DM-neutron to DM-proton couplings fn /fp .
In Fig. 5.3 we show the behavior of Θ as a function of ER both for small (left
panel) and large (right panel) DM masses, considering several targets, v0 = 220
km/s and vE = v (being v⊕ cos γ v ). In order to simplify the reading of the
figures, we take the limit vesc → ∞; one can figure out the effect of a finite escape
velocity as a target-dependent cut-off at high recoil energies, that sets in when
the minumum velocity required to have a recoil above threshold exceeds vesc + vE ,
the maximum escape velocity in the Earth frame. There, the function Θ loses its
meaning since the expected rate is zero both in the standard picture and in the
case considered here.
A first striking feature of Θ is its overall magnitude: depending on the DM
mass and on the experiment, we have in fact a suppression of the expected rate of
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
10-1
10-1
m Χ = 10 GeV
m Χ = 50 GeV
F
F
Na
O
10-2
65
Na
O
10-2
Ge
I
Xe
-3
10
QHER L
QHER L
Ca
W
Ca
10-3
I
Ge
0
5
10
15
25
10-5
0
CoGeNT 2nd bin
5
10
15
DAMA-I
XENON 8PE
XENON 4PE
CDMS
CRESST
W
DAMA-Na
DAMA-I
20
Recoil Energy ER @keVD
PICASSO
10-4
XENON 8PE
XENON 4PE
CDMS
CRESST
DAMA-Na
PICASSO
10-5
CoGeNT 2nd bin
Xe
10-4
20
25
Recoil Energy ER @keVD
Figure 5.3: The function Θ(ER ) (dashed lines), parametrizing the deviation
of the allowed regions and constraints with respect to the standard case of
contact interaction without isospin violation presented in Fig. 5.1. Each line
is for one of the target elements used in the experiments taken into account
in the text. The vertical lines indicate the low energy thresholds for the various experiments (apart from CoGeNT, for which we show the threshold of
the second energy bin, where much of the signal is recorded). For DAMA two
thresholds are shown, corresponding to the corrections due to the quenching
factors for sodium (qNa = 0.3, in the assumption of scattering mainly off Na)
and iodine (qI = 0.09, for the assumption of scattering mainly off I). For
XENON the 4 photoelectrons (4PE, published value [39]) and 8 photoelectrons (8PE, our conservative estimate) thresholds are shown, as discussed in
Sec. 5.3. We have shown also the lowest threshold for the PICASSO experiment, although we refer to the main text for a more precise interpretation of
its temperature dependent thresholds. The picture is reported for two values
of the DM mass, mχ = 10 GeV (left panel) and mχ = 50 GeV (right panel).
roughly 1 to 4 orders of magnitude. For nuclei in which the interaction is mostly
SD (r > 1), Θ provides a suppression of 1 to 3 orders of magnitude for the well
known reason that the rate does not carry any A2 factor, contrarily to the usual
SI case. Due to this fact, the SD part of the cross section is usually considered
negligible respect to the SI one, but this turns out to be false in our case, for nuclei
with large magnetic moment, since also the SI part is strongly suppressed. This
suppression is due to the interplay of the last two factors in Eq. (5.27), while the
first term plays only a marginal role given that A ∼ 2 Z roughly for all the target
nuclei. The second term, 2µ2p /mA ER , enhances Θ by 2 to 4 orders of magnitude
due to the presence of ER , whose typical scale is few (tens of) keV and therefore
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
66
very small compared to all the other mass scales involved. The last term GSI /I,
containing the velocity integrals, gives instead a suppression of roughly 6 orders
of magnitude. Given the measured rate, this overall suppression provided by the
Θ function will reflect in the fit pointing to higher interaction cross sections, with
respect to the standard case of Fig. 5.1. This fact will play a role in fitting the
relic abundance (Sec. 5.5).
The steep rise at low energies is always due to the long-range type of the
SI part of the interaction, being the SD one of a contact type; in the particular
case of magnetic dipole moment interaction considered here, the differential cross
section diverges as 1/ER . At energies higher than the steep rise, for nuclei with
large magnetic moment (r > 1), the interaction rapidly becomes of a contact type
and therefore the function Θ exhibits a plateau whose value is ∼ (λ̄nuc /A λp )2 .
Instead, for the other nuclei for which r < 1, two regimes are possible, depending
on the ratio vmin /vE . These reflect the two trends assumed by the function1
ζ(vmin ) ≡ I1 (vmin )/I(vmin ), namely a plateau ζ ∼ 1.8 v02 for vmin vE and a rise
2
ζ ∼ vmin
for vmin vE , with an intermediate value of ζ(vE ) ∼ 2.8 v02 .
Light DM particles require a higher minimum velocity to recoil compared to
heavier ones. For the range of DM masses we are interested in, and for the range of
recoil energies relevant for the direct DM search experiments, the value of vmin /vE
is controlled by the ratio mA /mχ . We will see that, depending on this ratio, the
−1
, while the function
function ΘSI can assume a constant behavior or scale as ER
ΘSD is always constant.
5.2.1
Light Dark Matter
For DM particles much lighter than the target nuclei, we have that vmin > vE
already at low energies for all the targets considered, and therefore
2mA + mχ
GSI (ηmin (ER ), t)
mA ER
2
' vmin (ER ) − ER
=
;
(5.29)
I(ηmin (ER ), t)
2mA mχ
2m2χ
h
2 i
accordingly, Θ simplifies to (µp /(A mp ))2 Z2 (mp /mχ )2 + 2/3 λ̄nuc /λp , displaying the plateau shown in the left panel of Fig. 5.3. As expected, the plateau
sets in earlier for heavy targets if the SI interaction dominates (Z2 (mp /mχ )2 2
2/3 λ̄nuc /λp ), while its starting point is independent on the target mass for SD
interactions. We notice that, for high enough recoil energies, Θ always gets to this
constant behavior; it is interesting though that it exists a regime in which this
happens at the low energies relevant for DM direct detection experiments. An
indicative mχ ∼ 10 GeV falls into this case, and as we will show in Sec. 5.4 can
fit the regions of the parameter space allowed by CoGeNT, DAMA-Na, CRESST-O
1
Here we ignore the time dependence, which gives only a negligible contribution.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
67
and CRESST-Ca2 .
For an experiment with a lower threshold above the steep rise, the differential
rate does not feature any dependence on ER and it is therefore similar to the rate
of spin-independent contact interactions, apart from a different functional dependence on mχ . Notice that due to this different dependence on the mass of the DM,
the rate is more suppressed for heavier DM particles, and therefore we expect the
favored regions in the (mχ , σp ) plane as well as the exclusion lines to raise and
slightly tilt at higher mχ (compared to the spin-independent contact interaction).
Considering now that most of the signal in a given detector comes generally
from the first energy bin3 , i.e. close to the low energy threshold, we can roughly
estimate, before doing any statistical analysis of the data, the shift of the allowed
regions and constraints compared to the standard picture. For example, the expected rate of DM scatterings off Na nuclei is reduced by a factor ∼ 10−2 with
respect to the standard case (see Fig. 5.3), and therefore the DAMA favored region,
when assumed that most of the signal comes from DM scattering on Na, is shifted
up by ∼ 102 in the (mχ , σp ) plane. Taking now the DAMA-Na allowed region as
benchmark, we can see how the other favorite regions and exclusion lines move
with respect to it. CoGeNT moves a factor ∼ 5 up, making the agreement with
DAMA almost perfect. Concerning CRESST, the fit at small DM masses is due
equally to O and Ca [37] in the standard scenario, from which we conclude that
the cross section for DM magnetic dipole interaction with Ca is ∼ 1.5 times bigger
than that with O; the overall effect is an increase in σp , that lifts the favored region
to better agree with DAMA and CoGeNT. In detail we expect that the CRESST
allowed region moves a factor ∼ 4 towards the DAMA-Na ballpark. Finally, CDMS
and XENON move up by a factor ∼ 6.5 and ∼ 8.5 respectively, roughly irrespective of the choice of the threshold. This improves once again the agreement with
the other experiments. For PICASSO the situation is different due to the special
experimental setup in which the low energy threshold is a function of the temperature, and therefore one should be careful especially if the differential cross section
is energy dependent. However, since the scattering in the PICASSO experiment
is dominated by the dipole-dipole interaction which is of a contact type, a simple
rescaling is again applicable, and we expect that the constraints will become a
factor 2.5 more stringent (see left-panel of Fig. 5.3), compared to the standard
picture.
5.2.2
Heavy Dark Matter
For higher DM masses, where for instance one can find the regions of the parameter
space allowed by DAMA-I and CRESST-W, the effect of the long-range interaction
2
DAMA and CRESST are multi-target detectors and allowed regions at large DM mass
correspond to scattering on I for DAMA and W for CRESST, while at small DM mass
regions correspond to scattering on Na for DAMA and both O and Ca for CRESST.
3
In CoGeNT, the larger part of the signal comes instead from the second energy bin.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
68
turns to be very evident. As the minimal velocity for a given nuclear recoil vmin
becomes smaller than vE , the function ζ(vmin ) changes behavior. For vmin vE
(i.e. mχ mA )4
2mA + mχ
GSI (ηmin (ER ), t)
2
.
(5.30)
' 1.8 v0 − ER
I(ηmin (ER ), t)
2mA mχ
−1
In this regime, for the nuclei whose interaction is SI, Θ scales as ER
for the
whole range of the recoil energies relevant for the direct detection experiments. If
ER increases sufficiently enough, vmin becomes eventually larger than vE and Θ
assumes the constant behavior we described in the previous section. The difference
in Θ between the low and high DM masses we depicted in Fig. (5.3) is that in the
first case the plateau sets in at small recoil energies (smaller than the energy
threshold of most experiments), whereas in the second case, the plateau sets in at
large recoil energies (more than 25 keV). Instead for the other nuclei with large
magnetic moment, the interaction is SD and as in the case of light DM the plateau
manifests itself at low recoil energy.
Due to the pronounced energy dependence in this case, the allowed regions
and the constraints will shift in the (mχ , σp ) plane considerably more than in the
case of low DM mass (still compared to the standard contact spin-independent
interaction). By taking again the DAMA-Na allowed region as a benchmark from
the right panel of Fig. (5.3), we estimate that both DAMA-I and CRESST-W shift
more than one order of magnitude up in σp . The same happens for CDMS and
XENON. On the other hand PICASSO, like DAMA-Na, does not change its behavior
respect to the light DM case: this is due to the fact that 19 F and 23 Na enjoy mostly
spin-dependent interactions, that depend only slightly on the DM mass.
To summarize, taking into account the whole DM mass range, we see that,
apart from an overall shift upwards in σp , there is a general trend of the various
experiments to “gather”, getting closer to each other, with respect to the standard
case. This behavior is not homogeneous in the DM mass and is more pronounced
for masses above ∼ 25 GeV.
While this is due mainly to the dipole-charge interaction, also the dipole-dipole
interaction contributes to a better agreement between the experiments. In fact, the
increase in the cross section in DAMA favors a lower value for the DM magnetic
moment λχ , and therefore the DAMA region shifts down towards CoGeNT and
CRESST. The overlap becomes then almost perfect, for a Maxwellian velocity
distribution, especially with velocity dispersion 220 km/s. On the other hand also
the bound by PICASSO lowers, but even this enhanced constraint excludes only a
minor part of the overlap zone. The overall effect is to favor a better fit compared
to the standard case, with a large agreement of all the experiments.
4
This is generally true provided we consider a recoil energy ER within our interest:
from a minimum threshold value of a few keV up to 25 − 30 keV.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
69
Even though this is true in general for any DM mass, we do not expect anyway
to get a good agreement for higher masses: in that region in fact the XENON
experiment rules out the signal featured by other experiments by several orders of
magnitude in the standard picture (Fig. 5.1), and the shifts that take place in our
case are not big enough to change this situation.
Finally we comment on the dependence of the function Θ from the halo model
considered, and in particular from the choice of the dispersion velocity v0 . As
we have seen, the transition between the two regimes discussed above is driven
by the ratio vmin /vE , and it is only important for the nuclei which experience SI
interactions with a DM particle. Therefore in this case we expect that for larger
values of vE ≈ v0 , the plateau behavior of Θ sets in at higher recoil energies (or
equivalently at smaller DM masses). This would make the shifts more pronounced
even for light DM. In Sec. 5.4 and Figs. 5.4 and 5.5 we present the exact numerical
results.
5.3
Data sets and analysis technique
In this section we discuss the techniques used to analyze the various data sets. In
particular we adopt the approach of [106], and we summarize below the details of
how we perform the fits to data and constraints from null results experiments.
For DAMA, CoGeNT and CRESST, we test the null hypothesis (absence of
signal on top of estimated background for CRESST and absence of modulation for
DAMA and CoGeNT). From this we infer:
i) the confidence level for the rejection of the null hypothesis (we find 8 ÷ 9σ
for DAMA, 1 ÷ 2σ for CoGeNT, and 4σ for CRESST);
ii) the domains in the relevant DM parameter space (defined by the DM mass
mχ and the DM magnetic dipole moment λχ ) where the values of the likelihood function depart for more than nσ from the null hypothesis, and thus
the corresponding evidence of the DM signal. We use n = 7, 8, n = 1, and
n = 3, 4 for DAMA, CoGeNT, and CRESST, respectively [34].
Our statistical estimator
is the likelihood function of detecting the observed
Q
number of events L = i Li , where the index i indicates the i-th energy bin in
DAMA and CoGeNT, and the i-th detector in CRESST. For DAMA and CoGeNT
Li are taken to follow a Gaussian distribution and for CRESST, since in this case
the number of events in each sub-detector is low, a Poissonian one. Defining Lbg
as the likelihood of absence of signal, we assume the function ỹ = −2 ln Lbg /L to
be distributed as a χ2 -variable with one degree of freedom for a given value of the
DM mass (notice that for DAMA and CoGeNT ỹ reduces to ỹ = χ2bg − χ2 ). From
the ỹ function, the interval on λχ where the null hypothesis (i.e. λχ = 0) can be
excluded at the chosen level of confidence are extracted: 7σ (outer region) or 8σ
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
70
(inner region) for DAMA, 1σ for CoGeNT and 3σ (outer region), 4σ (inner region)
for CRESST. We then plot allowed regions in the (mχ , λχ ) plane.
We derive constraints from CDMS and XENON100 with a similar likelihood
function λ = −2 ln L/Lbg ; here L is the likelihood of detecting the observed number of events (2 for CDMS and 3 for XENON100), while in Lbg the DM signal is
not included. Both likelihoods are taken as Poissonian variables and λ is assumed
to follow a χ2 -distribution. For PICASSO we instead derive constraints by using a
∆χ2 method of the data points shown in fig. 5 of [42]. Bounds are conservatively
shown at 5σ C.L.
Concerning the experimental data sets and statistical methods, we refer to
[106]; for completeness we summarize below for each experiment the most important ingredients used in the analysis.
DAMA: We use the entire set of DAMA/NaI [31] and DAMA/LIBRA [32]
data, corresponding to a cumulative exposure of 1.17 ton×yr. We analyze
exp
the modulation amplitudes Smk
reported in Fig. 6 of Ref. [32] requiring
that the DM contribution to the unmodulated component of the rate, S0 ,
does not exceed the corresponding experimental value S0exp in the 2 ÷ 4 keV
energy range. We compute
y = −2 ln L ≡ χ2 (λχ , mχ ) =
8
exp
X
Smk − Smk
σk2
2
k=1
2
(S0 − S0exp )
θ(S0 − S0exp ) , (5.31)
+
2
σ
exp
where σk and σ are the experimental errors on Smk
and S0exp , respectively.
The last term in Eq. (5.31) implements the upper bound on S0 by penalizing
the likelihood when S0 exceeds S0exp with the Heaviside function θ. The
detector energy resolution
is parametrized by a Gaussian function of width
√
σres (E) = E(0.448/ E + 0.0091) [137], using for the quenching factor the
central values quoted by the collaboration, namely qNa = 0.3 and qI = 0.09
[138]. We don’t take into account the possibility of a nonzero channeling
fraction [46].
CoGeNT: We consider the time-series of the data, treating the measured
total rate as a constraint. Similarly to the analysis done on the DAMA data,
we define
y = −2 ln L ≡ χ2 (λχ , mχ ) =
2
exp
16
S̃
−
S̃
X m1,k
m1,k
k=1
σk2
+
16
X
exp
S̃m2,k − S̃m2,k
σk2
2
k=1
exp
31
S0j − S0j
X
j=1
σj2
2
+
exp
θ(S0j − S0j
) ; (5.32)
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
71
R
here S̃mk = 1/∆tk ∆tk Smk cos [2π(t − φ)/τ ] dt, with ∆tk the temporal bin
exp
exp
exp
exp
of the data, and S̃mk
= Rmk
− hRmk
i, where Rmk
is the total rate (taken
exp
from Fig. 4 of Ref. [36]) and hRmk i is its annual average. The subscripts 1
and 2 in Eq. (5.32) refer to the first and second energy bins. The total rate
in the 0.9 − 3.0 keVee energy-bin is computed by subtracting the rate in the
0.5 − 0.9 keVee bin to the rate in the 0.5 − 3.0 keVee bin, with a Gaussian
exp
propagation of the errors. S0j
and σj denote the experimental counts and
the corresponding errors as given in Ref. [36] (31 energy bins in the interval
0.4 − 2 keVee ), after removal of the L-shell peaks but without removing any
other background. The total fiducial mass is 330 g, the energy resolution is
given by a Gaussian with width taken from [139], and the quenching factor
1.12 [140].
below 10 keV is described by the relation E = 0.2 ER
CRESST: We compute the expected DM signal in each of the 8 CRESST detector modules. The acceptance regions and the number of observed events
are provided in Table 1 of Ref. [37], and we derive background events according to estimates in Sec. 4 of Ref. [37]. A likelihood-ratio test yields a
4.1σ C.L. evidence for the best-fit of a DM signal over the background-only
hypothesis, in good agreement with the result quoted by the collaboration.
We use the published value of 730 kg×days for the exposure and assume
an even contribution among the different modules5 (each module accounts
therefore for an exposure of 730/8 kg×days); we consider moreover a constant efficiency.
CDMS: We use the “standard” 2009 CDMS II results based on Ge data [38];
these are obtained employing conservative nuclear recoil selection cuts and
assuming an energy threshold of 10 keV. The total exposure is 612 kg×days
and we take the efficiency from the black curve of Fig. 5 in Ref. [141] with
q ' 1 as quenching factor.6 In spite of an expected background of 0.9 ± 0.2
events, two signal events were found in the 10 − 100 keV energy interval [38]
(we use these numbers to derive the constraints).
XENON100: We use the results presented in Ref. [39], with an exposure of
100.9 days in a fiducial volume of 48 kg. After all the cuts, three events
were reported in the DM signal region in spite of an expected background of
1.8±0.6 events. We model the data using a Poissonian distribution of photoelectrons, with a single-photoelectron resolution equal to 0.5. The shape of
the Leff function is very crucial for such a low number of photoelectrons and
for small DM masses. Following [106] we try to enclose a possible (but not
5
This is the same analysis performed in [106], although a value of 400/9 kg days for
the exposure was erroneously reported in the text of the published version.
6
In the case of light DM, one can perform a similar analysis by using combined data
from the CDMS and EDELWEISS experiments (see Fig. 1 in Ref. [142]), with basically
the same results.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
72
exhaustive) uncertainty on the bounds derived for XENON100 by adopting
two different approaches:
i) we adopt as threshold the published value of 4 photoelectrons and the
nominal central value of Leff as shown in Fig. 1 of Ref. [39], which relies
heavily on linear extrapolation below 3 keVnr ;
ii) more conservatively, we raise the threshold for the photomultipliers to
8 photoelectrons: this value is the lowest one for which the analysis is
nearly independent on the shape of Leff below 3 keVnr .
Notice that these two approaches are not exhaustive of all the possible assumptions one can do to determine the XENON100 response to light DM
(for further discussion and considerations, see e.g. Refs. [43, 44]). It appears
therefore still preliminary, given these large uncertainties, to assume the
bounds quoted by the collaboration as strictly firm. The 8 photoelectrons
bound is less dependent on the Leff extrapolation, and therefore, conservatively, we consider it as more appropriate.
Finally, we follow Eqs. (13–16) in Ref. [143] to compute the expected signal.
We derive upper bounds for both CDMS and XENON as mentioned at the
beginning of this section.
PICASSO: The PICASSO experiment, located at SNOLAB [42], is very different from the ones discussed above; it is in fact based on the superheated
droplet technique, a variant of the bubble chamber technique, to search for
DM recoiling on 19 F in a C4 F10 target. The experimental procedure consists
in measuring the acoustic signal released by the nucleation of a bubble as
a function of the temperature T . Details of the detector principle can be
found in [144, 145]. Since bubble formation is only triggered above a certain
energy threshold Eth (T ), the spectrum of the particle-induced energy depositions can be constructed by varying the temperature. We compute the
predicted DM rate as a function of Eth (T ) from Eq. (3) of [42] and we compare such prediction with the experimental rate shown in Fig. 5 by using a
∆χ2 method. In this analysis we adopt two reference values of the parameter
a(T ) which describes the steepness of the energy threshold; namely we take
a = (2.5, 7.5) in order to encapsulate as much as possible the experimental
uncertainties [42]. We checked our result against the one of the collaboration
given in Fig. 7 of Ref. [42], and we found an excellent agreement. Remaining
coherent with our choice of being conservative we show here the result at
5σ, as we did also for the other exclusion experiments.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
CDMS
10-17
MB Halo Hv0 =220 kmsL
H5ΣL
SO H5Σ
PICAS
L
10-18
N
XE
DAMA H8Σ,7ΣL
ON
DM Magnetic Dipole Λ Χ @e cmD
WDM
73
CRESST H4Σ,3ΣL
L
H5Σ
CoGeNT H1ΣL
-19
10
10
102
DM Mass m Χ @GeVD
Figure 5.4: DM magnetic dipole moment λχ as a function of the Dark Matter
mass mχ . The galactic halo has been assumed in the form of an isothermal
sphere with velocity dispersion v0 = 220 km/s and local density ρ0 = 0.3
GeV/cm3 . Notations are the same as in Fig. 5.1; to match the two figures,
one has to note that the role of σp is played here by αEM λ2χ . The orange strip
shows the values for (mχ , λχ ) that fit the relic abundance ΩDM assuming a
completely thermal DM production (see Sec. 5.5).
5.4
Fit to the direct detection experiments
We analyze the direct detection data sets by using a standard isothermal halo
model, which basically implies a truncated Maxwell-Boltzmann velocity distribution function (see discussion in Sec. 5.1.3). Since the response of direct detection
experiments is quite sensitive to the DM distribution in the galactic halo [33],
especially in the scenario we are interested to study, we take into account uncertainties on the velocity dispersion v0 , as discussed in Ref. [33]. We will use the
three values v0 = 170, 220, 270 km/s, which bracket the uncertainty in the local
rotational velocity. Let us notice that the value of the local DM density ρ0 is correlated to the adopted value of v0 , as discussed e.g. in Ref. [33]. This corresponds
to the model denoted as A0 in [33], and we adopt here the case of minimal halo,
which implies lower values of the local DM density (since a fraction of the galactic
potential is supported by the disk/bulge). In turn, this implies the adoption of
ρ0 = 0.18, 0.30, 0.45 GeV/cm3 for v0 = 170, 220, 270 km/s, respectively [33].
Fig. 5.4 shows the constraints and the favored regions coming from DM direct
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
L
10-18
N
NO
XE
H5Σ
L
ΣL
SO H5
10-18
S
PICA
DAMA H8Σ,7ΣL
CoGeNT H1ΣL
CRESST H4Σ,3ΣL
10-19
102
10
DM Mass m Χ @GeVD
L
10-19
5Σ
NH
CRESST H4Σ,3ΣL
NO
CoGeNT H1ΣL
XE
DAMA H8Σ,7ΣL
MB Halo Hv0 =270 kmsL
L
SO H5Σ
PICAS
DM Magnetic Dipole Λ Χ @e cmD
WDM
L
DM Magnetic Dipole Λ Χ @e cmD
MB Halo Hv0 =170 kmsL
S H5Σ
WDM
CDMS H5Σ
10-17
CDM
10-17
74
10
102
DM Mass m Χ @GeVD
Figure 5.5: DM magnetic dipole moment λχ as a function of the Dark Matter
mass mχ . The galactic halo has been assumed in the form of an isothermal
sphere with velocity dispersion v0 = 170 km/s and local density ρ0 = 0.18
GeV/cm3 (left panel); v0 = 270 km/s and local density ρ0 = 0.45 GeV/cm3
(right panel). Notations are the same as in Fig. 5.1; to match the two figures,
one has to note that the role of σp is played here by αEM λ2χ . The orange
strip shows the values for (mχ , λχ ) that fit the relic abundance ΩDM , in the
assumption of thermal DM production (see Sec. 5.5).
detection experiments in the (mχ , λχ ) plane. The galactic halo has been assumed
to be in the form of an isothermal sphere with velocity dispersion v0 = 220 km/s
and local density ρ0 = 0.3 GeV/cm3 . The solid green contours denote the regions
compatible with the DAMA annual modulation effect [31, 32], in absence of channeling [46]. The short-dashed blue contour refers to the region derived from the
CoGeNT annual modulation signal [36], when the bound from the unmodulated
CoGeNT data is taken into account. The dashed brown contours denote the regions
compatible with the CRESST excess [37]. For all the data sets, the contours refer to
regions where the absence of modulation can be excluded with a C.L. of 7σ (outer
region), 8σ (inner region) for DAMA, 1σ for CoGeNT, and the absence of an excess
can be excluded at 3σ (outer region), 4σ (inner region) for CRESST. Constraints
derived by the null result experiments are shown at 5σ as gray, magenta and red
dashed lines for CDMS, XENON100 and PICASSO respectively. For the XENON
detector, as discussed in the previous section, the constraints refer to thresholds of
4 and 8 photoelectrons, while for PICASSO we take into account the uncertainty
in the intrinsic energy resolution of the employed detection technique [42].
We can see that, as expected from the discussion in Sec. 5.2, for a given value
of mχ , all the experiments determine a DM-proton cross section σp = αEM λ2χ
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
75
that is higher than the one for the standard case of non isospin violating contact
interaction depicted in Fig. 5.1; to give a benchmark value, DAMA points now
to σp ∼ 1.5 × 10−38 cm2 , about 102 times above the standard case, as shown in
Sec. 5.2. Moreover we observe the expected gathering of the various experiments,
that get closer to each other with respect to the standard scenario, for an overall
better agreement, and with DAMA, CoGeNT and CRESST featuring a larger overlap
at low masses. Both the DAMA and CoGeNT regions point towards a DM mass
in the 10 GeV ballpark (more specifically, from about 7 up to about 12 GeV)
and DM magnetic dipole moment around 1.5 × 10−18 e cm without exceeding the
constraints, corresponding to an inverse mass energy scale of circa ΛM ∼ 10 TeV.
CRESST allows for even heavier DM masses, but still compatible with the range
determined by the other two experiments. CDMS and XENON, on the other hand,
exclude a smaller part of the parameter space with respect to the standard case.
These two experiments do not exclude the overlapping region once one accepts our
conservative choice of the XENON threshold at 8 photoelectrons. At 5σ it is seen
that PICASSO cannot exclude the common regions of the experiments reporting
a signal. Finally, we also notice the occurrence of the expected flattening of the
various experiments, meaning that the tilt featured in the standard case (higher
mass regions pointing to lower σp ) is very suppressed in the case of magnetic
moment interaction, as explained in Sec. 5.2.
These results depend on the galactic halo model assumed. The effect induced
by the variation in the DM dispersion velocity is shown in the two panels of
Fig. 5.5. In the case of v0 = 170 km/s, the regions are not significantly modified
as compared to the case v0 = 220 km/s, except for the overall normalization due
to the different values of the local DM density in the two cases. Larger dispersion
velocities, instead, lead to the second regime discussed in Sec. 5.2 also at small
DM masses (see right panel of Fig. 5.3). Due to the high energy dependence of
the differential rate in this regime, experiments that use heavy targets and/or
have high thresholds, get a suppression of the event rate, and point therefore to
relatively higher values of the interaction cross section compared to the standard
case. Since the DM signal is expected at lower recoil energies, smaller DM masses
are now favorite. The right panel of Fig. 5.5 shows in fact that the allowed regions
are shifted toward smaller DM masses for v0 = 270 km/s.
5.5
Relic Abundance
In addition to a satisfactory fitting of the direct detection data, the DM magnetic
dipole interaction can accommodate the annihilation cross section appropriate for
thermal production. Since it couples to the photon, the DM annihilates to any
charged SM particle-antiparticle pair, with mass lower than mχ . Considering a
DM particle with 10 GeV mass, that as we argued fits well the direct detection
¯ ss̄, cc̄,
experiments, the allowed primary channels are e+ e− , µ+ µ− , τ + τ − , uū, dd,
bb̄ and γγ.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
76
The DM annihilation cross sections to SM fermions and photons at tree level are
αEM Q2f Ncf βf 1
σχχ̄→f f¯(s) =
×
12
βχ s2
2 2
λχ s (3 − β 2 ) + 12m2χ s + 48m2f m2χ + d2χ s2 (3 − β 2 ) − 12m2χ s , (5.33)
λ4χ
σχχ̄→γγ (s) =
4πβχ
m4χ arcth(βχ )
s sβχ2
m2χ + −
−4
8
24
s
βχ
!
,
(5.34)
where βχ,f = (1 − 4m2χ,f /s)1/2 , β ≡ βχ βf , Qf is the charge of the fermion and
finally Nc` = 1 for leptons and Ncq = 3 for quarks. In the non-relativistic limit,
Eqs. (5.33) and (5.34) reduce to
hσχχ̄→f f¯ vrel i ' N αEM λ2χ · BR(f ) ,
1 4 2
hσχχ̄→γγ vrel i '
λ m ,
4π χ χ
(5.35)
(5.36)
P
where N = f Q2f Ncf = 20/3 accounts for the number of degrees of freedom of the
SM fermions into which the DM can annihilate; the branching ratios BR(f ) are
defined as BR(f ) = Q2f Ncf /N . As one can see, for the low values of λχ mχ pointed
by the direct detection experiments, the two photons final state is suppressed with
respect to the fermionic one. Using a DM energy density ΩDM h2 = 0.1126±0.0036
measured by WMAP [19], we show7 with an orange strip in the (mχ , λχ ) plane of
Figs. 5.4 and 5.5 the phase space where a thermal production of the magnetic DM
is possible.
It can be seen in Figs. 5.4 and 5.5 that the agreement between the allowed
regions and the relic abundance depends strongly on the dispersion velocity v0 . The
best agreement takes place for lower values of v0 . This is due to the fact that, as
commented already in Sec. 5.4, ρ0 is linked to the dispersion velocity v0 . By varying
v0 , ρ0 is modified accordingly, changing therefore the expected rate and the fitting
values for the interaction cross sections, or in other words λχ . As a consequence,
the favored regions and constraint lines of direct detection experiments move along
the vertical axis in the (mχ , λχ ) plane. For lower velocity dispersion, both the local
DM density and the expected rate decrease. Therefore, the favored regions will
point to a higher DM magnetic moment, getting closer to the relic abundance
strip. The value of the relic abundance is obviously independent on the local DM
density.
7
The figures are produced by solving numerically the Boltzmann equation
1
ṅtot + 3Hntot = − hσann vrel i n2tot − n2eq ,
(5.37)
2
where ntot is the total number density of particles and antiparticles, H is the Hubble
parameter and for hσann vrel i we use the more precise formula given in Eq. (3.8) of [146].
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
77
The fact that the magnetic DM (with only two parameters) provides a good fit
to the direct detection experiments and simultaneously can accommodate a thermal annihilation cross section (which is several orders of magnitude stronger than
the DM-nucleon cross section in the context of “standard” contact interaction) is
not easily met by other DM candidates. Although there are candidates that can
have a DM-nucleon cross section much lower than their thermal annihilation cross
section, these candidates have usually spin-dependent interactions and cannot fit
nicely the direct search experiments with a positive DM signal. For a typical
candidate with spin-independent contact interactions, and for a strength of interaction that leads to a DM-nucleon cross section pointed by the direct detection
experiments featuring a signal, the annihilation cross section is way too small to
produce this candidate thermally. In order to match, a suppression mechanism for
the direct detection event rate should take place. In the case of magnetic moment
DM, this mechanism exists and has two different reasons for the SI and the SD
parts of the interaction. As already explained in Sec. 5.2, this suppression is encoded into the function Θ in Eq. (5.26), that spans roughly from 10−3 to 10−1 for
a 10 GeV DM (see Fig. 5.3). This suppression is a result between two competing
−1
dependence of the
factors for the SI case: the enhancement provided by the ER
differential cross section, and the suppression provided by the kinetic integral; in
the SD case, the suppression is instead due to the lack of the A2 enhancement
usually present in the standard SI case.
Notice that in the case of electric dipole DM one faces a very different situation:
here the differential cross section has the same dependence on the velocity as in
a contact interaction (Eq. (5.7)), and therefore there is no kinetic suppression,
−1
while on the contrary the rate is enhanced by the ER
dependence. Moreover,
the annihilation to fermions is a p-wave process, Eq. (5.33), and therefore the
annihilation cross section is suppressed, making the value of dχ needed to fit the
relic abundance even bigger.
Furthermore, it is worth to point out that neither asymmetric/mixed DM [147],
nor oscillating DM [148–150] can improve the agreement between the relic density
and the allowed regions of direct DM searches in the context of magnetic DM
interaction.
5.6
Constraints
Given the qualitative agreement between the direct detection allowed regions and
the fit of the relic abundance, it is natural to ask whether constraints coming
from other searches could limit the parameter space of the magnetic moment DM.
This is even more pressing since the bounds coming from indirect DM searches are
on the verge of constraining the thermal annihilation rate for light DM particles.
We will discuss the constraints imposed by indirect searches, colliders and by the
observations of compact stars, and then identify the most stringent ones.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
CDMS
10-17
WDM
H5ΣL
DM Magnetic Dipole Λ Χ @e cmD
MB Halo Hv0 =220 kmsL
78
Γ-ray lines
B
ctic
Gala warf
D
CM
Σ
SO H5
S
PICA
L
10-18
DAMA H8Σ,7ΣL
L
N H5Σ
CoGeNT H1ΣL
10-19
CRESST H4Σ,3ΣL
XENO
10
102
DM Mass m Χ @GeVD
Figure 5.6: DM magnetic dipole moment λχ as a function of the Dark Matter
mass mχ . The galactic halo has been assumed in the form of an isothermal
sphere with velocity dispersion v0 = 220 km/s and local density ρ0 = 0.3
GeV/cm3 . Notations are the same as in Fig. 5.1. The orange strip shows
the values for (mχ , λχ ) that fit the relic abundance ΩDM , in the assumption
of thermal DM production (see Sec. 5.5). The shaded regions refer to the
constraints from γ-ray lines, CMB, galactic γ-rays and dwarf galaxies. These
constraints are only valid in the assumption of symmetric DM. The galactic
photons constraint enforces total annihilation of the DM into bb̄, and therefore
a less stringent constraint is expected for magnetic moment DM.
5.6.1
Epoch of reionization and CMB
Strong constraints are imposed on DM annihilations from considering the effect on
the generation of the CMB anisotropies at the epoch of recombination (at redshift
∼ 1100) and their subsequent evolution down to the epoch of reionization. The
actual physical effect of energy injection around the recombination epoch results
in an increased amount of free electrons, which survive to lower redshifts and affect
the CMB anisotropies [151–154]. Detailed constraints have been recently derived
in [155,156], based on the WMAP (7-year) and Atacama Cosmology Telescope 2008
data. The constraints are somewhat sensitive to the dominant DM annihilation
channel: annihilation modes for which a portion of the energy is carried away by
neutrinos or stored in protons have a smaller impact on the CMB; on the contrary
the annihilation mode which produces directly e+ e− is the most effective one.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
79
Usually the approach here is to consider 100% annihilation rate in a single final
state; anyway, in our case several annihilation channels are open, and therefore
we expect a smaller energy injection in the interstellar medium with respect to
the case of annihilation only into e+ e− . In Fig. 5.6 we reproduce the constraints
as obtained in Refs. [154–156] considering now all the channels, each with its
branching ratio defined in Sec. 5.5.
5.6.2
Present epoch γ-rays
For most of the DM annihilation modes, another relevant constraint is in fact
imposed by the indirect DM searches in the present epoch. The DM constraints
provided by the FERMI−LAT γ-ray data are particularly relevant as they are now
cutting into the thermal annihilation cross section for low DM masses (. 30 GeV)
and a variety of channels.
In particular, dwarf satellite galaxies of the Milky Way are among the most
promising targets for Dark Matter searches in γ-rays because of their large dynamical mass to light ratio and small expected background from astrophysical sources.
No dwarf galaxy has been detected in γ-rays so far and stringent upper limits are
placed on DM annihilation by applying a joint likelihood analysis to 10 satellite
galaxies with 2 years of FERMI−LAT data, taking into account the uncertainty
in the Dark Matter distribution in the satellites [157]. The limits are particularly
strong for hadronic annihilation channels, and somewhat weaker for leptonic channels as diffusion of leptons out of these systems is poorly constrained. In our case,
having both hadronic as well as leptonic annihilation channels, we expect again a
weaker constraint with respect to the pure hadronic annihilation (see e.g. figure 2
in Ref. [157]).
Other strong limits on annihilation channels are set by, for example, the γray diffuse emission measurement at intermediate latitudes, which probes DM
annihilation in our Milky Way halo [158–160]. In particular, the most recent limits
come from 2 years of the FERMI−LAT data in the 5◦ 6 b 6 15◦ , −80◦ 6 ` 6 80◦
region [160], where b and ` are the galactic latitude and longitude. Since bounds
on all annihilation channels are not available with the latest data, we report only
the most stringent one, coming from bb̄.
The last kind of constraints that we can set on DM annihilation in the case of
magnetic moment interaction comes from γ-ray lines. Indeed, given the possibility
of annihilation in two photons, we expect a line in the cosmic photon spectrum at
energies equal to the mass of the DM. Constraints on the annihilation cross section
into photons can be drawn from the latest FERMI−LAT data [161]. For instance,
the FERMI−LAT collaboration excludes at 95% the annihilation hσχχ̄→f f¯ vrel i <
0.5 × 10−27 cm3 /s for a DM mass mχ = 30 GeV in the case of an isothermal halo.
In our case we use the latest (preliminary) data on the flux from spectral lines
(shown in slide 39 of [162]) to constrain the annihilation cross section of Eq. (5.36)8 .
8
A similar study has been published in Ref. [90] for DM masses above 30 GeV.
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
80
We consider an all-sky region with the Galactic plane removed (|b| > 10◦ ), plus a
20◦ ×20◦ square region centered on the Galactic center [161]. Notice that, for small
DM masses and dipole moments, the one-loop contribution becomes comparable
with the tree level one; a rough estimate of the loop cross section in the non3 λ2 /4π, and therefore we expect it to become sizeable for
relativistic limit is αEM
χ
3/2
λχ mχ < αEM ' 6 × 10−4 . In the part of the parameter space probed by γ-ray
lines searches the loop correction is negligible.
We superimpose all these constraints in Fig. 5.6. We see that, apart from the
γ lines bound, they are somewhat stronger than the CMB one considered above.
We keep however the latter as it is less model dependent.
As shown in Fig. 5.5, a change in the DM local density modifies the direct
searches results. This does not happen for indirect searches, as they are not very
sensitive to ρ0 . Therefore the constraints shown in Fig. 5.6 also apply, unchanged,
to this case. For lower DM dispersion velocity v0 these constraints start to play
an important role in cutting the parameter space for direct detection searches.
5.6.3
Collider and other astrophysical constraints
The constraints we have addressed above apply to the case of symmetric (thermally
produced) DM. These constraints are not relevant in the case where DM is of
asymmetric nature. There are two extra type of constraints: constraints imposed
by collider searches, and constraints imposed by observations of compact stars
such as white dwarfs and neutron stars. The collider constraints are applicable
whether DM is symmetric or not, whereas the compact star constraints are valid
only for asymmetric DM.
The collider constraints emerge from the fact that for a given λχ and mχ that
fit the direct DM searches experimental data (and even provide the proper DM
annihilation cross section for thermal production), the cross section for production
of a pair of DM particles in colliders is completely fixed. The processes that
lead to constraints are mono-photon production (from initial or final state) plus
missing energy due to the pair of DM particles in e+ e− collisions at LEP, or monojet production plus missing energy in proton-antiproton collision in Tevatron, or
proton-proton collisions at LHC. In the case of magnetic DM these constraints
have been studied in [129, 130] where it is found that the upper bound on λχ is
safely above the range of values of λχ relevant for the direct search experiments
(for the range of mχ ∼ 10 GeV).
In the case of asymmetric DM, constraints can be imposed by compact star
observations based on the fact that a substantial number of captured DM particles
might lead to gravitational collapse and formation of a black hole that can destroy
the host star. The magnetic DM being a fermion evades the severe constraints
on asymmetric bosonic DM based on neutron stars [163–165]. The constraints on
DM self-interaction cross section with Yukawa interactions presented in [166] are
Effective Operators for Dark Matter Detection
Chapter 5. Magnetic Moment DM in Direct Detection Searches
81
avoided for several reasons: firstly DM-DM interactions scale as m2χ λ4χ (leading
to a typical DM-DM cross section of ∼ 10−43 cm2 which is much smaller than
the constraint). Secondly the constraints are not directly applicable because the
mediator of the magnetic DM is a massless photon (and not a massive mediator
necessary for the constraint) and the DM-DM interaction is repulsive. The latter
adds up to the effect of the Fermi pressure of the DM particles and therefore
the amount of particles needed for gravitational collapse cannot be accumulated
within the age of the Universe. A potential attractive photon interaction takes
place in the symmetric case, which will however lead also to annihilation of the
DM population inside the neutron star invalidating thus the constraint derived
from black hole formation.
Finally the magnetic DM evades constraints on the spin-dependent part of
the cross section imposed by observations of white dwarfs [167]. Although these
constraints are typically weaker than the ones derived from direct searches at the
range mχ ∼ 10 GeV, they could become stricter if spin-dependent interactions
scale as some (positive) power of the recoil energy. Since DM particles acquire
high velocities when entering the white dwarf, such constraints could in principle
exclude such a candidate. However, as it was pointed out in [122], the spindependent cross section does not scale with the recoil energy and therefore the
white dwarf constraints can be safely ignored.
Effective Operators for Dark Matter Detection
Chapter 6
Conclusions
The existence of two distinct dark components of the energy density of the Universe has been now firmly established. The so called Dark Energy behaves like
a pressureless fluid and influences the expansion rate of the Universe, while the
Dark Matter clumps like normal matter and affects many astrophysical and cosmological observables on different length and time scales. Under the assumption
that Dark Matter (DM) is actually made of particles, many experimental searches
are ongoing to understand its true nature. Indirect searches look for traces of DM
annihilation or decay in cosmic rays, while direct searches try to directly detect
DM particles measuring anomalous recoils of detector nuclei.
At present, the situation of direct detection searches is unclear and seems still
far from being settled. The DAMA/LIBRA, CoGeNT and CRESST−II experiments
detect events that can be attributed to DM-nuclei collisions, while all the other
searches find null evidence for Dark Matter. In particular, CDMS II, XENON10/100
and PICASSO impose severe constraints on the DM-nucleons cross sections, excluding much of the parameter space allowed by the experiments featuring a signal.
Moreover, the DAMA, CoGeNT and CRESST allowed regions do not coincide, thus
making difficult a simultaneous interpretation of their signal in terms of Dark Matter. In the analyses that are usually performed, though, assumptions are made in
order to reduce the general mathematical problem to a form that can be tackled
with our limited tools and knowledge. In this Thesis we addressed the problem of
a possible oversimplification, that might prevent us to understand the true nature
of Dark Matter.
First we pointed out the fact that, beside the few types of interaction customarily considered in model independent analyses, a large number of operators exist
that might change the phenomenology of the interaction. An example is given by
the flavor violating operators potentially arising when the DM is charged under the
Weak interaction. These operators are often neglected in model independent analyses. Another case of interest is the possibility of quantum interference between
different operators, that is not taken into account in the usual analyses which only
consider one operator at the time.
82
Chapter 6. Conclusions
83
With this in mind, we showed that interference within exchange processes
of two mediators of the DM-nucleus interaction can provide the right amount of
isospin violation needed to accommodate the DAMA, CoGeNT, CDMS and the 2011
XENON results. We checked whether interference can work between photon and
Higgs, Z and Z 0 , Z 0 and Higgs, and two Higgses. After the work illustrated in this
Thesis was done, anyway, newer XENON100 results became available which now
exclude most of the DAMA/CoGeNT overlapping region; moreover, the isospin violating DM paradigm has been found to be in tension also with other DM searches.
In the last part of this Thesis we considered a very peculiar type of Dark Matter,
namely a DM particle with a magnetic dipole moment that allows scattering with
the photon. This kind of interaction, being of long-range type, is in contrast with
the standard paradigm of contact interaction. The phenomenology of these two
kinds of scattering is very different due to the energy dependence of the scattering
amplitude introduced by the photon propagator. In fact, we were able to show
that this DM candidate accommodates all present direct detection experiments,
assuming a conservative estimate of the XENON100 low energy threshold. The
scale of the new physics responsible for the DM magnetic moment to arise has
been fitted to be ∼ 10 TeV, with a DM mass around 10 GeV. This study shows
once again that the complicated interpretation of the experimental data might
need us to relax some of the assumptions that are usually made.
The situation of DM searches remains unclear. The presence of many ongoing
experiments with continuously updated results makes it possible to test theoretical scenarios, but no model so far has been able to stand out as a truly viable
explanation of the experimental data. The difficulty generates principally from the
uncertainties involved: regarding direct searches, these come from poorly known
nuclear physics (still inaccurate estimation of the strange quark contribution to the
nucleon mass and spin, unknown nuclear form factors, the possible presence of the
channelling effect in DAMA, ...), unknown DM velocity distribution, and subtleties
related to the experimental detection efficiency close to the low energy threshold
(e.g. ambiguities in the correct statistical treatment of the threshold in XENON).
Regarding indirect searches, instead, the uncertainties originate from the a priori
unknown astrophysical background, poorly known propagation of cosmic rays in
our Galaxy, inaccuracies in the modeling of hadronization and parton showering
in Monte Carlo event generators used to simulate cosmic rays production from
DM annihilation/decay, and so on. This makes extremely difficult to understand
whether a signal detected by an experiment is due to Dark Matter, or it is instead
due to some unknown source of background, left aside the possibility of unknown
systematics in the detector that might affect the interpretation of the data.
Even if one of the experiments is wrong, however, many others remain that
point to a possible DM signal. DAMA, CoGeNT and CRESST, even if they are
not precisely in agreement one with the other and with other direct detection
experiments, all point to a DM particle with mass around ∼ 10 GeV. There is
actually no reason to assume that the DM is made by only one kind of particle,
Effective Operators for Dark Matter Detection
Chapter 6. Conclusions
84
and indeed the electron/positron excess measured by PAMELA and FERMI, and
the gamma line in the FERMI data all point to a DM in the hundreds of GeV
ballpark. Attempts to accommodate all the various experimental results might
suffer from the too strong assumptions usually made in modeling the dark sector.
Relaxing some of these assumptions, or finding new paradigms might open the
way to our understanding of the true nature of Dark Matter.
Effective Operators for Dark Matter Detection
Appendix A
DM Effective Interaction Terms
with SM Gauge Bosons
We list here all possible hermitian interaction terms of the scalar DM fields with
the SM gauge bosons. We require invariance under Lorentz, electromagnetic and
color gauge symmetries, with the low energy part of the dark sector featuring a
lightest neutral state and possibly also electrically charged states; moreover, we
take the DM to be charged under an extra U (1) global symmetry, protecting the
lightest state against decay. When writing the terms in hermitian form is not
suitable, we explicit it in the text. This classification covers, to the best of our
knowledge, all the possible effective interaction terms of the DM fields with SM
gauge bosons up to dimension (d) six in mass (see Sec. 3 for the list of interactions
with other particles).
To insure invariance under U (1)EM and SU (3)c , the photon and gluon fields
can only enter the Lagrangian in the form of field strength operators (Fµν and
Gaµν , respectively) and covariant derivatives
←
→ 1←
→
Dµ ≡ ∂µ − ieQAµ ,
2
Dµ ≡ ∂µ − ieQAµ ,
(A.1)
as defined in Sec. 3.2.2. Consider that the photon field strength is gauge-neutral
and hence its covariant derivative1 coincides with its ordinary derivative, while we
will not need here to apply the covariant derivative to the gluon field.
In each list we label some of the operators with a number or a symbol in
boldface, under the column indicated by n; these labels are used to construct
explicit relations among the operators, below the table where the operators appear.
When needed we have also combined operators directly in the tables. Note that
these numbers or symbols are just labels and therefore do not need to be ordered.
For a generic gauge group the covariant derivative of the field strength Gµν is Dρ Gµν =
∂ρ Gµν + [Gρ , Gµν ], where Gρ is the non-abelian gauge field.
1
85
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
A.1
86
Singlet’s interaction terms with SM
gauge bosons
We start with the list of effective interactions of an electromagnetically neutral
scalar field φ with the SM gauge bosons.
A.1.1
Interaction with gluons
The possible terms involving gluon fields are
n
d−4
2
2
term
φ∗ φ Gaµν Gµν
a
∗
a
φ φ Gµν G̃µν
a
µνρσ Ga .
where G̃µν
a ≡ε
ρσ
A.1.2
Interaction with photons only
In the same way the possible terms involving only photon fields are
n
d−4
term
2
φ∗ φ F µν Fµν
2
φ∗ φ Fµν F̃ µν
2
Jµ (∂ν F µν )
←
→
where F̃ µν ≡ εµνρσ Fρσ and Jµ ≡ i φ∗ ∂µ φ ≡ i [φ∗ (∂µ φ) − (∂µ φ∗ )φ].
A.1.3
Interaction with Electroweak gauge bosons W ,
Z and A
We can divide the remaining couplings into three classes with regard to the number
of derivatives.
With no derivatives
n
d−4
term
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
0
φ∗ φ Z µ Zµ
0
φ∗ φ W +µ Wµ−
2
φ∗ φ Z µ Zµ Z ν Zν
2
φ∗ φ Z µ Zµ W +ν Wν−
2
φ∗ φ Z µ Z ν Wµ+ Wν−
2
φ∗ φ W +µ Wµ− W +ν Wν−
2
φ∗ φ W +µ Wµ+ W −ν Wν−
87
With one derivative
We define
µ
v± ν ≡ W +µ Wν− ± Wν+ W −µ ,
(A.2)
on which the hermitian conjugation acts just exchanging the Lorentz indices. We
also introduce
+µ
) Wν− ± Wν+ (Dµ W −µ ) ,
w±
1 ν ≡ (Dµ W
w±
2ν
≡W
µ
+µ
(Dµ Wν− )
±
(Dµ Wν+ ) W −µ
,
(A.3)
(A.4)
µ
−
−
such that v+ ν , w+
i ν are hermitian while v ν , wi ν are anti-hermitian. These composite operators are electrically neutral, so the superscript sign does not represent
their electric charge. The operators satisfy the following relations:
µ
±
±
w±
1 ν + w2 ν = ∂µ v ν ,
n
d−4
→ +µ
←
→
±
−←
∓ W −µ Dµ Wν+ .
w±
1 ν − w2 ν = Wν Dµ W
(A.5)
term
0
Jµ Z µ
0
φ∗ φ (∂µ Z µ )
2
Jµ Z µ Z ν Zν
21
2
∂µ (φ∗ φ) Z µ Z ν Zν
22
2
φ∗ φ (∂µ Z µ ) Z ν Zν
23
2
φ∗ φ Z µ (∂µ Z ν ) Zν
2
Jµ Z µ W +ν Wν−
31
2
∂µ (φ∗ φ) Z µ W +ν Wν−
32
2
φ∗ φ(∂µ Z µ )W +ν Wν−
33
2
φ∗ φ Z µ ∂µ (W +ν Wν− )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
2
←
→
i φ∗ φ Z µ (Wν− Dµ W +ν )
1
2
2
2
i [φ∗ (∂µ φ) Z ν W +µ Wν− − (∂µ φ∗ )φ Z ν Wν+ W −µ ]
1+2
2
1−2
2
i ∂µ (φ∗ φ) Z ν v− ν
3
2
φ∗ (∂µ φ) Z ν W +µ Wν− + (∂µ φ∗ )φ Z ν Wν+ W −µ
4
2
φ∗ (∂µ φ) Z ν Wν+ W −µ + (∂µ φ∗ )φ Z ν W +µ Wν−
3+4
2
∂µ (φ∗ φ) Z ν v+ ν
4−3
2
i Jµ Z ν v− ν
9
2
i φ∗ φ (∂µ Z ν )v− ν
10
2
φ∗ φ (∂µ Z ν )v+ ν
5
2
i φ∗ φ Z ν w−
1ν
6
2
i φ∗ φ Z ν w−
2ν
5+6
2
6−5
2
7
2
i φ∗ φ Z ν (∂µ v− ν )
←
→
←
→
i φ∗ φ Z ν (W +µ Dµ Wν− − W −µ Dµ Wν+ )
8
2
φ∗ φ Z ν w+
2ν
7+8
2
8−7
2
φ∗ φ Z ν (∂µ v+ ν )
←
→
←
→
φ∗ φ Z ν (W +µ Dµ Wν− + W −µ Dµ Wν+ )
2
i εµνρσ Jµ Zν Wρ+ Wσ−
11
2
i εµνρσ ∂µ (φ∗ φ) Zν Wρ+ Wσ−
12
2
i εµνρσ φ∗ φ (∂µ Zν ) Wρ+ Wσ−
13
2
2
i εµνρσ φ∗ φ Zν ∂µ (Wρ+ Wσ− )
←
→
εµνρσ φ∗ φ Zν (Wρ+ Dµ Wσ− )
2
i φ∗ φ Fµν W +µ W −ν
2
i φ∗ φ F̃µν W +µ W −ν
88
i [φ∗ (∂µ φ) Z ν Wν+ W −µ − (∂µ φ∗ )φ Z ν W +µ Wν− ]
µ
Jµ Z ν v+ ν
µ
µ
µ
µ
µ
µ
φ∗ φ Z ν w+
1ν
µ
Not all the operators are independent. In fact the following linear combinations
amount to total derivatives:
(3 + 4) + (7 + 8) + 10
(1 − 2) + (5 + 6) + 9
11 + 12 + 13
21 + 22 + 2 ∗ 23
31 + 32 + 33.
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
89
With two derivatives
We can divide these terms in three lists, depending on which gauge bosons participate in the interaction.
Couplings to a photon and a Z boson
d−4
n
term
2
Jµ F µν Zν
2
Jµ F̃ µν Zν
1
2
∂µ (φ∗ φ) F µν Zν
2
2
φ∗ φ (∂µ F µν )Zν
3
2
φ∗ φ F µν (∂µ Zν )
1̃
2
∂µ (φ∗ φ) F̃ µν Zν
3̃
2
φ∗ φ F̃ µν (∂µ Zν )
where the following combinations are total derivatives and therefore vanish:
1+2+3
1̃ + 3̃, due to the fact that ∂µ F̃ µν = 0.
Couplings to two Z bosons
n
1
d−4
2
2
2
11
2
2∗1+2
2
3
2
4
2
12
2
2∗3+4
2
5
2
6
2
term
(∂ µ φ∗ )(∂
ν
µ φ) Z Zν
[(∂ µ ∂µ φ∗ )φ + φ∗ (∂ µ ∂µ φ)] Z ν Zν
−i [(∂ µ ∂µ φ∗ )φ − φ∗ (∂ µ ∂µ φ)] Z ν Zν
= (∂µ J µ ) Z ν Zν
∂ µ ∂µ (φ∗ φ) Z ν Zν
(∂µ φ∗ )(∂ν φ) Z µ Z ν
[(∂µ ∂ν φ∗ )φ + φ∗ (∂µ ∂ν φ)] Z µ Z ν
−i [(∂µ ∂ν φ∗ )φ − φ∗ (∂µ ∂ν φ)] Z µ Z ν
= (∂µ Jν ) Z µ Z ν
∂µ ∂ν (φ∗ φ) Z µ Z ν
∂ µ (φ∗ φ) (∂µ Z ν ) Zν
∂µ (φ∗ φ) (∂ν Z µ ) Z ν
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
7
2
∂µ (φ∗ φ) (∂ν Z ν ) Z µ
6+7
2
7−6
2
∂µ (φ∗ φ) ∂ν (Z µ Z ν )
←
→
∂µ (φ∗ φ) (Z µ ∂ν Z ν )
8
2
J µ (∂µ Z ν ) Zν
9
2
Jµ (∂ν Z µ ) Z ν
10
2
Jµ (∂ν Z ν ) Z µ
9 + 10
2
10 − 9
2
Jµ ∂ν (Z µ Z ν )
←
→
Jµ (Z µ ∂ν Z ν )
13
2
φ∗ φ (∂µ Z µ ) (∂ν Z ν )
14
2
φ∗ φ (∂µ Z ν ) (∂ν Z µ )
15
2
φ∗ φ (∂ µ Z ν ) (∂µ Zν )
16
2
φ∗ φ (∂µ ∂ν Z µ ) Z ν
17
2
φ∗ φ (∂ µ ∂µ Z ν ) Zν
2 ∗ (15 + 17)
2
φ∗ φ ∂ µ ∂µ (Z ν Zν )
2
φ∗ φ ∂µ ∂ν (Z µ Z ν )
13 + 14+
+2 ∗ 16
90
where the following combinations are total derivatives:
(2 ∗ 1 + 2) + 2 ∗ 5
(2 ∗ 3 + 4) + (6 + 7)
(6 + 7) + (13 + 14 + 2 ∗ 16)
2∗1+2+2∗5
2 ∗ 5 + 2 ∗ (15 + 17)
11 + 2 ∗ 8
12 + (9 + 10).
It is amusing to note that the following relations hold:
(2 ∗ 3 + 4) + 2 ∗ (6 + 7) + (13 + 14 + 2 ∗ 16) = ∂µ ∂ν (φ∗ φ Z µ Z ν )
and
2 ∗ 1 + 2 + 2 ∗ (2 ∗ 5) + 2 ∗ (15 + 17) = ∂ µ ∂µ (φ∗ φ Z ν Zν ).
Coupling to two W bosons plus photons
n
1
2
d−4
2
2
term
(∂ µ φ∗ )(∂
[(∂ µ ∂µ φ∗ )φ +
+ν W −
µ φ) W
ν
∗
µ
φ (∂ ∂µ φ)] W +ν Wν−
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
11
2
−i [(∂ µ ∂µ φ∗ )φ − φ∗ (∂ µ ∂µ φ)] W +ν Wν−
2∗1+2
2
∂ µ ∂µ (φ∗ φ) W +ν Wν−
3a
2
(∂µ φ∗ )(∂ν φ) W +µ W −ν
3b
2
(∂µ φ∗ )(∂ν φ) W +ν W −µ
3a + 3b
2
(∂µ φ∗ )(∂ ν φ) v+ ν
3a − 3b
2
4
2
(∂µ φ∗ )(∂ ν φ) v− ν = − 2i (∂µ J ν ) v− ν
[(∂µ ∂ν φ∗ )φ + φ∗ (∂µ ∂ν φ)] W +µ W −ν
12
2
(∂µ J ν ) v+ ν =
3a + 3b + 4
2
5a
2
5b
2
∂ µ (φ∗ φ) ∂µ (W +ν Wν− )
←
→
i ∂ µ (φ∗ φ) (W −ν Dµ Wν+ )
6a
2
∂ ν (φ∗ φ) w+
1ν
6b
2
i ∂ ν (φ∗ φ) w−
1ν
7a
2
∂ ν (φ∗ φ) w+
2ν
7b
2
i ∂ ν (φ∗ φ) w−
2ν
6a + 7a
2
6a − 7a
2
6b + 7b
2
∂ ν (φ∗ φ) (∂µ v+ ν )
←
→
←
→
∂ ν (φ∗ φ) (Wν− Dµ W +µ − W −µ Dµ Wν+ )
6b − 7b
2
8a
2
8b
2
J µ ∂µ (W +ν Wν− )
←
→
i J µ (W −ν Dµ Wν+ )
9a
2
J ν w+
1ν
9b
2
i J ν w−
1ν
10a
2
J ν w+
2ν
10b
2
i J ν w−
2ν
9a + 10a
2
9a − 10a
2
9b + 10b
2
J ν (∂µ v+ ν )
←
→
←
→
J ν (Wν− Dµ W +µ − W −µ Dµ Wν+ )
9b − 10b
2
i J ν (∂µ v− ν )
←
→
←
→
i J ν (Wν− Dµ W +µ + W −µ Dµ Wν+ )
13
2
φ∗ φ (Dµ W +µ ) (Dν W −ν )
91
= (∂µ J µ ) W +ν Wν−
µ
µ
µ
µ
−2 i [(∂µ ∂ν φ∗ )φ − φ∗ (∂µ ∂ν φ)] W +µ W −ν
∂µ ∂ν (φ∗ φ) W +µ W −ν
µ
µ
i ∂ ν (φ∗ φ) (∂µ v− ν )
←
→
←
→
i ∂ ν (φ∗ φ) (Wν− Dµ W +µ + W −µ Dµ Wν+ )
µ
µ
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
14
2
φ∗ φ (Dµ W +ν ) (Dν W −µ )
15
2
φ∗ φ (D µ W +ν ) (Dµ Wν− )
16a
2
φ∗ φ [(Dµ Dν W +µ ) W −ν + W +µ (Dµ Dν W −ν )]
2
2 i φ∗ φ [(Dµ Dν W +µ ) W −ν − W +µ (Dµ Dν W −ν )]
←
→
←
→
= i φ∗ φ ∂µ (W −ν Dν W +µ + W −µ Dν W +ν )
16b
−
= i φ∗ φ ∂ ν (w−
2 ν − w1 ν )
17a
2
φ∗ φ [(D µ Dµ W +ν ) Wν− + W +ν (D µ Dµ Wν− )]
17b
2
i φ∗ φ [(D µ Dµ W +ν ) Wν− − W +ν (D µ Dµ Wν− )]
←
→
= i φ∗ φ ∂ µ (W −ν Dµ Wν+ )
2 ∗ 15 + 17a
2
φ∗ φ ∂ µ ∂µ (W +ν Wν− )
13 + 14 + 16a
2
2 ∗ (14 − 13)
2
φ∗ φ ∂µ ∂ν (W +µ W −ν )
←
→
←
→
φ∗ φ ∂µ (W −ν Dν W +µ − W −µ Dν W +ν )
21
2
εµνρσ (∂µ φ∗ )(∂ν φ) Wρ+ Wσ−
2
22
2
23
2
24
92
+
= φ∗ φ ∂ ν (w+
2 ν − w1 ν )
i εµνρσ ∂
∗
+
−
µ (φ φ) ∂ν (Wρ Wσ )
R
by parts
−−−−−−→ 0
←
→ −
µνρσ
∗
+
ε
∂µ (φ φ) (Wρ Dν Wσ )
2
i εµνρσ Jµ ∂ν (Wρ+ Wσ− )
←
→
εµνρσ Jµ (Wρ+ Dν Wσ− )
2
εµνρσ φ∗ φ (∂µ Wρ+ )(∂ν Wσ− )
where the following combinations are total derivatives:
(2 ∗ 1 + 2) + 5a and 5a + (2 ∗ 15 + 17a)
11 + 8a
2 ∗ (3a + 3b + 4) + (6a + 7a)
12 + (9a + 10a)
−2 ∗ (3a − 3b) + (9b + 10b)
5b + 17b
7a − 6a + 2 ∗ (14 − 13)
7b − 6b + 16b
(6a + 7a) + 2 ∗ (13 + 14 + 16a)
2 ∗ 21 + 23
2 ∗ 24 + 22.
We also have
(2 ∗ 1 + 2) + 2 ∗ 5a + (2 ∗ 15 + 17a) = ∂ µ ∂µ (φ∗ φ W +ν Wν− ).
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
93
With three derivatives
The only possible SM gauge boson that can enter the interaction terms with three
derivatives is the Z boson.
n
d−4
term
2
φ∗ φ (∂ 2 ∂µ Z µ )
2
2
∂µ (φ∗ φ) (∂ 2 Z µ )
3
2
∂ µ (φ∗ φ) (∂µ ∂ν Z ν )
4
2
Jµ (∂ 2 Z µ )
5
2
J µ (∂µ ∂ν Z ν )
6
2
(∂ µ φ∗ )(∂µ φ) (∂ν Z ν )
7
2
[(∂µ φ∗ )(∂ν φ) + (∂ν φ∗ )(∂µ φ)] (∂ µ Z ν )
8
2
i [(∂µ φ∗ )(∂ν φ) − (∂ν φ∗ )(∂µ φ)] (∂ µ Z ν )
9
2
i εµνρσ (∂µ φ∗ )(∂ν φ) (∂ρ Zσ ) −−−−−−→ 0
10
2
[(∂ 2 φ∗ )φ + φ∗ (∂ 2 φ)] (∂µ Z µ )
11
2
−i [(∂ 2 φ∗ )φ − φ∗ (∂ 2 φ)] (∂µ Z µ )
12
2
[(∂µ ∂ν φ∗ )φ + φ∗ (∂µ ∂ν φ)] (∂ µ Z ν )
13
2
2 ∗ 6 + 10
2
i [(∂µ ∂ν φ∗ )φ − φ∗ (∂µ ∂ν φ)] (∂ µ Z ν )
7 + 12
2
∂µ ∂ν (φ∗ φ) (∂ µ Z ν )
8 − 13
2
(∂ µ Jν ) (∂µ Z ν )
−(8 + 13)
2
(∂µ J ν ) (∂ν Z µ )
14
2
[(∂ 2 φ∗ )(∂µ φ) + (∂µ φ∗ )(∂ 2 φ)] Z µ
15
2
i [(∂ 2 φ∗ )(∂µ φ) − (∂µ φ∗ )(∂ 2 φ)] Z µ
16
2
17
2
18
2
19
2
14 + 16
2
∂ µ [(∂µ φ∗ )(∂ν φ) + (∂ν φ∗ )(∂µ φ)] Z ν
15 − 17
2
i ∂ µ [(∂µ φ∗ )(∂ν φ) − (∂ν φ∗ )(∂µ φ)] Z ν
1
R
16 + 18
2
by parts
= (∂µ J µ ) (∂ν Z ν )
∂ µ ∂µ (φ∗ φ) (∂ν Z ν )
[(∂µ ∂ν φ∗ )(∂ µ φ) + (∂ µ φ∗ )(∂µ ∂ν φ)] Z ν
= ∂µ [(∂ ν φ∗ )(∂ν φ)] Z µ
i [(∂µ ∂ν φ∗ )(∂ µ φ) − (∂ µ φ∗ )(∂µ ∂ν φ)] Z ν
[(∂ 2 ∂µ φ∗ )φ + φ∗ (∂ 2 ∂µ φ)] Z µ
i [(∂ 2 ∂µ φ∗ )φ − φ∗ (∂ 2 ∂µ φ)] Z µ
∂ µ [(∂µ ∂ν φ∗ )φ + φ∗ (∂µ ∂ν φ)] Z ν
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
17 + 19
(14 + 16)+
(16 + 18)
(15 − 17)−
(17 + 19)
−(15 − 17)
−(17 + 19)
2
i ∂ µ [(∂µ ∂ν φ∗ )φ − φ∗ (∂µ ∂ν φ)] Z ν
2
∂ 2 ∂µ (φ∗ φ) Z µ
2
(∂ 2 Jµ ) Z µ
2
(∂µ ∂ν J µ ) Z ν
94
= −15 − 19
Not all these terms are independent, due to the vanishing of the following total
derivatives:
1 + 2 and 1 + 3
6 + 16
7 + (14 + 16)
12 + (16 + 18)
13 + (17 + 19)
2 + (7 + 12) and 3 + (7 + 12) and 3 + (2 ∗ 6 + 10)
(7 + 12) + (14 + 16) + (16 + 18) and (2 ∗ 6 + 10) + (14 + 16) + (16 + 18)
4 + (8 − 13) and (8 − 13) + ((15 − 17) − (17 + 19))
5 − (8 + 13) and −(8 + 13) + (−15 − 19)
5 + 11 and 11 + (−15 − 19).
A.1.4
Terms with four φ’s
The possible terms that are quartic in the fields φ only concern couplings to the
Weak gauge bosons:
n
d−4
term
2
(φ∗ φ)2 Z µ Zµ
2
(φ∗ φ)2 (∂µ Z µ )
2
(φ∗ φ)2 W +µ Wµ−
2
φ∗ φ J µ Zµ
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
A.2
95
Doublet’s interaction terms with SM
gauge bosons
We enlarge the lists provided in Appendix A.1 to include the interactions of an
electrically charged particle D± . The neutral state, that we call here D0 , has to
be identified with φ in the previous appendix. To obtain the full list of interaction
terms with the SM gauge bosons one should consider the terms presented here and
in the previous appendix.
A.2.1
Interaction with gluons only
n
d−4
2
2
A.2.2
term
∗
D± D± Gaµν Gµν
a
∗ ± a
±
D D Gµν G̃µν
a
Interaction with photons only
n
d−4
term
2
D
±∗
D± F µν Fµν
±∗
2
D± F µν F̃µν
→
∗←
i (D± Dµ D± )(∂ν F µν )
1
0
(D µ Dµ D± )D± + D± (D µ Dµ D± )
2
0
(D µ D± )(Dµ D± )
1+2∗2
0
∂ µ ∂µ (D± D± )
3
0
i [(D µ Dµ D± )D± − D± (D µ Dµ D± )]
→
∗←
= −i ∂ µ (D± Dµ D± )
4
2
(D µ Dµ D ν Dν D± )D± + D± (D µ Dµ D ν Dν D± )
5
2
(D µ Dµ D ν D± )(Dν D± ) + (Dν D± )(D µ Dµ D ν D± )
4i
2
5i
2
i [(D µ Dµ D ν Dν D± )D± − D± (D µ Dµ D ν Dν D± )]
6
2
7
2
2
D
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
i [(D µ Dµ D ν D± )(Dν D± ) − (Dν D± )(D µ Dµ D ν D± )]
∗
(D µ Dµ D± )(D ν Dν D± )
∗
(D µ D ν D± )(Dµ Dν D± )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
96
Not all the operators are independent. In fact the following linear combinations
amount to total derivatives:
(1 + 2 ∗ 2)
3
4+5
4i + 5i
5+6
5 + 7.
A.2.3
Interaction with Electroweak gauge bosons W ,
Z and A
With no derivatives
n
d−4
term
0∗
0
D D± Wµ∓ Z µ + h.c.
0
i [D0 D± Wµ∓ Z µ − h.c.]
∗
∗
2
D0 D± Wµ∓ Z µ Z ν Zν + h.c.
2
i [D0 D± Wµ∓ Z µ Z ν Zν − h.c.]
∗
∗
2
D0 D± Wµ∓ W +µ Wν− Z ν + h.c.
2
i [D0 D± Wµ∓ W +µ Wν− Z ν − h.c.]
∗
∗
2
D0 D± Wµ∓ Wν+ W −µ Z ν + h.c.
2
i [D0 D± Wµ∓ Wν+ W −µ Z ν − h.c.]
∗
∗
0
D ± D ± Z µ Zµ
0
D± D± W +µ Wµ−
2
D ± D ± Z µ Zµ Z ν Zν
2
D± D± Z µ Zµ W +ν Wν−
2
D± D± Z µ Z ν Wµ+ Wν−
2
D± D± W +µ Wµ− W +ν Wν−
2
D± D± W +µ W −ν Wµ+ Wν−
∗
∗
∗
∗
∗
∗
With one derivative
To ease the notation we divide the terms in two lists:
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
97
i) The first contains non-hermitian terms; each non-hermitian term gives rise
to two hermitian terms, namely its real and imaginary parts.
ii) In the second list all the terms are already hermitian.
n
d−4
term
2
∗
(∂ µ D0 ) D± Wµ∓
∗
D0 (D µ D± ) Wµ∓
∗
(∂ µ D0 ) D± Wµ∓ Z ν Zν
∗
D0 (D µ D± ) Wµ∓ Z ν Zν
∗
D0 D± Wµ∓ (∂ µ Z ν ) Zν
∗
(∂µ D0 ) D± Wν∓ Z µ Z ν
∗
D0 (Dµ D± ) Wν∓ Z µ Z ν
∗
D0 D± Wν∓ (∂µ Z µ ) Z ν
∗
D0 D± Wν∓ Z µ (∂µ Z ν )
∗
εµνρσ D0 D± Wµ∓ Zν (∂ρ Zσ )
∗
D0 D± Wµ∓ Zν F µν
∗
D0 D± Wµ∓ Zν F̃ µν
2
(∂ µ D0 )D± Wµ∓ W ±ν Wν∓
2
D0 (D µ D± ) Wµ∓ W ±ν Wν∓
2
D0 D± Wµ∓ (D µ W ±ν )Wν∓
2
D0 D± Wµ∓ W ±ν (D µ Wν∓ )
2
(∂µ D0 ) D± W ∓ν W ±µ Wν∓
2
D0 (Dµ D± ) W ∓ν W ±µ Wν∓
2
D0 D± (Dµ W ∓ν ) W ±µ Wν∓
2
εµνρσ D0 D± (Dµ Wν∓ )Wρ± Wσ∓
0
0
2
2
2
2
2
2
2
2
2
∗
∗
∗
∗
∗
∗
∗
∗
µ
±
We use here the operators v± ν , w±
1 ν and w2 ν defined in Sec. A.1.3.
n
d−4
0
0
term
∂µ
∗
(D± D± ) Z µ
→
∗←
i (D± Dµ D± ) Z µ
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
∗
2
∂µ (D± D± ) Z µ Z ν Zν
→
∗←
i (D± Dµ D± ) Z µ Z ν Zν
2
D± D± (∂µ Z µ )Z ν Zν
2
∂µ (D± D± ) Z µ W +ν Wν−
→
∗←
i (D± Dµ D± ) Z µ W +ν Wν−
2
2
∗
∗
∗
2
D± D± (∂µ Z µ )W +ν Wν−
←
→
∗
i D± D± Z µ (W +ν Dµ Wν− )
1
2
(Dµ D± )D± Z ν W +µ Wν− +
2
2
D± (Dµ D± )Z ν W +µ Wν− +
1+2
2
2−1
2
∂µ (D± D± ) Z ν v+ ν
→
∗←
µ
(D± Dµ D± ) Z ν v− ν
3
2
D± D± (∂µ Z ν ) v+ ν
4
2
D± D± Z ν w+
1ν
5
2
D± D± Z ν w+
2ν
4+5
2
4−5
2
D± D± Z ν (∂µ v+ ν )
←
→
←
→
∗
D± D± Z ν (Wν− Dµ W +µ − W −µ Dµ Wν+ )
6
2
i [(Dµ D± )D± Z ν W +µ Wν− −
7
2
i[D± (Dµ D± )Z ν W +µ Wν− −
6+7
2
7−6
2
i ∂µ (D± D± ) Z ν v− ν
→
∗←
µ
i (D± Dµ D± )Z ν v+ ν
8
2
i D± D± (∂µ Z ν ) v− ν
9
2
i D± D± Z ν w−
1ν
10
2
i D± D± Z ν w−
2ν
9 + 10
2
9 − 10
2
2
i D± D± Z ν (∂µ v− ν )
←
→
←
→
∗
i D± D± Z ν (Wν− Dµ W +µ + W −µ Dµ Wν+ )
→
∗←
εµνρσ (D± Dµ D± ) Zν Wρ+ Wσ−
11
2
i εµνρσ ∂µ (D± D± ) Zν Wρ+ Wσ−
12
2
i εµνρσ D± D± (∂µ Zν )Wρ+ Wσ−
2
98
∗
∗
D± (Dµ D± )Z ν Wν+ W −µ
∗
∗
(Dµ D± )D± Z ν Wν+ W −µ
∗
µ
∗
µ
∗
∗
∗
µ
∗
∗
D± (Dµ D± )Z ν Wν+ W −µ ]
∗
∗
(Dµ D± )D± Z ν Wν+ W −µ ]
∗
µ
∗
µ
∗
∗
∗
µ
∗
∗
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons
13
∗
2
i εµνρσ D± D± Zν ∂µ (Wρ+ Wσ− )
←
→
∗
εµνρσ D± D± Zν (Wρ+ Dµ Wσ− )
2
i D± D± W +µ W −ν Fµν
2
i D± D± W +µ W −ν F̃µν
2
99
∗
∗
The following combinations are total derivatives:
(1 + 2) + 3 + (4 + 5)
(6 + 7) + 8 + (9 + 10)
11 + 12 + 13.
With two derivatives
For simplicity and readability we list here only some terms, the rest being derived
from these by permitting the derivates to act on the various fields in different
order. When below a term the symbol ‘∗’ appears this means that the remaining
terms obtained by simply changing position of the derivatives are not displayed
here but should be taken into account. One has also to pay attention to the fact
that a partial derivative turns into a covariant one when it acts on a charged field.
n
d−4
2
term
(∂ µ ∂
±
∓ ν
0∗
µ D ) D Wν Z
2
2
+ h.c.
*
∗
i [(∂ µ ∂µ D0 ) D± Wν∓ Z ν
2
*
− h.c.]
0∗
2
(∂ µ ∂ν D ) D± Wµ∓ Z ν + h.c.
2
*
2
i [(∂ µ ∂ν D0 ) D± Wµ∓ Z ν − h.c.]
2
2
∗
*
∗
εµνρσ (∂µ D0 ) (Dν D± ) Wρ∓ Zσ + h.c.
2
2
2
*
0∗
i [εµνρσ (∂µ D ) (Dν D± ) Wρ∓ Zσ − h.c.]
*
0∗
2
(∂µ D ) D± Wν∓ F µν + h.c.
2
*
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 100
2
2
∗
i [(∂µ D0 ) D± Wν∓ F µν − h.c.]
*
0∗
2
(∂µ D ) D± Wν∓ F̃ µν + h.c.
2
*
2
i [(∂µ D0 ) D± Wν∓ F̃ µν − h.c.]
2
∗
*
[(D µ Dµ D
±∗
∗
) D± + D± (D µ Dµ D± )] Z ν Zν
1
2
2
2
(D µ D± ) (Dµ D± ) Z ν Zν
1+2∗2
2
∂ µ ∂µ (D± D± ) Z ν Zν
2
i [(D µ Dµ D± ) D± − D± (D µ Dµ D± )] Z ν Zν
→
∗←
= −i ∂ µ (D± Dµ D± ) Z ν Zν
2
∗
∗
∗
∗
∗
D± D± (∂ µ Z ν ) (∂µ Zν )
∗
∗
3
2
[(Dµ Dν D± ) D± + D± (Dµ Dν D± )] Z µ Z ν
4
2
(Dµ D± ) (Dν D± ) Z µ Z ν
3+2∗4
2
∂µ ∂ν (D± D± ) Z µ Z ν
2
i [(Dµ Dν D± ) D± − D± (Dµ Dν D± )] Z µ Z ν
→
∗←
= −i ∂µ (D± Dν D± ) Z µ Z ν
∗
∗
∗
∗
∗
∂µ (D± D± ) (∂ν Z µ )Z ν
→
∗←
i (D± Dµ D± ) (∂ν Z µ )Z ν
31
2
32
2
33
2
34
2
33 + 31
2
33 − 31
2
34 + 32
2
34 − 32
2
∂µ (D± D± ) ∂ν (Z µ Z ν )
←
→
∗
∂µ (D± D± ) (Z µ ∂ν Z ν )
→
∗←
i (D± Dµ D± ) ∂ν (Z µ Z ν )
→
←
→
∗←
i (D± Dµ D± ) (Z µ ∂ν Z ν )
35
2
D± D± (∂µ Z µ ) (∂ν Z ν )
36
2
D± D± (∂µ Z ν ) (∂ν Z µ )
37
2
D± D± (∂µ ∂ν Z µ ) Z ν
2
D± D± ∂µ ∂ν (Z µ Z ν )
35 + 36
+2 ∗ 37
35 − 36
2
38
2
∗
∂µ (D± D± ) Z µ (∂ν Z ν )
→
∗←
i (D± Dµ D± ) Z µ (∂ν Z ν )
∗
∗
∗
∗
∗
←
→
∗
D± D± ∂µ (Z µ ∂ν Z ν )
←
→
∗
εµνρσ ∂µ (D± D± ) (Zρ ∂ν Zσ )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 101
2
→
←
→
∗←
i εµνρσ (D± Dµ D± ) (Zρ ∂ν Zσ )
39
2
εµνρσ D± D± (∂µ Zρ )(∂ν Zσ )
5
2
[(D µ Dµ D± ) D± + D± (D µ Dµ D± )] W +ν Wν−
6
2
(D µ D± ) (Dµ D± ) W +ν Wν−
5+2∗6
2
∂ µ ∂µ (D± D± ) W +ν Wν−
40
2
41
2
i [(D µ Dµ D± ) D± − D± (D µ Dµ D± )] W +ν Wν−
→
∗←
= −i ∂ µ (D± Dµ D± ) W +ν Wν−
42
2
43
2
∗
∗
∗
∗
∗
∗
∗
∗
2
∂ µ (D± D± ) ∂µ (W +ν Wν− )
←
→
∗
i ∂ µ (D± D± ) (W +ν Dµ Wν− )
→
∗←
i (D± D µ D± ) ∂µ (W +ν Wν− )
→
←
→
∗←
(D± D µ D± ) (W +ν Dµ Wν− )
7
2
[(Dµ Dν D± ) D± + D± (Dµ Dν D± )] W +µ W −ν
8a
2
(Dµ D± ) (Dν D± ) W +µ W −ν
8b
2
(Dµ D± ) (Dν D± ) W +ν W −µ
7 + 8a + 8b
2
(8a − 8b)/2
2
7i
2
∂µ ∂ν (D± D± ) W +µ W −ν
→
∗←
µ
∂µ (D± D ν D± ) v− ν
→
∗←
µ
i ∂µ (D± D ν D± ) v+ ν =
51
2
51i
2
52
2
52i
2
53
2
53i
2
54
2
54i
2
∗
∗
∗
∗
∗
∗
∗
2 i [(Dµ Dν D± ) D± − D± (Dµ Dν D± )] W +µ W −ν
∗
(Dµ D± )D± W +µ (Dν W −ν )
∗
+ D± (Dµ D± ) (Dν W +ν )W −µ
∗
i [(Dµ D± )D± W +µ (Dν W −ν )
∗
− D± (Dµ D± ) (Dν W +ν )W −µ ]
∗
(Dµ D± )D± (Dν W +ν )W −µ
∗
+ D± (Dµ D± ) W +µ (Dν W −ν )
∗
i [(Dµ D± )D± (Dν W +ν )W −µ
∗
− D± (Dµ D± ) W +µ (Dν W −ν )]
∗
(Dµ D± )D± (Dν W +µ )W −ν
∗
+ D± (Dµ D± ) W +ν (Dν W −µ )
∗
i [(Dµ D± )D± (Dν W +µ )W −ν
∗
− D± (Dµ D± ) W +ν (Dν W −µ )]
∗
(Dµ D± )D± W +ν (Dν W −µ )
∗
+ D± (Dµ D± ) (Dν W +µ )W −ν
∗
i [(Dµ D± )D± W +ν (Dν W −µ )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 102
∗
− D± (Dµ D± ) (Dν W +µ )W −ν ]
∗
−(51i + 52i)
2
∂ ν (D± D± ) w+
1ν
→ν ± −
∗←
±
(D D D ) w1 ν
→
∗←
i (D± D ν D± ) w+
1ν
52i − 51i
2
i ∂ ν (D± D± ) w−
1ν
53 + 54
2
53 − 54
2
51 + 52
2
51 − 52
2
∗
∗
−(53i + 54i)
2
∂ ν (D± D± ) w+
2ν
→ν ± −
∗←
±
(D D D ) w2 ν
→
∗←
i (D± D ν D± ) w+
2ν
54i − 53i
2
i ∂ ν (D± D± ) w−
2ν
61
2
D± D± (Dµ W +µ ) (Dν W −ν )
62
2
D± D± (Dµ W +ν ) (Dν W −µ )
63
2
D± D± (D µ W +ν ) (Dµ Wν− )
64a
2
D± D± [(Dµ Dν W +µ ) W −ν + W +µ (Dµ Dν W −ν )]
2
2 i D± D± [(Dµ Dν W +µ ) W −ν − W +µ (Dµ Dν W −ν )]
←
→
←
→
∗
= i D± D± ∂µ (W −ν Dν W +µ + W −µ Dν W +ν )
∗
∗
∗
∗
∗
∗
64b
∗
∗
−
= i D± D± ∂ ν (w−
2 ν − w1 ν )
65a
2
D± D± [(D µ Dµ W +ν ) Wν− + W +ν (D µ Dµ Wν− )]
65b
2
i D± D± [(D µ Dµ W +ν ) Wν− − W +ν (D µ Dµ Wν− )]
←
→
∗
= i D± D± ∂ µ (W −ν Dµ Wν+ )
2 ∗ 63 + 65a
2
D± D± ∂ µ ∂µ (W +ν Wν− )
61 + 62 + 64a
2
2 ∗ (62 − 61)
2
D± D± ∂µ ∂ν (W +µ W −ν )
←
→
←
→
∗
D± D± ∂µ (W −ν Dν W +µ − W −µ Dν W +ν )
9
2
εµνρσ (Dµ D± )(Dν D± ) Wρ+ Wσ−
10a
2
εµνρσ [(Dµ D± ) D± (Dν Wρ+ ) Wσ−
10b
2
i εµνρσ [(Dµ D± ) D± (Dν Wρ+ ) Wσ−
11a
2
εµνρσ [(Dµ D± ) D± Wρ+ (Dν Wσ− )
11b
2
i εµνρσ [(Dµ D± ) D± Wρ+ (Dν Wσ− )
−(10a + 11a)
2
∗
∗
∗
∗
+
= D± D± ∂ ν (w+
2 ν − w1 ν )
∗
∗
∗
−D± (Dµ D± ) Wρ+ (Dν Wσ− )]
∗
∗
+D± (Dµ D± ) Wρ+ (Dν Wσ− )]
∗
∗
−D± (Dµ D± ) (Dν Wρ+ ) Wσ− ]
∗
∗
+D± (Dµ D± ) (Dν Wρ+ ) Wσ− ]
→
∗←
εµνρσ (D± Dµ D± ) ∂ν (Wρ+ Wσ− )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 103
11a − 10a
2
10b + 11b
2
10b − 11b
2
12
2
2
2
2
2
2
←
→
∗
εµνρσ ∂µ (D± D± ) (Wρ+ Dν Wσ− )
R
by parts
∗ ±
µνρσ
±
+
−
iε
∂µ (D D ) ∂ν (Wρ Wσ ) −−−−−−→
→
←
→
∗←
i εµνρσ (D± Dµ D± ) (Wρ+ Dν Wσ− )
∗
εµνρσ D± D± (Dµ Wρ+ )(Dν Wσ− )
∗
∂µ (D± D± ) Zν F µν
∗
D± D± (∂µ Zν )F µν
→
∗←
i (D± Dµ D± ) Zν F µν
0
∗
∂µ (D± D± ) Zν F̃ µν
→
∗←
i (D± Dµ D± ) Zν F̃ µν
The following combinations are total derivatives:
2 ∗ 9 + 10a + 11a
11a − 10a + 2 ∗ 12
(3 + 2 ∗ 4) + (33 + 31)
(33 + 31) + (35 + 36 + 2 ∗ 37)
These terms enjoy the relation:
∗
(3 + 2 ∗ 4) + 2 ∗ (33 + 31) + (35 + 36 + 2 ∗ 37) = ∂µ ∂ν (D± D± Z µ Z ν )
(33 − 31) + (35 − 36)
2 ∗ 39 + 38
(5 + 2 ∗ 6) + 41
41 + (2 ∗ 63 + 65a)
These terms enjoy the relation:
∗
(5 + 2 ∗ 6) + 2 ∗ 41 + (2 ∗ 63 + 65a) = ∂ µ ∂µ (D± D± W +ν Wν− )
42 − 65b
43 − 40
2 ∗ (7 + 8a + 8b) + (51 + 52) + (53 + 54)
(51 + 52) + (53 + 54) + 2 ∗ (61 + 62 + 64a)
These terms enjoy the relation:
∗
(7 + 8a + 8b) + (51 + 52) + (53 + 54) + (61 + 62 + 64a) = ∂µ ∂ν (D± D± W +µ Wν− )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 104
(51 − 52) + (53 − 54) + (8a − 8b)/2
−(51i + 52i) − (53i + 54i) + 7i
(51 + 52) − (53 + 54) − 2 ∗ (62 − 61)
(52i − 51i) − (54i − 53i) − 64b.
With three derivatives
Only one Weak boson field at the time can enter the terms with three derivatives.
The interaction terms involving the photon and the W are
n
d−4
term
(∂ µ ∂
2
ν 0∗
±
∓
µ ∂ D ) D Wν
+ h.c.
2
*
2
i [(∂ µ ∂µ ∂ ν D0 ) D± Wν∓ − h.c.]
∗
2
*
εµνρσ
2
(∂µ
∗
D0 ) (D
±
∓
ν D ) (Dρ Wσ )
R
by parts
+ h.c. −−−−−−→ 0
R
∗
by parts
i εµνρσ [(∂µ D0 ) (Dν D± ) (Dρ Wσ∓ ) − h.c.] −−−−−−→ 0
2
The interaction terms involving the photon and the Z are instead
n
1
d−4
term
∗
D± D± (∂ 2 ∂
2
µZ
µ)
∗
2
2
∂µ (D± D± ) (∂ 2 Z µ )
3
2
4
2
5
2
∂ µ (D± D± ) (∂µ ∂ν Z ν )
→
∗←
i (D± Dµ D± ) (∂ 2 Z µ )
→
∗←
i (D± D µ D± ) (∂µ ∂ν Z ν )
6
2
(D µ D± )(Dµ D± ) (∂ν Z ν )
7
2
[(Dµ D± )(Dν D± ) + (Dν D± )(Dµ D± )] (∂ µ Z ν )
8
2
i [(Dµ D± )(Dν D± ) − (Dν D± )(Dµ D± )] (∂ µ Z ν )
9
2
i εµνρσ (Dµ D± )(Dν D± ) (∂ρ Zσ ) −−−−−−→ 0
10
2
∗
∗
∗
∗
∗
∗
R
∗
∗
by parts
∗
[(D2 D± )D± + D± (D2 D± )] (∂µ Z µ )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 105
∗
∗
11
2
−i [(D2 D± )D± − D± (D2 D± )] (∂µ Z µ )
→
∗←
= i ∂µ (D± D µ D± ) (∂ν Z ν )
12
2
[(Dµ Dν D± )D± + D± (Dµ Dν D± )] (∂ µ Z ν )
13
2
2 ∗ 6 + 10
2
i [(Dµ Dν D± )D± − D± (Dµ Dν D± )] (∂ µ Z ν )
7 + 12
2
8 − 13
2
−(8 + 13)
2
∂µ ∂ν (D± D± ) (∂ µ Z ν )
→
∗←
i ∂ µ (D± Dν D± ) (∂µ Z ν )
→
∗←
i ∂µ (D± D ν D± ) (∂ν Z µ )
14
2
[(D2 D± )(Dµ D± ) + (Dµ D± )(D2 D± )] Z µ
15
2
i [(D2 D± )(Dµ D± ) − (Dµ D± )(D2 D± )] Z µ
∗
∗
∗
∗
∗
∂ µ ∂µ (D± D± ) (∂ν Z ν )
∗
∗
∗
∗
∗
∗
∗
[(Dµ Dν D± )(D µ D± ) + (D µ D± )(Dµ Dν D± )] Z ν
16
2
17
2
18
2
19
2
14 + 16
2
∂ µ [(Dµ D± )(Dν D± ) + (Dν D± )(Dµ D± )] Z ν
15 − 17
2
16 + 18
2
i ∂ µ [(Dµ D± )(Dν D± ) − (Dν D± )(Dµ D± )] Z ν
17 + 19
2
i ∂ µ [(Dµ Dν D± )D± − D± (Dµ Dν D± )] Z ν
2
∂ 2 ∂µ (D± D± ) Z µ
2
→
∗←
i ∂ 2 (D± Dµ D± ) Z µ
2
→
∗←
i ∂µ ∂ν (D± D µ D± ) Z ν
(14 + 16)+
(16 + 18)
(15 − 17)−
(17 + 19)
−(15 − 17)
−(17 + 19)
∗
= ∂µ [(D ν D± )(Dν D± )] Z µ
∗
∗
i [(Dµ Dν D± )(D µ D± ) − (D µ D± )(Dµ Dν D± )] Z ν
∗
∗
[(D2 Dµ D± )D± + D± (D2 Dµ D± )] Z µ
∗
∗
i [(D2 Dµ D± )D± − D± (D2 Dµ D± )] Z µ
∗
∗
∗
∗
∗
∗
∂ µ [(Dµ Dν D± )D± + D± (Dµ Dν D± )] Z ν
∗
∗
∗
= −15 − 19
The following linear combinations are total derivatives:
1 + 2 and 1 + 3
6 + 16
7 + (14 + 16)
12 + (16 + 18)
13 + (17 + 19)
2 + (7 + 12) and 3 + (7 + 12) and 3 + (2 ∗ 6 + 10)
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 106
(7 + 12) + (14 + 16) + (16 + 18) and (2 ∗ 6 + 10) + (14 + 16) + (16 + 18)
4 + (8 − 13) and (8 − 13) + ((15 − 17) − (17 + 19))
5 − (8 + 13) and −(8 + 13) + (−15 − 19)
5 + 11 and 11 + (−15 − 19).
A.2.4
Terms with four D’s
With no derivatives
n
d−4
term
0∗
0∗
∗
2
D D0 (D D± Wµ∓ + D0 D± Wµ± ) Z µ
2
i D0 D0 (D0 D± Wµ∓ − D0 D± Wµ± ) Z µ
∗
∗
∗
∗
∗
2
D 0 D 0 D ± D ± Z µ Zµ
2
D0 D0 D± D± W +µ Wµ−
2
D0 D0 D± D± W ∓µ Wµ∓ + D0 D0 D± D± W ±µ Wµ±
2
i (D0 D0 D± D± W ∓µ Wµ∓ − D0 D0 D± D± W ±µ Wµ± )
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
2
D± D± (D0 D± Wµ∓ + D0 D± Wµ± )Z µ
2
i D± D± (D0 D± Wµ∓ − D0 D± Wµ± )Z µ
∗
∗
∗
∗
∗
2
D ± D ± D ± D ± Z µ Zµ
2
D± D± D± D± W +µ Wµ−
∗
∗
With one derivative
As in A.2.3 we separate the list of terms in two parts, the first with non-hermitian
terms (all the hermitian terms can then be derived from these by taking their real
and imaginary parts), and the second with hermitian terms.
The non-hermitian terms are:
n
d−4
term
0∗
∗
2
(∂µ D )D0 D0 D± W ∓µ
2
D0 (∂µ D0 )D0 D± W ∓µ
2
D0 D0 D0 (Dµ D± )W ∓µ
2
(∂µ D0 )D± D± D± Wµ∓
2
D0 (D µ D± )D± D± Wµ∓
2
D0 D± (D µ D± )D± Wµ∓
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 107
∗
∗
D0 D± D± (D µ D± )Wµ∓
2
The hermitian terms are:
d−4
n
2
term
∂µ
∗
∗
(D0 D0 ) D± D± Z µ
∗
2
Jµ D± D± Z µ
2
D0 D0 ∂µ (D± D± ) Z µ
→
∗
∗←
i D0 D0 (D± Dµ D± ) Z µ
2
2
2
∗
∗
∗
∗
∂µ (D± D± ) D± D± Z µ
→
∗←
∗
i (D± Dµ D± ) D± D± Z µ
→
∗←
where Jµ = i D0 ∂µ D0 as defined in Appendix A.1.
With two derivatives
In the following we mark with a ♠ those operators that do not describe interactions
of the doublet with the SM particles, constituting therefore doublet self-interaction
terms. We list them here because they combine with other terms to yield total
derivatives vanishing upon integration by parts. We refer to these combinations
at the bottom of the table. All the terms appearing here are included in the list
of the doublet’s self-interaction terms (3.52).
n
♠4
5
d−4
2
2
2
7
2
8
2
♠ 2∗7+8
2
9
2
21
2
term
∗
∂ µ (D0 D0 ) ∂
± ∗ D± )
µ (D
→
∗
∗←
i ∂ µ (D0 D0 ) (D± Dµ D± )
→
∗←
i J µ (D± Dµ D± )
∗
∗
D0 D0 (D µ D± )(Dµ D± )
∗
∗
∗
D0 D0 [(D µ Dµ D± ) D± + D± (D µ Dµ D± )]
∗
∗
D0 D0 ∂ µ ∂µ (D± D± )
∗
∗
∗
i D0 D0 [(D µ Dµ D± ) D± − D± (D µ Dµ D± )]
→
∗
∗←
= −i D0 D0 ∂ µ (D± Dµ D± )
∗
∗
∗
[(D µ Dµ D± )D± + D± (D µ Dµ D± )] D± D±
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 108
2
23
2
♠ 21 + 2 ∗ 23
2
2
∗
∗
∗
i [(D µ Dµ D± )D± − D± (D µ Dµ D± )] D± D±
→
∗←
∗
= −i ∂ µ (D± Dµ D± ) D± D±
∗
∗
(D µ D± )(Dµ D± ) D± D±
∗
∗
∂ µ ∂µ (D± D± ) D± D±
→
→
∗←
∗←
(D± D µ D± ) (D± Dµ D± )
The following linear combination is a total derivative no longer depending on the
SM fields:
4 + (2 ∗ 7 + 8)
and therefore is a ♠ type of operator. We also have the following combination
which is a total derivative involving the SM fields:
5 − 9.
A.3
Triplet’s interaction terms with SM
gauge bosons
We enlarge further the list of operators given in the two previous appendices
allowing the contemporary presence of a positive and a negative charged state,
beside the neutral one, denoted respectively T + , T − and T 0 . The full list of
interaction terms with the SM gauge bosons is now given by the terms presented
in this appendix plus those listed in Appendix A.1 and A.2, provided one identifies
the same electric charge components: φ = D0 = T 0 and D± = T ± . While in the
case of the doublet only one charged component was present, i.e. either D+ or D− ,
now both electric charges appear at the same time as T + and T − . Therefore in
considering the terms in Appendix A.2 both signs have to be taken into account.
In the same way, whenever a ± or ∓ appear in the terms listed below, both the
possibilities have to be considered.
A.3.1
Interaction with Electroweak gauge bosons W ,
Z and A
With no derivatives
n
d−4
0
0
term
∗
∗
T + T − W +µ Wµ+ + T − T + W −µ Wµ−
∗
∗
i (T + T − W +µ Wµ+ − T − T + W −µ Wµ− )
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 109
∗
∗
2
(T + T − W +µ Wµ+ + T − T + W −µ Wµ− ) Z ν Zν
2
i (T + T − W +µ Wµ+ − T − T + W −µ Wµ− ) Z ν Zν
∗
∗
∗
∗
(T + T − Wµ+ Wν+ + T − T + Wµ− Wν− ) Z µ Z ν
2
∗
∗
i (T + T − Wµ+ Wν+ − T − T + Wµ− Wν− ) Z µ Z ν
2
∗
∗
2
(T + T − W +µ Wµ+ + T − T + W −µ Wµ− ) W +ν Wν−
2
i (T + T − W +µ Wµ+ − T − T + W −µ Wµ− ) W +ν Wν−
∗
∗
With one derivative
n
14
d−4
2
14i
2
15
2
15i
2
14 + 15
2
15 − 14
2
14i − 15i
2
−(14i + 15i)
2
2
2
term
∗
∗
[(Dµ T + )T − W +ν Wν+ + T − (Dµ T + ) W −ν Wν− ] Z µ
∗
∗
i [(Dµ T + )T − W +ν Wν+ − T − (Dµ T + ) W −ν Wν− ] Z µ
∗
∗
[(Dµ T − )T + W −ν Wν− + T + (Dµ T − ) W +ν Wν+ ] Z µ
∗
∗
i [(Dµ T − )T + W −ν Wν− − T + (Dµ T − ) W +ν Wν+ ] Z µ
∗
∗
[Dµ (T + T − ) W +ν Wν+ + Dµ (T − T + ) W −ν Wν− ] Z µ
→
→
∗←
∗←
[(T + Dµ T − ) W +ν Wν+ − (T − Dµ T + ) W −ν Wν− ] Z µ
→
→
∗←
∗←
i [(T + Dµ T − ) W +ν Wν+ + (T − Dµ T + ) W −ν Wν− ] Z µ
∗
∗
i [Dµ (T + T − ) W +ν Wν+ − Dµ (T − T + ) W −ν Wν− ] Z µ
∗
∗
[T + T − W +ν Wν+ + T − T + W −ν Wν− ] (∂µ Z µ )
∗
∗
i [T + T − W +ν Wν+ − T − T + W −ν Wν− ] (∂µ Z µ )
∗
∗
24
2
[(Dµ T + )T − W +µ Wν+ + T − (Dµ T + ) W −µ Wν− ] Z ν
24i
2
25
2
i [(Dµ T + )T − W +µ Wν+ − T − (Dµ T + ) W −µ Wν− ] Z ν
25i
2
24 + 25
2
25 − 24
2
24i − 25i
2
−(24i + 25i)
2
31
2
31i
2
32
2
∗
∗
∗
∗
[(Dµ T − )T + W −µ Wν− + T + (Dµ T − ) W +µ Wν+ ] Z ν
∗
∗
i [(Dµ T − )T + W −µ Wν− − T + (Dµ T − ) W +µ Wν+ ] Z ν
∗
∗
[Dµ (T + T − ) W +µ Wν+ + Dµ (T − T + ) W −µ Wν− ] Z ν
→
→
∗←
∗←
[(T + Dµ T − ) W +µ Wν+ − (T − Dµ T + ) W −µ Wν− ] Z ν
→
→
∗←
∗←
i [(T + Dµ T − ) W +µ Wν+ + (T − Dµ T + ) W −µ Wν− ] Z ν
∗
∗
i [Dµ (T + T − ) W +µ Wν+ − Dµ (T − T + ) W −µ Wν− ] Z ν
∗
∗
[T + T − (Dµ W +µ )Wν+ + T − T + (Dµ W −µ )Wν− ] Z ν
∗
∗
i [T + T − (Dµ W +µ )Wν+ − T − T + (Dµ W −µ )Wν− ] Z ν
∗
∗
[T + T − W +µ (Dµ Wν+ ) + T − T + W −µ (Dµ Wν− )] Z ν
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 110
32i
2
31 + 32
2
32 − 31
2
31i + 32i
2
32i − 31i
2
33
2
33i
2
∗
∗
i [T + T − W +µ (Dµ Wν+ ) − T − T + W −µ (Dµ Wν− )] Z ν
∗
∗
[T + T − Dµ (W +µ Wν+ ) + T − T + Dµ (W −µ Wν− )] Z ν
←
→
←
→
∗
∗
[T + T − (W +µ Dµ Wν+ ) + T − T + (W −µ Dµ Wν− )] Z ν
∗
∗
i [T + T − Dµ (W +µ Wν+ ) − T − T + Dµ (W −µ Wν− )] Z ν
←
→
←
→
∗
∗
i [T + T − (W +µ Dµ Wν+ ) − T − T + (W −µ Dµ Wν− )] Z ν
∗
∗
[T + T − W +µ Wν+ + T − T + W −µ Wν− ] (∂µ Z ν )
∗
∗
i [T + T − W +µ Wν+ − T − T + W −µ Wν− ] (∂µ Z ν )
∗
∗
2
εµνρσ [T + T − (Dµ Wν+ )Wρ+ + T − T + (Dµ Wν− )Wρ− ] Zσ
2
i εµνρσ [T + T − (Dµ Wν+ )Wρ+ − T − T + (Dµ Wν− )Wρ− ] Zσ
∗
∗
Not all these operators are independent, in fact the following linear combinations
amount to total derivatives:
(24 + 25) + (31 + 32) + 33
(24i − 25i) + (31i + 32i) + 33i.
With two derivatives
We use here the notation introduced in Sec. A.2.3: we list only some terms, the
rest being derived from these by permitting the derivates to act on the various
fields in different order. When below a term the symbol ‘∗’ appears this means
that the remaining terms obtained by simply changing position of the derivatives
are not displayed here but should be taken into account. One has also to pay
attention to the fact that a partial derivative turns into a covariant one when it
acts on a charged field.
n
d−4
2
term
(D µ D
± ∗ )T ∓ W ±ν W ±
µT
ν
+ h.c.
2
*
2
i [(D µ Dµ T ± )T ∓ W ±ν Wν± − h.c.]
2
2
∗
*
(Dµ Dν T
±∗
)T ∓ W ±µ W ±ν + h.c.
2
*
2
i [(Dµ Dν T ± )T ∓ W ±µ W ±ν − h.c.]
2
∗
*
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 111
∗
2
εµνρσ (Dµ T ± )(Dν T ∓ ) Wρ± Wσ± + h.c.
2
*
2
i εµνρσ [(Dµ T ± )(Dν T ∓ ) Wρ± Wσ± − h.c.]
∗
2
A.3.2
*
Terms with four T ’s
With no derivatives
n
d−4
term
∗
∗
(T 0 T 0 T ± T ∓
2
∗
∗
∗
∗
+ T 0 T 0 T ± T ∓ ) Z µ Zµ
∗
∗
i (T 0 T 0 T ± T ∓ − T 0 T 0 T ± T ∓ ) Z µ Zµ
2
∗
∗
∗
∗
2
(T 0 T 0 T ± T ∓ + T 0 T 0 T ± T ∓ )W +µ Wµ−
2
i (T 0 T 0 T ± T ∓ − T 0 T 0 T ± T ∓ )W +µ Wµ−
∗
∗
∗
∗
∗
∗
∗
2
T 0 T 0 (T + T − W +µ Wµ+ + T − T + W −µ Wµ− )
2
i T 0 T 0 (T + T − W +µ Wµ+ − T − T + W −µ Wµ− )
∗
∗
∗
∗
∗
∗
2
T ± T ± (T 0 T ∓ Wµ± + T 0 T ∓ Wµ∓ )Z µ
2
i T ± T ± (T 0 T ∓ Wµ± − T 0 T ∓ Wµ∓ )Z µ
∗
∗
∗
∗
∗
2
T + T + T − T − Z µ Zµ
2
T + T + T − T − W +µ Wµ−
2
T ± T ± (T + T − W +µ Wµ+ + T − T + W −µ Wµ− )
2
i T ± T ± (T + T − W +µ Wµ+ − T − T + W −µ Wµ− )
∗
∗
∗
∗
∗
∗
∗
∗
With one derivative
As in A.2.3 we ease the notation by dividing the terms in two lists:
i) The first one contains non-hermitian terms; each non-hermitian term gives
rise to two hermitian terms, namely its real and imaginary parts.
ii) In the second list all the terms are already hermitian.
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 112
d−4
n
2
(∂µ
2
T 0 (Dµ T + )T 0 T − Z µ
2
T 0 T + T 0 (Dµ T − )Z µ
2
(∂µ T 0 )T ± T ± T ∓ Wµ±
2
T 0 (D µ T ± )T ± T ∓ Wµ±
2
T 0 T ± (D µ T ± )T ∓ Wµ±
2
T 0 T ± T ± (D µ T ∓ )Wµ±
d−4
n
term
∗
∗
T 0 )T + T 0 T − Z µ
2
2
2
2
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
term
∂µ
∗
∗
(T + T + ) T − T − Z µ
→
∗←
∗
i (T + Dµ T + ) T − T − Z µ
∗
∗
T + T + ∂µ (T − T − ) Z µ
→
∗
∗←
i T + T + (T − Dµ T − ) Z µ
With two derivatives
Using the notation introduced in section A.2.4 we mark with a ♠ those operators
that do not actually describe interactions of the triplet with SM particles, but only
interactions between the DM fields themselves. All the terms appearing here are
included in the list of the triplet’s self-interaction terms (3.91).
n
14
d−4
2
term
∂ µ (T 0 T 0 ) (D
µT
+ ∗ )T − ∗
+∗
−∗
+ h.c.
2
i [∂ µ (T 0 T 0 ) (Dµ T
15
2
15i
2
♠ 14 + 15
2
15 − 14
2
∗
∗
∂ µ (T 0 T 0 ) T + (Dµ T − ) + h.c.
∗
∗
i [∂ µ (T 0 T 0 ) T + (Dµ T − ) − h.c.]
∗
∗
∂ µ (T 0 T 0 ) ∂µ (T + T − ) + h.c.
∗
∗
i [∂ µ (T 0 T 0 ) ∂µ (T + T − ) − h.c.]
→ ∗
∗←
∂ µ (T 0 T 0 ) (T + Dµ T − ) + h.c.
→ ∗
∗←
i [∂ µ (T 0 T 0 ) (T + Dµ T − ) − h.c.]
14i
♠ 14i + 15i
2
15i − 14i
2
)T
− h.c.]
Effective Operators for Dark Matter Detection
Appendix A. DM Effective Interaction Terms with SM Gauge Bosons 113
∗
∗
17
2
T 0 T 0 (D µ T + )(Dµ T − ) + h.c.
17i
2
i [T 0 T 0 (D µ T + )(Dµ T − ) − h.c.]
31
2
[(D µ Dµ T + )T + + T + (D µ Dµ T + )] T − T −
32
2
33
2
i [(D µ Dµ T + )T + − T + (D µ Dµ T + )] T − T −
→
∗←
∗
= −i ∂ µ (T + Dµ T + ) T − T −
♠ 31 + 2 ∗ 33
2
∂ µ ∂µ (T + T + ) T − T −
♠ 34
2
35
2
36
2
2
∂ µ (T + T + ) ∂µ (T − T − )
→
∗
∗←
i ∂ µ (T + T + ) (T − Dµ T − )
→
∗←
∗
i (T + D µ T + ) ∂µ (T − T − )
→
→
∗←
∗←
(T + D µ T + ) (T − Dµ T − )
37
2
T + T + (D µ T − )(Dµ T − )
38
2
T + T + [(D µ Dµ T − ) T − + T − (D µ Dµ T − )]
♠ 2 ∗ 37 + 38
2
T + T + ∂ µ ∂µ (T − T − )
39
2
i T + T + [(D µ Dµ T − ) T − − T − (D µ Dµ T − )]
→
∗
∗←
= −i T + T + ∂ µ (T − Dµ T − )
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
(D µ T + )(Dµ T + ) T − T −
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
The following combinations are total derivatives not involving SM fields and therefore ♠ objects:
(31 + 2 ∗ 33) + 34
34 + (2 ∗ 37 + 38)
These terms enjoy the relation:
∗
∗
(31 + 2 ∗ 33) + 2 ∗ 34 + (2 ∗ 37 + 38) = ∂ µ ∂µ (T + T + T − T − ).
The operators forming total derivatives and involving SM fields are:
35 − 39
36 − 32.
Effective Operators for Dark Matter Detection
Appendix B
Nucleus Interaction with Higgs
and Gauge Bosons
In DM direct detection experiments, one is concerned with the scattering of DM
particles off the nuclei forming the detector. Since the energy release is very small
(up to the MeV scale for DM as heavy as hundreds or thousands of GeV), the
interaction occurs with the whole of the nucleus at once, in a coherent fashion.
Starting from the DM-quarks Lagrangian, one has therefore to step up writing the
relative effective interaction with the nucleon N and finally arrive at the nucleus
level.
B.1
Nucleons coupling to gauge bosons
The photon-quarks interaction Lagrangian is, for up and down type quarks,
¯ µ d Aµ ,
Lγq = Qu ūγ µ u Aµ + Qd dγ
(B.1)
with Qu = 2/3 and Qd = −1/3 the quarks electric charges and with the quarks
written here as Dirac fermions.
The Z-quarks interaction Lagrangian is, with the same notation,
LZq = ū vZ,u γ µ + aZ,u γ µ γ 5 u Zµ + d¯ vZ,d γ µ + aZ,d γ µ γ 5 d Zµ ,
(B.2)
with
vZ,u
vZ,d
g
1 4
2
=+
− sin θW ,
2 cos θW 2 3
g
1 2
2
=−
− sin θW ,
2 cos θW 2 3
g
1
,
2 cos θW 2
g
1
=+
.
2 cos θW 2
aZ,u = −
(B.3)
aZ,d
(B.4)
For the case of a conserved vector current, only the valence quarks of the
nucleon share its charge and therefore the vector boson coupling to a nucleon is
114
Appendix B. Nucleus Interaction with Higgs and Gauge Bosons
115
promptly computed by simply summing their charges. This amounts to say that,
at first order in αEM , the photon interacts with the electric charge of the proton
p while it does not interact with the neutron1 n, since Qp = 2Qu + Qd = 1 while
Qn = Qu + 2Qd = 0. The same happens for the vector interaction with the Z
boson, for which the charge of the nucleon is simply the sum of the charges of its
valence quarks: writing
LZN = N̄ vZ,N γ µ + aZ,N γ µ γ 5 N Zµ ,
(B.5)
one has therefore vZ,p = 2vZ,u + vZ,d and vZ,n = vZ,u + 2vZ,d (see Eq. (4.12)).
What has been said so far concerns the vector charges, which are conserved by
virtue of the conservation of the vector currents and for this reason only the valence
quarks contribute. For all the other interactions, the situation is more complicated:
in fact, both the valence and the sea quarks (as well as, in some cases, the gluons
inside the nucleon) are responsible for the interaction. For each quark current Jq
one has to determine the quantity hN |Jq |N i; the computation or measure of the
single contributions is not easy and is plagued by sizable uncertainties.
The nucleons axial-vector coupling to the Z boson can be written as aZ,N =
P
(N )
(N )
with ∆q ≡ hN |q̄γµ γ 5 q|N isµ , sµ being the nucleon spin four-vector
q aZ,q ∆q
(N )
(see e.g. Ref. [168]). Numerical values for the ∆q coming from lattice computations can be found e.g. in [49, 113], from which the values displayed in Eq. (4.21)
are deduced.
B.2
Nucleons coupling to the Higgs boson
√EW ) in unitary gauge, with vEW =
Parametrizing the Higgs doublet as H = (0, h+v
2
246 GeV, the SM quark-Higgs Lagrangian reduces to the form
X mq
Lhq =
q̄q h .
(B.6)
v
EW
q
For the processes of interest in DM direct detection, the matrix element at the
nucleon level will depend therefore on the quantities
(N )
fT q ≡
hN |mq q̄q|N i
,
mN
(B.7)
which determine the amount of mass of the nucleon N carried by the respective
(N )
quark flavor. The fT q are measured or computed on the lattice for the light quarks
q` = u, d, s, with sizable uncertainties (see e.g. [49]). Actually quarks heavier than
1
Being composite particles, protons and neutrons feature nevertheless an anomalous
magnetic moment that accounts for higher order electromagnetic interactions.
Effective Operators for Dark Matter Detection
Appendix B. Nucleus Interaction with Higgs and Gauge Bosons
116
the QCD compositeness scale, namely qh = c, b, t, are integrated out and therefore
contribute to the Higgs coupling to nucleons via the loop-induced Higgs-gluons
coupling; this is effectively described by the substitution [111]
mqh q̄h qh → −
αs a a
G G .
12π µν µν
(B.8)
The gluons contribution to the nucleon scalar current can be inferred by computing
the trace of the energy-momentum tensor of QCD, namely
Θµµ =
X
mq q̄q +
q
β(αs ) a a
G G ,
4αs µν µν
(B.9)
where β(αs ) is the QCD beta function. The gluon contribution is due to the trace
anomaly [111]. Since at zero momentum (i.e. for nucleon momenta much smaller
than the nucleus mass) one expects that mN = hN |Θµµ |N i, one gets
−
X (N )
1 9αs
(N )
hN |Gaµν Gaµν |N i = 1 −
fT q` ≡ fT G ,
mN 8π
q
(B.10)
`
after expanding β(αs ) in powers of αs and retaining only the lowest order.
N
Finally, the DM-nucleon matrix element will be proportional to vmEW
f ūN uN ,
where
X (N ) X (N )
2 X (N ) X (N ) 2 (N )
f≡
fT q =
fT q` +
f
=
fT q` + fT G .
(B.11)
27 q T G
9
q
q
q
`
h
`
This can be also obtained from the effective nucleon-Higgs Lagrangian
LhN =
mN
f N̄ N h ,
vEW
(B.12)
where N denotes the nucleon field. By estimates in [112, 113], f ' 0.3.
B.3
Interaction with the nucleus
From the nucleon interactions computed in the previous sections one can finally
determine the interaction with the whole nucleus. The details of the calculation
are subtle and strictly depend on the form of the interaction. In particular, one
can distinguish two classes of interactions: the spin-independent and the spindependent ones. The former are the interactions which do not depend on the spin
of the nucleus, while the latter do. These two kinds of interaction behave very
differently one from the other: in fact, while in the spin-independent case the
contributions of the nucleons sum up coherently, in the spin-dependent one they
approximately cancel between pairwise nucleons with opposite spins. The spindependent interaction therefore only counts the nucleons which actually carry the
Effective Operators for Dark Matter Detection
Appendix B. Nucleus Interaction with Higgs and Gauge Bosons
117
nucleus spin, that is roughly speaking only one. The spin-independent amplitude
counts instead all the nucleons taking part in the interaction and is therefore enhanced respect to the spin-dependent one, that is for this reason usually considered
negligible. For example the photon interacts with the Z protons of the nucleus,
while the Higgs boson, as we have shown in the previous section, interacts equally
with all its A nucleons, regardless of their nature.
In going from the nucleon to the nuclear level one has to include the relevant
form factor, which takes into account the loss of coherence of the interaction as
the exchanged energy grows (for a recent complete treatment of the various form
factors that can arise in different types of interaction, see [134]). The best known
case is the one of spin-independent interaction, for which the customary choice
is the Helm form factor [135]. This is extracted from electron-nucleus scattering
data and maps the charge distribution of the nucleus. For mediators of spinindependent interaction different from the photon one usually assumes that the
charge distribution faithfully represents the distribution of matter (i.e. of nucleons)
in the nucleus, so that the interaction amplitude merely needs to be normalized
to the number of nucleons taking part in the interaction, as mentioned above.
We show now the nature of the nucleon-Higgs and nucleon-vector interactions
introduced in the previous sections and used in Sec. 4, in the limit of zero exchanged
momentum.
The positive energy solution to the Dirac equation, in Weyl representation,
can be approximated in the extreme non-relativistic limit by
s
√
ξ
usN ' m N
.
(B.13)
s
ξN
Neglecting the nucleons momenta one has then
s† r
ūsN 0 urN ' 2m ξN
0 ξN ,
ūsN 0 γ µ urN '
s† r
2m ξN
0 ξN
~0
!
,
ūsN 0 iγ 5 urN ' 0 ,
0
s
µ 5 r
,
ūN 0 γ γ uN '
4m ~sNsr
(B.14)
(B.15)
s† ~
σ r
where ~sNsr ≡ ξN
0 2 ξN is the nucleon spin operator. Therefore the scalar and vector
nucleon currents, respectively N̄ N and N̄ γ µ N , give rise to a spin-independent
interaction; the pseudoscalar current N̄ iγ 5 N is suppressed in the limit of zero
momentum, while the axial-vector current N̄ γ µ γ 5 N is spin-dependent and for this
reason it has been neglected in Chapter 4.
Effective Operators for Dark Matter Detection
Appendix C
Corrections to the Articles
Related to This Thesis
In this Thesis we made a few minor changes to the text of the articles [I] and [III].
We report here track of the modifications, for a better comparison with the articles.
We stress that none of these modifications change in any case the conclusions drawn
in the articles, being due in many cases only to typos or misprints which do not
affect the subsequent results.
• In Sec. 3.2.2, the ‘hermitianized’ covariant derivative has been defined as
←
→ 1←
→
Dµ ≡ ∂µ − ieQAµ
2
(C.1)
(with the addition of the color term −igs T a Gaµ when it is applied to quark
fields). The 12 factor multiplying the partial derivative is needed in order
for this operator to be gauge covariant, but it is absent in the definition
reported in [I]. Also, the strong coupling constant gs was omitted in [I].
• The Lagrangian terms in (3.30), (3.31) and (3.32) are not hermitian. As
pointed out in Sec. A.2.3, each of them gives rise to two hermitian operators,
respectively their real and imaginary parts. This information can be also
found in the Appendix of Ref. [I], but not in the main text where the terms
(3.31), (3.32) seem to be understood as hermitian.
• In the self-interaction Lagrangian (3.52), the term
s
∗
(D µ Dµ D+ )(D ν Dν D+ )
Λ2
(C.2)
was erroneously written in [I] with partial derivatives instead of covariant
derivatives. Nevertheless, in the Appendix of the same paper it is reported
correctly.
118
Appendix C. Corrections to the Articles Related to This Thesis
119
• In Ref. [I], the DM couplings to proton and neutron are quoted as |fp | '
0.14×10−5 GeV−2 and |fn | ' 0.10×10−5 GeV−2 . In both cases it should actually read 10−4 instead of 10−5 . The correct values are reported in Sec. 4.1,
where the more precise value for the nucleon mass mp = 939 MeV (intended
as an average of the proton and neutron mass) has been used.
• In Ref. [I], the Z boson coupling to nucleons was erroneously written as
icZ,N N̄ γ µ N Zµ , where the imaginary constant i must be removed. In
Sec. 4.2.2 and in Appendix B we provide the right definition.
• In Ref. [III], the quoted values for the axial-vector couplings of the Z boson
to proton and neutron, ap and an , are double the true value, and have been
corrected in Eq. (4.21).
Effective Operators for Dark Matter Detection
Bibliography
[I] E. Del Nobile and F. Sannino, “Dark Matter Effective Theory,” Int. J. Mod.
Phys. A 27, 1250065 (2012) [arXiv:1102.3116 [hep-ph]].
[II] E. Del Nobile, C. Kouvaris and F. Sannino, “Interfering Composite Asymmetric Dark Matter for DAMA and CoGeNT,” Phys. Rev. D 84, 027301
(2011) [arXiv:1105.5431 [hep-ph]].
[III] E. Del Nobile, C. Kouvaris, F. Sannino and J. Virkajarvi, “Dark Matter
Interference,” Mod. Phys. Lett. A 27, 1250108 (2012) [arXiv:1111.1902 [hepph]].
[IV] E. Del Nobile, C. Kouvaris, P. Panci, F. Sannino and J. Virkajarvi, “Light
Magnetic Dark Matter in Direct Detection Searches,” JCAP 1208, 010
(2012) [arXiv:1203.6652 [hep-ph]].
[1] E. Aprile et al. [XENON100 Collaboration], arXiv:1207.5988 [astro-ph.CO].
[2] J. Kumar, D. Sanford and L. E. Strigari, Phys. Rev. D 85, 081301 (2012)
[arXiv:1112.4849 [astro-ph.CO]].
[3] H. B. Jin, S. Miao and Y. F. Zhou, arXiv:1207.4408 [hep-ph].
[4] [CMS Collaboration], Phys. Lett. B 716, 30 (2012) [arXiv:1207.7235 [hepex]]. See also the statement on the CMS webpage.
[5] [ATLAS Collaboration], Phys. Lett. B 716, 1 (2012) [arXiv:1207.7214 [hepex]]. See also the statement on the ATLAS webpage.
[6] C. a. D. Group [Tevatron New Physics Higgs Working Group and CDF
Collaboration and D0 ], arXiv:1207.0449 [hep-ex].
[7] S. P. Martin, arXiv:hep-ph/9709356.
[8] C. Csaki, J. Hubisz and P. Meade, arXiv:hep-ph/0510275.
120
Bibliography
121
[9] F. Sannino, Acta Phys. Polon. B 40, 3533 (2009) [arXiv:0911.0931 [hep-ph]].
[10] P. P. Giardino, K. Kannike, M. Raidal and A. Strumia, arXiv:1207.1347
[hep-ph].
[11] D. Carmi, A. Falkowski, E. Kuflik, T. Volansky and J. Zupan,
arXiv:1207.1718 [hep-ph].
[12] N. Arkani-Hamed, K. Blum, R. T. D’Agnolo and J. Fan, arXiv:1207.4482
[hep-ph].
[13] LHCb website.
[14] L. Canetti, M. Drewes, T. Frossard and M. Shaposhnikov, arXiv:1208.4607
[hep-ph].
[15] M. Milgrom, arXiv:0801.3133 [astro-ph].
[16] B. Famaey and S. McGaugh, Living Rev. Rel. 15, 10 (2012) [arXiv:1112.3960
[astro-ph.CO]].
[17] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267, 195 (1996)
[arXiv:hep-ph/9506380].
[18] D. Larson et al., Astrophys. J. Suppl. 192, 16 (2011) [arXiv:1001.4635 [astroph.CO]].
[19] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18
(2011) [arXiv:1001.4538 [astro-ph.CO]].
[20] J. Beringer et al. [Particle Data Group], Phys. Rev. D 86, 010001 (2012).
See also the PDG website.
[21] O. Adriani et al. [PAMELA Collaboration], Nature 458, 607 (2009)
[arXiv:0810.4995 [astro-ph]].
[22] M. Ackermann et al. [Fermi LAT Collaboration], Phys. Rev. Lett. 108,
011103 (2012) [arXiv:1109.0521 [astro-ph.HE]].
[23] O. Adriani et al. [PAMELA Collaboration], Phys. Rev. Lett. 105, 121101
(2010) [arXiv:1007.0821 [astro-ph.HE]].
[24] M. Cirelli, M. Kadastik, M. Raidal and A. Strumia, Nucl. Phys. B 813, 1
(2009) [arXiv:0809.2409 [hep-ph]].
[25] E. Nardi, F. Sannino and A. Strumia, JCAP 0901, 043 (2009)
[arXiv:0811.4153 [hep-ph]].
[26] C. Weniger, JCAP 1208, 007 (2012) [arXiv:1204.2797 [hep-ph]].
Effective Operators for Dark Matter Detection
Bibliography
122
[27] E. Tempel, A. Hektor and M. Raidal, JCAP 1209, 032 (2012)
[arXiv:1205.1045 [hep-ph]].
[28] R. Bernabei et al., Phys. Lett. B 424, 195 (1998).
[29] A. K. Drukier, K. Freese and D. N. Spergel, Phys. Rev. D 33, 3495 (1986).
[30] K. Freese, J. A. Frieman and A. Gould, Phys. Rev. D 37, 3388 (1988).
[31] R. Bernabei et al. [DAMA Collaboration], Eur. Phys. J. C 56, 333 (2008)
[arXiv:0804.2741 [astro-ph]].
[32] R. Bernabei et al. [DAMA Collaboration and LIBRA Collaboration], Eur.
Phys. J. C 67, 39 (2010) [arXiv:1002.1028 [astro-ph.GA]].
[33] P. Belli, R. Cerulli, N. Fornengo and S. Scopel, Phys. Rev. D 66, 043503
(2002) [arXiv:hep-ph/0203242].
[34] P. Belli, R. Bernabei, A. Bottino, F. Cappella, R. Cerulli, N. Fornengo and
S. Scopel, Phys. Rev. D 84, 055014 (2011) [arXiv:1106.4667 [hep-ph]].
[35] C. E. Aalseth et al. [CoGeNT collaboration], Phys. Rev. Lett. 106, 131301
(2011) [arXiv:1002.4703 [astro-ph.CO]].
[36] C. E. Aalseth et al., Phys. Rev. Lett. 107, 141301 (2011) [arXiv:1106.0650
[astro-ph.CO]].
[37] G. Angloher et al., Eur. Phys. J. C 72, 1971 (2012) [arXiv:1109.0702 [astroph.CO]].
[38] Z. Ahmed et al. [CDMS-II Collaboration], Science 327, 1619 (2010)
[arXiv:0912.3592 [astro-ph.CO]].
[39] E. Aprile et al. [XENON100 Collaboration], Phys. Rev. Lett. 107, 131302
(2011) [arXiv:1104.2549 [astro-ph.CO]].
[40] T. Schwetz and J. Zupan, JCAP 1108, 008 (2011) [arXiv:1106.6241 [hepph]].
[41] M. Farina, D. Pappadopulo, A. Strumia and T. Volansky, JCAP 1111, 010
(2011) [arXiv:1107.0715 [hep-ph]].
[42] S. Archambault et al. [PICASSO Collaboration], Phys. Lett. B 711, 153
(2012) [arXiv:1202.1240 [hep-ex]].
[43] R. Bernabei, P. Belli, A. Incicchitti and D. Prosperi, arXiv:0806.0011 [astroph].
[44] J. I. Collar, arXiv:1106.0653 [astro-ph.CO].
Effective Operators for Dark Matter Detection
Bibliography
123
[45] J. I. Collar, arXiv:1103.3481 [astro-ph.CO].
[46] R. Bernabei et al., Eur. Phys. J. C 53, 205 (2008) [arXiv:0710.0288 [astroph]].
[47] N. Bozorgnia, G. B. Gelmini and P. Gondolo, JCAP 1011, 019 (2010)
[arXiv:1006.3110 [astro-ph.CO]].
[48] B. Feldstein, A. L. Fitzpatrick, E. Katz and B. Tweedie, JCAP 1003, 029
(2010) [arXiv:0910.0007 [hep-ph]].
[49] J. R. Ellis, K. A. Olive and C. Savage, Phys. Rev. D 77, 065026 (2008)
[arXiv:0801.3656 [hep-ph]].
[50] M. W. Goodman and E. Witten, Phys. Rev. D 31, 3059 (1985).
[51] J. D. Lewin and P. F. Smith, Astropart. Phys. 6, 87 (1996).
[52] J. L. Feng, J. Kumar, D. Marfatia and D. Sanford, Phys. Lett. B 703, 124
(2011) [arXiv:1102.4331 [hep-ph]].
[53] R. Foadi, M. T. Frandsen and F. Sannino, Phys. Rev. D 80, 037702 (2009)
[arXiv:0812.3406 [hep-ph]].
[54] T. A. Ryttov and F. Sannino, Phys. Rev. D 78, 115010 (2008)
[arXiv:0809.0713 [hep-ph]].
[55] M. T. Frandsen and F. Sannino, Phys. Rev. D 81, 097704 (2010)
[arXiv:0911.1570 [hep-ph]].
[56] R. Lewis, C. Pica and F. Sannino, Phys. Rev. D 85, 014504 (2012)
[arXiv:1109.3513 [hep-ph]].
[57] M. Taoso, G. Bertone and A. Masiero, JCAP 0803, 022 (2008)
[arXiv:0711.4996 [astro-ph]].
[58] R. Massey, T. Kitching and J. Richard, Rept. Prog. Phys. 73, 086901 (2010)
[arXiv:1001.1739 [astro-ph.CO]].
[59] M. Kamionkowski, arXiv:astro-ph/9809214.
[60] K. G. Begeman, A. H. Broeils and R. H. Sanders, Mon. Not. Roy. Astron.
Soc. 249, 523 (1991).
[61] A. Klypin, H. Zhao and R. S. Somerville, Astrophys. J. 573, 597 (2002)
[arXiv:astro-ph/0110390].
[62] G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405, 279 (2005) [arXiv:hepph/0404175].
Effective Operators for Dark Matter Detection
Bibliography
124
[63] M. Markevitch et al., Astrophys. J. 567, L27 (2002) [arXiv:astroph/0110468].
[64] D. Clowe, A. Gonzalez and M. Markevitch, Astrophys. J. 604, 596 (2004)
[arXiv:astro-ph/0312273].
[65] D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall,
C. Jones and D. Zaritsky, Astrophys. J. 648, L109 (2006) [arXiv:astroph/0608407].
[66] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez and M. Bradac,
Astrophys. J. 679, 1173 (2008) [arXiv:0704.0261 [astro-ph]].
[67] M. Colless et al. [2DFGRS Collaboration], Mon. Not. Roy. Astron. Soc. 328,
1039 (2001) [arXiv:astro-ph/0106498].
[68] M. Tegmark et al. [SDSS Collaboration], Astrophys. J. 606, 702 (2004)
[arXiv:astro-ph/0310725].
[69] V. Springel et al., Nature 435, 629 (2005) [arXiv:astro-ph/0504097].
[70] C. Tao, arXiv:1110.0298 [astro-ph.CO].
[71] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese and A. Riotto, Phys.
Rev. Lett. 97, 071301 (2006) [arXiv:astro-ph/0605706].
[72] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese and A. Riotto, Phys.
Rev. D 71, 063534 (2005) [arXiv:astro-ph/0501562].
[73] S. Cole et al. [2dFGRS Collaboration], Mon. Not. Roy. Astron. Soc. 362,
505 (2005) [arXiv:astro-ph/0501174].
[74] W. J. Percival et al. [SDSS Collaboration], Mon. Not. Roy. Astron. Soc. 401,
2148 (2010) [arXiv:0907.1660 [astro-ph.CO]].
[75] D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer and
E. L. Wright, Astrophys. J. 473, 576 (1996) [arXiv:astro-ph/9605054].
[76] D. J. Fixsen, Astrophys. J. 707, 916 (2009) [arXiv:0911.1955 [astro-ph.CO]].
[77] C. L. Bennett et al., Astrophys. J. Suppl. 192, 17 (2011) [arXiv:1001.4758
[astro-ph.CO]].
[78] A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116,
1009 (1998) [arXiv:astro-ph/9805201].
[79] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999) [arXiv:astro-ph/9812133].
Effective Operators for Dark Matter Detection
Bibliography
125
[80] M. Kowalski et al. [Supernova Cosmology Project Collaboration], Astrophys.
J. 686, 749 (2008) [arXiv:0804.4142 [astro-ph]].
[81] P. Scott, arXiv:1110.2757 [astro-ph.CO].
[82] M. Tegmark, M. Zaldarriaga and A. J. S. Hamilton, Phys. Rev. D 63, 043007
(2001) [arXiv:astro-ph/0008167].
[83] M. Beltran, D. Hooper, E. W. Kolb and Z. C. Krusberg, Phys. Rev. D 80,
043509 (2009) [arXiv:0808.3384 [hep-ph]].
[84] Q. H. Cao, C. R. Chen, C. S. Li and H. Zhang, JHEP 1108, 018 (2011)
[arXiv:0912.4511 [hep-ph]].
[85] S. Chang, A. Pierce and N. Weiner,
[arXiv:0908.3192 [hep-ph]].
JCAP 1001,
006 (2010)
[86] W. Shepherd, T. M. P. Tait and G. Zaharijas, Phys. Rev. D 79, 055022
(2009) [arXiv:0901.2125 [hep-ph]].
[87] Y. Bai, P. J. Fox and R. Harnik, JHEP 1012, 048 (2010) [arXiv:1005.3797
[hep-ph]].
[88] M. Beltran, D. Hooper, E. W. Kolb, Z. A. C. Krusberg and T. M. P. Tait,
JHEP 1009, 037 (2010) [arXiv:1002.4137 [hep-ph]].
[89] K. Cheung, P. Y. Tseng and T. C. Yuan, JCAP 1101, 004 (2011)
[arXiv:1011.2310 [hep-ph]].
[90] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and
H. B. P. Yu, Nucl. Phys. B 844, 55 (2011) [arXiv:1009.0008 [hep-ph]].
[91] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and
H. B. P. Yu, Phys. Rev. D 82, 116010 (2010) [arXiv:1008.1783 [hep-ph]].
[92] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T. M. P. Tait and
H. B. P. Yu, Phys. Lett. B 695, 185 (2011) [arXiv:1005.1286 [hep-ph]].
[93] W. Y. P. Keung, I. Low and G. Shaughnessy, Phys. Rev. D 82, 115019
(2010) [arXiv:1010.1774 [hep-ph]].
[94] J. M. Zheng, Z. H. Yu, J. W. Shao, X. J. Bi, Z. Li and H. H. Zhang, Nucl.
Phys. B 854, 350 (2012) [arXiv:1012.2022 [hep-ph]].
[95] M. R. Buckley, Phys. Rev. D 84, 043510 (2011) [arXiv:1104.1429 [hep-ph]].
[96] K. Cheung, P. Y. Tseng and T. C. Yuan, JCAP 1106, 023 (2011)
[arXiv:1104.5329 [hep-ph]].
Effective Operators for Dark Matter Detection
Bibliography
126
[97] P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, Phys. Rev. D 84, 014028 (2011)
[arXiv:1103.0240 [hep-ph]].
[98] P. J. Fox, R. Harnik, J. Kopp and Y. Tsai, Phys. Rev. D 85, 056011 (2012)
[arXiv:1109.4398 [hep-ph]].
[99] X. Gao, Z. Kang and T. Li, arXiv:1107.3529 [hep-ph].
[100] J. Goodman and W. Shepherd, arXiv:1111.2359 [hep-ph].
[101] Y. Mambrini and B. Zaldivar, JCAP 1110, 023 (2011) [arXiv:1106.4819
[hep-ph]].
[102] A. Rajaraman, W. Shepherd, T. M. P. Tait and A. M. Wijangco, Phys. Rev.
D 84, 095013 (2011) [arXiv:1108.1196 [hep-ph]].
[103] Z. H. Yu, J. M. Zheng, X. J. Bi, Z. Li, D. X. Yao and H. H. Zhang, Nucl.
Phys. B 860, 115 (2012) [arXiv:1112.6052 [hep-ph]].
[104] K. Cheung, P. Y. Tseng, Y. L. Tsai and T. C. Yuan, JCAP 1205, 001 (2012)
[arXiv:1201.3402 [hep-ph]].
[105] P. J. Fox, R. Harnik, R. Primulando and C. T. Yu, Phys. Rev. D 86, 015010
(2012) [arXiv:1203.1662 [hep-ph]].
[106] N. Fornengo, P. Panci and M. Regis, Phys. Rev. D 84, 115002 (2011)
[arXiv:1108.4661 [hep-ph]].
[107] Z. Ahmed et al. [CDMS-II Collaboration], Phys. Rev. Lett. 106, 131302
(2011) [arXiv:1011.2482 [astro-ph.CO]].
[108] J. Angle et al. [XENON10 Collaboration], Phys. Rev. Lett. 107, 051301
(2011) [arXiv:1104.3088 [astro-ph.CO]].
[109] C. Savage, G. Gelmini, P. Gondolo and K. Freese, Phys. Rev. D 83, 055002
(2011) [arXiv:1006.0972 [astro-ph.CO]].
[110] S. Chang, J. Liu, A. Pierce, N. Weiner and I. Yavin, JCAP 1008, 018 (2010)
[arXiv:1004.0697 [hep-ph]].
[111] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Phys. Lett. B 78, 443
(1978).
[112] J. Giedt, A. W. Thomas and R. D. Young, Phys. Rev. Lett. 103, 201802
(2009) [arXiv:0907.4177 [hep-ph]].
[113] H. Y. Cheng and C. W. Chiang, JHEP 1207, 009 (2012) [arXiv:1202.1292
[hep-ph]].
Effective Operators for Dark Matter Detection
Bibliography
127
[114] CMS Collaboration, “Search for Narrow Resonances using the Dijet Mass
Spectrum in pp Collisions at sqrt s of 8 TeV”, CMS-PAS-EXO-12-016. See
also the CMS Exotica Public Physics Results webpage.
[115] Y. G. Kim, K. Y. Lee and S. Shin, JHEP 0805, 100 (2008) [arXiv:0803.2932
[hep-ph]].
[116] G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher and
J. P. Silva, Phys. Rept. 516, 1 (2012) [arXiv:1106.0034 [hep-ph]].
[117] H. Y. Cheng, Phys. Lett. B 219, 347 (1989).
[118] J. Bagnasco, M. Dine and S. D. Thomas, Phys. Lett. B 320, 99 (1994)
[arXiv:hep-ph/9310290].
[119] M. Pospelov and T. ter Veldhuis, Phys. Lett. B 480, 181 (2000) [arXiv:hepph/0003010].
[120] A. L. Fitzpatrick and K. M. Zurek, Phys. Rev. D 82, 075004 (2010)
[arXiv:1007.5325 [hep-ph]].
[121] K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell and M. Kamionkowski,
Phys. Rev. D 70, 083501 (2004) [Erratum-ibid. D 73, 089903 (2006)]
[arXiv:astro-ph/0406355].
[122] V. Barger, W. Y. Keung and D. Marfatia, Phys. Lett. B 696, 74 (2011)
[arXiv:1007.4345 [hep-ph]].
[123] S. Chang, N. Weiner and I. Yavin, Phys. Rev. D 82, 125011 (2010)
[arXiv:1007.4200 [hep-ph]].
[124] W. S. Cho, J. H. Huh, I. W. Kim, J. E. Kim and B. Kyae, Phys. Lett. B
687, 6 (2010) [Erratum-ibid. B 694, 496 (2011)] [arXiv:1001.0579 [hep-ph]].
[125] J. H. Heo, Phys. Lett. B 693, 255 (2010) [arXiv:0901.3815 [hep-ph]].
[126] S. Gardner, Phys. Rev. D 79, 055007 (2009) [arXiv:0811.0967 [hep-ph]].
[127] E. Masso, S. Mohanty and S. Rao, Phys. Rev. D 80, 036009 (2009)
[arXiv:0906.1979 [hep-ph]].
[128] T. Banks, J. F. Fortin and S. Thomas, arXiv:1007.5515 [hep-ph].
[129] J. F. Fortin and T. M. P. Tait, Phys. Rev. D 85, 063506 (2012)
[arXiv:1103.3289 [hep-ph]].
[130] V. Barger, W. Y. Keung, D. Marfatia and P. Y. Tseng, Phys. Lett. B 717,
219 (2012) [arXiv:1206.0640 [hep-ph]].
Effective Operators for Dark Matter Detection
Bibliography
128
[131] J. M. Cline, Z. Liu and W. Xue, Phys. Rev. D 85, 101302 (2012)
[arXiv:1201.4858 [hep-ph]].
[132] Particle Data Group webpage of Atomic and Nuclear Properties.
[133] N. J. Stone, “Table of Nuclear Magnetic Dipole and Electric Quadrupole
Moments” released by the IAEA Nuclear Data Services.
[134] A. L. Fitzpatrick, W. Haxton, E. Katz, N. Lubbers and Y. Xu,
arXiv:1203.3542 [hep-ph].
[135] R. H. Helm, Phys. Rev. 104, 1466 (1956).
[136] N. Fornengo and S. Scopel, Phys. Lett. B 576, 189 (2003) [arXiv:hepph/0301132].
[137] R. Bernabei et al. [DAMA Collaboration], Nucl. Instrum. Meth. A 592, 297
(2008) [arXiv:0804.2738 [astro-ph]].
[138] R. Bernabei et al., Phys. Lett. B 389, 757 (1996).
[139] C. E. Aalseth et al. [CoGeNT Collaboration], Phys. Rev. Lett. 101, 251301
(2008) [Erratum-ibid. 102, 109903 (2009)] [arXiv:0807.0879 [astro-ph]].
[140] P. S. Barbeau, J. I. Collar and O. Tench, JCAP 0709, 009 (2007)
[arXiv:nucl-ex/0701012].
[141] Z. Ahmed et al. [CDMS-II Collaboration and CDMS Collaboration], Phys.
Rev. D 83, 112002 (2011) [arXiv:1012.5078 [astro-ph.CO]].
[142] Z. Ahmed et al. [CDMS Collaboration and EDELWEISS Collaboration],
Phys. Rev. D 84, 011102 (2011) [arXiv:1105.3377 [astro-ph.CO]].
[143] E. Aprile et al. [XENON100 Collaboration], Phys. Rev. D 84, 052003 (2011)
[arXiv:1103.0303 [hep-ex]].
[144] R. E. Apfel, Nucl. Inst. and Meth. 162 (1979) 603–608.
[145] H. Ing, R. Noulty, T. McLean, Radiation Measurements 27 (1997) 1–11.
[146] P. Gondolo and G. Gelmini, Nucl. Phys. B 360, 145 (1991).
[147] A. Belyaev, M. T. Frandsen, S. Sarkar and F. Sannino, Phys. Rev. D 83,
015007 (2011) [arXiv:1007.4839 [hep-ph]].
[148] M. Cirelli, P. Panci, G. Servant and G. Zaharijas, JCAP 1203, 015 (2012)
[arXiv:1110.3809 [hep-ph]].
[149] S. Tulin, H. B. Yu and K. M. Zurek, JCAP 1205, 013 (2012)
[arXiv:1202.0283 [hep-ph]].
Effective Operators for Dark Matter Detection
Bibliography
129
[150] M. R. Buckley and S. Profumo, Phys. Rev. Lett. 108, 011301 (2012)
[arXiv:1109.2164 [hep-ph]].
[151] S. Galli, F. Iocco, G. Bertone and A. Melchiorri, Phys. Rev. D 80, 023505
(2009) [arXiv:0905.0003 [astro-ph.CO]].
[152] T. R. Slatyer, N. Padmanabhan and D. P. Finkbeiner, Phys. Rev. D 80,
043526 (2009) [arXiv:0906.1197 [astro-ph.CO]].
[153] G. Huetsi, A. Hektor and M. Raidal, Astron. Astrophys. 505, 999 (2009)
[arXiv:0906.4550 [astro-ph.CO]].
[154] M. Cirelli, F. Iocco and P. Panci, JCAP 0910, 009 (2009) [arXiv:0907.0719
[astro-ph.CO]].
[155] G. Hutsi, J. Chluba, A. Hektor and M. Raidal, Astron. Astrophys. 535, A26
(2011) [arXiv:1103.2766 [astro-ph.CO]].
[156] S. Galli, F. Iocco, G. Bertone and A. Melchiorri, Phys. Rev. D 84, 027302
(2011) [arXiv:1106.1528 [astro-ph.CO]].
[157] M. Ackermann et al. [Fermi-LAT collaboration], Phys. Rev. Lett. 107,
241302 (2011) [arXiv:1108.3546 [astro-ph.HE]].
[158] M. Cirelli, P. Panci and P. D. Serpico, Nucl. Phys. B 840, 284 (2010)
[arXiv:0912.0663 [astro-ph.CO]].
[159] M. Papucci and A. Strumia, JCAP 1003, 014 (2010) [arXiv:0912.0742 [hepph]].
[160] G. Zaharijas, A. Cuoco, Z. Yang and J. Conrad [Fermi-LAT collaboration
and Fermi-LAT collaboration and Fermi-LAT coll], PoS IDM2010, 111
(2011) [arXiv:1012.0588 [astro-ph.HE]].
[161] A. A. Abdo et al., Phys. Rev. Lett. 104, 091302 (2010) [arXiv:1001.4836
[astro-ph.HE]].
[162] Talk given by A. Morselli at the 7th International workshop on Dark Side
of the Universe.
[163] C. Kouvaris and P. Tinyakov, Phys. Rev. Lett. 107, 091301 (2011)
[arXiv:1104.0382 [astro-ph.CO]].
[164] S. D. McDermott, H. B. Yu and K. M. Zurek, Phys. Rev. D 85, 023519
(2012) [arXiv:1103.5472 [hep-ph]].
[165] T. Guver, A. E. Erkoca, M. H. Reno and I. Sarcevic, arXiv:1201.2400 [hepph].
Effective Operators for Dark Matter Detection
Bibliography
130
[166] C. Kouvaris, Phys. Rev. Lett. 108, 191301 (2012) [arXiv:1111.4364 [astroph.CO]].
[167] C. Kouvaris and P. Tinyakov, Phys. Rev. D 83, 083512 (2011)
[arXiv:1012.2039 [astro-ph.HE]].
[168] G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys.
Commun. 180, 747 (2009) [arXiv:0803.2360 [hep-ph]].
Effective Operators for Dark Matter Detection