Download Nuclear matrix elements from QCD

Nuclear matrix
elements from QCD
William Detmold
MIT
“Nuclear Aspects of Dark Matter Searches”, INT, Seattle, Dec 8-12 2014
The intensity frontier
The intensity frontier
Particle physics: over the next decade(s) many experiments
address the intensity frontier
The intensity frontier
Particle physics: over the next decade(s) many experiments
address the intensity frontier
Targets are nuclei (C, Fe, Si, Ar, Ge, Xe, Pb, CHx, H2O, steel)
Extraction of neutrino mass hierarchy and mixing parameters at LBNF requires knowing energies/fluxes to high accuracy
The intensity frontier
Particle physics: over the next decade(s) many experiments
address the intensity frontier
Targets are nuclei (C, Fe, Si, Ar, Ge, Xe, Pb, CHx, H2O, steel)
Extraction of neutrino mass hierarchy and mixing parameters at LBNF requires knowing energies/fluxes to high accuracy
Nuclear axial & transition form factors
Nuclear structure in neutrino DIS
~10% uncertainty on oscillation parameters [C Mariani, INT workshop 2013]
The intensity frontier
Particle physics: over the next decade(s) many experiments
address the intensity frontier
Targets are nuclei (C, Fe, Si, Ar, Ge, Xe, Pb, CHx, H2O, steel)
Extraction of neutrino mass hierarchy and mixing parameters at LBNF requires knowing LBNE,
energies/fluxes to high accuracy
δCP Sensit
Nuclear axial & transition form factors
Nuclear structure in neutrino DIS
~10% uncertainty on oscillation parameters [C Mariani, INT workshop 2013]
[U Mosel FSNu town meeting]
The intensity frontier
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http://www.hep.ucl.ac.uk/darkMatter/
The intensity frontier
Laboratory searches for new physics
Dark matter detection: nuclear recoils as signal
Nuclear matrix elements of exchange current
µ2e conversion expt: similar requirements &
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will need precise nuclear matrix elements
with
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quantified uncertainties
to
discern what it is
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The!
http://www.hep.ucl.ac.uk/darkMatter/
The intensity frontier
Laboratory searches for new physics
Dark matter detection: nuclear recoils as signal
Nuclear matrix elements of exchange current
µ2e conversion expt: similar requirements &
b
a
l
i
m
r
If(when) we detect dark matter or µ→e, we e
F
!
t
a
!
t
n
e
will need precise nuclear matrix elements
with
fully
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i
r
e
p
x
e
!
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quantified uncertainties
to
discern what it is
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The!
http://www.hep.ucl.ac.uk/darkMatter/
Nuclear physics is the new flavour physics!
Must be based on the Standard Model
Precision nuclear physics
Precision nuclear physics
We need to develop the tools for precision predictions
Precision nuclear physics
We need to develop the tools for precision predictions
Exploit effective degrees of freedom
Xe
Ge
Ar
Si
N
Z
Precision nuclear physics
Exploit effective degrees of freedom
Establish quantitative control through linkages between different methods
QCD forms a foundation
determines few body interactions & matrix elements
Match existing EFT and many body techniques onto QCD
Z
Ge
Ar
Si
3
Xe
3
We need to develop the tools for precision predictions
Shell model, coupled cluster, configuration-interaction
Exact many body:
GFMC, NCSM,
lattice EFT
3
3
QCD
Density
Functional,
Mean field
N
Quantum Chromodynamics
Lattice QCD: tool to deal with quarks and gluons
Formulate problem as functional integral over quark and gluon d.o.f. on R4
hOi =
Z
hOi =
Z
dAµ dqdq̄
O[q, q̄, A]e
SQCD
dAµ dqdq̄ O[q, q̄, A]e
SQCD
Discretise and compactify system
Finite but large number of d.o.f (1010)
Integrate via importance sampling
(average over important configurations)
Undo the harm done in previous steps
Spectroscopy
How do we measure the proton mass?
Create three quarks at a source: and annihilate the three quarks
at sink far from source
QCD adds all the quark anti-quark pairs and gluons
automatically: only eigenstates with correct q#’s propagate
time
Spectroscopy
Correlation decays exponentially with distance
C(t)
=
X
n
Zn exp( En t)
all eigenstates with q#’s of proton
at late times
! Z0 exp( E0 t)
Ground state mass revealed through “effective mass plot”

C(t)
M (t) = ln
C(t + 1)
t!1
! E0
QCD spectrum
After 30 years of developments
😍 Ground state hadron spectrum
reproduced
[summary by A Kronfeld, 1209.3468]
QCD spectrum
After 30 years of developments
bbb
5500
cbb
Experiment
Lattice QCD (Brown et al., 2014)
😍 Ground state hadron spectrum
reproduced and predicted
cbb
5000
ccb
ccb
ccc
M/MeV
nb · 3000
4500
bb
bb
bb
bb
4000
cb
cb
cb
cb
cb
cb
cc
3500
cc
cc
cc
b
b
3000
b
b
b
b
b
b
c
2500
c
c
c
c
2000
c
c
c
[Z Brown et al 1409.0497]
FIG. 15. Our results for the masses of charmed and/or bottom baryons, compared to the experimental results where availab
[8, 10, 12]. The masses of baryons containing nb bottom quarks have been o↵set by nb · (3000 MeV) to fit them into this pl
Note that the uncertainties of our results for nearby states are highly correlated, and hyperfine splittings such as M⌦⇤b M
can in fact be resolved with much smaller uncertainties than apparent from this figure (see Table XIX).
[summary by A Kronfeld, 1209.3468]
QCD spectrum
😍 Precise isospin mass splittings in QCD+QED
Figure 2:
10
experiment
QCD+QED
prediction
M [MeV]
8
6
D
4
cc
2
N
CG
0
[S. Borsanyi et al. [BMWc] 1406.4088]
BMW 2014
HCH
Figure 2: Mass splittings in channels that are stable under the strong and electromagnetic interactions. Both
these interactions are fully unquenched in our 1+1+1+1 flavor calculation. The horizontal lines are the experimental values and the grey shaded regions represent the experimental error [29]. Our results are shown by red
Nuclear Spectra
QCD for Nuclear Physics
QCD (+EW) describes nuclear physics Can compute the mass of lead nucleus ... in principle In practice: a hard problem
At least two exponentially difficult challenges
Noise: probabilistic method so statistical uncertainty grows exponentially with A
Contraction complexity grows factorially
QCD for Nuclear Physics
Quarks need to be tied together in all possible ways
Ncontractions = Nu!Nd!Ns! !
!
!
!
!
Managed using algorithmic trickery [WD & Savage, WD & Orginos; Doi & Endres]
Study up to N=72 pion systems, A=5 nuclei
Light nuclei
Light hypernuclear spectrum @ 800 MeV
0
-20
1+
1+
0+
0+
1+
-40
1+
2
DE HMeVL
-60
1+
2
3+
2
-80
-100
1+
2
3+
2
0+
0+
0+
0+
-120
0+
-140
-160
d
nn
3
He
4
He
nS
3
LH
3
LHe
3
SHe
4
LHe
H-dib
nX
4
LLHe
[NPLQCD Phys.Rev. D87 (2013), 034506 ]
Heavy quark universe
[Barnea et al. 1311.4966]
Combining LQCD and nuclear EFT (pionless EFT)
For heavy quarks, even spectroscopy requires QCD matching:
0
-50
DE @MeVD
-100
3
-200
In a world @ mπ = 800 MeV
Tuni
n
g
Valid
at
ion
-150
LQCD
EFT
nn
d
3
Heê3 H
4
He
3
Pred
ictio
n
Heê Li
Li
5
Equally important for matrix elements
5
6
Nuclear Structure
External currents and nuclei
External currents and nuclei
Current-nucleus interaction
Born approximation – interacts with a single
nucleon
⇠ |A hN |J|N i|2
External currents and nuclei
Current-nucleus interaction
which is two orders lower than the contribution from the impulse
is the origin of the enhancement suggested in Ref. [1]. The isos
strange and heavier quarks do not contribute to the non-deriva
and, as such, are not expected to be enhanced in WIMP-nucleus
the WIMP-nucleus interactions quantitatively, nuclear matrix e
need to be calculated.
Born approximation – interacts with a single
nucleon
⇠ |A hN |J|N i|2
known from expt/LQCD
FIG. 1: Some of the diagrams contributing to nuclear -terms. The
order contribution to the single-nucleon -term in PT. The middl
(“D2 -terms” contributions from Eq. (7)) panels show contributions to
leading order in KSW power counting [13–15]. The crossed box corre
light-quark mass matrix.
Ideally, one would simply determine the matrix element of the L
in the ground state of a given nucleus, at the relevant momentum
ing the intermediate matchings in Eq. (3) and in Eq. (5). This wo
from the hadronic EFT to all orders in perturbation theory, and
trix elements directly from QCD. While such formidable calcula
accomplished, the forward matrix element of the scalar-isoscalar o
in light nuclei, albeit with significant uncertainties, by combinin
culations of the binding energies with the corresponding experim
the ground state of a nucleus with Z protons and N neutrons,
(gs)
(gs, )
EZ,N = EZ,N + Z,N , where
Z,N
= hZ, N (gs)| mu uu + md dd |Z, N (gs
External currents and nuclei
Current-nucleus interaction
which is two orders lower than the contribution from the impulse
is the origin of the enhancement suggested in Ref. [1]. The isos
strange and heavier quarks do not contribute to the non-deriva
and, as such, are not expected to be enhanced in WIMP-nucleus
the WIMP-nucleus interactions quantitatively, nuclear matrix e
need to be calculated.
Born approximation – interacts with a single
nucleon
⇠ |A hN |J|N i|2
known from expt/LQCD
Interact non-trivially with multiple nucleons
FIG. 1: Some of the diagrams contributing to nuclear
-terms. The
order contribution to the single-nucleon -term in PT. The middl
(“D2 -terms” contributions from Eq. (7)) panels show contributions to
2
leading order in KSW power counting [13–15]. The crossed box corre
light-quark mass matrix.
⇠ |A hN |J|N i + ↵ hN N |J|N N i + . . . |
Ideally, one would simply determine the matrix element of the L
in the ground state of a given nucleus, at the relevant momentum
ing the intermediate matchings in Eq. (3) and in Eq. (5). This wo
from the hadronic EFT to all orders in perturbation theory, and
trix elements directly from QCD. While such formidable calcula
accomplished, the forward matrix element of the scalar-isoscalar o
in light nuclei, albeit with significant uncertainties, by combinin
culations of the binding energies with the corresponding experim
the ground state of a nucleus with Z protons and N neutrons,
(gs)
(gs, )
EZ,N = EZ,N + Z,N , where
Z,N
= hZ, N (gs)| mu uu + md dd |Z, N (gs
External currents and nuclei
Current-nucleus interaction
which is two orders lower than the contribution from the impulse
is the origin of the enhancement suggested in Ref. [1]. The isos
strange and heavier quarks do not contribute to the non-deriva
and, as such, are not expected to be enhanced in WIMP-nucleus
the WIMP-nucleus interactions quantitatively, nuclear matrix e
need to be calculated.
Born approximation – interacts with a single
nucleon
⇠ |A hN |J|N i|2
known from expt/LQCD
Interact non-trivially with multiple nucleons
FIG. 1: Some of the diagrams contributing to nuclear
-terms. The
order contribution to the single-nucleon -term in PT. The middl
(“D2 -terms” contributions from Eq. (7)) panels show contributions to
2
leading order in KSW power counting [13–15]. The crossed box corre
light-quark mass matrix.
⇠ |A hN |J|N i + ↵ hN N |J|N N i + . . . |
unknown/poorly known!
which is two orders lower than the contribution from the impulse approximation. This term
Ideally, one would simply determine the matrix element of the L
is the origin of the enhancement suggested in Ref. [1]. The isoscalar interactions with the
in the ground state of a given nucleus, at the relevant momentum
strange and heavier quarks do not contribute to the non-derivative interaction with pions
ing the intermediate matchings in Eq. (3) and in Eq. (5). This wo
and, as such, are not expected to be enhanced in WIMP-nucleus interactions. To determine
from the hadronic EFT to all orders in perturbation theory, and
the WIMP-nucleus interactions quantitatively, nuclear matrix elements of these operators
trix elements directly from QCD. While such formidable calcula
need to be calculated.
accomplished, the forward matrix element of the scalar-isoscalar o
in light nuclei, albeit with significant uncertainties, by combinin
culations of the binding energies with the corresponding experim
the ground state of a nucleus with Z protons and N neutrons,
(gs)
(gs, )
EZ,N = EZ,N + Z,N , where
Z,N
= hZ, N (gs)| mu uu + md dd |Z, N (gs
FIG. 1: Some of the diagrams contributing to nuclear -terms. The left panel shows the leading
External currents and nuclei
Current-nucleus interaction
which is two orders lower than the contribution from the impulse
is the origin of the enhancement suggested in Ref. [1]. The isos
strange and heavier quarks do not contribute to the non-deriva
and, as such, are not expected to be enhanced in WIMP-nucleus
the WIMP-nucleus interactions quantitatively, nuclear matrix e
need to be calculated.
Born approximation – interacts with a single
nucleon
⇠ |A hN |J|N i|2
known from expt/LQCD
Interact non-trivially with multiple nucleons
FIG. 1: Some of the diagrams contributing to nuclear
-terms. The
order contribution to the single-nucleon -term in PT. The middl
(“D2 -terms” contributions from Eq. (7)) panels show contributions to
2
leading order in KSW power counting [13–15]. The crossed box corre
light-quark mass matrix.
⇠ |A hN |J|N i + ↵ hN N |J|N N i + . . . |
unknown/poorly known!
is two orders lower than the contribution from the impulse approximation. This term
Second term may bewhich
significant
Ideally, one would simply determine the matrix element of the L
is the origin of the enhancement suggested in Ref. [1]. The isoscalar interactions with the
in the ground state of a given nucleus, at the relevant momentum
strange and heavier quarks do not contribute to the non-derivative interaction with pions
ing the intermediate matchings in Eq. (3) and in Eq. (5). This wo
and, as such, are not expected to be enhanced in WIMP-nucleus interactions. To determine
from the hadronic EFT to all orders in perturbation theory, and
the WIMP-nucleus interactions quantitatively, nuclear matrix elements of these operators
trix elements directly from QCD. While such formidable calcula
need to be calculated.
accomplished, the forward matrix element of the scalar-isoscalar o
in light nuclei, albeit with significant uncertainties, by combinin
culations of the binding energies with the corresponding experim
the ground state of a nucleus with Z protons and N neutrons,
(gs)
(gs, )
EZ,N = EZ,N + Z,N , where
May shift cross sections
May scale differently with Z and A
Leads to significant uncertainty
Z,N
= hZ, N (gs)| mu uu + md dd |Z, N (gs
FIG. 1: Some of the diagrams contributing to nuclear -terms. The left panel shows the leading
Groupe de Physique Théorique, Centre de Recherches Nucléaires, Institu
Particles, Centre National de la Recherche Scientifique, Université Lo
Cedex 2, France
(August 12, 2013)
1.0
iv:nucl-th/9603039v1 26 Mar 1996
Nuclear uncertainties
reducti
[Martinez-Pinedo et al., Phys. Rev. C53, 2602 (1996)]
We have calculated
1.0 the Gamow-Teller matrix elements of
Gamow-Teller transitions 64
in decays
nucleiof nuclei in the mass range A = 41–50. In all the
cases the valence space of the full0.77
pf -shell is used. Agreement
are a0.77
stark example of problems
0.8
with the experimental results demands
0.744
0.744 the introduction of an
average quenching factor, q = 0.744 ± 0.015, slightly smaller
0.6
Well measured
but statistically compatible
with the sd-shell value, thus indicating that the present number is close to the limit for large
0.4
A.
Best nuclear structure calculations
T(GT) Exp.
R(GT) Exp.
norma
whose
0.8
In p
Gamow
tion of
0.6
reactio
dividua
0.4
experim
are systematically off by 20–30%
proced
PACS number(s): 0.2
21.10.Pc, 25.40.Kv, 27.40.+z
0.2
model
Points correspond to different nuclei
up to A
Large range of nuclei The observed Gamow
0.0
0.0
be
Our
0.0
0.2
0.4
0.6
0.8
1.0
0.0 Teller
0.2 strength
0.4 appears
0.6 to 0.8
1.0
(30<A<60)
spectrum
is
systematically
smaller than what isT(GT)
theoretically
exR(GT) where
Th.
Th.
the pf
pectedeleon the basis
model independent
“3(N −Z)” values
FIG. 1. Comparison
of the experimental matrix
FIG.of2.theComparison
of the experimental
of
antoin
well described
s
workThas
to the subject
X
ments R(GT ) with the theoretical calculations sum
basedrule.
on Much
the sums
(GTbeen
) withdevoted
the correspondig
theoretical minima
value
T“free-nucleon”
(GT
⇠
h ·problem
⌧ ii!f operator.
in transithe last fifteen
[1–4].
The) heart
ofGamow-Teller
the
the “free-nucleon” Gamow-Teller operator. Each
basedyears
on the
Kuo B
QRPA,
shell-model,...
byis defining
thea point
reduced
transition
tion is indicated by a point in the x-y plane, can
withbe
thesummed
Eachup
sum
indicated by
x-y plane, with
the
fin the
details
theoretical value given by the x coordinate of probability
the point
astheoretical value given by the x coordinate of the point
Follo
and the experimental
value for
by the
experimental value by #
the y coordinate. beta de
Correct
it ybycoordinate.
“quenching” ! and"the
2
⟨f || k σ k tk± ||i⟩
gA
2
daught
√
⟨στ ⟩ ,
⟨στ ⟩ =
,
axial charge in nuclei ... B(GT ) = g
2Ji + 1
V
perime
TABLE I. Experimental and theoretical M (GT ) matrix elements. The experimental data have been taken from [19]. Iβ +
0h̄ω ca
(1)
e the branching ratios . All other quantities explained in the text.
erage q
ocess
2Jnπ , 2Tnπ
Q
Iβ + IIsϵ the observed
log quenching
ft
)
transit
and asking:
due toMa (GT
renormal-
Nuclear matrix elements
For deeply bound nuclei, use the techniques as for single hadron
matrix elements
Σ
permutations
3 pt function
2 pt function
At large time separations gives ground-state matrix element of
current
For near threshold states, need to be careful with volume effects
Calculations of matrix elements of currents in light nuclei just
beginning for A<5
( )
6
To optimize the re-use of light-quark propagators in the
4
production, calculations were performed for UQ (1) fields
2
Background
field
method
with ñ = 0, 1, 2, +4. Four field strengths were found
0
to be sufficient for this initial investigation. With three
0
5
10
/
degenerate flavors of light quarks, and a traceless electriccharge matrix, there are no contributions from coupling
of the B field to sea quarks at leading order in the elecFIG. 1: The correlator ratios R(B
tric charge. Therefore, the magnetic moments presented
Hadron/nuclear
two-point
functions
are dis- slice for the various states (p, n, d
here
are complete calculations
(there
are no missing
+1, 2, +4. Fits to the ratios are als
connected
contributions).
modified
in presence of fixed eternal fields
The ground-state energy of a non-relativistic hadron
of Eg:
massfixed
M , and
chargemodified
Q e in a uniform
magnetic
field is
B field:
exponential
behaviour
aligned and anti-aligned with th
|Q e B|
E(B) = M +
2M
!
2⇡ M 0 |B|2
µ·B
2⇡
M 2 Tij Bi Bj
+ ... , (3)
where
the spectroscopy
ellipses denote terms
are cubic
and enable
higher
QCD
withthat
multiple
fields
in the magnetic field, as well as terms that are 1/M
extraction of coefficients of response
suppressed [19, 20]. The first contribution in eq. (3) is
the hadron’s rest mass, the second is the energy of the
Eg: magnetic
moments,
polarisabilities,
…
lowest-lying
Landau level,
the third
is from the interaction of its magnetic moment, µ, and the fourth and fifth
restricted
EM fields
(axial,
terms Not
are from
its scalarto
andsimple
quadrupole
magnetic
polarizabilities,
twist-2,…)
M 0,M 2 , respectively (Tij is a traceless symmetric tensor [21]). The magnetic moment term is only
present for particles with spin, and M 2 is only present
for j
1. In order to determine µ using lattice QCD
calculations, two-point correlation functions associated
will be split by spin-dependent in
B
ference, E (B) = E+j
E Bj , can
correlation functions that we con
of E (B) that is linear in B dete
Explicitly, the energy di↵erence i
large time behaviour of
(B)
R(B) =
Cj
C
(0)
j (t)
(0)
Cj (t)
(t) C
(B)
j (t)
t!
Each term in this ratio is a correla
quantum numbers of the nuclear s
sidered, which we compute using t
As discussed in Ref. [14], subtra
from the correlation functions cal
her
/M
is
the
acfth
armnly
ent
CD
ted
±j
10
15
20
Nuclear
magnetic
moments
/
( )
FIG. 1: The correlator ratios R(B) as a function of time
slice for the various states (p, n, d, 3 He, and 3 H) for ñ =
+1, 2, +4. Fits to the ratios are also shown.
( )
B
alignedMagnetic
and anti-aligned
withfrom
the magnetic
field, E±j
,
moments
spin splittings
will be split by spin-dependent interactions, and the dif
(B) (B) (B) B (B)B
ference,
be extracted
j , can
E E ⌘ E=+jE+j E Ej =
2µ|B|
+ |B|3 +from
. . . the
correlation functions that we consider. The component
of E (B) that is linear in B determines µ via Eq. (3).
Extract
splittings
from ratios
of
Explicitly,
the energy
di↵erence
is determined
from the
correlation
large time
behaviourfunctions
of
( )
(3)
5
R(B) =
(B)
Cj (t)
(B)
C j (t)
(0)
C j (t)
(0)
Cj (t)
t!1
! Ze
E (B) t
12
10
8
6
4
2
0
4
p
n
3
2
1
0
4
d
3
2
1
0
4
.
(4)
( )
on
d is
0
3
3
He
2
1
Each term
in this
is asingle
correlation
function with the
Careful
toratio
be in
exponential
quantum numbers of the nuclear state that is being conof compute
each correlator
sidered,region
which we
using the methods of Ref. [3].
As discussed in Ref. [14], subtracting the contribution
from the correlation functions calculated in the absence
of a magnetic field reduces fluctuations in the ratio, enabling a more precise determination of the magnetic mo-
( )
nd
ree
icng
ected
dis-
2
0
0
12
10
8
6
4
2
0
3
0
H
5
10
/
15
20
[NPLQCD 1409.3556, PRL to appear]
Nuclear magnetic moments
3
Energy shift vs B
4
3
p
H
n
0.1
2
0.0
[
( )
]
d
-0.1
0
p
n
-2
3
He
-0.2
3
He
0.1
FIG. 3: The magnetic moments of the proton, neutron,
deuteron, 3 He and triton. The results of the lattice QCD calculation at a pion mass of m⇡ ⇠ 806 MeV, in units of lattice
nuclear magnetons, are shown as the solid bands. The inner
bands corresponds to the statistical uncertainties, while the
outer bands correspond to the statistical and systematic uncertainties combined in quadrature, and include our estimates
of the uncertainties from lattice spacing and volume. The red
dashed lines show the experimentally measured values at the
physical quark masses.
0.0
( )
d
-0.1
3
H
-0.2
0
1
2
3
4
| |
|B|
FIG. 2: The calculated E (B) of the proton and neutron
(upper panel) and light nuclei (lower panel) in lattice units
of the magnetic moment using the above form. These
results agree with previous calculations [14] within the
uncertainties. In the more natural units of lattice nuclear magnetons (LNM), 2Me N , where MN is the mass
[NPLQCD 1409.3556, PRL to appear]
of the nucleon at the quark masses of the lattice cal-
Nuclear magnetic moments
3
Energy shift vs B
n
0.1
p
H
3
4
p
2
H
d
]
n
0.0
0
2
d
]
[
( )
3
4
3
-0.1
p
n
3
[
-2
-0.2
p3 He
n
( )
0.0
-0.1
3
n
d
p
He
3
H
2
]
-0.2
1
2
| |
|B|
3
[
0
0
3
He
FIG. 3: The magnetic moments of the3 proton, neutron,
3-2
deuteron, He and triton. The results of the lattice QCD calculation at a pion mass of m⇡ ⇠ 806 MeV, in units of lattice
nuclear magnetons, are shown as the solid bands. The inner
3
H
bands corresponds to the statistical
uncertainties, while the
outer bands correspond to the statistical andQCD
systematic
@ mπ un= 800 MeV
certainties combined in quadrature, and include
our estimates
Experiment
of the uncertainties from lattice spacing and volume. The red
FIG. 3: The
magnetic moments
of theat proton,
neutron,
dashed lines show
the
experimentally
measured
values
the
3
d
deuteron,
He and triton. The results of the lattice QCD calphysical
quark
masses.
0.1
4
He
4
0d
FIG. 2: p The calculated E (B) of the proton and neutron
n
3
(upper panel) and light nuclei (lower panel)
-2H in lattice units
3
3
n
p
d
culation at a pion mass of m⇡ ⇠ 806 MeV, in units of lattice
nuclear magnetons,
are 3.21(3)(6)
shown as 1.22(4)(9)
the solid -2.29(3)(12)
bands. The
inner
-1.98(1)(2)
3.56(5)(18)
μ moment tousing
of the
magnetic
above form.
These
bands
corresponds
thethe
statistical
uncertainties,
while the
results agree with previous calculations [14] within the
outer bands
correspond
to theunits
statistical
and
systematic ununcertainties. In
Inunits
the more
natural
of
lattice
nuof appropriate
nuclear magnetons (heavy MN)
e
certainties
combined
in
quadrature,
include
3
clear magnetons (LNM), 2M
MN and
is the
mass our estimates
He , where
[NPLQCD 1409.3556, PRL to appear]
of the
uncertainties
frommasses
lattice
andcalvolume. The red
of the
nucleon
at the quark
ofspacing
the lattice
N
Nuclear magnetic moments
3
Numerical values are surprisingly
interesting
4
3
p
H
n
3
n
]
d
0
n
3
-2
3
3H
3
He
H
QCD @ mπ = 800 MeV
Experiment
]
FIG. 3: The magnetic moments of the proton, neutron,
3
2
d
deuteron,
and triton. The results of the lattice QCD calLattice results appear to suggest culation at He
3
3
n
p
d
a pion mass of m⇡ ⇠ 806 MeV, in units of lattice
heavy quark nuclei are shell-model
nuclear magnetons, are shown as the solid bands. The inner
-1.98(1)(2) 3.21(3)(6)
1.22(4)(9) -2.29(3)(12) 3.56(5)(18)
d
μ
0
bands
corresponds
to
the
statistical
uncertainties, while the
like!
outer bands correspond to the statistical and systematic unIn units of appropriate nuclear magnetons (heavy MN)
n
certainties
combined in
3 quadrature, and include our estimates
3
He
H
[NPLQCD 1409.3556, PRL to appear]
-2
of the uncertainties from lattice
spacing and volume. The red
[
p
He
2
[
Shell model expectations
µd = µp + µn
µ
3 H = µp
µ
3 He = µn
J=0p
n n
p
4
p
Nuclear magnetic moments
Numerical values are surprisingly
interesting
]
0.2
3
H
0.0
-0.2
d
-0.4
3
-0.6
3
3
He
H
FIG. 4: The di↵erences between the nuclear magnetic moQCD @ mπ = 800 MeV
ments and the predictions of the naive shell-model. The
3
Experiment
results of the lattice QCD calculation at a pion mass of
m⇡ ⇠d 806 MeV, in units of lattice nuclear magnetons, are
2
Lattice results appear to suggest shown as the solid bands.
The inner
band corresponds
to
3
3
d
the statistical uncertainties, while the outer bands correspond
heavy quark nuclei are shell-model
to the statistical
and
systematic -0.34(2)(9)
uncertainties combined
0.01(3)(7)
0.45(4)(16) in
δμ
0
like!
quadrature, including estimates of the uncertainties from lattice spacing and volume. The red dashed lines show the exDifference from NSM expectation
n
perimentally measured
3 di↵erences.
H
[
]
n
0.4
[
Shell model expectations
µd = µp + µn
µ
3 H = µp
µ
3 He = µn
J=0
n n
p
4
p
0.6
p
-2
He
[NPLQCD 1409.3556, PRL to appear]
Nuclear sigma terms
One possible DM interaction is through scalar exchange GF X (q)
L=
aS (
2 q
!
)(q q)
Accessible via Feynman-Hellman theorem At hadronic/nuclear
level
✓
Lagrange density in Eq. (2) matches onto
h
i
h
i
1
1
†
†
†
†
†
L ! GF
h0|qq|0i Tr aS ⌃ + aS ⌃ + hN |qq|N iN N Tr aS ⌃ + aS ⌃
4
4
◆
⇣
h from the impulse
i
⌘
which is two orders lower
approximation.
This
term
1 than3 the contribution
†
†
†
†
hN
|q⌧
q|N
i
N
N
Tr
a
⌃
+
a
⌃
4N
a
N
+
...
S
S,⇠
S
is the origin of the enhancement
suggested in Ref.
[1]. The
isoscalar interactions
with the(3)
4
strange and heavier quarks do not contribute to the non-derivative interaction with pions
as such,
are notbreaking
expectedscale
to be⇤enhanced
WIMP-nucleus
interactions.matrix
To determine
at theand,
chiral
symmetry
, which in
describes
the single-hadron
elements
the associated
WIMP-nucleus
interactions
quantitatively,
nuclear
matrix ⌃
elements
of these operators
and the
interactions
at LO
in the chiral
expansion.
is the exponentiated
pion
to be
calculated.
field, need
and N
is the
nucleon field,
Contributions:
⌃ = exp
2i
M
f⇡
!
0
, M =
p
⇡ p
⇡0/ 2
!
, N =
⇣
†
⇠a†S ⇠
+
⇡ / 2
⇡
1
2
†
⌘
p
n
!
,
p
(4)
f⇡ = 132 MeV is the pion decay constant, aS,⇠ =
⇠ aS ⇠ +
with ⇠ = ⌃, and the
ellipsis denotes higher-order interactions including those involving more than one nucleon.
FIG. 1: Eq.
Some
of in
the the
diagrams
contributing
nuclear
-terms. The
panel
the leading
Expanding
(3)
number
of pion tofields
(neglecting
theleft
shift
in shows
the WIMP
mass
order
to the single-nucleon
in PT. The
(pion-exchange)
induced
bycontribution
the chiral condensate),
the LO-term
contributions
to middle
the interactions
are and right
658
0.0139(56)(70)
d
0.057(20)(14)
m⇡ 0.0515(81)(71)
d
MN =
MN
(12)
N = mhN | uu + dd |N i = m
dm
2 dm⇡
(two-flavor) nuclear -term can be written as
is the nucleon
-term and |N i is the single-nucleon
state. The first term in Eq.0 (11) is the
0
0
⇡ d
noninteracting single-nucleon contribution to
-term,
while
the second
termmcor-10 the nuclear
-20
=
A
+
=
A
BZ,N ,
Z,N
N
BZ,N
N
-5
-20
2 dm⇡
responds to the corrections due to interactions between the nucleons, including-40the possibly
-30
enhanced-10contributions from MECs. It is useful
to define the ratio
-60
where -40
-80
1-50 m⇡ scalar
d
-15
Previous work
suggested
dark
matter
couplings
nuclei
d (13)to m
BZ,N
⇡ d
-100
Z,N =
-60
=
mhN
|
uu
+
dd
|N
i
=
m
M
=
MN
A N 2 dmN⇡
N
dm et 300
2 dm
-70
-20
O(50%)
MECs -120
[Prezeau
al 2003]
0 100 200arising
300 400 500 from
600 700 800
0
100 have
200 300 400
500 600 700 800uncertainty
0 100 200
400 500
600 700⇡ 800
mp HMeVL
mp HMeVL
mp HMeVL
to quantify the deviations
from the impulse approximation.
In addition to representing deis the nucleon -term and |N i is the single-nucleon state. The first term in Eq. (11)
viations of nuclear -terms from the impulse approximation, this quantity also describes the
noninteracting single-nucleon
contribution
the nuclear bounds
-term, while the second term
Quark
mass WIMP-nucleus
dependence
of nuclear
binding
energies
3 He to
FIG.of4:theThe
nuclear contributions
to the scattering
deuteron (left
panel),
(middle
panel) and 4 He (right
deviation
scalar-isoscalar
matrix
element
from
the
impulse
responds to the corrections due to interactions between the nucleons, including the po
such
contributions
contributions
approximation
at zero from
momentum
transfer,
panel) -terms
nuclear
interactions.
The inner
outerIt shaded
correspond
enhanced
fromand
MECs.
is usefulregions
to define
the ratio to the
statistical and total (statistical combined with systematic) uncertainties, respectively.
hZ, N (gs)| uu + dd|Z, N (gs)i
1 m⇡ d
1 .=
B(14)
!
Z,N =
Z,N
Z,N
A N 2 dm⇡
A hN | uu +3 dd|N i 4
Z,N
HMeVL
He
sB4
sB3He HMeVL
sBd HMeVL
Nuclear sigma terms
the nuclear -terms of the deuteron, He and He are shown in Fig. 5. For each nucleus,
to quantify
deviations
from
the 10%
impulse
approximation.
In addition to representin
the nuclear interactions
modify
the the
-term
by less
than
of the
impulse approximation
Lattice
calculations
+
physical
point
suggest
such
III. LIGHT NUCLEI FROM LATTICE
QCD AND
THEIR
nuclear
-terms
from
the-TERMS
impulse
approximation,
quantity
contribution for both pionviations
massesofconsidered,
and
by less
than 2%
at the lighterthis
pion
mass,also describe
contributions are O(10%) or less for light nuclei (A<4)
as can be seen in Fig. 6. deviation of the scalar-isoscalar WIMP-nucleus scattering matrix element from the im
ds4He H%L
ds3He H%L
dsd H%L
approximation
zero momentum
Lattice QCD has evolved to the stage
where the at
binding
energies oftransfer,
the lightest nuclei and
hypernuclei0 have been determined at a small 0number of relatively heavy pion masses
in the
0
hZ,
N (gs)|
uu + extensively
dd|Z, N (gs)i
limit of isospin symmetry.
Further, the mass
of
the
nucleon
has
been
explored
=
1 .
-2
-2
Z,N
-1
A hN | uu +
i
over a large range of
light-quark masses, with calculations now being performed
at dd|N
the phys-4
-4
ical value -2
of the pion mass. These sets of calculations, along with the experimental values
-6
-6
of the masses
of
the
light
nuclei,
are
sufficient
to
arrive
at
a
first
QCD
determination
of the
III.
LIGHT
NUCLEI
FROM
LATTICE
QCD AND
THEIR -TERMS
-3
-8
-8
nuclear -terms for these nuclei at a small number of pion masses. This work provides an es-10
-10
-4 modifications to the impulse approximation
timate of the
0 has
100 200
300 400for
500
600
800 900whereWIMP-nucleus
0 100
200 300 400
500 600 700
0 100 200 300 400 500 600 700 800
900
Lattice
QCD
evolved
toscalar-isoscalar
the700stage
the
binding
energies
of 800
the900lightest nucle
2
mpdetermined
HMeVL
mp HMeVL. In particular,
p HMeVL
interactions in light nuclei
these
can
be usedatto
explore
the conjechypernuclei
haveresults
been
a small
number
ofmrelatively
heavy pion masses i
tured enhancement of MEC contributions
thesesymmetry.
interactions,
and tothe
investigate
the nucleon
size of has been explored exten
limit of to
isospin
Further,
mass of the
FIG. 5: Percentage
modifications
to the
the
impulse
approximation
to[NPLQCD
the
deuteron
the uncertainties
introduced
by the
usea of
impulse
approximation
incontribution
phenomenological
over
large
range
of light-quark
masses,
with
calculations
now
being
PRD performed
to(left
appear] at the
4 He (right panel) -terms.
analyses.
panel), 3 He (middle panel) ical
and value
of the pion mass. These sets of calculations, along with the experimental v
QCD for nuclear physics
Nuclei are under serious study directly from QCD
Spectroscopy of light nuclei and exotic nuclei (strange,
charmed, …)
Nuclear properties/matrix elements
Prospect of a quantitative connection to QCD makes this a very exciting time for nuclear physics
Critical role in current and upcoming intensity frontier experimental program
Learn many interesting things about nuclear
physics along the way
fin