Download Strong Electroweak Symmetry Breaking

Nikhef Institute of Subatomic Physics
and
Utrecht University
Strong Electroweak Symmetry Breaking
Generating Masses Dynamically
Author:
Jory Sonneveld
Supervisor:
Prof. Eric Laenen
May 20, 2014†
Front page image1
The Higgs need not be an elementary particle; it could, for instance, be a top and an antitop
quark condensate. In top condensate models, this phenomenon occurs because of a new strong
force. As a result of this force, there is an attraction between top quarks; this interaction gives
them a mass. The tops form a condensate and in this way break the electroweak symmetry
to give particles their mass.
† Note:
This thesis was originally printed on July 6, 2012. Minor corrections have been made
since then; to accurately reflect the last update, the date on the front page has been changed.
Please contact the author if there is anything unclear to you or if you found a mistake. For
contact information see phys.onmybike.nl.
This work is available under the Creative Commons ShareAlike (CC BY-SA) license. See
for more information creativecommons.org .
1
Figure modified and taken from The Particle Zoo .
I
Acknowledgments
I would like to express my gratitude to my supervisor Professor Dr. Eric Laenen for his inspiration
and patient explanations of the topics in particle physics that I found hard to grasp by myself. I
have learned much from him while he enthusiastically told me about this area of research and found
his corrections and questions very helpful. I also want to thank Professor Francesco Sannino and Dr.
Mads Frandsen for their inspiring conversations with me, and Professor Fawzi Boudjema and Dr. Diego
Guadagnoli for the good questions they asked me. I would like to give special thanks to Alex Kieft, a
bachelor student who always asked exactly the right questions exposing my weaknesses regarding the
topic of this thesis. Thanks to him I understand the physics of strong electroweak symmetry breaking
much more thoroughly. Furthermore, I am greatly indebted to Dr. Pierre Artoisenet for his great
help with using FeynRules and making MadGraph look as simple as it is not. After many hours of his
patience I could I (we) succeeded in obtaining what I needed from the program. Finally, I would like
to thank Damien, again Pierre, Sander, and all others of the theory group for their readiness to help
answer my questions. My thanks also go to Jan and Lisa who greatly improved my presentation, as
well as Robbert and Philipp for both their help in physics and their pleasant company. From other
fields of expertise I want to thank Herman, Wicher, Laurens, and Aliza for improving the language in
and readability of my thesis.
II
Abstract
One viable model of electroweak symmetry breaking through strong dynamics is topcolorassisted technicolor (TC2). The underlying basic model, the Nambu–Jona-Lasinio (NJL)
model, is studied and compared to the Standard Model. Custodial symmetry is shown to
hold in the Standard Model and is explored in the NJL model. Several variants of technicolor
and topcolor based on the NJL model are briefly discussed, before the phenomenology of TC2
is investigated. Quantitative results of the effect of TC2 on the asymmetry of top, which has
been measured at the Tevatron in 2011, are given as a function of the mass of several new
composite particles that appear in the TC2 effective theory.
III
Contents
1 Introduction
1.1 Higgs mechanism: a simple example . . . . . . . . . .
1.2 Standard Model and the Higgs mechanism . . . . . . .
1.2.1 Hypercharges . . . . . . . . . . . . . . . . . . .
1.2.2 The Bosonic Masses . . . . . . . . . . . . . . .
1.2.3 The couplings g and g 0 . . . . . . . . . . . . . .
1.2.4 Fermion Masses . . . . . . . . . . . . . . . . . .
1.3 Superconductors and spontaneous symmetry breaking
1.4 Chiral Symmetry . . . . . . . . . . . . . . . . . . . . .
1.5 Problems with the Standard Model . . . . . . . . . . .
1.5.1 Mass Renormalization . . . . . . . . . . . . . .
1.5.2 Naturalness and the Hierarchy Problem . . . .
1.5.3 Other Problems . . . . . . . . . . . . . . . . . .
1.6 LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
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2 Custodial Symmetry
2.1 The ρ parameter and precision measurements . . . . . . . . . . . . . . . . . .
2.2 Custodial symmetry from the Higgs potential . . . . . . . . . . . . . . . . . .
2.3 The Peskin-Takeuchi Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
24
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25
28
3 The
3.1
3.2
3.3
3.4
3.5
3.6
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31
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53
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57
57
Nambu–Jona-Lasinio Model
Top condensation versus Standard Model Higgs
The Gap Equation . . . . . . . . . . . . . . . .
Evidence of Bound States . . . . . . . . . . . .
Auxiliary Fields . . . . . . . . . . . . . . . . . .
Gauge Boson Masses . . . . . . . . . . . . . . .
Custodial symmetry in the NJL model . . . . .
4 Technicolor
4.1 From QCD to Technicolor . . .
4.2 Minimal Model of Susskind and
4.3 Extended Technicolor (ETC) .
4.4 Walking Technicolor . . . . . .
4.5 Other models . . . . . . . . . .
4.6 Problems with Technicolor . . .
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Weinberg
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IV
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mechanism
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5 Topcolor and Topcolor-Assisted Technicolor
5.1 Dynamics of topcolor . . . . . . . . . . . . . .
5.2 Topcolor-Assisted Technicolor . . . . . . . . .
5.3 Top Seesaw . . . . . . . . . . . . . . . . . . .
5.4 Phenomenology of Top Condensate Models .
5.4.1 Top Seesaw Phenomenology . . . . . .
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59
60
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62
63
64
6 Forward-Backward Asymmetry of the Top Quark
6.1 Effective Lagrangian . . . . . . . . . . . . . . . . .
6.2 Mass matrices and mass eigenstates . . . . . . . .
6.3 Vector Resonances . . . . . . . . . . . . . . . . . .
6.4 Experimental Results . . . . . . . . . . . . . . . . .
6.5 Analytical Result . . . . . . . . . . . . . . . . . . .
6.6 Numerical results using MadGraph . . . . . . . . .
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7 Discussion and Outlook
81
Appendices
85
A Group Theory: Lie Algebras
85
B Gap Equation in BCS Theory
B.1 Physical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
89
91
C The Goldberger-Treiman relation
96
D (Non-)Linear Sigma Model
D.1 The linear σ model in the Higgs Lagrangian . . . . . . . . . . . . . . . . . . .
D.2 The nonlinear σ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
99
100
E Phenomenology of a General Model
101
F Feynrules and MadGraph
F.1 Feynrules . . . . . . . . .
F.2 Checking the model . . .
F.3 MadGraph and Madevent
F.4 MadAnalysis . . . . . . .
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107
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110
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References
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113
V
Chapter 1
Introduction
The exact way particles in the Standard Model obtain mass is a question to which various
answers exist but none has been shown to be true in experiment. One possibility in the
Standard Model is that particles obtain mass through spontaneous symmetry breaking at
the scale of the electroweak force. Spontaneous symmetry breaking can be understood with
a “Mexican hat” depicting a potential for a particle (see figure 1.1) where a ball (particle)
that is initially placed at the tip of the hat (maximum potential, i.e. at high energies) and is
symmetric under rotations takes up a specific value when it tips off the top of the hat into the
rim; the picture is then no longer invariant under rotations. Electroweak symmetry breaking
(EWSB), also called the Higgs mechanism, is the process of spontaneous symmetry breaking
through which gauge bosons in gauge theories acquire their mass. In this mechanism, they do
so through “eating” or absorbing so-called massless Nambu–Goldstone bosons. The simplest
implementation of this mechanism is addition of an extra Higgs field to the Standard Model.
Here, symmetry is broken once the added Higgs field takes up a nonzero vacuum expectation
value, or, in analogy with the Mexican hat, chooses a side of the hat (with the rim the
“non-zero vacuum”). This mechanism implies the existence of a Higgs boson, but this field
need not necessarily be an elementary particle [1, p5]. In many condensed matter systems,
like superconductors, symmetries are spontaneously broken by an additional field; this field,
however, is generated dynamically. An example is the bound state of two electrons in a
superconductor [2, p201]. Although analyses of results from experiments yielded an upper
bound for the Higgs boson of the Standard Model, nothing is known about this particle and
its existence has not yet been verified [3, p5]. It may very well be that the Higgs boson does
not exist and particles acquire their mass through some other mechanism. The Standard
Model and the Higgs mechanism are introduced in this chapter; implementations of so-called
“strong” electroweak symmetry breaking are discussed in the remaining chapters.
1.1
Higgs mechanism: a simple example
Spontaneous symmetry breaking occurs when a physical vacuum does not respect the symmetry of the originally symmetric Lagrangian [4, p67]. If this occurs for a non-zero expectation
value of φ for a potential V (φ), a mass is generated for a gauge boson through the process
of spontaneous symmetry breaking; this is known as the Higgs mechanism [5, p692]. This
mechanism had an earlier application in the field of superconductivity (see section 1.3). In the
simplest example of the Higgs mechanism, the abelian Higgs model, a massive scalar known
1
Figure 1.1: Graph of the Mexican Hat potential. A ball on the top of the hat would be rotationally
symmetric; this symmetry is lost when it rolls down into the rim of the hat and picks a particular
“side” or spot in the rim. Figure by RupertMillard (own work by uploader, with gnuplot) [Public
domain], via Wikimedia Commons.
as the Higgs boson appears. This model uses the Lagrangian [4, p74]
1
L = |Dµ φ|2 − µ2 |φ|2 − |λ|(φ∗ φ)2 − Fµν F µν
4
(1.1)
with the complex scalar field
φ=
φ1 ± iφ2
√
,
2
(1.2)
the covariant derivative
Dµ ≡ ∂µ + iqAµ ,
(1.3)
Fµν ≡ ∂ν Aµ − ∂µ Aν .
(1.4)
and the field strength tensor
Following further the computations of [4], the Lagrangian density (1.1) is invariant under the
U (1) rotations
φ → φ0 = eiθ φ
(1.5)
with θ the space-time–independent phase parametrizing the rotations. The Lagrangian density is also invariant under the local gauge transformations
φ(x) → φ0 (x) = eiqα(x) φ(x),
Aµ (x) →
A0µ (x)
= Aµ (x) − ∂µ α(x).
(1.6)
(1.7)
In this equation q denotes a charge, which is chosen for convenience as will become clear later,
and α(x) is the phase or parameter for the transformations. For µ2 > 0, the potential has a
minimum at φ = 0 and the exact symmetry of (1.1) is preserved. For µ2 < 0 the potential
has minima at
2
µ2
v2
|φ| 0 = − |λ| ≡ .
(1.8)
2
2
We now rewrite the Lagrangian in terms of displacements from the physical vacuum, choosing
the physical vacuum1 as
v
hφi0 = √ ,
(1.9)
2
1
Note that we do not typically choose a negative vacuum expectation value, although a negative solution
for hφi0 exists here.
2
and a shifted field
φ0 = φ − hφi0
(1.10)
which can be parametrized as
1
φ = eiζ/v √ (v + h)
2
1
' √ (v + h + iζ)
2
(1.11)
(1.12)
with h the quantum fluctuations around the vacuum expectation value v. Note that the
original two degrees of freedom of the complex field φ are retained: one in the h of the term
v + h and one in ζ. The above equations can now be combined to form the Lagrangian
Lsmall oscillations =
1
1
[(∂µ h)(∂ µ h) + 2µ2 h2 ] + [(∂ µ ζ)(∂µ ζ)]
2
2
1
q2v2
− Fµν F µν + qvAµ (∂ µ ζ) +
Aµ Aµ + . . .
4
2
(1.13)
(1.14)
(1.15)
The Lagrangian can be expressed more neatly if we rewrite the terms containing Aµ and ζ
(excluding the Fµν term) as
q2v2
1
1 µ
µ
Aµ + ∂µ ζ
A + ∂ ζ ,
(1.16)
2
qv
qv
and make the gauge transformation
Aµ → A0µ = Aµ +
1
∂µ ζ,
qv
(1.17)
which is just the phase rotation of the scalar field:
1
φ → φ0 = e−iζ(x)/v φ(x) = √ (v + h)
2
(1.18)
yielding a Lagrangian of the following form:
Lsmall
oscillations
1
q 2 v 2 0 0µ
1
= [(∂µ h)(∂ µ h) + 2µ2 h2 ] − Fµν F µν +
Aµ A + const.
2
4
2
(1.19)
Now we have an h-field with (mass)2 = -2µ2 = 2|λ|v 2 > 0 and a massive vector field A0µ with
mass = qv, while our ζ-field has vanished.
According to Goldstone’s theorem, Goldstone bosons appear after spontaneous breakdown
of continuous symmetries [6, p564]. There is one Nambu–Goldstone boson (scalar, massless
particle) corresponding to every generator of the symmetry group that is broken by the
vacuum-expectation value of the field [6, p564]. It is these Goldstone bosons that can give
particles their mass. The field ζ was such a massless Goldstone boson, which is in this case
said to be “eaten” by the massless photon Aµ so that the photon would become a massive
vector boson A0µ .
3
~ the helicity. Figure taken from university
Figure 1.2: Helicity of a particle; ~k denotes the momentum, S
of Nebraska (http://physics.unl.edu/∼ tgay/content/CPE.html).
The massive scalar h is known as the Higgs boson. The former four degrees of freedom,
which were contained in the two scalars φ and φ∗ and the two helicity2 states of Aµ (a massless
particle, which thus travels at the speed of light c so that it has no degree of freedom in this
longitudinal direction ([7])), have after spontaneous symmetry breaking turned into four new
degrees of freedom: the Higgs boson and the three helicity states of the new massive gauge
field A0µ (three, because this particle is massive and travels at a speed less than c - it thus has
degrees of freedom in all spatial directions). The latter three will later in the Standard Model
represent the longitudinal components of the massive gauge bosons W + , W − , and Z 0 s [8].
The counting of massless Goldstone bosons can be made simpler by looking at the number
of generators of the original symmetry group and the group it is broken to3 . If the original
group was G with n(G) generators, and it is broken to a subgroup H with n(H) generators,
then the number of massless Goldstone bosons that will appear are n(G) − n(H) [2, p199].
For example, if G = SU (2) ⊗ U (1) is broken to H = U (1), then n(G) − n(H) = 3 + 1 − 1 = 3
massless Goldstone bosons appear [2, p238] (as they do in the Standard Model, on which
more in the next section). In this case, a U (1) symmetry was broken to leave no subgroup,
and thus only 1 − 0 = 1 Nambu-Goldstone boson appeared, which then gave the new gauge
boson its longitudinal component.
1.2
Standard Model and the Higgs mechanism
The Standard Model is a gauge theory with the gauge group SU (3)c ⊗ SU (2)L ⊗ U (1)Y (see
appendix A about groups and symmetries)4 in which color charge, weak isospin (T3 )5 , electric
charge (Q), and weak hypercharge6 (Y ) are conserved. The particles appearing in this model
2
Helicity defines the handedness of a particle, i.e. whether it is left- or right- handed; this observable
depends on the frame of reference [5, p47]. See figure 1.2.
3
For more on group theory, see appendix A
4
Strictly speaking, gauge invariance is not really a symmetry, but it merely shows that there is a redundancy
in the degrees of freedom used. In this respect, there is no such thing as spontaneously breaking a gauge
symmetry [2, p241].
5
Weak isospin is the weak analogue of the strong isospin, which in turn is a number representing a nucleon
state in a representation of isospin doublets; in this way, a nucleon can be represented as a linear combination
of a proton and a neutron state. Strong isospin is conserved under the strong interactions, meaning that the
strong interactions are invariant under a rotation in isospin space. The weak analogue, weak isospin, has a
conserved quantum number for the weak interaction, in which there are not nucleon but quark and lepton
doublets [7, app. C].
6
Weak hypercharge is related to the weak isospin by the weak equivalent of the Gell-Mann–Nishijima
formula Q = T3 + 12 Y [7, app. C].
4
are depicted in figure 1.3. The electric charge Q is the generator that the massless photon
couples to. We want this quantity to be invariant; it is defined as [2, p363]:
1
Q = T3 + Y
2
(1.20)
Here T denotes the generator of the isospin symmetry group SU (2) and Y denotes the generator of hypercharge symmetry group U (1)Y . The generators of SU (2) obey the Lie algebra
[T a , T b ] = iεabc Tc .
(1.21)
The sign εabc denotes the antisymmetric Levi-Civita symbol. In their fundamental representation the generators of SU (2) take the form of the Pauli matrices: Ta = σa /2 [9, p3]. The
Standard Model Lagrangian is ([10])
LSM = Lkinetic + LHiggs + LY ukawa ,
(1.22)
with the kinetic term describing the dynamics of the spinor fields ψ
Lkinetic = iψ(∂ µ γµ )ψ,
(1.23)
with ψ ≡ ψ † γ 0 , and ψ the three fermion generations which will be later represented by an
index i.
To impose gauge invariance under the three symmetry groups, we replace the derivative
by a covariant derivative containing the fields of the three interactions corresponding to the
symmetry groups:
Lkinetic = iψ(Dµ γµ )ψ;
(1.24)
with
Dµ = ∂ µ + i
gs µ
g0
g
Ga La + i Wbµ σb + i B µ Y,
2
2
2
(1.25)
where we define
• La are the Gell-Mann matrices, where ta ≡ La /2 the generators of SU (3);
• σb are the Pauli matrices, with Ta ≡ iσb /2 the generators of SU (2);
• gs , g, and g 0 are the strong SU (3), the electroweak isospin SU (2), and hypercharge U (1)
couplings, respectively;
• Gµa , Wbµ , and B µ are the eight gluon fields, the three weak interaction bosons that form
a triplet, and the single hypercharge boson, respectively.
The fermion fields ψ then transform as
ψ(x) → eiπ
a σ /2
a
eiY β(x) ψ(x),
(1.26)
with Y the generator of U (1) and π a the parameters of the SU (2) transformations. The
abelian U (1)Y and the nonabelian SU (2)L have the Yang-Mills terms Lgauge = − 41 (Gaµν Gaµν +
Hµν H µν ) with Hµν = ∂µ Bν − ∂ν Bµ and Gaµν = ∂µ Wνa − ∂ν Wµa + gεabc Wµb Wµc , from which it
can be seen that the gauge bosons self-interact [9, p5].
5
Figure 1.3: The Standard Model of Elementary Particles.
Figure by MissMJ [CC-BY-3.0
(www.creativecommons.org/licenses/by/3.0)], via Wikimedia Commons.
6
The SU (2)L × U (1)Y gauge theory is the partially unified theory of the weak and electromagnetic interactions of Glashow, Weinberg, and Salam [5, p700]. Through spontaneous
breaking of the symmetry of these groups to U (1)EM , the electromagnetic gauge group, the
W and Z bosons can acquire their mass. This works again according to the Higgs mechanism of the previous section; for this to occur, first a Higgs scalar field is to be added to the
Lagrangian [4, p108]:
LHiggs = (Dµ φ)† (Dµ φ) − µ2 φ† φ − λ(φ† φ)2 ,
(1.27)
from which the mass of the bosons comes from the interaction term (which contains a covariant
derivative) after spontaneous symmetry breaking. This is essentially a generalization of the
abelian Higgs model to a nonabelian gauge group.
Each fermion generation consists of five representations, here given in the interaction basis
denoted by I in which their Higgs couplings (i.e. the strengths of interaction with the Higgs
particle, which will emerge as in the previous section) are diagonalized [10, p12], [5, p714]:
I
u
1. Left-handed quarks QILi , which are a triplet under SU (3), a doublet
under
d Li
SU (2), and have hypercharge Y = 13 ;
2. Right handed “up” quarks (i.e. up, charm, top) uIRi , which are an SU (3) triplet, SU (2)
singlet, and have hypercharge Y = 43 ;
3. Right handed “down” quarks (i.e. down, strange, bottom) dIRi , which are an SU (3)
triplet, SU (2) singlet, and have hypercharge Y = −2
3 ;
4. Left-handed leptons
LILi ,
which are an SU (3) singlet, SU (2) doublet
hypercharge Y = −1;
ν
e
I
, and have
Li
5. Right-handed “electron” leptons LIRi , which are an SU (3) singlet, SU (2) singlet, and
have hypercharge Y = −2. The name is given because only the electron, muon, and
tau are included; right-handed neutrinos are not.
Right-handed neutrinos are not included because, as we shall see, they would then obtain a
mass similar to their lepton counterpart (e.g. to the electron for the electron-neutrino) even
though from experiments we know their mass to be very small [5, p715]. The hypercharges
are merely given here, but will be explained further in the next section. Remember that
hypercharge must be conserved.
1.2.1
Hypercharges
To determine the form of φ(x), we must take into account the various symmetries we impose
on the Lagrangian, such as Lorentz invariance7 , the correct dimensions (in our case 4)8 , but
0
7
ν
µν
A Lorentz transformation is written as xµ → x µ = Λµ
is Lorentz invariant if
ν x ; a term T
−1 ρ
−1 σ µν
ρσ
(Λ )µ (Λ )ν T = T [5, p36].
8
A
R theory with a coupling constant with negative dimensions is not renormalizable [5, p80]. Since the action
S = d4 xL is dimensionless, and in terms of powers of mass dim x ≡ [x] = −1, by looking at the Lagrangian,
which must then have [L] = 4, we see [ψ] = 23 and [Aµ ] = [φ] = 1.
7
above all, invariance under the Standard Model symmetries SU (3)c ⊗ SU (2)L ⊗ U (1)Y . Take,
for example, a term from the Standard Model Lagrangian that is to end up giving us the
mass of the electron (from a Yukawa interaction; see section 1.2.4 for more on this): f ψ̄L φeR ,
with f a coupling. Since we are only looking at electroweak symmetry breaking, we only
need to take into account the electroweak symmetry SU (2)L ⊗ U (1)Y . SU (2)L invariance
tells us then that φ transforms as an isospin doublet under SU (2), such
that upon symmetry
0
breaking which is a result of φ taking the vacuum expectation value
we have [2, p363]:
v
0
e = f vēL eR .
(1.28)
f ψ̄L φeR → f (ν̄, ē)L
v R
In order for our electron eL to have charge Q = −1 (which it has by definition), we need
to have 21 Y = − 21 , since we know that the weak isospin gives the left-handed lepton SU (2)
doublet T3 = 21 for νL and T3 = − 21 for eL (this follows from the form of the third SU (2)L
generator, which contains the Pauli matrix σ3 ); this gives Q(eL ) = − 21 − 12 = −1. This
then holds for the entire doublet ψL , since the generators of U (1) are numbers, not matrices:
the hypercharge of ψL is then -1, from which it follows that the hypercharge of ψ̄L = 1.
The right-handed component, eR , does not listen to the weak isospin generators (since those
generators only couple to left-handed particles), so it has T3 (eR ) = 0; it then follows that
Q(eR ) = 0 + (−1) = −1 which is what we want; then we must have Y (eR ) = −2, as listed on
page 7.
In order for the entire term on the left-hand side of equation 1.28 to be invariant under
U (1)Y , and for φ to transform as a singlet under U (1)Y , φ must have a hypercharge Yφ = 1
(since then YēL + YeR + Yφ = 1 − 2 − 1 = 0 . It then follows that the first entry of the doublet
φ(x) has charge Q = T3 + 12 Y = 12 + 21 = 19 ; the second entry has Q = − 12 + 21 = 0. We can
then write φ as
+ φ
,
(1.29)
φ(x) =
φ0
with φ+ (+, because of charge +1)and φ0 (0, because of charge 0) complex scalar fields, so
that φ(x) has 4 degrees of freedom. Three of these become Nambu–Goldstone bosons and
combine with the gauge bosons to give them mass; the fourth degree of freedom will be the
Higgs boson itself. The field φ transforms as a singlet under SU (3)color [9, p6].
1.2.2
The Bosonic Masses
Let φ(x) now take a vacuum expectation value of v =
potential V
(φ† φ)
=
µ2 (φ† φ)
+
q
−µ2
|λ| ,
which is the minimum of the
|λ|(φ† φ)2 :
φ+
φ0
1
→√
2
0
v
≡ hφi0 .
(1.30)
Precisely here, where h0|φ|0i =
6 0,10 electroweak symmetry is spontaneously broken: SU (2)L ⊗
U (1)Y → U (1)EM . Note that v is a minimum only if µ2 < 0; if this were not the case, the
9
In some literature one may find Q = T3 + Y (see e.g. [5, p702]); this is a result of the way the couplings
are written in the Lagrangian. Writing g instead of g2 would result in this second form of the equation for the
charge quantum number.
10
Note that |0i denotes a vacuum state of an unperturbed theory. The ground state or vacuum of an
interacting theory is different from this free theory ground state, and will be denoted by |Ωi [5, p82]. For
8
minimum would be 0. Furthermore, it is necessary to take the absolute value of λ, because
if this coefficient were negative, the potential would not be bounded from below and there
would be no minimum. The generators of SU (2)L ⊗ U (1)Y do not leave this vacuum where
v is assumed invariant:
√ 0√
0 1
v/ 2
(1.31)
σ1 hφi0 =
=
1 0
v/ 2
0
√ 0√
0 −i
−iv/ 2
σ2 hφi0 =
(1.32)
=
i 0
v/ 2
0
0√
0√
1 0
σ3 hφi0 =
=
(1.33)
0 −1
v/ 2
−v/ 2
Y hφi0 = 1 hφi0 .
(1.34)
The generator corresponding to electric charge, however, does leave the vacuum invariant, so
that the photon associated with electric charge will not acquire a mass [4, p109]:
1
0√
1 0
Q hφi0 = (σ3 + Y ) hφi0 =
= 0.
(1.35)
0 0
v/ 2
2
As before, we are free to write φ in terms of fluctuations about its expectation value, as long
as the original 4 degrees of freedom are retained:
−
~→
0
i ξ·2σ 1
√
.
(1.36)
φ(x) = e
2 v + h(x)
Now h0|h(x)|i = 0. Choosing the U (1) gauge in such a way that the Lagrangian stays
invariant, and using that φ(x) transforms as φ → φ0 = U φ with U a unitary Hermitian
matrix, we have:
~σ
1
0
0
−i ξ·~
2
φ→φ =e
(1.37)
φ= √
2 v + h(x)
where h(x) is now the only degree of freedom remaining in this gauge [9, p7]. The gauge
fields transform as
0
(1.38)
Wbµ σb → Wb µ σb ≡ Wbµ σb ; B µ Y → B µ Y.
Writing out the Higgs Lagrangian,
g
g0
0
0
LHiggs = ((∂µ + i σb Wµb + i Y Bµ )φ0 )† · (h.c.) + V (φ † φ0 )
2
2
1
1
0
µ
2
b 2
0 v + h(x) (σb Wµ )
=
∂µ h∂ h + g
+
v + h(x)
8
8
1
1 02 2 2
1
g Y Bµ (v + h)2 + µ2 (v + h)2 + |λ|(v + h)4 +
8
2
4
1 0
0
b
µ
gg 0 v + h(x) σb Wµ Y B
+
v + h(x)
8
1 0
0
µa
g g 0 v + h(x) Y Bµ σa W
v + h(x)
8
+other interactions,
(1.39)
simplicity, it is not necessary to take into account interactions (and thus also renormalization - see section
1.5.1.
9
we can read off the masses of the bosons from the squared terms. First, we compute, using
equations 1.31-1.33, the φ† σa φ terms:
√ √ (v + h)/ 2
φ† σ 1 φ =
=0
(1.40)
0 (v + h)/ 2
0
√ √ −i(v + h)/ 2
†
φ σ2 φ =
=0
(1.41)
0 (v + h)/ 2
0
√ 1
0 √
†
φ σ3 φ =
= − (v + h)2 .
0 (v + h)/ 2
−(v + h)/ 2
2
(1.42)
Then we see that the (Wµ )2 -term is
1 2
g
8
=
=
=
0 v + h(x)
σa Wµa σb W µb
0
v + h(x)
=
1 2
0
c
a
µb
0 v + h(x) [δab I + iabc σ ] Wµ W
g
v + h(x)
8
h
i
2
2
2
1 2
g (v + h(x))2 Wµ1 + Wµ2 + Wµ3
+
8
a µb
0
3
0 v + h(x) iab3 σ Wµ W
v + h(x)
h
2 i
1 2
2
2
g (v + h(x))2 Wµ1 + Wµ2 + Wµ3
+
8
3 1 µ2
0
3
2
µ1
0 v + h(x) iσ Wµ W − iσ Wµ W
v + h(x)
h
i
1 2
2
2
2
g (v + h(x))2 Wµ1 + Wµ2 + Wµ3
,
8
(1.43)
where in the second line the identity σ a σ b = δab I + iabc σ c has been used. Now the masses
can be derived from the squared terms:
•
1 2 2
2µ h
•
1 02 2 2
µ
8 g Y v Bµ B
•
1 2 2
8g v
→ h has mass µ (through self-interaction);
h
Wµ1
2
→ Bµ has mass
+ Wµ2
2
+ Wµ3
g0 v
2
2 i
(since Yφ = 1);
→ Wµ has mass
gv
2 .
However, the particles that are actually observed are not the Wµa or Bµ particles. The
definitions of the measurable particles, which are the Wµ+ , Wµ− , Z 0 , and Aµ (photon) can
be found through rearranging the Lagrangian. The interaction term of Bµ and W µ is (using
again equations 1.31-1.33)
1 0
1
0
µa
2 g g 0 v + h(x) Y Bµ σa W
= − (v + h)2 g 0 gBµ W µ3 ,
(1.44)
v
+
h(x)
8
4
10
so that, combining equations 1.39, 1.43, and 1.44,
1
1 0
1
1
∂µ h∂ µ h + g 2 Y 2 Bµ2 (v + h)2 + µ2 (v + h)2 + |λ|(v + h)4
2
8
2
4
h
i
1 2
2
2
2
+ g (v + h)2 Wµ1 + Wµ2 + Wµ3
+
8
g0g
−(v + h)2
Bµ W µ3 + W µ3 Bµ + other interactions
4
1 2 2 v2 2
= − µ h +
g (W12 + W22 ) + (g 0 Bµ − gWµ3 )2 +
2
8
other interactions
i
1
v2 h 2
0
≡ − µ 2 h2 +
2g (Wµ+ Wµ− ) + (g 2 + g 2 )Zµ2 +
2
8
other interactions.
LHiggs =
(1.45)
From this we can define the observable bosons:
Wµ± =
Zµ0 =
Aµ =
Wµ1 ∓ iWµ2
√
;
2
−g 0 Bµ + gWµ3
p
;
g2 + g02
gBµ + g 0 Wµ3
p
.
g2 + g02
(1.46)
(1.47)
(1.48)
The zero in Zµ0 is to stress that it is a neutral boson, as opposed to the charged Wµ± bosons.
p
mW
Now we know the masses of the W and Z bosons: mW = gv
and
m
=
g 2 + g 0 2 v2 ≡ cos
Z
2
θW .
Here θW is the mixing angle; it is so called because it denotes the mixing of the W and B
bosons that are composites of the Z and A bosons. The mixing angle θW is defined as follows:
g0
g
p
cos θW = p
,
sin
θ
=
.
W
g2 + g02
g2 + g02
(1.49)
The relationships between the W and B and Z and A bosons are then
0 Wµ3
Z
cos θW − sin θW
=
;
Aµ
sin θW
cos θW
Bµ
or, using the inverse matrix,
Wµ3
cos θW
=
− sin θW
Bµ
1.2.3
sin θW
cos θW
Z0
Aµ
(1.50)
.
(1.51)
The couplings g and g 0
To see what the coupling constants g and g 0 mean, let us look at an interaction term between
the Aµ boson and a lepton in Lkinetic from equations 1.24 and 1.25, using the definition of
11
Aµ from equation 1.48, and using LL ≡
I
ν
e
and LR ≡ eR :
L
I
Llepton,int = LRi iγ µ (ig 0 Bµ Yijl,R )LIRj + LLi iγ µ (ig 0 Bµ Yijl,L + igσa Wµa )LILj +
L∂µ L {+other interactions}
0
I
µ 0
µ g
= −eR γ g Bµ (−2) eR − ν e
γ
Bµ (−1) +
2
iL
ν I
g
a
+ L∂µ L + . . .
σa Wµ
e jL
2
g0
Aµ cos θW eL +
2
!I
g
1 0
ν
3
Wµ
+
e
2 0 −1
= eR γ µ g 0 Aµ cos θW eR + eL γ µ
−
LAµ −electron
ν e
I
iL
jL
1
W , W 2}
L∂µ L {+interactions of ν, Z, and
+ ...
0
g
g
g
Aµ eR + eL γ µ p
Aµ e L +
= eR γ µ g 0 p
02
2
2
2
g +g
g + g02
g
eL γ µ Aµ sin θW eL
2
0
g
g
µ 0
µg
p
e
γ
= eR γ g p
A
e
+
Aµ e L +
µ
L
R
02
2
2
2
g +g
g + g02
=
=
=
=
=
=
=
g0
g
eL γ µ p
Aµ e L +
2 g2 + g02
gg 0
p
Aµ [eR γµ eR + eL γµ eL ]
g2 + g02
1
1
1
1
gg 0
p
Aµ (1 + γ5 )eγµ (1 + γ5 )e + (1 − γ5 )eγµ (1 − γ5 )e
2
2
2
2
g2 + g02
gg 0
1 †
1 †
†
†
p
Aµ e (1 + γ5 ) γ0 γµ (1 + γ5 )e + e (1 − γ5 ) γ0 γµ (1 − γ5 )e
4
4
g2 + g02
i
gg 0
1h †
2
†
2
p
A
e
γ
γ
(1
+
γ
)
e
+
e
γ
γ
(1
−
γ
)
e
µ
0
µ
5
0
µ
5
4
g2 + g02
gg 0
1
2
2
p
1
+
2γ
+
γ
+
1
−
2γ
+
γ
e
A
ēγ
µ
µ
5
5
5
5
4
g2 + g02
gg 0
1
p
Aµ ēγµ [4]e
02
2
4
g +g
0
gg
p
ēγµ eAµ .
2
g + g02
(1.52)
Identifying Aµ as the photon, we can now set the coupling √ gg
2
0
g +g 0 2
12
≡ e.
1.2.4
Fermion Masses
We now know how, according to the Standard Model, the Z and W bosons acquire their
mass. But what about the quarks and leptons? Some interaction of the added scalar field
φ(x) and the fermions is needed. For that, we should add an interaction term like ψ¯α φ to the
Lagrangian, e.g. QαL φ. This term is, however, not a Lorentz scalar (it carries a spinor index),
has the wrong dimensions ( 32 + 1 6= 4), and is not invariant under SU (3)C ⊗ SU (2)L ⊗ U (1)H ,
which we see by counting the hypercharges (see page 7) Y = −1
3 + 1 6= 0. We rather choose
the interaction of φ(x) with fermions given by the Yukawa coupling [10] which does obey the
necessary symmetries:
LY ukawa = Yij ψi φψj
(1.53)
= Yij ψLi φψRj +
Yij∗ ψRi φψLj
(1.54)
≡ Yij ψLi φψRj + h.c.
= Yijd QLi φdRj + Yiju QLi φ̃uRj + Yijl LLi φlRj + h.c..
(1.55)
Here, Y is the Yukawa coupling constant, and i,j are flavor indices that matter for QCD.
The meaning of φ̃ in the last line will become clear later; what matters is that this is the
form we look for in order to be able to read off the fermion masses. This Lagrangian should
be invariant under SU (2)Y ⊗ U (1)L as before; we can indeed check that the hypercharges of
−2
the first term indeed add up to Y = −1
3 + 1 + 3 = 0 and that the dimensions are correct:
[LY ukawa ] = [ψ̄] + [φ] + [ψ] = 23 + 1 + 23 = 4, which is what we expected.
In order for this Lagrangian to be Lorentz invariant, we choose φ(x) then again as in
equation 1.36 and demand it to transform as in equation 1.37. The fermion field transforms
as ψ → ψ 0 = e−i
follows:
~σ
ξ·~
2
ψ. Then the transformation of the term with the d-quark will be as
L0Y ukawa,quarks,dR
LY ukawa,d−quark
0
0
= Yijd QILi φ0 dIRj + h.c.
I
0
= Yijd u0 d0
φ0 dIRj + h.c.
iL
I
~σ
~σ
~σ
ξ·~
ξ·~
ξ·~
d
= Yij u d
ei 2 e−i 2 ei 2 ×
iL
~σ
ξ·~
1
0
√
e−i 2 dIRj + h.c.
2 v + h(x)
I 1 0
d
√
= Yij u d
dIRj + h.c.
Li
2 v + h(x)
I v
= Yijd dLi √ dIRj + h.c. + interaction terms
2
I
= Mijd dLi dIRj + h.c. + interaction terms.
(1.56)
Here Mijd is a mass matrix that we want in diagonal form. In order to obtain that, we will
introduce unitary matrices V d which obey
VLd† VLd = 1,
(VRd )ij dIRj
I
dLi (VLd† )ij
VLd M d VRd†
(1.57)
≡ dRi ,
(1.58)
≡ dLj ,
(1.59)
=
13
d
Mdiag
,
(1.60)
with dL and dR the down quark in its mass eigenstate. This gives
I
LY ukawa,d−quark = Mijd dLi Mijd dIRj + h.c. + interaction terms
= dLi VLd† VLd (Mijd )VRd† VRd dIRj + h.c. + interaction terms
= dLi (Mijd )diag dRj + h.c. + interaction terms.
(1.61)
Now the down quark has acquired a mass. What about the up quark? We cannot use the
same mechanism; in order to obtain a mass for the up quark we should redefine φ as
φ̃ = iσ2 φ∗
+∗ 0 −i
φ
= i
i 0
(φ0 )∗
+∗ 0 1
φ
=
−1 0
(φ0 )∗
(φ0 )∗
=
.
−(φ+ )∗
(1.62)
This should give a proper term in the Yukawa potential. Indeed, the mass dimension of
4
Yiju QLi φ̃uRj is again as before; the hypercharges add up to Y = −1
3 − 1 + 3 = 0. Also, it must
be a Lorentz scalar, and transform in the same way as φ did:
i
∗
~
φ̃ = iσ2 φ∗ → iσ2 e 2 ξ·~σ φ
i~ ∗
≈ iσ2 1 − ξ · ~σ φ∗
2
i
∗
= iσ2 1 − ξj σj σ2 σ2 φ∗
2
i
∗
= σ2 1 − ξj σj σ2 φ̃
2
i
=
1 + ξj σj φ̃
2
≈
i
~
e 2 ξ·~σ φ̃,
(1.63)
where in the second line the property (σi )2 = I and in the third line the relation σ2 σj∗ σ2 = σj
([11, p279]) have been used. Now φ̃ can be redefined as
i
v
+
h
ξ·σ
φ̃ = e 2
(1.64)
0
where the vacuum expectation value of φ̃ has been used:
D E
1
1
(v + h)∗
v+h
∗
φ̃ = hiσ2 φ i0 = √
=√
0
0
0
2
2
14
(1.65)
so that the up quark term of the Yukawa potential we earlier forgot about becomes:
LY ukawa,quarks,u = Yiju QILi φ̃uIRj + h.c.
I 1 v + h d
√
= Yij u d
uRj + h.c.
0
Li
2
= uILi Miju uIRj + h.c. + interaction terms
= uILi VLu† VLu (Miju )VRu† VRu uIRj + h.c. + interaction terms
= uLi (Miju )diag uRj + h.c. + interaction terms;
(1.66)
and the up quark has acquired a mass.
In the Standard Model, as we have seen, particles gain a mass through an added Higgs
potential. In some alternative models, however, particles gain their mass dynamically, or
through interactions. Two examples of a model in which electroweak symmetry is broken
dynamically are chiral symmetry breaking and superconductors [12, p13].
1.3
Superconductors and spontaneous symmetry breaking
The abelian Higgs Model described above in section 1.1 is actually also known as the LandauGinzburg superconductor [3, p8]. This is a phenomenological model of superconductivity [9,
p12]. In superconductors, the electromagnetic symmetry U (1)EM is spontaneously broken by
a Cooper pair condensate, and subsequently photons acquire a mass. Another way to explain
superconductivity than with the Landau-Ginzburg model is with Bardeen-Cooper-Schrieffer
theory, in which photons acquire their mass dynamically. Dynamically means the mass of the
photons is not read from the photon propagator, but rather taken from the self-energies of a
two-point interaction which then gives photons a mass.
Superconducting was thought to be related to Bose-Einstein condensation, even though
the electrons conducting are fermions [2, p270]. This was correct: electrons condensate into
Cooper pairs behaving as bosons. When an external magnetic field is turned on around a potential conductor, it is absent from the conducting material when it becomes superconducting
below a critical temperature (see figure 1.4 ); this is known as the Meissner effect.
Below a critical temperature Tc determined by the superconducting material, the free
energy is minimized by a non-zero value taken by the condensate of electrons, which now acts
as a boson; this causes spontaneous symmetry breaking from which the gauge field of the
Cooper pairs gains a mass [2, p271]. The state at which the material starts superconducting
is comparable to our ground state or vacuum that is assumed at the electroweak scale and
causes spontaneous symmetry breaking there. In this process, one or more particles, in the
superconductor case the photon, acquire a mass, just as the W and Z bosons acquired masses
in the Higgs mechanism (while the photon remained massless) [3, p7]. In the Lagrangian this
is visible by adding a massless spinless scalar φ just like before, which couples to the photon
[3, p8]:
1
1
1
L = − Fµν F µν + (∂µ φ)2 + e2 f 2 (Aµ )2 − ef Aµ ∂ µ φ
4
2
2
1
1
1 µ 2
= − Fµν F µν + e2 f 2 (Aµ −
∂ φ)
4
2
ef
1
1
≡ − Fµν F µν + m2γ (Bµ )2
4
2
15
(1.67)
Figure 1.4: The Meissner effect. Below a critical temperature Tc the conducting material becomes
superconducting and the magnetic field is excluded from the superconductor. Figure by Piotr Jaworski;
via Wikimedia commons.
with f a coupling constant, mγ the mass of the photon and Bµ the massive photon field. If
2
ns is the density of electrons, the so-called Meissner-mass of the photon is then m2γ = q4mnes ,
with q and me the charge and mass of the electron, respectively [12, p14]. This result is
2 = g 2 v 2 . The Meissner mass results
comparable to the relativistic mass of the W -boson: MW
4
from the wave function whose modulus squared is |ψ|2 = n2s ≡ nC , the latter the density
of
pairs; this is comparable to the vacuum expectation value of the Higgs field H:
Cooper
2
2
|H| = v [12, p13].
The spontaneous symmetry breaking in superconductors as described by Bardeen, Cooper,
and Schrieffer11 was later taken to particle physics [2, p272]. In a BCS superconductor, the
electron has a so-called “mass-gap” of the amount of a Majorana-mass: its mass is slightly
increased by this amount with spontaneous symmetry breaking. Something similar happens
in chiral symmetry breaking in QCD, where the up, down, and strange quarks (which are
very light) acquire larger “constituent quark masses” [3, p7].
1.4
Chiral Symmetry
Without the Higgs sector in the Standard Model, the quark bilinear ūL uR + d¯L dR 6= 0
spontaneously breaks the electroweak symmetry and gives rise to a W -boson mass of MW =
gfπ
2 ∼ 29 MeV, with fπ ' 93 MeV the pion decay constant [13, p12]; this is much less than
the real W -boson mass, but nevertheless quarks can already generate mass on their own.
This will be explored further in chapter 3. Because quarks have nonzero masses, the chiral
symmetry is not exact but approximate, with an approximately massless Goldstone boson
[14, p184], the pion.
In the approximation where the masses of the quarks are taken as very light and are
ignored, the kinetic Lagrangian of the quarks enjoys a so-called chiral symmetry. The La11
For more on the theory of Bardeen, Cooper, and Schrieffer, or BCS theory, see appendix B.
16
grangian then looks like:
¯ Dd
/ + di
/ ≡ Q̄iDQ,
/
Lquarks = ūiDu
(1.68)
u
with Q =
and D the ordinary Standard Model covariant derivative. This kinetic part is
d
symmetric under SU (2)L⊗
SU (2)R ⊗ U (1)L ⊗ U
(1)
R [5, p668]. When writing the Lagrangian
1+γ 5 u
1−γ 5 u
and QR = 2
, currents associated with this symmetry
in terms of QL = 2
d
d
group can be found using Noether’s theorem. The current associated with a symmetry group
G with fields ϕ transforming under the representation R of G is given by [2, p73]:
JµA =
δL
δL A A
δϕa =
θ Tab ϕb ,
δ∂µ ϕa
δ∂µ ϕa
(1.69)
where the T A are the generators of the groupP
G in the representation R. In general, for a
θ·T
transformation matrix R = e
with θ · T = A θA T A (a real, antisymmetric matrix), the
A ϕ [2, p72].
field ϕ transforms as ϕa → Rabϕb ' (1 + θA T A )ab ϕb , so that δϕa = θA Tab
b
In this case, there are four symmetry groups with which we can associate four conserved
currents. The generators of U (1) are 1, and the generators of SU (2) are τa = σa /2, with σ the
Pauli matrices [5, p668]. The infinitesimal transformations are thus δϕa = θϕa and θA τ A ϕa
with ϕa ∈ QR , QL . The four currents associated with the four symmetries are thus [5, p668]:
jLµ = Q̄L γ µ QL ,
jLµa = Q̄L γ µ τ a QL ,
jLµ = Q̄R γ µ QR ,
jLµa = Q̄L γ µ τ a QL ,
(1.70)
where the parameters θA have been set to 1. From these currents we can find the baryon
current and isospin current by adding the left- and right-handed parts, and the axial vector
currents by subtracting those parts. They are:
µ
= j µ = Q̄γ µ Q (baryon number current),
jLµ + jR
µa
= j µa = Q̄γ µ τ a Q (isospin current),
jLµa + jR
µ
= j µ5 = Q̄γ µ γ 5 Q (axial vector current),
jLµ − jR
µa
= j µa5 = Q̄γ µ γ 5 τ a Q (axial vector current).
jLµa − jR
(1.71)
The isospin current j µa is also called the vector current, as opposed to the axial vector current
j µa5 .
The chiral symmetry can be expressed and found in a different way as well, namely starting
not with a Lagrangian in terms of left- and right-handed fields, so that among the generators
of the symmetry groups there is one involving γ 5 . This is discussed in a review on technicolor
by Hill and Simmons [3] and by Weinberg in his book on quantum field theory [14]. It can
most easily be seen in rewriting the generators used above. Let us give the generators above of
(S)U (1/2)L/R the name TL/R,A ; then the symmetry group SU (2)L ⊗ SU (2)R ⊗ U (1)L ⊗ U (1)R
has a subgroup SU (2) ⊗ U (1), and the generators can also be written as two other generators
TA = TL,A + TR,A and XA = TL,A − TR,A [14, p.183].
/ = ψ̄L i∂ψ
/ L + ψ̄R i∂ψ
/ R , is symmetric under
The kinetic Lagrangian of fermions, L = ψ̄i∂ψ
the chiral symmetry U (1)L ⊗ U (1)R ([3, p9]):
ψL → e−iθ ψL ;
ψR → e−iω ψR ;
17
(1.72)
where conserved fermion number corresponds to θ = ω and an axial or γ 5 symmetry corresponds to θ = −ω, with resulting (Noether) vector (jµ ) and axial vector (jµ5 ) currents,
corresponding to the two generators TA and XA ,
jµ =
=
jµ5 =
=
=
δL δψ
δ
δψ
/
=
ψ̄i∂ψ
δ∂µ ψ δθ
δ∂µ ψ
δθ
δ
ψ̄iγµ (−iθψ) = ψ̄γµ ψ;
δθ
δ
δL δψ
δ
/
=
ψ̄i∂ψ
[δψL + δψR ]
δ∂µ ψ δω
δ∂µ ψ
δω
δ
ψ̄iγµ [(iωψL ) + (−iωψR )] = ψ̄γµ [ψR − ψL ]
δω
1
ψ̄γµ [1 + γ5 − (1 − γ5 )]ψ = ψ̄γµ γ5 ψ.
2
(1.73)
(1.74)
Adding a mass term to the Lagrangian would break the chiral symmetry to a symmetry
corresponding to fermion number conservation U (1)L+R [3, p9], since mψ̄ψ = m(ψ̄L ψR +
ψ̄R ψL ) giving ∂µ jµ5 = ∂µ ψ̄γ µ γ5 ψ = −2imψ̄γ5 ψ 6= 0. The original chiral symmetry can be
preserved through adding a potential Φ according to the abelian Higgs mechanism, with which
the fermion will be given a mass. This potential transforms under the U (1)L ⊗ U (1)R chiral
symmetry as
Φ → e−i(θ−ω) Φ
(1.75)
and the Lagrangian changes to
1
/ + |∂Φ|2 + M 2 |Φ|2 − λ|Φ|4 − g(ψ̄L ψR Φ + ψ̄R ψL Φ∗ ).
L = ψ̄i∂ψ
2
(1.76)
In this case, the vector current stays the same (because δΦ = 0 for θ = ω), but the axial
vector current (for which θ = −ω) changes:
δL δψ
δL δΦ
δL δΦ∗
+
+
δ∂µ ψ δω δ∂µ Φ δω
δ∂µ Φ∗ δω
δ
δ
= ψ̄γµ γ5 ψ + ∂µ Φ∗ (−i(−2ω)Φ) + ∂µ Φ (+i(−2ω)Φ∗ )
δω
δω
←
− −
→
= ψ̄γµ γ5 ψ + 2iΦ∗ (∂µ − ∂µ )Φ.
jµ5 =
(1.77)
Chiral symmetry is spontaneously broken when, as before, Φ assumes a vacuum expectation
M
value of hΦi0 = √v2 = √
, or the minimum of the potential VΦ . If we now parametrize small
λ
oscillations around this vacuum as Φ =
our potential becomes [3, p10]
√1 (v +h(x))eiφ(x)/f
2
so that the Lagrangian describing
1
LΦ = |∂Φ|2 − V (|Φ|) = |∂Φ|2 + M 2 |Φ|2 − λ|Φ|4
2
r
λ
1
1
v2
1
=
(∂h)2 − M 2 h2 −
M h3 − λh4 + 2 (∂φ)2 + 2 h2 (∂φ)2 +
2
2
8
2f
2f
√
2M
h(∂φ)2 + Λ,
λf 2
18
(1.78)
4
with Λ = −M
2λ a negative vacuum energy density or cosmological constant (of which we
can always add one, since we can start with any vacuum energy). The field φ is a massless
√
v2
2
Nambu–Goldstone boson; the field h has a mass 2M . Renormalizing the term 2f
2 (∂φ)
requires that f = v [3, p11] with f a decay constant; apart from conventional factors, the
decay constant f is always equivalent to the vacuum expectation value [3, p11]. By taking
e.g. M → ∞ and λ → ∞, fluctuations can be suppressed. We can then set Φ = √f2 eiφ/f .
Then the axial current becomes
←
− −
→
jµ5 = ψ̄γµ γ5 ψ + 2iΦ∗ (∂µ − ∂µ )Φ
f2
i
i
= ψ̄γµ γ5 ψ + 2i (− ∂µ φ − ∂µ φ)
2
f
f
= ψ̄γµ γ5 ψ + 2f ∂µ φ.
(1.79)
The mass m of the fermion as well as the coupling
of the massless pseudoscalar Nambu–
√
Goldstone boson to iψ̄γ5 ψ with strength g = 2m/f (known as the “Goldberger-Treiman
relation”) are now visible in the Lagrangian in equation (1.76):
1
/ + (∂φ)2 −
Lfermions = ψ̄i∂ψ
2
1
/ + (∂φ)2 −
= ψ̄i∂ψ
2
gf
√ (ψ¯L ψR eiφ/f + ψ¯R ψL e−iφ/f )
2
gf
g
√ ψ̄ψ − i √ φψ̄γ5 ψ + h.o.t.
2
2
(1.80)
The Goldberger-Treiman relation holds in QCD with g equal to gA , the axial coupling
constant of a pion to a nucleon; m the mass of a nucleon; and f the pion decay constant
fπ , thus showing a pion may be a Nambu–Goldstone boson [3, p11]. Chiral symmetry is
thus a symmetry of the strong interaction that is spontaneously broken, as Nambu and JonaLasinio had suggested in 1960 [5, p668]. The essence of the Goldberger-Treiman argument is
separating the infinite number of diagrams (as there are always infinitely many possibilities
with given externals) into those with a pole in the complex plane and those without a pole
but with a cut [2, p207]. The Goldberger-Treiman relation is satisfied experimentally to 5%
accuracy [5, p672]. More on the Goldberger-Treiman relation can be found in appendix C.
A procedure of chiral symmetry breaking similar to the one for abelian groups holds in
the nonabelian chiral SU (2)L ⊗ SU (2)R which then breaks down to a diagonal subgroup
SU (2)R [14, p182]. Now there is no Higgs potential added and the symmetry is broken
dynamically through quarks forming a condensate [15, p3]. For the Lagrangian in quantum
chromodynamics with the covariant derivative as in equation 1.25,
LQCD = iψ(∂ µ γµ )ψ + . . . ,
(1.81)
the field ψ transforms under chiral SU (2) ⊗ SU (2) as
~V · ~σ +iγ5 iθ~A · ~σ
ψ → ψ 0 = eiθ
2
2
.
(1.82)
Here σ are the Pauli matrices. Using Noether’s theorem, this leads to the vector and axial
19
vector currents [14, p183]
δL δψ
δ
δψ
/
=
ψ̄i∂ψ
V
δ∂µ ψ δθ
δ∂µ ψ
δθV
δ
~σ
~σ
= ψ̄iγµ V (iθ~V · ψ) = iψ̄γµ ψ;
δθ
2
2
δL δψ
δ
δψ
/
=
=
ψ̄i∂ψ
δ∂µ ψ δθA
δ∂µ ψ
δθA
δ
~σ
~σ
= ψ̄iγµ A (iγ5 θ~A · ψ) = iψ̄γµ γ5 ψ.
δθ
2
2
Vµ =
Aµ
(1.83)
(1.84)
(1.85)
The SU (2) ⊗ SU (2) chiral symmetry is taken to be spontaneously broken. It is easier to show
that it is not broken; however, if it were exact and unbroken this would require any onehadron state to be degenerate with another state of opposite parity (and equal spin, baryon
number, and strangeness), and such parity doubling has not been observed [14, p184]. The
discovery of spontaneously broken symmetry in the strong interaction ultimately led to the
concepts of quarks and gluons [2, p207].
1.5
Problems with the Standard Model
Notice that the mass of the Higgs, or more precisely, the Higgs mass parameter, at the
vacuum expectation value of v is equal to µ2 = λv 2 and thus increases with λ [13, p6]; this
may pose a problem. The Standard Model is then actually only valid up to a cut-off scale
Λ; above this parameters such as λ grow beyond control if they are not largely fine-tuned12 .
This is precisely because the Higgs mass operator has no symmetry protecting it from large
corrections (bosons do, as they do not have mass above the electroweak, or Fermi, scale), so
that the electroweak scale is violated. This problem is called the hierarchy problem [13, p7].
To understand this better, something needs to be said about renormalization.
1.5.1
Mass Renormalization
When renormalizing a theory that is valid up to a cutoff momentum of Λ, the pole of the
propagator and thus the mass is shifted. This can be seen from the diagram 1.5(a) which
R Λ d4 q
i
, where the cutoff Λ implies that all four momentum components
is ∼ −iλ
(2π)4 q 2 −m2 +iε
integrated over up to Λ, and from diagram 1.5(b), which is something like:
2
(−iλ)
Z
ΛZ Λ
d4 p d4 q
i
i
i
;
4
4
2
2
2
2
(2π) (2π) p − m + iε q − m + iε (p + q + k)2 − m2 + iε
(1.86)
which depends on k 2 and quadratically on the cutoff Λ (this latter fact can be seen by counting
powers of p and q in the integrand). It can thus be written as a series expanded around k 2 = 0:
to obtain the coefficients of a term k 2a in this series, we can differentiate the above integral
12
Fine-tuning means that a parameter in a model needs to be adjusted in orders of magnitude. A parameter
that is largely fine-tuned is many orders of magnitude greater or smaller than the model in which it appears.
This is the opposite of naturalness, where the parameters in a theory are all of the order 1 [2, p404]. See also
section 1.5.2.
20
p
q
k
k
q
k
(a) First mass correction
k
(b) Another mass correction
Figure 1.5: Mass corrections
2a times and subsequently set k 2 = 0. This decreases the powers of p and q in the integrands;
then the first coefficient is a constant depending quadratically on the cutoff, and the second
coefficient of the series, that of k 2 , is only logarithmically dependent on the cutoff Λ. Then
any coefficients of terms of order k 4 and higher are cutoff independent as the cutoff goes to
infinity ([2, p159]), giving a new inverse propagator k 2 −m2 +a+bk 2 with a quadratically and
b logarithmically cutoff dependent. The two diagrams in figure 1.5 represent the quantum
fluctuations of the “bare” propagator. This propagator is now thus renormalized to:
k2
1
1
→
.
2
2
−m
(1 + b)k − (m2 − a)
(1.87)
Hence the mass is renormalized to m2P ≡ (m2 − a)(1 + b)−1 , which is called the physical mass
and is what we actually observe [2, p159]. Then m2 is the so-called bare mass of the particle
for which this propagator holds; the physical mass is thus the bare mass shifted by quantum
fluctuations.
This shift in mass is different for bosons than for fermions. For bosons, quantum fluctuations give a shift of δµ ∝ Λ2 /µ, while for fermions this shift is δm ∝ log(Λ/m) (where
mP ≡ m + δm) [2, p165]. Weisskopf explained this in terms of quantum statistics [2, p166]:
fermions push away virtual fermions fluctuating in the vacuum creating a cavity in the vacuum charge distribution around it, so that its self-energy or mass correction is less singular
(and thus its mass less shifted, as mass is shifted by singularities in the self-energy) than
without quantum statistics. A boson does the opposite, so that its mass correction diverges
more than that of the fermion. This is known as the Weisskopf phenomenon.
1.5.2
Naturalness and the Hierarchy Problem
In theoretical physics one expects that dimensionless ratios of parameters are of order 10−3 to
103 , or of “order unity” [2, p404]. ’t Hooft formulated this expected “naturalness” as follows:
a small parameter would be natural if a symmetry emerges when this parameter goes to zero.
In this respect, small fermion masses are natural as the chiral symmetry emerges when their
mass approaches zero. No symmetry emerges when we set the mass of a scalar field such as
the Higgs field to zero; this is called the hierarchy problem. So how exactly does the Higgs
obtain such large corrections?
Because the Standard Model has many different representations (see also as listed on page
7), it can be expected that these are just the result of one grand unified theory (GUT) that
breaks down to SU (2) ⊗ SU (3) ⊗ U (1) at some energy or mass scale MGUT . By looking at the
21
renormalization group of the coupling constants of the three groups of the Standard Model,
this scale can be calculated and is estimated to be 1014−15 GeV. This yields a very large ratio
of MGUT /MEW , with MEW the electroweak unification scale ∼ 102 GeV. The physical mass
of the Higgs particle would be of order MEW . This is because the electroweak symmetry is
broken by the VEV (vacuum expectation value) of the Higgs, which is defined as λv = µ2 ,
with the latter the squared Higgs mass operator; notice that we assume naturalness of the
parameter λ here. This VEV is determined experimentally by the known W -boson mass, as
the VEV gives rise to this mass in the Standard Model [13, p7]. Muon decay measurements
enable us to determine GF , the so-called Fermi coupling, which is related to the VEV v
through [16, p6]
√
v = ( 2GF )−1/2 ≈ 246 GeV.
(1.88)
Then Weisskopf (see p. 21) shows that the mass of a boson µ20 is shifted by quantum correc2
tions of δµ20 ∼ f 2 Λ2 ∼ f 2 MGUT
with f a dimensionless coupling constant. Λ is substituted
2
by the natural mass scale MGUT
. We know, however, that the mass of the Higgs particle
should be of order MEW ; this gives the problem of how we obtain µ2 = µ20 + δµ20 ∼ MEW
[2, p403]. It is possible to make a fine-tuned and unnatural cancellation, but how it would
happen naturally remains a question. All other corrections in the Standard Model are of
order log Λ [13, p4]; only the Higgs mass operator has corrections of the order of Λ2 ! Thus
the Standard Model does not explain how the Higgs mass operator is of the order of the
electroweak scale, or of order MEW .
1.5.3
Other Problems
There are problems other than the hierarchy and naturalness problems that emerge in the
Standard Model. One is the so-called strong CP problem: the electric dipole moment of
a neutron is very small, which is something the standard model cannot explain [13, p8].
Furthermore, the Standard Model can fit but not explain the number of matter generations
and their mass spectra; this is also called the flavor problem. There is also no equal unification
of forces in the Standard Model, as pictured in figure 1.6. Next, the mass of the neutrino
is also not explained by the Standard Model, but may be natural with a so-called seesaw
extension13 . Finally, dark matter and the matter-antimatter symmetry of the universe go
unexplained in the Standard Model.
1.6
LHC
These are exciting times at the LHC and for particle physics phenomenology. This year’s
measurements may lead to significant insights. Here a short word on why all the excitement
about the LHC.
13
A seesaw extension adds a right-handed neutrino to the Standard Model and gives it a mass M without
breaking the Standard Model symmetry groups. The mass of this right-handed neutrino is larger than the
scale at which Standard Model symmetries break, which explains
why
we have not seen it. Then a mass matrix
0 m
appears with a small mass m << M so that it looks like
. This matrix then has a large eigenvalue
m M
2
M and a small eigenvalue m
, where the latter is the tiny mass of the left-handed neutrino, suppressed by the
M
factor m/M . The eigenvalues of this matrix affect each other in such a way that if one goes up, the other goes
down – hence the name seesaw [2, p410].
22
Figure 1.6: The Standard Model forces do not unify at a single point. This can be solved by a theory
such as technicolor. Figure taken from http://www.interactions.org/imagebank/images/OT0082M.jpg
The significance of a measurement is defined as
√
Nsignal
σsignal · luminosity
=p
significance = p
∼ L.
Nbackground
σbackground · luminosity
(1.89)
Here L denotes the luminosity. This is only correct for large number of events, which is
the case for the LHC. This means
that the signal of measurements at the end of 2012 are
q
√
L2012
S2012 = Snow × L = 3 × Lnow if the energy of the center of mass does not change,
which itqwill however
q from 7 to 8 TeV. Then the signal will increase more than by S2012 ≥
Snow ×
L2012
Lnow
≥3
20
5
∼ 6σ. Only 5 σ suffices by convention to call something a discovery.
23
Chapter 2
Custodial Symmetry
Any symmetry that protects a mass scale, meaning they take care that radiative corrections
do not induce a nonzero value of the mass parameter, is called a custodial symmetry [3, p15].
Thus, if a parameter is tuned to a small value, e.g. the electron mass me in QED is set to
zero, the chiral U (1)L ⊗ U (1)R symmetry forbids a mass to be generated by perturbative
radiative corrections (a perturbative power series multiplies a nonzero bare electron mass1 ).
Many scalar particles, such as the Higgs boson, have no such custodial symmetry [3, p15].
There are three cases, however, where this custodial symmetry holds for scalars [3, p15]:
1. Nambu–Goldstone bosons with low masses due to their spontaneously broken chiral
symmetry;
2. Composite scalars that form only at a strong scale like ΛQCD : the mass obtains only
additive renormalizations of the order of that scale;
3. Scalars associated with fermionic particles, as in supersymmetry.
It is the second of these that is used in technicolor. So-called “Little Higgs Models” use the
first, where a Higgs boson is a low-mass pseudo-Nambu-Goldstone boson, similar to the pion
in QCD.
2.1
The ρ parameter and precision measurements
In the Standard Model, the parameter commonly known as ρ, which is
2
MW
2 cos2 θ
MZ
W
≡ ρ, is
equal to 1 at tree level [17, p481]. This is a consequence of the form of the Higgs potential.
Quite accurate measurements of the W and Z boson masses and weak mixing angle result in
a value of ρ close to ρ = 1. This result of custodial symmetry, and thus custodial symmetry
itself, must thus hold in any candidate replacement for the Standard Model. In some models
other than the Standard Model, such as Little Higgs Models in which the Higgs boson acts as
a massless Nambu-Goldstone boson, custodial symmetry does no longer hold, which in this
case means ρ 6= 1 at tree level diagrams [18, p309]. Custodial symmetry can be seen as a test
1
The bare mass is the mass that appears in the Lagrangian. This is also the mass of a particle when doing
calculations at tree level, where the pole is exactly at the bare mass. However, when loop corrections are taken
into account, this pole shifts, and the mass is renormalized; in this way, the calculations of the mass approach
the physical mass that is actually measured.
24
of any new theory. The ρ-parameter is restricted by experiments; if your parameter agrees
with experiment, you can be sure not to violate electroweak precision measurements.
2.2
Custodial symmetry from the Higgs potential
The Higgs potential has, next to the symmetries of the Standard Model, a global SO(4) =
SU (2)L ⊗ SU (2)R symmetry [11, p279]. This can easily be seen by rewriting the Higgs
potential. Using the Higgs doublet:
+ φ
φ1 + iφ2
H≡
=
,
(2.1)
φ0
φ3 + iφ4
the term φ† φ can be written as:
φ† φ = φ21 + φ22 + φ23 + φ24 ,
(2.2)
from which the global SO(4) symmetry is now manifest. If we reconstruct a potential while
imposing this symmetry, we could represent the potential by a matrix Φ that consists of φ
and φ∗ :
φ0∗ φ+
∗
(2.3)
Φ = iτ2 H , H =
−φ+∗ φ0
As we have seen in equation 1.63, if H transforms as a doublet, then iτ2 H ∗ also transforms
as a doublet [11, p279]. The Lagrangian can now be written as:
1
(2.4)
LHiggs = Tr[(Dµ Φ)† Dµ Φ] + µ2 Tr[Φ† Φ] − λTr[Φ† ΦΦ† Φ].
2
Notice that this Lagrangian is similar to the one that breaks chiral symmetry in equation
1.76. A mere assumption here is again that the parameter µ2 is less than zero so that it
spontaneously breaks symmetry. The other couplings are dimensionless, so they cannot take
care of electroweak symmetry breaking [13, p5]. The Lagrangian above is invariant under the
transformation:
(2.5)
Φ → UL ΦUR† .
Here UL ∈ SU (2)L and UR ∈ SU (2)R . This invariance can be seen in for example the
potential µ2 Tr[Φ† Φ] − λTr[Φ† ΦΦ† Φ]:
V [UL ΦUR† ] = µ2 Tr[UR Φ† UL† UL ΦUR† ] − λTr[UR Φ† UL† UL ΦUR† UR Φ† UL† UL ΦUR† ]
= µ2 Tr[UR† UR Φ† Φ] − λTr[UR† UR Φ† UR† UR Φ† Φ]
= µ2 Tr[Φ† Φ] − λTr[Φ† ΦΦ† Φ],
where in the second line the properties U † U = I of unitary matrices and Tr[ABC] =
Tr[CAB] = Tr[BCA] of the trace have been used. The covariant derivative Dµ in the Lagrangian is:
1
1
Dµ Φ = ∂µ Φ + igWµa τa Φ − ig 0 Bµ Φτ3 .
(2.6)
2
2
with a ∈ {1, 2, 3}. For this covariant derivative, the SU (2)L symmetry is gauged through the
weak gauge bosons Wµa and the hypercharge generator is set as the third generator of SU (2)R
[13, p5]. It can be shown that
Φ
1 − τ3
= (0, H),
2
and
25
Φ
1 + τ3
= (iτ2 H ∗ , 0).
2
(2.7)
This is more easily seen if we write Φ as a linear combination of matrices. Using a slightly
different Higgs field H [13, p5],
π2 + iπ1
H=
,
(2.8)
σ − iπ3
then we have now for M :
1
M = √ (σ1 + i~τ · ~π ) = (iτ2 H ∗ , H).
2
(2.9)
Now if we set g and g 0 to zero, the Lagrangian is obviously invariant under the transformation
in 2.5; this can be derived from the calculations preceding equation 2.6. Now if ∂Φ →
∂UL ΦUR† , we would also want this to hold for the other two terms of the covariant derivative;
in other words, in order for the Lagrangian to be symmetric under SU (2)L ⊗ SU (2)R , we
want Dµ Φ → UL Dµ ΦUR† . For the second term, we then have (omitting the constants)
Wµa τa Φ → UL Wµa τa ΦUR†
0
= Wµa τa UL ΦUR† ,
(2.10)
(2.11)
from which the transformation of W can be derived:
0
0
Wµa τa = UL Wµa τa UL† .
(2.12)
Thus, W transforms as a triplet under SU (2)L (because of the three matrices τa ) and as a
singlet under SU (2)R . This does not work for the last term in the Lagrangian, which breaks
the SU (2)R symmetry:
Bµ Φτ3 → UL Bµ Φτ3 UR†
=
Bµ0 UL ΦUR† τ3
=
(e.g.) UL Bµ UL† UL ΦUR† τ3 .
(2.13)
We see that, since τ3 and UR† do not commute, this term will not preserve the full symmetry.
To further explore the SU (2)L ⊗ SU (2)R symmetry, g 0 will be set to zero. When Φ then takes
its vacuum expectation value
v
1 0
hΦi0 = √
,
(2.14)
2 0 1
– or, in the notation of equation 2.7, hΦi0 = hσi – it is not invariant under SU (2) (see
equations 1.31-1.33); the symmetry is now broken down to a single SU (2) group:
hΦi → U hΦi U † .
(2.15)
In other words, SU (2)L ⊗ SU (2)R → SU (2)V , with the latter a diagonal subgroup [13, p6],
which is equivalent to our earlier SU (2)L ⊗ U (1)Y → U (1)EM because of our explicit choice
of the hypercharge generator in this case. This means that the W field, as calculated before
for SU (2)L ⊗ SU (2)R , will transform under SU (2), or equivalently under O(3) [11, p280], as
26
(W a τa )µ → U † (W a τa )µ U . Then the mass of the W -field can be read from the squared W
term in the original Lagrangian; this term now becomes
"
† #
1
1
1
Tr
igWµa τa hΦi
igWµa τa hΦi
=
2
2
2
11 2
v2
1 0
=
g Tr (Wµa )2 (τa )2
0 1
24
2
g2v2
(Wµa )2 .
8
=
(2.16)
We can now see that MW = gv
2 . Rewriting Wµ in the Lagrangian and noting that, while we
0
0
have taken g = 0, we must have sin θW = √ 2g 0 2 = 0 and cos θW = √ 2g 0 2 = 1, we can
g +g
rewrite W in terms of
W µ±
=
Wµ1 ∓iWµ2
√
2
and
Wµ3
g +g
= cos θW Zµ + sin θW Aµ :
(Wµ )2 = 2W µ+ Wµ− + (Wµ3 )2
= 2W µ+ Wµ− + (cos θW Zµ + sin θW Aµ )2
= 2W µ+ Wµ− + Zµ2 .
(2.17)
Then it is easy to see that MZ = gv
2 = MW so that ρ = 1.
0
If now g is not set to zero, the difference in mass between the W and Z bosons will come
only from the third term of the Lagrangian that breaks the symmetry [11, p280]. The terms
in the Lagrangian contributing to the boson masses will be (using I for the 2 × 2 identity
matrix):
1
1
1
1
1
Tr[( igWµa τa hΦi − g 0 Bµ hΦi τ3 )† ( igWµa τa hΦi − ig 0 Bµ hΦi τ3 )] =
2
2
2
2
2
i
1 v2 1 h
†
=
Tr gWµa τa I − g 0 Bµ Iτ3
gWµa τa I − g 0 Bµ Iτ3
2 2 4
v2 2
1
02
2
0
a
0
a
21
2
=
g (Wµ ) + Tr[−gg (Wµ τa IBµ τ3 ) − g gBµ τ3 Wµ τa )] + g (Bµ ) Tr[(τ3 ) ]
8
2
2
i
h
2
v
0
=
g 2 (Wµ )2 − gg 0 Wµa Bµ Tr[τa τ3 ] + g 2 (Bµ )2
8 "
"
#
#
X
v2 2
0
=
g (Wµ )2 − gg 0 Wµa Bµ Tr δa3 I + i
a3c τc + g 2 (Bµ )2
8
c
h
i
2
v
0
=
g 2 (Wµ )2 − 2gg 0 Wµ3 Bµ + g 2 (Bµ )2
8 v2g2
g0
+
−µ
3
2
=
2Wµ W + (Wµ − Bµ ) ,
(2.18)
8
g
where in the sixth line Tr[τi ] = 0 was used. It can be seen that the W3 field is replaced with
0
Wµ3 − gg Bµ , which is similar to what was found in section 1.2.2. Therefore, the relationship
between the masses of the W and Z bosons is preserved, and ρ = 1 as desired. This means ρ is
constrained to this value before including symmetry-breaking effects; this residual symmetry
is the so-called custodial SU (2) [11, p280]. From the above calculations it is now visible
that this SU (2) custodial symmetry, or the ρ-parameter being unity at tree level, is not
27
unique to the breaking of electroweak symmetry by a single scalar field (the Higgs field) [5,
p719]. Custodial symmetry is not respected by the masses of fermions in the Standard Model.
Corrections at levels higher than tree-level give corrections to the value of ρ. The largest effect
comes from the heaviest known fermion, the top quark ([11, p280]):
m
2
3GF
t
ρ−1∼ √
m2t = 0.01
.
175GeV
8 2π 2
(2.19)
To conclude, the custodial SU (2) symmetry is actually the global SU (2)R acting on righthanded fermions, and which is broken by the U (1)Y and Higgs-Yukawa couplings. The custodial symmetry ensures that when electroweak symmetries are broken (so the Higgs or some
other effective particle takes a vacuum expectation value), there remains an approximate
global SU (2) symmetry, a subgroup of SU (2)L ⊗ SU (2)R [3, p67].
2.3
The Peskin-Takeuchi Parameters
The Peskin and Takeuchi parameters are three measurable quantities, namely S, T , and U ,
that parametrize possible new physics contributions to electroweak radiative corrections, or, to
be more precise, oblique corrections, which are vacuum-polarization diagrams that contribute
to four-fermion scattering processes2 [20, p382]. These parameters are named after Peskin
and Takeuchi, who, followed soon by others, proposed these parameters in 1990 [21, 22]. The
electroweak gauge group is assumed, and thus no additional gauge bosons beyond γ, Z and
W , and the non-oblique radiative corrections are neglected as it is assumed that new couplings
to light fermions are suppressed. Also, the energy scale at which the new physics appears
is assumed to be large compared to the electroweak scale [19, p63]. The self-energies of the
photon, Z-, and W-bosons and the mixing of Z-boson and photon are then the parameters of
the oblique corrections. For X, Y ∈ {γ, Z, W }, any self-energy diagram ΠXY (q 2 ) of a vacuum
polarization amplitude can be defined by [20, p383]:
Z
µ
(x)JYν (0)
(2.20)
ig µν ΠXY (q 2 ) + (q µ q ν terms) ≡ d4 xe−iqx JX
and it can be written as
ΠXY (q 2 ) = ΠXY (0) + q 2 Π0XY (q 2 ).
(2.21)
The last term is defined as Π0 (q 2 ) ≡ dΠ
dq . This self-energy diagram is depicted in figure
2.1. These self-energies (Πγγ , ΠγZ , ΠZZ , ΠW W ) can then be expressed into six parameters
using the definitions above; three of these can be eliminated because they depend on the
SU (2) ⊗ U (1) gauge couplings and the Higgs-boson vacuum expectation value [20, p386]3 .
Some of the most accurately measured parameters of electroweak interactions at the time,
which are still often used as input for the S, T , and U parameters, are the fine-structure
2
For four-fermion scattering processes, there are three types of radiative corrections: vacuum polarization
corrections, vertex corrections, and box corrections. The first of these radiative corrections are called oblique,
because they affect only mixing and propagation of gauge bosons, and do not depend on the fermion types in
the initial or final states [19, p61].
3
For a theory without custodial symmetry, the lowest order expressions would have contained ρ as a fourth
parameter, and a fourth observable would be needed to fix this parameter [20, p386].
28
I
J
=
(q)
IJ
Figure 2.1: A vacuum polarization amplitude involving gauge bosons I and J, which can each be a
photon, Z-, or W -boson.
constant4 α, the Fermi coupling5 GF , and the mass of the Z-boson MZ .
The Peskin-Takeuchi parameters are defined, when assuming the new physics scale is large
compared to the electroweak scale, as follows [19, p63]:
c2w − s2w 0
0
2 2
0
ΠZγ (0) − Πγγ (0)
(2.22)
αS = 4sw cw ΠZZ (0) −
sw cw
ΠW W (0) ΠZZ (0)
αT =
−
(2.23)
2
MW
MZ2
(2.24)
αU = 4s2w Π0W W (0) − c2w Π0ZZ (0) − 2sw cw Π0Zγ (0) − s2w Π0γγ (0)
Here Π0 (0) = dΠ
dq |q = 0, and cw and sw are cos θW and sin θW , respectively. If the scale of
new physics is allowed to be close to the electroweak scale, however, the approximation that
was used to obtain the equations 2.23-2.24 no longer holds. Three new parameters are then
added, and new physics may not deviate much from the accepted values of S, T , and U ; in
other words, it may be hidden close to the electroweak scale [19, p67].
The S-parameter is an isospin symmetric observable that measures the ultraviolet part of
the momentum dependence of Π33 and can be thought of as the measure of the size of a new
fermion sector [20, p389]. When new fermion doublets are added to the Standard Model, the
S-parameter may change; this is what happens in technicolor in such a way that it violates
the known value of the S-parameter. Technicolor is discussed in chapter 4.
The T -parameter obtains contributions from effects that violate custodial isospin symmetry: it is sensitive to the difference between the loop corrections to the Z boson vacuum polarization function and the W boson vacuum polarization function. Hence, the T -parameter is a
measure of the total weak-isospin breaking of such a new sector [20, p389]. The T -parameter
can thus be seen as the ρ-parameter in a new jacket. This is visible from the following
definition of T [19, p64]:
ρ = 1 + δρSM + αT.
(2.25)
The U -parameter is not used very often, as it is small compared to the other two parameters. If the new physics preserves custodial isospin symmetry, T and U vanish.
4
The fine-structure constant can be determined from quantum-electrodynamic measurements: it is equal
2
to α = 4πεe 0 h̄c [5, pxxi].
5
The Fermi coupling GF actually stems from the time when the weak force was still explained by four
fermions interacting at a single vertex. In fact,√this process is mediated by a W -boson. The Fermi constant is
GF
2 g2
indeed proportional to the W -mass: (h̄c)
[19, p4]. The Fermi coupling can be measured through
3 = 8 M2
W
muon decay, or µ− → e− + ν̄e +νµ ,
µ+ → e+ +νe + ν̄µ , for which the tree level decay width is Γµ =
3m2
µ
with ∆ a power series in α and is to lowest order 1 + α∆1 with ∆1 = −8x − 12x2 log x + 8x3 + 5M 2
W
2
me
with x = m2 [19, p37].
µ
29
5
G2
F mµ
∆,
192π 3 m4
+ O m4µ ,
W
The Peskin-Takeuchi parameters have been constrained by precise measurements of LEP
at CERN and SLAC at Stanford ([19, p64]):
S = 0.16 ± 0.14
(0.10)
T
= 0.21 ± 0.16
(+0.10)
U
= 0.25 ± 0.24
(+0.01)
Although the S- and T -parameters are good probes of new physics, they remain indirect
results from measurements. Later, when discussing the phenomenology of composite Higgs
models, direct tests of these models will be discussed as well.
30
Chapter 3
The Nambu–Jona-Lasinio Model
Analogous to the Cooper pair electron condensate in BCS theory, a top quark condensate
could form as a result of spontaneous symmetry breaking of the standard model; this condensate could then serve to give particles mass dynamically through interactions (meaning
one starts without a bare mass). New strong gauge dynamics could then replace the role of
phonons in superconductors in topcolor and technicolor [13, p12], which are theories based
on the Nambu–Jona-Lasinio Model. A simple version of a “composite-Higgs” theory that
generalizes to a more realistic version can be achieved through adding a four-fermion interaction term, Gt (ψ̄Lia tRa )(t̄bR ψLib ), to the Lagrangian. When replacing the Standard Model Higgs
Lagrangian with a four-fermion interaction of dimension six, the theory is still renormalizable.
Here i runs over the SU (2)L indices; a, b run over SU (3) color indices; and Gt is a coupling
constant of the order Λ12 , with Λ the energy scale up to which this Lagrangian holds (the fourfermion interaction is essentially a low-energy effective interaction of strong gauge dynamics).
This four-fermion interaction is what Nambu and Jona-Lasinio used when they took the idea
of Cooper pairs from the theory of superconductivity and published their Dynamical Model of
Elementary Particles in 1960 [23]. The top quark is in this model assumed to acquire its own
mass dynamically through the two-point interaction mt̄t; the bottom quark mass is assumed
zero in this approximation, so no interaction term of this particle alone is added [9, p22].
3.1
Top condensation versus Standard Model Higgs mechanism
In the Nambu–Jona-Lasinio model, there is no more Higgs potential. This means there is no
more minimum of a potential that is taken on as a vacuum expectation value and subsequently
gives gauge bosons their mass. In this version of a top condensation model, or a model where
top quarks condense and gain a mass dynamically, as well as give other particles their mass,
the breaking of the electroweak symmetry or Higgs mechanism is slightly more complicated.
If the Lagrangian is rewritten using auxiliary fields, however, it can be made to look like the
Standard Model Lagrangian with Higgs particle (but without the Higgs potential). In that
case, the now composite top condensate takes a vacuum expectation value and gives particles
mass through the Higgs mechanism. The top-Higgs is then an excitation of this condensate
field. This will be treated in section 3.4.
At high energies, a force called topcolor (see also chapter 5) has an extra symmetry that
serves as a gauge group of this force. This symmetry is broken by some field, let us call
31
it Φt , that takes on a vacuum expectation value hΦt i = M , with M the top-gluon mass.
Top-gluons are, like gluons for the strong force in quantum chromodynamics, the mediators
of the topcolor force in top condensate models. After this symmetry of topcolor is broken,
the effective theory contains a four-fermion interaction as described above. As a result of
this four-fermion interaction, the top quark, which was set as massless to start with, gains a
mass dynamically through interactions. This can be seen from the gap equation that will be
discussed below in section 3.2.
Another result of this four-fermion interaction is that fermions condense just like electrons
condense in BCS theory and quarks condense in QCD. These condensates can then be treated
as effective particles, similar to mesons in QCD. How do we know of the existence of these
effective particles? This can be seen by examining the propagators of effective particles. Since
1
propagators have the form ∼ p2 −m
2 , the mass of an effective particle can be read from the
propagator. For example, when looking at the propagator of tt̄ → tt̄, it can be seen that this
1
is ∼ p2 −4m
2 , so that an effective scalar particle of twice the top quark mass exists. If one
t
examines propagators of tγ 5 t̄ or t(1 − γ 5 )b̄, one finds massless particles that can be seen as
(pseudo-) Nambu–Goldstone bosons. This is further explained in section 3.3.
Then still another question remains: how do the gauge bosons, the W and Z, obtain
their mass? For this, one looks at the self-energy of the gauge bosons. The propagator
1
, so that the coefficients of terms proportional to
including self-energy Π looks like ∼ p12 1−Π
1
in the self-energy Π give rise to a mass in the gauge boson. One will see that in the
p2
self-energy one of the Nambu–Goldstone bosons, found through examining propagators of
effective particles, is contained in the self-energy of a gauge boson, and thus the gauge boson
has “eaten” a Nambu–Goldstone boson - just like one would expect in the Higgs mechanism.
This procedure is shown in section 3.5.
3.2
The Gap Equation
Through inserting the physical mass directly into the Lagrangian, an equation from which this
mass can be calculated is obtained; this equation is called the gap equation. This is similar
to the gap equation obtained in condensed matter physics. In condensed matter physics, an
order parameter1 ∆ is introduced to distinguish phases in phase transitions by quantifying
the order of the system [24, p8]. The order parameter is, in BCS theory with two fermion
fields φ↓ and φ↑ , and coupling constant of a four-fermion interaction of V0 , defined as follows
([24, p284])2 :
h∆(x, τ )i = V0 hφ↓ (x, τ )φ↑ (x, τ )i .
(3.1)
In BCS theory, by looking at the pole of a fermionic single particle excitation (as visible from
the effective propagator of a fermion) it can be seen that a minimum of ∆ is needed to make
an elementary excitation for a single fermion, and thus the minimum amount of energy needed
to break up a Cooper pair is 2∆ [24, p284]; this so-called energy gap gives the gap equation
(see appendix B and figure 3.4) its name. The form of the gap equation in BCS-theory is:
∆=
V0 X
h̄∆
−
,
h̄βV
(h̄ωn )2 + (εk − µ)2 + |∆|2
k,n
1
2
The order parameter is that quantity which changes through a phase transition [2, p270].
An explanation and derivation of the gap equation in BCS theory can be found in appendix B.
32
(3.2)
=
Figure 3.1: The exact Dyson’s equation for an interacting Green’s function. The double line is the
interacting Green’s function, and the single line the noninteracting Green’s function. Σ corresponds
to the self-energy diagrams.
where the right hand side depends on ∆ as well, so that this is a so-called self-consistent
equation: it is recursive.
In top condensation models, this gap will correspond to the Higgs mass and consist of
twice the top mass. The induced top quark mass (or, equivalently, mass gap), which is similar
to the energy gap above, can then be found through the gap equation [25, p1648]. In the
Nambu–Jona-Lasinio model, the gap equation can be derived from requiring the self-energy3
of the top quark to be zero on shell4 [9, p23]. The reason that the self-energy is zero on shell is
that the physical mass has been directly inserted into the Lagrangian instead of just the bare
mass of the top quark; the latter (bare mass) would need corrections from the self-energy,
whereas for the former (physical mass) we demand these corrections to be absent.
The gap equation can for our case be derived from Dyson’s equation, in which the interacting Green’s function (propagator) can be expressed as a sum of corrections to the noninteracting Green’s function using self-energy (one-particle irreducible) diagrams [24, p164,
165]: with G the renormalized propagator (taking interactions into account) and G0 the bare
propagator, it is G = G0 + G0 ΣG with Σ the self-energy. Dyson’s equation is depicted in
figure 3.1. It is a recursive equation, and therefore sometimes called self-consistent.
Treating the top quark as a particle obtaining its mass dynamically, the propagator of the
top quark can be approximated by inserting a physical mass in the bare propagators (where
the bare mass is set to zero, since we start with a massless top quark):
1 1 1 1 1 1
1 1
1
+ m + m m + ··· =
,
m =
p
p
p
p
p
p
p
1
−
p
−
/ / / / / /
/
/ m
p
/
(3.3)
which is again Dyson’s equation. This equation is depicted in figure 3.2. The self-energy of
the top quark can then be approximated in a similar way, through top quark mass insertions
into the one-particle irreducible diagrams. This is depicted in figure 3.3. Combining now that
we want the self-energy to be zero, as argued above, with this figure, we see that our demand
can be satisfied if the sum of merely the first two diagrams is zero, as depicted in figure 3.4.
This can also be derived from the Lagrangian. The full Lagrangian used is:
X
1 X (i)a 2
/ k−
L =
Ψ̄k iDΨ
(Fµν ) + LI
(3.4)
4
i
k
gs µ
g µ
g0 µ
µ
µ
/ = D γµ = ∂ + i Ga La + i Wb σb + i B Y γµ ,
D
(3.5)
2
2
2
with LI the four-fermion interaction as below. We have, taking only the third generation and
neglecting the rest, the following relevant part of the Lagrangian:
L0 + LI = t̄iγ µ ∂µ t + b̄iγ µ ∂µ b + Gt (ψ̄L tR )(t̄R ψL ),
3
4
The self-energy consists of the contributions from quantum corrections.
A particle on shell has a four-momentum that obeys the equation p2 = m2 .
33
(3.6)
which can be rewritten as L = L0 0 + L0 I with
L0 0 = t̄iγ µ ∂µ t + b̄iγ µ ∂µ b − mt t̄t
0
LI
= mt t̄t + Gt (ψ̄L tR )(t̄R ψL ).
(3.7)
(3.8)
Here the physical mass of the top quark is inserted by hand. This results in a two-point
interaction between a top and antitop quark with a coupling of mt . This interaction vertex
is denoted with a cross in diagrams 3.2-3.4.
The interaction from the Lagrangian part in equation 3.8 can be rewritten using
Gt (ψ̄L tR )(t̄R ψL ) =
=
=
=
=
=
1
Gt (ψ̄L (1 + γ5 )tR )(t̄R (1 − γ5 )ψL )
4
1
Gt (ψ̄(1 + γ5 )(1 + γ5 )t)(t̄(1 − γ5 )(1 − γ5 )ψ)
16
1
Gt (ψ̄(2 + 2γ5 )t)(t̄(2 − 2γ5 )ψ)
16
1
Gt [(t̄(1 + γ5 )t)(t̄(1 − γ5 )t)
4
+(b̄(1 + γ5 )t)(t̄(1 − γ5 )b)
1
Gt [(t̄t)(t̄t) + (t̄γ5 t)(t̄t) − (t̄t)(t̄γ5 t)
4
(t̄γ5 t)(t̄γ5 t) + (b̄(1 + γ5 )t)(t̄(1 − γ5 )b)
1
Gt [(t̄t)(t̄t) + (t̄iγ5 t)(t̄iγ5 t)
4
+(b̄(1 + γ5 )t)(t̄(1 − γ5 )b)) ,
(3.9)
from which we can see that the self-energy diagrams derived from the last two terms vanish,
as they give traces with γ5 , which gives zero [2, p475], and because we consider only diagrams
of leading order in Nc . This ’t Hooft mathematical large number of colors limit5 is one in
which all mesons made by quark bilinears become stable and non-interacting; this state is
then expected to exist also when the number of colors is reduced to three [15, p4].
Taking this limit in Nc yields the fact that from the two choices in connecting lines in the
diagram of the first correction to the top propagator, which is the second diagram depicted
in figure 3.4, only one diagram contributes. This diagram is depicted in detail in figure 3.5;
only the diagram in 3.5(a) contributes because it is of leading order in Nc . If we now add
to this contributing diagram a factor of γ5 , we see that, since we have only one vertex, only
one of γ5 appears
h
iin the trace, and one γ5 in a trace gives zero: it would be proportional
R d4 l
iγ5
to (2π)4 Tr /l−m . Furthermore, if we look at the last term in the interaction LI , we see
t
that it does not contribute to the terms of leading order in Nc either; see figure 3.6. That
is why we have the sum of the two diagrams in figure 3.4 equal to zero; these correspond to
mt t̄t + 41 Gt (t̄t)(t̄t).
When not inserting the physical mass into the Lagrangian by hand, we can also obtain
the gap equation. The equation relating the induced mass that we call now m, bare mass m0 ,
5
QCD is a theory without any parameters, so it is difficult to expand [2, p377]. A parameter was invented
in order to expand QCD: a number of colors N. Gerard ’t Hooft worked out the implications of this limit; see
[26, p378].
34
+
+
x
x
x
+ ...
=
Figure 3.2: An approximation of a renormalized propagator assuming the mass is gained dynamically
through an interaction which is denoted by crosses. For the top quark propagator, the crosses are
mass insertions and are the interactions that give the top quark a mass.
Figure 3.3: An approximation of the self energy of the top quark using one-particle irreducible diagrams
and insertions of the top quark mass mt . Figure taken from [9, p22].
x
=
+
0
Figure 3.4: These diagrams can be derived from the Lagrangian 3.9. The gap equation for the top
quark mass originates from the terms in the Lagrangian mt t̄t+ 14 Gt (t̄t)(t̄t). If the sum of these diagrams
that are of leading order Nc is zero, this would be sufficient to obtain a zero self-energy. Figure taken
from [9, p23].
35
t
t
t
t
t
t
t
t
t
t
t
t
(a) First correction to top propa- (b) Second choice of connecting
gator: first choice.
lines in first correction to top
propagator.
Figure 3.5: Two choices for connecting lines in the first diagram for the correction to the top propagator. The second choice (figure 3.5(b)) does not contribute since the color is fixed through the
connection of all the lines, and thus there is no sum over the number of colors Nc . The first choice has
a loop not connected directly to the propagator, and has thus a sum over the number of colors and is
of leading order in Nc .
and self-energy is Σ [23, p348]
m − m0 = Σ(p, m, G, Λ)|γ·p+m=0 6 .
(3.10)
After setting the bare mass to zero and using that the self-energy is the expectation value of
the fields [24, p173-181], we obtain a gap equation as follows [25, p1648]:
1
m = Σ = − (2)G ht̄ti .
4
(3.11)
The factor two is a symmetry factor. This equation is depicted in figure 3.4 to first order,
or in the one-loop approximation; we consider here only the leading order in the number of
colors Nc . A similar result can be obtained using a mean field approximation in BCS theory
[7]. Setting Σ|p2 =m2t , we have mt tt̄ = − 41 Gt t̄tt̄t. If we then approximate by using mean field
theory7 , so that t̄tt̄t = 2 ht̄ti t̄t [7], we obtain mt = − 12 Gt ht̄ti. See also section 3.4 for a similar
approximation in the Nambu–Jona-Lasinio model.
Using the Bjorken-and-Drell metric (+1, −1, −1, −1) (see also [28, p277, 278]), an amount
of Nc colors, and applying the appropriate Feynman rules for the interactions mt tt̄ (denoted
by crosses) and − 14 Gt t̄tt̄t (the four-fermion interaction), we obtain from diagram 3.4 the
6
The 4 × 4 γ-matrices are defined as γ 0 =
I
0
0
−I
and γ i =
0
−σ i
σi
0
, with i ∈ {1, 2, 3} and σ i
the Pauli matrices [2, p90].
7
Mean field theory is also known as Landau-Ginzburg theory or self-consistent field theory. It is widely
used to study critical phenomena, as critical exponents easily emerge from it. Mean field theory is based on a
series expansion of the free energy in the order parameter of the phase transition of interest [27, p145].
36
b
b
t
t
t
t
Figure 3.6: Correction to the top propagator using the third term in the Lagrangian (b̄(1 + γ5 )t)(t̄(1 −
γ5 )b)). This is not of leading order in Nc , as the colors are fixed: by the connected lines.
following expression8 :
mt =
=
=
=
Z
d4 p
i
1
− Gt (Nc )(−1)Tr
2
(2π)4 (p
/ − mt + iε)
Z
1
d4 p Tr p
/ + mt
Gt Nc i
4
2
2
(2π) p − m2t + iε)
Z
d4 p
4mt
1
Gt Nc i
4
2
2
(2π) p − m2t
Z
i
1
2Gt Nc mt
d4 l 2
;
(2π)4
l − m2t
(3.12)
where in the fourth line the property Tr γµ = 0 was used. The self-consistency of this equation
is manifest; the top quark mass mt appears on the right hand side as well and can be divided
out if it is not zero. Equation 3.12 can be evaluated with Wick rotation and a momentum8
Here, p
/ = p · γ as in Dirac’s notation.
37
space cutoff Λ [25, p1648]:
G−1
t
=
Wick rotation
=
l0 →il4
=
=
introduce cutoff Λ
→
=
=
=
=
'
Z
Nc
1
i d4 l 2
4
8π
l − m2t
Z
Z −i∞
1
iNc
dl0 d3~l
4
2
2
~
8π i∞
l0 − l − m2t
Z
Z ∞
1
iNc
dl4 d3~l
(i)
4
2
~
8π
−l4 − l2 − m2t
−∞
Z
Nc
1
d4 l 2
8π 4
l + m2t
Z
Nc Λ
2π 2 l3
dl
8π 4 0
l2 + m2t
2
Z
l + m2t
Nc Λ
m2t
dl
l
−
4π 2 0
l2 + m2t
l2 + m2t
Z
lm2t
Nc Λ
dl
l
−
4π 2 0
l2 + m2t
Nc Λ m2t
2
2
2
−
log Λ + mt − log mt
4π 2 2
2
2
Λ
Nc
2
2
+1
Λ − mt log
8π 2
m2t
Nc
Λ2
2
2
Λ − mt log 2 .
8π 2
mt
(3.13)
Here we used an invariant cutoff of l2 = Λ2 and a change of variables il0 → il4 . The
integral in equation 3.12 could alternatively be obtained using Cauchy’s residue theorem and
a noninvariant cutoff |~l| = Λ. The result is then ([23, p348], [7])
(
"
#)
1/2
2
1 −1
Nc
Λ
Λ
G = 2 Λ(m2t + Λ)1/2 − m2t log
.
+1
+
2 t
8π
mt
m2t
2
Nc Λ
A solution for mt exists when log mΛ2 > 0, or, equivalently, −G−1
t + 8π 2 ≥ 0. This then
t
implies that the coupling must be sufficiently strong in order for the top quark to gain a
2
mass: N8π
2 ≡ Gc ≤ Gt . For such value of the coupling, a nonzero solution for mt exists and
cΛ
thus the top gains a mass. For lower values of Gt , only a zero mass solves the equation. For
a strong enough coupling, the nonzero solution for the top quark mass corresponds to the
lowest energy state, or the vacuum9 [9, p23].
The fine-tuning problem from the Standard Model arises here, too. To obtain a mass
much smaller than the cut-off momentum, or mt Λ, the parameters Gt and Λ must be
2
chosen such that Gt /Gc = Gt / N8πc Λ = 1 + δ with δ 1 [9, p23].
9
This can be seen when using an auxiliary top-Higgs field in the Lagrangian and computing an effective
potential of this effective (auxiliary) top-Higgs field. When trying to find the mininum of this effective potential
by setting its derivative to zero, again two solutions exist; of these only one is a real minimum (this can be seen
by investigating the second derivative, which will be positive for a mininum) and corresponds to the nonzero
solution of the top quark mass [9, p38, 39].
38
+
+
+
...
Figure 3.7: An approximation of the four-fermion interaction in the scalar channel.
3.3
Evidence of Bound States
Quark-antiquark bound states show up as poles in propagators of composite fields [9, p24].
1
Effective particles with effective masses can be seen if their propagators of the form ∼ p2 −m
2
are calculated. There are four such composite field propagators, one of them a scalar. The
other three are a pseudoscalar and two charged composite fields, which, as we will see later,
are the massless Goldstone bosons. The composite fields are given by:
scalar:
t̄t
t̄γ 5 t
pseudoscalar:
charged:
t̄(1 − γ 5 )b
charged:
b̄(1 + γ 5 )t.
(3.14)
Looking at the scalar channel of the four-fermion interaction, we will see that the theory
predicts a scalar bound state with a mass of 2mt . Using the Lagrangian with the interaction
written as in equation 3.9, which we rewrite here for convenience:
LI = Gt ψ̄L tR t̄R ψL =
1
Gt [(t̄t)(t̄t) + (t̄iγ5 t)(t̄iγ5 t)
4
+(b̄(1 + γ5 )t)(t̄(1 − γ5 )b)) ,
(3.15)
we see that the scalar channel of the four-fermion interaction as approximated in figure 3.7
yields [9, p25] the following two-point correlation function10 :
Z
2
Γs (p ) =
d4 xeipx h0|T [t̄t](x)[t̄t](0)|0iconnected
(
)
Z
Gt
1
i
i
i
= (2)
+ (4)(−1)( Gt )2 Nc
d4 lTr
/l − mt /l + p
4
4
(2π)4
/ − mt
+...
≡
=
=
Gt
2
(3.16)
!
iGt Nc Is
iGt Nc Is 2
1−
+ −
+ ...
2
2
1
Gt
c Is
2 1 − − iGt N
2
Gt
iGt Nc Is −1
1+
.
2
2
(3.17)
10
A two-point correlation function or a two-point Green’s function is hΩ|T (ϕ(y)ϕ(x))|Ωi which is the propagator of the field ϕ. In free theory (without interactions), this is the Feynman propagator: h0|T (ϕ(x)ϕ(y))|0ifree =
R dp ieip·(x−y)
DF (x − y) = (2π)
4 p2 −m2 +iε [5, p82, 83].
39
In the third line an integral Is was defined for convenience. In the fourth line a geometric
series is summed; the integral Is can be evaluated and amounts to:
(
)
Z
d4 l
i
i
Is =
Tr
/l − mt /l + p
(2π)4
/ − mt
Z
d4 l Tr /l + mt /l + p
/ + mt
= −
(2π)4 l2 − m2t (p + l)2 − m2t
2
2
Z
/
/
Tr
l
p
+
l
+
m
4
/
t
d l
= −
2
4
2
2
(2π) l − mt (p + l) − m2t
Z
d4 l (p + l)2 − m2t + 2m2t − p2 − pl
= −4
(2π)4
l2 − m2t (p + l)2 − m2t
Z
Z
1
p2 − 2m2t + pl
d4 l
d4 l
,
+
4
= −4
(2π)4 l2 − m2t
(2π)4 l2 − m2t (p + l)2 − m2t
(3.18)
where the last term can be rewritten using partial fraction decomposition:
Z
Z
lp
d4 l
1
1
1
d4 l
=
−
2
2
2
4
4
2
2
2
(2π) l − mt (p + l) − mt
2
(2π) l − mt
(p + l)2 − m2t
#
p2
.
+
l2 − m2t (p + l)2 − m2t
(3.19)
The integrals above can be calculated, but in order to do that a shift in momentum is needed.
This is not allowed when a cutoff is involved, because it may then give rise to extra terms
called surface terms (see [29, p458, 459]). The second of the above integrals on the right hand
side can be written as an integral independent of p and a surface term [9, p91]:
Z
Z
d4 l
1
d4 l
1
ip2
,
(3.20)
=
−
(2π)4 (p + l)2 − m2t
(2π)4 l2 − m2t
32π 2
The extra term arises because an integral which is linearly or more than linearly divergent,
is not invariant under shift in momentum (whereas a convergent or logarithmically divergent
integral is). The effect of this extra term is that it may in some cases shift the pole that we
will find for the scalar channel; it will not, however, shift the pole for the Goldstone bosons
(where we expect to find a pole at p2 = 0) [30, p2878]. Combining equations 3.17-3.20, the
scalar integral Is now becomes
Z
d4 l
1
d4 l p2 − 2m2t − 21 p2
1
= −4
+
4
2
2
4
4
2
2
(2π) l − mt
(2π)
l − mt
(p + l)2 − m2t
1 ip2
+4( )
2 32π 2
Z
1
1
d4 l
1
2
2
= −4
+ 2(p − 4mt )
2
4
2
2iNc Gt
(2π) l − mt (p + l)2 − m2t
+ surface terms,
Z
Is
40
(3.21)
where in the last line the gap equation from equation 3.12 was used. We now neglect the
extra constant terms, assuming their contributions are not physically relevant [9, p25]. The
scalar amplitude reads now
iGt Nc Is −1
Γs (p ) =
1+
2
−1
Z
i
1
1
1
2
2
4
= − Gt Gt Nc (p − 4mt )
d l 2
2
(2π)4
l − m2t (p + l)2 − m2t
Z
−1
−1
d4 l
1
1
=
i
(2π)4 l2 − m2t (p + l)2 − m2t
2Nc (p2 − 4m2t )
2
Gt
2
(3.22)
From this it can be seen that there is a pole at p2 = 4m2t or at p = 2mt ; this is precisely
what we were looking for, namely an effective mass of the top condensate of 2mt . Although
it seems that the binding energy here is absent so that this is a loosely bound state, this
is not the case; corrections from subleading N and other interactions are not yet taken into
account, and the result is subject to renormalization [3, p171]. The integral in equation 3.22
can be evaluated using ([2, p.151])
1
=
ab
Z
1
dx
0
1
,
[xa + (1 − x)b]2
(3.23)
so that, using the following algebraic expression:
x((p + l)2 ) + (1 − x)l2 = l2 x + p2 x + 2xpl + l2 − xl2
= l2 + 2xpl + x2 p2 − x2 p2 + p2 x
= (l + xp)2 + x(1 − x)p2 ,
41
(3.24)
the integral in equation 3.22 becomes
Z
1
i
1
d4 l 2
2
4
(2π)
l − mt (p + l)2 − m2t
Z 1 Z
i
=
dx
(2π)4 0
Z 1 Z
i
dx
=
(2π)4 0
Z 1 Z
i
=
dx
(2π)4 0
Z 1 Z
i
l→l−xp
dx
=
(2π)4 0
Z 1 Z
i
≡
dx
(2π)4 0
Z 1 Z
i
Wick rotate; l0 =il4
dx
=
(2π)4 0
=
=
cutoff
→
=
=
'
−1
(2π)4
Z
1
Z
dx
0
=
1
d4 l 2
2
2
x((p + l) − mt ) + (1 − x)(l2 − m2t )
1
d4 l 2
2
x((p + l) ) + (1 − x)l2 − m2t
1
d4 l 2
(l + xp)2 ) + x(1 − x)p2 − m2t
1
d4 l 2
2
l + x(1 − x)p2 − m2t
d4 l
1
l2
− m̃2t
Z
d3~lh
idl4
d4 l
2
1
−l42 − ~l2 − m̃2t
i2
1
l2
+ m̃2t
2
Z 1 Z
2π 2 l3
−1
dx
dl
2
(2π)4 0
l2 + m̃2t
#
Z Λ " 2
Z
l(l + m̃2t )
lm̃2t
−1 1
dx
dl
2 −
2
8π 2 0
0
l2 + m̃2t
l2 + m̃2t
#
Z
Z Λ "
l
−1 1
lm̃2t
dx
dl 2
−
2
8π 2 0
l + m̃2t
0
l2 + m̃2t
Z 1 Λ2 + m̃2t
1
−1
1
2
dx log
+ m̃t
− 2
16π 2 0
m̃2t
Λ2 + m̃2t
m̃t
Z 1
2
−1
Λ
dx log 2 .
2
16π 0
m̃t
(3.25)
Note that the shift in integration made in the fifth line above is not always allowed, and may
give rise to surface terms. Then the scalar channel amplitude from equation 3.22 amounts to
−1
Z
i
1
1
1
2
2
2
4
Γs (p ) = − Gt −Gt Nc (4mt − p )
d l 2
2
(2π)4
l − m2t (p + l)2 − m2t
−1
Z 1
1
Λ2
2
2 −1
=
dx log 2
(4mt − p )
2Nc
16π 2 0
m̃t
−1
Z 1
1
1
Λ2
2
2
(3.26)
= −
(4mt − p )
dx log
2Nc
16π 2 0
m2t − x(1 − x)p2
where we emphasize once more that a pole exists at |p| = 2mt , implying a scalar bound state
with a mass of 2mt . A top-Higgs would then have such a mass, which could be slightly less
42
because of the binding energy.
The pseudoscalar and charged channels can be calculated in a similar way. Their bound
states are
hT [t̄iγ5 t](x)[t̄(−i)γ5 t](0)i and T [t̄(1 − γ5 )b](x)[b̄(1 + γ5 )t](0) .
In the pseudoscalar channel the integral Is in equation 3.17 is renamed Ip and changes to:
)
(
Z
i
i
d4 l
Tr
iγ
iγ5
Ip = =
/l − mt 5 /l + p
(2π)4
/ − mt
Z
d4 l Tr /lγ5 /lγ5 + /lγ5 p
/γ5 + m2t
=
(2π)4 l2 − m2t (p + l)2 − m2t
Z
Tr −l2 − lp + m2t
d4 l
=
(2π)4 l2 − m2t (p + l)2 − m2t
Z
d4 l (l + p)2 − m2t − lp − p2
= −4
(2π)4 l2 − m2t (p + l)2 − m2t
Z
Z
1
1
d4 l
d4 l
1
2
= −4
+
4p
2
(2π)4 l2 − mt
(2π)4 l2 − m2t (p + l)2 − m2t
Z
1
d4 l
lp
.
(3.27)
+ 4
2
4
2
(2π) l − mt (p + l)2 − m2t
By looking at the difference with the scalar channel integral, we see that the pseudoscalar
correlation function is given by
1
Γp (p ) =
2Nc p2
2
Z
i
d4 l
1
1
2
4
2
(2π) l − mt (p + l)2 − m2t
−1
.
(3.28)
This expression has a pole at p2 = 0, indicating that the pseudoscalar is massless. This is
then one of our massless Nambu-Goldstone bosons.
Likewise, the integral Ic for the charged channel is
)
(
Z
i
i
d4 l
Ic =
Tr
(1 − γ5 )
(1 + γ5 )
/l − mt
/l + p
(2π)4
/ − mb
2
Z
/
/
/
/
/
Tr
l
+
l
p
−
l
γ
l
γ
−
l
γ
p
γ
4
/
/
5
5
5
5
d l
= −
(2π)4
l2 − m2t (p + l)2
Z
d4 l l2 + pl + pl + p2 − pl − p2
= −4
(2π)4
l2 − m2t (p + l)2
Z
Z
d4 l
1
d4 l
p(l + p)
= −4
+
4
,
(3.29)
2
4
4
2
(2π) l − mt
(2π) (l + p)2 (l2 − m2t )
where in the third line Trγ m γ ν = 4η µν has been used. From this the charged amplitude can
43
be found:
2
Γc (p ) =
=
=
=
=
=
Gt
+ (2)(−1)i
4
Gt
4
2
Nc Ic + . . .
!
2
2
2
1 − i( Gt Nc Ic + −iGt Nc Ic + . . .
4
4
−1
2
Gt
1 − −i Gt Nc Ic
4
4
Gt
iGt Nc Ic −1
1+
4
2
Z
Gt
d4 l
1
1 − 2iGt Nc
4
2
4
(2π) l − m2t
−1
Z
d4 l
p(l + p)
+ 2iGt Nc
(2π)4 (l + p)2 (l2 − m2t )
Z
−1
p(l + p)
d4 l
1
i
.
8Nc
(2π)4 (l + p)2 (l2 − m2t )
Gt
4
(3.30)
Notice that in the first line a symmetry factor of 2 appears, as opposed to 4 for the scalar
channel; this is because we have top and bottom quark instead of two tops. In the last line
the gap equation has been used again. This expression is proportional to p12 , because the
integral is proportional to p2 . That can be seen as follows:
Z
d4 l
p(l + p)
i
=
4
(2π) (l + p)2 (l2 − m2t )
Z
Z
p2
p·l
d4 l
d4 l
= i
+
i
(3.31)
2
4
4
2
2
2
(2π) (l + p) (l − mt )
(2π) (l + p) (l2 − m2t )
where the first integral on the right hand side is obviously proportional to p2 , and the second,
following the lines of equations 3.25, 3.23, and 3.24, amounts to:
Z
d4 l
p·l
=
i
4
2
(2π) (l + p) (l2 − m2t )
Z 1
Z
p·l
d4 l
=
i
dx
4
2
(2π) 0
[(l + p) x + (1 − x)(l2 − m2t )]2
Z
Z
1
d4 l
p·l
=
i
dx
(2π)4 0
[(l + px)2 − x(1 − x)p2 − (1 − x)m2 ]2
Z 1
Z
p · (l − px)
d4 l
l→l−xp
dx 2
=
i
(2π)4 0
[l − x(1 − x)p2 − (1 − x)m2 ]2
Z
Z 1
Z 1 Z
d4 l
xp2
d4 l
p·l
≡
i
dx
+
i
,
dx
4
2
2
2
4
2
(2π) 0
[l − m̃ ]
(2π) [l − m̃2 ]2
0
(3.32)
where the first integral on the right hand side in the last line is again obviously proportional
44
to p2 (and some additional factors including p), giving ([31]):
Z 1
Z
xp2
d4 l
=
i
dx
(2π)4 0
[l2 − m̃2 ]2
Z 1
−ip2 x
Λ2
Λ2
dx
= i
log 1 +
− 2
.
16π 2
Λ + m̃2
m̃2
0
(3.33)
The second integral on the right hand side in the last line of equation 3.32 is odd in l and
yields zero (see also [29, p458]). Thus the pseudoscalar and charged amplitudes have poles at
p2 = 0. These poles indicate the massless Goldstone bosons that are to give the gauge bosons
their mass (the gauge bosons will, as before, “eat” these Goldstone bosons). This mass can
be calculated with the self-energy diagrams of the vector bosons. Their explicit masses will
still depend on the cut-off Λ, but a relation between MW and MZ can be found, which in
turn puts a constraint on Λ when comparing this relation to experimental values.
3.4
Auxiliary Fields
The same scalar particle of mass 2m can also be found through using auxiliary fields11 . One
introduces an auxiliary field h into the Lagrangian, but gives it no kinetic term, as follows:
/ + chψ̄ψ − m2 h2 .
L = iψ̄ Dψ
(3.34)
Here c is the equivalent of Gt above. The kinetic term for h is absent, but is actually needed
in order for the theory to be renormalizable. But, because there is no kinetic term, there is
no propagation, and then we can have
δL
= cψ̄ψ − 2m2 h = 0.
δh
(3.35)
If now the auxiliary field is integrated out of the action, the theory becomes renormalizable.
As is visible in 3.35, a mass of twice the fermion mass is already present. That this is the
fermion mass can be seen from a four-fermion interaction that is now generated from the
Lagrangian 3.34 as follows:
/ + c2 (ψ̄ψ)2 .
L = iψ̄ Dψ
(3.36)
Let us now apply this technique to the Nambu–Jona-Lasinio model. Similar to how mean
field theory is used in BCS-theory to approximate a Cooper pair, a top quark condensate that
breaks electroweak symmetry can be approximated by a field which we now call Ht . We can
then add this field to the Lagrangian in the form of an auxiliary field , using the identity
Z
1 =
DHt† DHt exp(m2t0 Ht − gt0 t̄R ψL |m−2
t0 |Ht − gt0 t̄R ψL )
Z
Z
≡
DHt† DHt exp
d4 x[m2t0 Ht† Ht − Ht† gt0 t̄R ψL − gt0 ψ̄L tR Ht +
2
gt0
ψ̄L tR t̄R ψL ] ,
(3.37)
m2t0
11
An auxiliary field is a non-propagating field that is added to the Lagrangian without changing the dynamics,
as it has itself no kinetic term and thus no independent dynamics [5, p517]. It can be integrated out of the
action to obtain the original Lagrangian.
45
just as is done in equation B.10. Here mt0 is the bare mass of this auxiliary field, and gt0 a
coupling constant of this effective Yukawa term that is later changed per generation mix (e.g.
to gtuπ for the coupling of the t and u quarks to a pion). If the contents of the exponent of
equation 3.37 are then subtracted from the contents of the exponent in the partition function
with the original action we obtain an effective Lagrangian [25, p1650]:
Z
Z
Z
†
/ + Gt ψ̄L tR t̄R ψL −
Z =
Dψ̄Dψ DHt DHt exp
d4 x[ψ̄iDψ
2
gt0
†
2
mt0 Ht Ht + gt0 (ψ̄L tR Ht + h.c.) − 2 ψ̄L tR t̄R ψL ]
mt0
Z
Z
Z
/ − m2t0 Ht† Ht +
=
Dψ̄Dψ DHt† DHt exp
d4 x[ψ̄iDψ
gt0 (ψ̄L tR Ht + h.c.)] .
(3.38)
gt0
. It is
Here the couplings are related to Gt of the Nambu–Jona-Lasinio model as Gt = m
t0
now evident that when the field Ht takes a vacuum expectation value, which means that
the two tops forming this field condensate, the top and subsequently other fermions (after
mixing of quarks through diagonalizing mass matrices - see section 6.2) gain a mass. In this
2 < 0, a vacuum
Nambu-Jona–Lasinio model, the neutral auxiliary field Ht0 takes, for MH
t
v+h
expectation value so that the real part of this field becomes √2 with h parametrizing the
quantum fluctuations around the VEV as in the Standard Model [3, p170]. This changes
the Yukawa term in the Lagrangian with the auxiliary field (here expressed in terms of the
renormalized coupling constant g̃):
v+h
g̃ ψ̄L ψR Ht + h.c. → g̃ ψ̄L ψR √ + h.c.
2
(3.39)
This then allows us to read off the mass of the particle that forms a bound state as from a
2
term m(ψ̄L ψR + ψ̄R ψL ): we then have m = √12 vg̃, or 21 v 2 = m
[3, p170] with m the mass
g̃ 2
of the particle that forms a condensate. Comparing this to m2h = v 2 λ̃, which one can obtain
in a similar manner, one obtains 2λ̃/g̃ 2 m2 = m2h , or mh = 2m using the renormalization
group equation results from [3, p169]. These renormalization group results also yield the
Pagels-Stokar relation [3, p171]
1 2
Nc
v = m2 /g̃ 2 = m2
ln(Λ/m2 ).
2
16π 2
(3.40)
This gives a relationship between the electroweak vacuum expectation value and the mass of
the particle forming a condensate. As an example, one could take the fermion to be the top
quark. Then with a number of colors Nc = 3 and a cutoff of Λ = 3.438 × 1013 one obtains a
vacuum expectation value of v = 246 GeV. This is a high cutoff, but lower than the Planck
scale (O(1019 GeV)). In top condensation models, usually the dynamics is either at a very
high energy scale predicting the correct top mass as done here, albeit with finetuning, or the
new dynamics is at a more natural TeV scale - but then the top quark is too heavy to come
out of those models [3, p110].
46
t
b
Figure 3.8: The first contribution [Πµν
W ]1 to the self-energy of the W boson; this diagram contains no
four-fermion interactions.
+
+
...
Figure 3.9: The next order contributions [Πµν
W ]2 to the self-energy of the W boson, now including
four-fermion interactions.
3.5
Gauge Boson Masses
The W and Z boson masses can be calculated by finding the pole of their renormalized
(dressed) propagators. This happens according to the Higgs mechanism, in which the Goldstone bosons related to symmetries are eaten by the gauge bosons which then become massive.
The poles of the renormalized propagators can be found by computing their self-energies Πµν
W
and Πµν
Z . We start with the self-energy of the W boson. We demand that the self-energies
12
fulfill the Ward identity pµ Πµν
W = 0 to impose gauge invariance . The W boson self-energy
can be rewritten [9, p26]:
µν 2
µ ν
2
Πµν
(3.41)
W = (g p − p p )ΠW (p ).
This self-energy can be split into a four-fermion-interaction-independent part and a part
µν
µν
dependent on this interaction: Πµν
W = [ΠW ]1 + [ΠW ]2 , as depicted in figures 3.8 and 3.9.
/ of the Standard
We can rewrite the weak and strong couplings from the kinetic term ψ̄ Dψ
Model Lagrangian; remember that the covariant derivative is here given by
Dµ = ∂ µ + i
gs µ
g0
g
Ga La + i Wbµ σb + i B µ Y
2
2
2
where sums over a and b are implicit. Using W ±µ =
√1 (W µ
1
2
∓ iW2µ ) we can write
g
g 1
1
i
µ
µ
µ
µ
µ
µ
√
√
√
√
(W+ + W− )σ1 +
(W+ − W− )σ2
i (W1 σ1 + W2 σ2 ) = i
2
2 2
2
2
g
= i √ (W+µ T + W−µ T − )
2
with
1
T ± = (σ1 ± iσ2 ).
2
12
(3.42)
(3.43)
The Ward identity is the diagrammatic expression of the conservation of electric current. It states that
if εµ (k)Mµ (k) is the amplitude of a QED process involving an external photon with momentum k, and we
replace εµ with kµ , this amplitude vanishes: kµ Mµ (k) = 0. It thus imposes symmetry on quantum mechanical
amplitudes. The Ward identity follows from electric current conservation, which is in turn a consequence of
gauge invariance [5, p244, 238].
47
If we ignore quarks other than top and bottom quarks, we now have for the weak coupling
terms (using only left coupling as in the Standard Model):
g
g
tL
µ +
µ −
µ
†
†
+
Q̄L iγµ i √ (W+ T W− T )QL = i √ W+
tL , bL γ0 iγµ T
bL
2
2
g
tL
µ
†
†
−
+ i √ W−
tL , bL γ0 iγµ T
bL
2
g
g
µ +
µ −
≡ i √ W+ Jµ + i √ W− Jµ .
(3.44)
2
2
tL
bL
tL
0
+
−
Note that T
=
and T
=
so that Jµ+ = t̄L iγµ bL and
bL
0
bL
tL
Jµ− = b̄L iγµ tL . Remember also that the matrices PL = 21 (I − γ5 ) and PR = 12 (I − γ5 ), with I
the identity matrix, are projection operators in the sense that PL ψ = ψL and PR ψ = ψR . As
a result, we have the following combining equations 3.42 - 3.44:
g
g
Q̄L iγµ i (W1µ σ1 + W2µ σ2 )QL = Q̄L iγµ i √ (W+µ T + W−µ T − )QL
2
2
g
g
µ +
= i √ W+ Jµ + i √ W−µ Jµ−
2
2
g µ +
= i √ W+ Jµ + W−µ Jµ−
2
g µ
= i √ W+ t̄L iγµ bL + W−µ b̄L iγµ tL
2
ig
= Q̄L iγµ √ [Wµ+ PL γµ PL + Wµ− PL γµ PL ]QL .
2
(3.45)
We can now replace the weak part of the Lagrangian with the expression
g
ig
i (W1µ σ1 + W2µ σ2 ) = √ [Wµ+ PL γµ PL + Wµ− PL γµ PL ].
2
2
(3.46)
With the rewritten Lagrangian we obtain for the first diagram in figure 3.8 the following [9,
p28]:
"
#
Z
ig 2
d4 l
i
i
µν
µ
ν
√
γ PL
[ΠW ]1 =
(−1)Nc Tr PL
γ
/l + p
/l − mt
(2π)4
2
/
Z
1
d4 l Tr 12 (I − γ5 )(/l − p
ig 2
/)γ µ 2 (I − γ5 )(/l + mt )γ ν
√
=
(2π)4
(l2 − p2 )(l2 − m2t )
2
Z
ig 2
d4 l
1
1
√
=
4 l2 − p2 l2 − m2
(2π)
2
t
1
µ
ν
µ
ν
×
2Tr (/l − p
/)γ /lγ − 2Tr γ5 (/l − p
/)γ /lγ
4
Z
d4 l
ig 2
1
1
√
=
4
2
2
2
(2π) l − p l − m2t
2
1
× [lρ lσ − pρ lσ ] · 2[4(η ρµ η σν − η ρσ η µν + η ρν η µσ ) − 4iερµσν ].
4
(3.47)
48
After some more computations of integrals similar to those made before in section 3.2, which
will not be shown here, one obtains [9, p28]:
[Πµν
W ]1
µν 2
µ ν
= (g p − p p )
−m2t g µν
ig
√
2
ig
√
2
2
2
Z
1
Z
dx2x(1 − x)
2Nc
0
Z
1
Z
dx(1 − x)
2Nc
0
d4 l
1
4
2
(2π) (l − D)2
d4 l
1
.
4
2
(2π) (l − D)2
(3.48)
with D = −x(1 − x)p2 + (1 − x)m2t . Because pµ (g µν p2 − pµ pν ) = 0, the above fulfills the Ward
identity for mt = 0.
For mt 6= 0, four-fermion interactions must also be taken into account (since this interaction generates the top mass). By demanding that the self-energy obeys the Ward identity
µν
and thus the second term in [Πµν
W ]1 to vanish for mt 6= 0, we can guess the form of [ΠW ]2 :
[Πµν
W ]2
=
pµ pν
m2t 2
p
ig
√
2
2
Z
1
Z
dx(1 − x)
2Nc
0
d4 l
1
.
4
2
(2π) (l − D)2
(3.49)
Doing a direct calculation, we can use the expression from the charged amplitude since
the diagrams used there from figure 3.7 are the same as in figure 3.9. Then the self-energy
[Πµν
W ]2 amounts to
#
"
Z
ig
i
i
d4 l
µν
µ
[ΠW ]2 = √ Nc
γ
(−1)Tr
iΓc (p2 )
4
/
/
(2π)
l +p
2
/ l − mt
"
#
Z
i
ig
d4 l
i
× √ Nc
(−1)Tr
γν
/l + p
(2π)4
2
/ /l − mt
Z
µ /
4
d4 l Tr[mt γ µ (/l + p
ig 2 2
/)]
/)]
2 d l Tr[mt γ ( l + p
√
Nc
=
iΓ
(p
)
.
c
(2π)4 (l + p)2 (l2 − m2t )
(2π)4 (l + p)2 (l2 − m2t )
2
(3.50)
We now have
Πµν
W
= (g µν p2 − pµ pν )ΠW (p2 )
Z 1
Z
ig 2
1
d4 l
µν 2
µ ν
= (g p − p p ) √
2Nc
dx2x(1 − x)
4
2
(2π) (l − D)2
2
0
Z
Z
m2 1
d4 l
1
− 2t
dx(1 − x)
.
4
2
p 0
(2π) (l − D)2
(3.51)
The singularity above at p2 = 0 will shift the mass of the W boson away from the original
mass of zero. This can be seen as follows: the p12 term dominates, and the self-energy can be
1
written as Π = pA2 + B. Then the renormalized propagator is proportional to p12 1−Π
; upon
1
combining these two expressions, we obtain a propagator of the form (1−B)p2 −A . Expecting a
1
2
propagator of the form p2 −m
2 , we see that the coefficient A of the term singular in p is now
the mass, or in other words has shifted the mass away from zero. This is evident from the
1
self-energy part [Πµν
W ]2 in equation 3.50, which contains Γc and thus a singularity in p2 ; the
Nambu-Goldstone boson from Γc has thus been “eaten” by the W boson.
49
From equation 3.51 we can compute the self-energy:
Z 1
m2t
Λ2
ig 2
2
dx(2x − 2 )(1 − x) log 2 − 1
ΠW (p ) = √
2Nc
p
D
2
0
(3.52)
where we used
Z
d4 l
1
4
2
(2π) (l − D)2
Z
Λ
=
0
Z Λ
dl l(l2 − D)
dl
lD
+
2 (l2 − D)2
2 (l2 − D)2
8π
8π
0
0
D
log(l2 − D)|Λ
|Λ
0 −
2(l2 − D) 0
2
D
D
Λ −D
−
+
log
D
2(Λ2 − D) −D
2
Λ
D
log
−1 −
−1
2
D
2(Λ − D)
2
Λ
log
− 1.
D
Z
=
=
=
=
'
dl
l3 2π 2
(2π)4 (l2 − D)2
Λ
(3.53)
Then the renormalized W propagator has the form [9, p28]
µ ν
Z
4
d xe
ipx
µ ν
p p
i(gµν − p p2p )
i(gµν − p2 )
+
−
= 2
Ω|T {Wµ (x)Wµ (0)}|Ω = 2
.
p − p2 ΠW (p2 )
p − m20 − Σ
(3.54)
The mass of the W ± boson is given by the pole of this dressed propagator when it is on shell,
2 .
or when p2 = MW
3.6
Custodial symmetry in the NJL model
Without doing further computations, we will try to derive a relationship between the W and
Z masses. The propagator of the W boson can be written in the form [25, p1650]
1 W −1
pµ pν
1
2
2 2
¯
D (p) = −(gµν − 2 ) 2 2 p − f (p ) ,
(3.55)
g 2 µν
p
ḡ (p )
where f¯ and ḡ are functions of p2 and mt , and ḡ depends on the weak coupling g; both still
2 = p2 , be written as
depend on Λ. The W-boson mass can then on shell, where MW
2
2 ¯2
2
MW
= p2 = ḡ 2 (MW
)f (MW
).
(3.56)
2 for p2 = 0. For a process
From equation 3.55 it can be seen that the function f¯2 (p2 ) ∝ MW
of two fermions to two other fermions mediated by a W boson but dominated by mass, the
entire process can be approximated by a four-fermion interaction with an effective coupling
gW
of the Fermi constant GF ∼ M
. From this one can see that the function f¯ is related to the
W
Fermi constant [25, p1650]:
GF
1
√ = ¯2 .
(3.57)
8f (0)
2
50
This can already constrain Λ. The functions ḡ and f¯ are given by [9, p29]
2
Z 1
1
1
Nc
Λ
−1
=
+
dx 2x(1 − x) log
ḡ 2 (p2 )
g 2 (4π)2 0
D
2
Z 1
Λ
2 2
2 Nc
¯
f (p ) = mt
−1 .
dx (1 − x) log
(4π)2 0
D
(3.58)
(3.59)
√1
2GF
= (246 GeV)2 and mt = 173 GeV, we can combine this with equations
2 = p2 = 0 so that D = (1 − x)m2 . Using
3.59 and 3.57, where we evaluate f¯ on shell at MW
t
R1
Nc = 3 and 0 y log ay dy = 14 (2 log a + 1), this would give
Since we know
1
√
2GF m2t
=
246
173
2
=
4f¯2 (0)
m2t
Z 1
12
Λ2
−1
dx (1 − x) log
(4π)2 0
(1 − x)m2t
2 Z 0
Λ 1
3
−dy y log
−1
4π 2 1
m2t y
2 Z 1
3
Λ 1
dy y −1 + log
4π 2 0
m2t y
3
1 1
Λ2
1
− + log 2 +
4π 2
2 2
4
mt
3
1
Λ
− + log
.
2
4π
4
mt
=
y=1−x
=
=
=
=
(3.60)
Now we have an expression for Λ:
#
246 2 4π 2
1
+
Λ = mt exp
4
173
3
"
2 2 #
1
246
4π
= 173 exp
+
4
173
3
"
≈ 8 × 1013 GeV.
(3.61)
2 , and after that evaluated at M 2 = 0, so it
However, this was first set on shell for p2 = MW
W
should be treated with caution. In addition, remember that this model is an approximation
in which, among other things, mb = 0 and all other quark generations are ignored.
Similar redefinitions of the expressions for the Z boson can be made. Here, one must keep
in mind that this is a mix of the W3 and B particles from the Lagrangian and thus a matrix
will appear in the self-energy. One can here, in a similar way as was done for the W boson,
obtain a relation of the Z boson with newly defined functions:
MZ2 = (f (MZ2 ))2 [(g(MZ2 ))2 + (g 0 (MZ2 ))2 ],
(3.62)
where g(p2 ), and g 0 (p2 ) are functions of the couplings g and g 0 , respectively, and all still
depend on the cutoff Λ and mt . These functions are defined after taking into account the
51
gauge boson self-energies and putting the Lagrangian into its original form again. With this,
a relation between the masses of the Z and W bosons can be found:
2 )
2 (g(M 2 ))2 + (g 0 (M 2 ))2 f¯2 (MW
MW
Z
Z
=
.
(3.63)
2 ))2
MZ2
(ḡ 0 (MW
f 2 (MZ2 )
The difference between f 2 and f¯2 is again the correction to the ρ parameter as with custodial
symmetry [9, p31]. Remember the ρ parameter was defined in section 2.1 by:
ρ=
2
MW
= 1 + δρ,
MZ2 cos θW
δρ 1.
(3.64)
Furthermore, the weak mixing angle is defined as
g
.
cos θW = p
2
g + g02
This means that now we would have
2 (g(M 2 ))2 + (g 0 (M 2 ))2 MW
Z
Z
=ρ
2 ))2
MZ2
(ḡ 0 (MW
and thus
δρ = 1 −
2 )
f¯2 (MW
.
f 2 (MZ2 )
(3.65)
(3.66)
(3.67)
For the top mass we have restrictions from increasingly precise measurements13 . Λ, however,
is a free parameter and can be constrained by the definition of the ρ parameter above. In
this way, anything else defined in terms of Λ in the NJL model and models derived from it
can be constrained by custodial symmetry. Since there is no potential in this theory, a real
symmetry cannot be shown for the NJL model. However, from the expression 3.67 we can
show that δρ 1.
13
The LHC is sometimes called top factory because at the energies reached many tops are produced.
52
Chapter 4
Technicolor
In the previous chapter it was explained that in the Nambu–Jona-Lasinio (NJL-) Model, the
W - and Z-bosons obtain extra degrees of freedom similar to how they do with the Standard
Model Higgs boson. In the NJL-model these extra degrees of freedom come from fermion
bilinears that form condensates as a result of an effective four-fermion interaction. One of
these quark bilinears is then a so-called top-Higgs field (a scalar field), or t̄t, whose corrections
have a pole at p2 = 4m2t and is thus an effective particle with a mass mtop−Higgs = 2mt .
This result is subject to renormalization when effects of other interactions are included [3,
p171]. Three other bilinears, the pseudoscalar t̄iγ5 t and two charged bilinears t̄(1 − γ5 )b and
b̄(1 − γ5 )t, have poles at p2 = 0 and are thus effective particles with zero mass. These are
then the massless Nambu–Goldstone bosons which give the gauge bosons their extra degrees
of freedom (i.e. these are “eaten” by the gauge bosons). Remember that when the gauge
bosons are massless, they carry only 2 degrees of freedom each; they each have the required
three degrees of freedom once they eat the Nambu-Goldstone bosons and become massive [2,
p237]. The W -boson contains (has eaten) a top and bottom; these now constitute some of
its degrees of freedom. This means there must simultaneously exist processes where tops and
bottoms occur on their own and where a W -boson containing them emerges as well. This is
indeed possible, as processes with both quark bilinears and single quarks are valid processes
for similar energy levels.
Top Condensates and the W boson When the existence of the W boson was still
unknown, there was a belief that there was a four-fermion interaction that later turned out
to be mediated by a W-boson between two pairs of fermions, so that it was a two-fermionboson interaction. The Nambu–Jona-Lasinio Model on which technicolor models are based
makes use of such a low-energy effective four-fermion interaction; at higher energies these
may perhaps be two-fermion-boson interactions, with the boson a technifermion.
Technicolor is a strongly interacting theory of electroweak symmetry breaking that, because
of limited knowledge of strongly interacting gauge theories, is modeled on QCD [13, p10]. In
this theory, electroweak symmetry is hidden and mass is generated through new gauge interactions. In contrast to the Standard Model, this theory avoids the hierarchy problem where
parameters or couplings in the theory are vastly larger than what we expect by analyzing
measurements from experiment: technicolor obeys naturalness so that all parameters in the
theory are of “order unity”. The “beautiful strong naturalness” of QCD is hoped to be
53
imitated in technicolor theories with composite scalar bosons [3, p14]. A dynamically generated Higgs, or, in other words, a Higgs generated only through existing interactions (and not
through adding terms by hand), solves the Λ2 divergences by replacing Λ with MEW , which is
the scale of compositeness [13, p12]; Λ is thus no longer relevant for the Higgs mass operator
at the Planck scale, as the Higgs is a low energy effective particle.
Models with a Higgs boson are often called weakly coupled, because their ultraviolet cutoff is at a very high energy, or ΛU V λEW ∼ 4πv [32, p2]. Strongly coupled models have
ΛU V ∼ ΛEW , and give rise not to a Higgs boson but to vector resonances1 with masses
of the order of the electroweak cut-off. The first strongly coupled model of EWSB was
Technicolor, where an asymptotically free gauge theory at the ultraviolet scale with rapid
confinement and chiral symmetry breaking in infrared divergences (thus very much like QCD)
breaks electroweak symmetry at the electroweak scale. The NJL-model can be used as an
approximation to describe the dynamical symmetry breaking in technicolor. A cut-off of the
order of the rho-meson mass M ∼ mρ from the NJL-approximation is used, and the emerging
mass gap is related to the pion decay constant.
Technicolor is thus a non-abelian strongly interacting gauge theory. Any fermion condensate model where the force underlying a fermion condensate that drives electroweak symmetry
breaking is due to a strongly interacting gauge theory is one of the technicolor models [13,
p9]. If one replaces the Higgs Lagrangian LHiggs by a kinetic term of technifermions and
its field tensors, − 41 Fµν F µν + iQ̄T γµ Dµ QT + . . . (the dots represent terms that may have
to be added to avoid anomalies), the expected scale will be of the order of a few TeV or
less. For technicolor, just like for QCD, the chiral flavor symmetries break when a Q̄T QT
condensate forms [13, p10]. For any new gauge interaction to be added to the Standard
Model Lagrangian, it is sufficient that it is asymptotically free and has a global symmetry
that contains the SU (2)L ⊗ U (1)Y symmetry, which is then broken (dynamically) to U (1)EM ,
or the electromagnetic gauge group [13, p12].
4.1
From QCD to Technicolor
The simplest technicolor (TC) model is a scaled up version of QCD. It is an SU (NT C )
nonabelian gauge theory with two Dirac (techni)fermions transforming according to the fundamental representation of the gauge group [13, p13]. If the number of colors in technicolor
is greater than two, or NT C > 2, the flavor symmetries are and are subsequently broken as
follows: SU (2)L ⊗ SU (2)R ⊗ U (1)V → SU (2)V ⊗ U (1)V , just as in chiral symmetry breaking.
The scale of the theory is chosen such that the technicolor pion decay constant is equal to
the VEV of the Higgs: FπT C = 246 GeV.
The analogy between technicolor and QCD can be seen in the properties of technicolor.
If technicolor is a gauge group SU (NT ), it has NT2 − 1 gauge bosons (techni-gluons) and has
techniquarks QiT L and QiT R , with i the flavor index [3, p20]. There are NT f flavors of these,
so that there is a chiral symmetry: SU (NT f )L ⊗ SU (NT f )R ⊗ [U (1)A ] ⊗ U (1)Q , where the
axial anomaly (or chiral anomaly, i.e. the fact that the chiral current is not conserved) breaks
U (1)A , the so-called chiral group. In QCD, this global chiral symmetry of the Lagrangian
is SU (2)L ⊗ SU (2)R ⊗ [U (1)A ] ⊗ U (1)B , where A stands for axial and B stands for baryon
number [3, p20]. Just like in QCD, a condensate of these technifermions forms with a VEV
1
A resonance is a short-lived subatomic particle, or an excited state of the vacuum, that cannot be observed
directly [5, p101].
54
[3, p20]:
Q̄iLT QjRT ≈ Λ3T C δij .
(4.1)
Then the techniquarks acquire constituent masses mdynamic ∼ ΛT C from the mass gap as in
the NJL model. Furthermore, NT2 f − 1 Nambu–Goldstone bosons appear, which will have a
common decay constant of FT ∼ ΛT C . In the NJL model, Nf fermions with each Nc colors
condense whereafter quarks dynamically gain a mass and result in a common decay constant
fπ for the Nf2 − 1 Nambu–Goldstone bosons. In the NJL model the quark condensate is given
by [3, p21]
Nc
Q̄iL QjR ≈ Λ2 δij 2 m0 ,
(4.2)
8π
with m0 the constituent quark mass. In order to dynamically break symmetries, SU (2)L ⊗
U (1)Y must always be a subgroup of the chiral group.
In QCD, the pion decay constant f is defined by
a5 b 0|jµ |π = if pµ δab ,
(4.3)
a
with jµa5 = ψ̄γµ γ 5 τ2 ψ. Here f ∝ 93 MeV. This is then similar for technicolor, where the
a
quarks are replaced by techniquarks to get the techniquark axial current j̃µa5 = Q̄T γµ γ 5 τ2 QT .
The technipion decay constant is then given by
a5 b (4.4)
0|j̃µ |π̃ = iFT pµ δab ,
with the technipion decay constant of the order of the electroweak scale, or FT ∝ vweak ≡
1 √1
= √v2 = 175 GeV with v = 246 GeV the Higgs VEV.
23/4 GF
A technicolor gauge group SU (NT C ) with ND electroweak left-handed techniquark doublets and 2ND right-handed techniquark singlets will form a chiral condensate pairing lefthanded with right-handed fermions. This condensate then in turn breaks electroweak symmetry and (2ND )2 − 1 Nambu–Goldstone Bosons, which are in this case technipions π̃, appear
with decay constants FT [3, p23]. Using a set of scaling rules that are used to characterize
QCD [3, p21],
p
f ∼ Nc λQCD
Q̄i Qj ∼ δij Nc Λ3QCD
m0 ∼ ΛQCD ,
(4.5)
with m0 the constituent quark mass, one can set out to approximate similar values for technicolor,qa theory
based
on QCD. FT can be estimated from the QCD pion decay constant:
FT ∼
NT C
3
ΛT C
ΛQCD
f . From the kinetic term of techniquarks, a chiral Lagrangian that
describes the techniquarks with a nonlinear σ model2 can be written:
/ T L + Q̄T R iDQ
/ TR →
Q̄T L iDQ
FT2
Tr[(Dµ U )† (Dµ U )]
4
(4.6)
√
one can see that with ND doublets of these techniquarks the scale changes
ND FT .
√ to v =
3
The mass gap is now, using the relations in equation 4.5, mT C ∼ m0 v f √N N , with m0 the
T D
constituent quark mass in QCD (which can be approximated as a third of the neutron mass)
[3, p25].
2
See appendix D for more on the (non)linear sigma model.
55
4.2
Minimal Model of Susskind and Weinberg
One example of a technicolor model is the Minimal Model of Susskind and Weinberg. It
predicts techniquarks, techniaxions, technimesons, and techni-jets (at very high energies),
and is symmetric under the gauge group SU (NT ) × SU (3) × SU (2)L × U (1)Y . At least one
flavor doublet of color singlet technifermions, say (T, B), is included which forms two chiral
weak doublets (T, B)L,R [3, p24]. Here (T, B)L have isospin one half and transform as a
double under SU (2)L and will have hypercharge Y = 0, and (T, B)R become a pair of singlets
under SU (2)L and have hypercharge Y = 1, −1. The doublets can carry a technicolor index
a to indicate one of the NT copies of the technifermion doublet. Techni-vector mesons are the
lowest-lying resonances that can provide an obvious signal of the minimal-model physics [3,
p33]. Vector meson dominance, or VMD, allows the coupling between techni-fermion mesons
and ordinary fermions (direct coupling is absent). Vector mesons such as the technicolor
analog of the ρ-meson ρT are phenomenologically important for technicolor because of the
decay to weak gauge bosons and technipions. Technipions are the Nambu–Goldstone bosons
that appear in technicolor theories. For example, the QCD process ρ → ππ has a technicolor
analog of ρ → W W . Technipion pair production also receives large contributions from vector
mesons [3, p29]. Just like in the Nambu–Jona-Lasinio Model, all three Nambu-Goldstone
bosons are absorbed into the longitudinal modes of the electroweak gauge bosons.
4.3
Extended Technicolor (ETC)
Various particles are predicted by technicolor that are not observed in nature. To solve
that problem, the decay of techniquarks into the light observed quarks and leptons must be
provided for by an extension of technicolor; this precisely what Extended Technicolor does
[3, p20]. Extended technicolor also addresses the flavor problem, so it provides for quark
and lepton masses and weak mixing angles, as well as CP-violation [3, p57]. The model of
extended technicolor is symmetric under the gauge group GET C , which has generators that
form a Lie group [3, p58], and which is a unification of SU (3) color, SU (NT C ) technicolor and
flavor symmetries [1, p7]. Extended technicolor, although it also addresses the flavor problem
described above, has problems of its own. It cannot explain the large top quark mass. For
this, topcolor was invented in the 1990s [1, p15, 16].
4.4
Walking Technicolor
Extended technicolor has trouble to explain the heavy quark and lepton masses. This is where
walking technicolor comes to the rescue. Coupling constants of theories have different values
at different energy scales. Renormalization is used to describe the flow of couplings, and the
changing of the coupling with energy is called the running coupling. A function often called
γ describes a shift in field normalization with a change in energy scale; this function is also
sometimes called the anomalous dimension [5, p411, p. 427]. The anomalous dimension is
actually γ(λ∗ ) in the equation for the propagator for values of momentum p for which the
running coupling λ̄(p) is close to the fixed point λ∗ [5, p427]:
G(p) ≈ C
1
p2
56
1−γ(λ∗ )
(4.7)
Figure 4.1: A running QCD coupling and a walking coupling, both as a function of momentum. Note
that the QCD coupling tends to zero at high energies, which is why the theory is called asymptotically
free. The lower plot shows the evolving beta function as a function of a walking coupling. Figure
taken from [13].
A fixed point is a point where the beta function, or the function describing the running of the
coupling, vanishes [2, p340]; at this point the theory is scale-invariant. In walking technicolor,
the coupling does not run to the ultraviolet fixed point of technicolor but “walks” slowly, so
that the fermion-antifermion condensate is enhanced with respect to its earlier value [13, p20]
(see figure 4.1). A theory with a conformal fixed point, or a fixed point where the theory
is not only scale-invariant but also conformally invariant3 , can achieve such “walking” [13,
p20]. Technicolor adds a full generation of technifermions to the existing Standard Model
generations. As a result, the S-parameter (from the Peskin-Takeuchi parameters) is about
1.65, which clearly excludes technicolor [19, p66]. Walking technicolor has smaller or even
negative values of S, so this theory is not excluded through the Peskin-Takeuchi parameters.
4.5
Other models
Another example of a technicolor model is the Farhi-Susskind model in which the full generation of quarks and leptons is imitated. This is a more general TC model with a low energy
spectrum containing numerous pseudo-Nambu-Goldstone bosons. There are three electrically
0
neutral orthogonal color singlet objects [3, p38]: the P 0 , P 0 , and ηT0 . Of these, the former
receives mass through QCD instantons and ETC effects, and the first two could be visible
0
in gg → P 0 → γγ and a similar decay for P 0 . Much more can be detected, among which
color-triplet and -octet pseudo-Nambu-Goldstone bosons; production and detection at e+ e−
colliders is also possible [3, p42-56].
4.6
Problems with Technicolor
Technicolor is, like QCD, asymptotically free, and therefore solves the naturalness and hierarchy problems [1, p6]. Extended technicolor also addresses the problems of flavors, such
as why there are multiple generations and the explicit breaking of quark and lepton chiral
symmetries [1, p6]. However, technicolor itself is also threatened by problems. The flavorchanging neutral current problem, in which quark and lepton and technipion masses 10-1000
times too small are obtained, is solved by walking technicolor, which has a slowly running
3
Conformal invariance is, at the classical level, a consequence of scale and Poincaré invariance. A quantum
field theory, however, has a regularization prescription that introduces scale into the theory. This scale then
breaks conformal symmetry, except at the renormalization-group fixed points [33, p99].
57
gauge coupling [1, p12]. Furthermore, electroweak precision measurements, which are defined
by the parameters S, T , and U [20, p390], are violated by technicolor. The parameter S, a
measure of the splitting between MW and MZ induced by weak-isospin conserving effects, is
experimentally found to be negative, but calculated to be positive for Nc -color technidoublets
[1, p13].
58
Chapter 5
Topcolor and Topcolor-Assisted
Technicolor
The top quark is the heaviest particle found until now (the Higgs has not yet been found)
with a mass of 172.9 ± 1.5 GeV/c. It forms the third generation of quarks together with the
bottom quark. Topcolor is a model of electroweak symmetry breaking in which a top quark
and an antitop quark form a top quark condensate. Excitations of this condensate field would
then act effectively as a Higgs boson, also called the top-Higgs. This is analogous to BCS
theory in superconductivity. BCS Theory, or Bardeen-Cooper-Schrieffer Theory, describes
superconductivity as a microscopic effect of pairs of electrons (fermions) that condensate into
boson-like states. Unlike in technicolor, there are no new (techni)quarks in topcolor, as the
top quark itself accounts for electroweak symmetry breaking. Topcolor rescues Technicolor
from some difficulties in ’Topcolor-assisted Technicolor’. It is interesting to note that the
original ideas of the phenomenology of topcolor come from Bardeen, Hill, and Lindner [25],
with Bardeen the son of the Bardeen from BCS theory.
Topcolor is also called a top condensate model. The Yukawa coupling in this theory is
of order unity so that strong coupling dynamics is possible. The Yukawa interaction is an
interaction between a scalar field and a Dirac field to describe the strong nuclear force between
nucleons (fermions) mediated by pions (pseudoscalar mesons) and to describe the coupling
between the Higgs field and the massless quark and electron fields. The basis of top quark
condensate models, namely the Nambu–Jona-Lasinio Model, is discussed in chapter 3. The
NJL-model is an approximation to new strong dynamics that causes electroweak symmetry
breaking.
To see whether the top condensate is one that could really replace the Higgs as a top-Higgs,
let us look at he quantum numbers that it has. The top-Higgs field t̄t is electrically neutral,
so the electromagnetic gauge is not broken. Since the top-Higgs field can be rewritten as
t̄t = t̄L tR + t̄R tL , and the hypercharges of tL , tR are 31 , 43 , respectively (which will be negative
for the antiparticles), one can see that the top-Higgs field has a hypercharge of Y = ±1, just
like the Higgs field and its charge conjugate [7]. Furthermore, the condensate is an isospin
doublet and has isospin I = 21 since this is the isospin carried by tL , t̄L (the right-handed
particles are, as before, unaffected by isospin) [7]. This satisfies our expectations of the
quantum numbers of a different (effective) particle replacing the Higgs scalar.
The four-fermion interaction term that was introduced in chapter 3 can be rewritten using
59
the Fierz identities1 as follows [34, p419]:
−
A
λA i
g2
g2
a
b i
µλ
(
ψ̄
t
)
(
t̄
ψ
)
=
(
ψ̄
γ
ψ
)(
t̄
γ
tR ) + O(1/Nc ).
ra i Rb
iL µ
R
M2 L
M2
2 L
2
(5.1)
A massive color octet vector boson exchange induces exactly this interaction term at lower
energies [3, p111], so QCD could be treated as a subgroup of SU (3)3 ⊗ SU (3)1/2 with the
first acting on the third generation of quarks and the second group on the first and second
generations. However, something is needed to prevent a bottom quark condensate from
forming.
Topcolor in a nutshell In a top condensation model, at some scale M that is associated
with a new force called topcolor, the top forms a bound state denoted by Ht ∼ t̄t. Topcolor
acts on the top and bottom quarks only. When integrating out massive topcolor gauge bosons,
the leading operator for strong topcolor is, similar to what was done in the NJL model (see
also chapter 6 and equations 3.37, 3.38), [35, p5]
− g 2 t̄t
1
g2
1
t̄t
→
t̄tt̄t ∼ g t̄tHt − m2H Ht2 ,
2
2
2
p −M
M
2
(5.2)
Where Ht = Mg 2 t̄t and mH = M
g for an energy scale M . The doublet Ht then develops a
vacuum expectation value [36, p5]. Excitations of this field Ht are the effective top-Higgs
particles, acting like a Higgs boson.
5.1
Dynamics of topcolor
In minimal topcolor, the third generation (t and b) couple to an SU (3) ⊗ U (1) gauge group
which we will give the index 3, and the first two (u, d, c and s) couple to another copy of
this gauge group which we will give the index 1/2. The symmetry of topcolor SU (3)1/2 ⊗
SU (3)3 ⊗ U (1)1/2 ⊗ U (1)3 is then broken down by a vacuum expectation value of a scalar
field hΦi to SU (3)QCD ⊗ U (1)Y , giving the so-called colorons a mass [34, p3]. This field Φ
is thus charged under all these groups, and breaks the groups down to a diagonal subgroup
which comes from both original groups2 .
Let us study SU (3)1/2 ⊗ SU (3)3 → SU (3)QCD , where this is broken by the field Φ. The
charges under the first two of these groups are then as follows [34, p3]:
• Left-handed quarks (uL , dL ), (cL , sL ) : (3, 1)
• Left-handed quarks (tL , bL ) : (1, 3)
• Right-handed quarks uR , dR , cR , sR , bR : (3, 1)
• Right-handed quark tR : (1, 3)
• “Higgs” field Φ : (3, 3̄)
1
Fierz identities are interchange relations of Dirac bilinears.
(ū1L γ µ u2L )(ū3L γµ u4L ) = −(ū1L γ µ u4L )(ū3L γµ u2L ) [5, p805].
2
One important Fierz identity is
The same happens in QCD where in chiral symmetry breaking SU (2)L ⊗ SU (2)R
with the latter group coming from both original groups.
60
hq q̄i=vQCD
→
SU (2)v,diag ,
• All leptons transform as singlets
In order to cancel anomalies3 , another electroweak singlet quark must be added [34, p3]:
• Left-handed quark QR : (1, 3)
• Right-handed quark QL : (3, 1).
The Φ field has nine components. When this field takes the vacuum expectation value of
diag(M , M ), eight components of Φ are absorbed by colorons, the massive gauge bosons of the
residual global SU (3)0 . This is similar to the Standard Model gauge groups of SU (2)L ⊗U (1)Y
that is broken to U (1)EM , where the photon and the Z boson are linear combinations of the
original bosons: now the colorons and the gluons as we know them are linear combinations of
the gauge bosons of the two original groups. Let us call the gauge bosons of the original groups
Aµ1/2 and Aµ3 , and the gauge bosons of the SU (3)QCD group Gµ , and the gauge bosons of the
residual global4 symmetry Hµ (which are orthonormal to the gluons). These last two are then
linear combinations of the first two, and the other way around: for some angle θ [34, p4] , (in
order to have the coefficients of the linear combination normalized, since cos2 θ + sin2 θ = 1)
we have:
Aµ1/2 = cos θGµ − sin θHµ
Aµ3
= sinθGµ + cos θHµ
(5.3)
(5.4)
The new quark Q must be sufficiently massive such as to not influence dynamical symmetry
breaking [34, p3]. In order to have the top quark direction favored for the condensate to form,
the angle must be small: θ 1. Such is also necessary because there are only two quarks
coupling to one, and four quarks coupling to the other gauge group; this balance needs to
be restored. What is wanted is that the original gauge bosons of the first two generations
contain much of the gluons, and the original gauge group of the third generation contains
much of the colorons.
Electroweak symmetry is broken by the vacuum expectation value of the effective topHiggs field hHt i = fπ , with fπ the top-pion decay constant. The top-pion decay constant is
estimated using the Pagels-Stokar formula5 .
3
An anomaly is a symmetry that holds at the classical level but is broken at the quantum level, or, in
other word, broken by quantum corrections. Any new theory should be anomaly free or have, just like in the
Standard Model, its anomalies cancelled out. For example, for a diagram
of three vertices with each vertex
P
i, i ∈ {a, b, c}, carrying a coupling xi , the following should vanish: A L−Rfermions Tr(xa (xb xc − xc xb ), which
means the charges of the left-handed fermions under the gauge group considered should cancel with the charges
of the right-handed fermions in this gauge group. For example, for SU (3)QCD , the xi would be the Gell-Mann
generators.
4
This must be a residual global symmetry, since the bosons are massive. For a nonabelian gauge transformation (and SU (3) is nonabelian), a gauge field transformation carries a space-time dependent derivative,
which is a local symmetry. Mass terms break the symmetry for transformations where this derivative does not
vanish, so that the local symmetry vanishes and only a global symmetry is left.
5
The Pagels-Stokar formula is obtained through renormalization group flow. This can be seen as√follows:
when the top Higgs takes on a vacuum expectation value so that the real part of Ht becomes (v + ht )/ 2, with
√
ht the top Higgs field representing fluctuations, we see in equation 5.2 that t takes on a mass of mt = gv/ 2.
16π 2
If the coupling g is then renormalized to the coupling g̃ 2 = g 2 /ZH = Nc log(Λ
2 /µ2 ) [3, p169] , with ZH the
√
renormalized coefficient of the kinetic term of the auxiliary field Ht , the relationship mt = gv/ 2 becomes the
2
2
Nc
Pagels-Stokar formula: 12 v 2 = m2t /g̃ 2 = m2t 16π
2 log(Λ /µ ), with µ the new energy scale and Λ the cutoff [3,
p171].
61
Problems A problem with topcolor is that the electroweak symmetries
broken dynam√ are−1/2
ically by the top quark mass. This would mean that fπt = vW = (2 2GF )
≈ 175GeV.
Nc
Λ2
2
For a top-pion decay constant given by the Pagels-Stokar formula, fπt = 16π2 mt (log m
2 + k)
t
and for Λ ∼ 1.5TeV, k = 1, the top mass would be 900 GeV [36, p1]. This would mean either
fine-tuning or new dynamics is needed. In topcolor-assisted technicolor electroweak symmetry
is broken also by the technicolor-Higgs-field hHi = fT , with fT the technipion decay constant.
2 = f 2 + f 2.
The electroweak vacuum expectation value is then vew
π
T
5.2
Topcolor-Assisted Technicolor
In Topcolor-Assisted Technicolor, or TC2, there are two sources of electroweak symmetry
breaking, or three if QCD is included [35, p3]. The model postulates a dynamical quark mass
component for the top quark, (1 − ε)mt , and a fundamental mass component εmt mt [3,
p111]. The latter is generated by Extended Technicolor or a fundamental Higgs boson, and the
former is generated by Topcolor dynamics. For a pure top condensate model we have ε = 0;
for nonzero ε the ht̄ti condensate does not do all of the electroweak symmetry breaking. This
would help the fact that the top quark is too light to produce the full electroweak condensate;
it would need to be of the order of 600 GeV in order for electroweak symmetry breaking to
be identified naturally with a t̄t condensate [3, p134] 6 . The extended technicolor component
of the top mass is expected to be of the order of the bottom mass, or εmt ∼ mb . There will
appear three Nambu–Goldstone bosons from the ETC symmetry breaking, and three from
top condensation. These form orthogonal linear combinations of pseudo-Nambu–Goldstone
bosons π̃ a , a ∈ {1, 2, 3}, which are absorbed in the longitudinal degrees of freedom of the
W ± and Z bosons (π ± = π 1 ± iπ 2 go into W ± , and π 0 = π 3 goes into Z 0 ). The π̃ a are the
so-called top-pions, and have a mass proportional to ε. Because the decay mode t → π̃ + + b
is absent, the top-pions are phenomenologically forbidden to have a mass below 165 GeV [3,
p112]. Topcolor-assisted technicolor is discussed in more detail in chapter 6.
5.3
Top Seesaw
Another top condensation model is Top Seesaw, in which a seesaw mechanism is used to tune
the physical mass of the top quark from half of the mass gap of 600 GeV to its experimental
value of 175 GeV. If the top quark is 300 GeV to start with, it is heavy enough to account
for electroweak symmetry breaking through condensation; however, it is not this heavy, so it
cannot account for electroweak symmetry breaking on its own. The tuning of the top quark
mass in order for it to agree with experimental values is done through a new mixing angle.
The predictivity of the top quark mass is then given up [3, p134]. Top Seesaw does not use
technicolor; it replaces technicolor entirely with topcolor.
In the Top Seesaw Model, two particles are added, namely isosinglets of vector-like quarks
χL and χR , which are analogs of tR . The bound state Higgs boson that then forms is t̄L χR
with a mass or the order of 1 TeV. Although it seems this would be ruled out by the PeskinTakeuchi parameter constraints, the χ-fermions contribute to a large T -parameter. The χR
6
2
This one can see from the Pagels-Stokar formula from which this was calculated in [3, p109]: fπ2 = vwk
=
√
−1/2
the vacuum expectation value
+ k) (compare also to equation 3.40), with vwk = ( 2GF )
2
Nc
m2t (log M
16π 2
m2
t
of 246 GeV (see also equation 1.88), k ∼ 1 associated with the high energy theory and M the natural cut-off
of 1 TeV.
62
Figure 5.1: Decay of a composite Higgs particle through a Standard Model fermion loop. Figure taken
from [13].
quark has a hypercharge of 43 , so it is indistinguishable from a right-handed top quark. The
χ̄R tL
Higgs doublet field is then
. The masses of χ come from mixing of tR and χL :
χ̄R bL
¯ tL
0
µ
tR
which is then diagonalized.
χL
mt Mχ )
χR
5.4
Phenomenology of Top Condensate Models
Most top condensate models include, as described above, a top-Higgs state. One would expect
a top-Higgs to be of mass mHt = 2mt ' 350GeV from the large N /NJL approximation;
however, contributions from QCD interactions may lower this, and subleading N contributions
may also contribute giving a top-Higgs mass ranging from 200 to 600 GeV [37, p3]. A topHiggs with a mass of less than 300 GeV is excluded if the top-pions have a mass of at least 150
GeV [37, p2]. A top-Higgs decays as in figure 5.1, and couples more strongly to the top quark
than a Standard Model Higgs [37, p2]. Top Higgs decay modes through vector bosons such as
W W and ZZ are also possible. If the branching ratio of Ht → W W/ZZ is sufficiently large,
a top-Higgs could be detected at the LHC [37, p4]. This branching ratio is model-dependent,
as it depends on the top-pion mass. The top pion mass is expected to be less than of the
order of one TeV because the particle is in the electroweak symmetry breaking sector, but its
exact mass in topcolor-assisted technicolor depends on the amount of top-quark mass arising
from the technicolor sector [37, p4].
There are two ways to produce vector boson pairs at hadronic or leptonic colliders: annihilation of a light fermion/antifermion pair and gluon fusion. Through Vector Meson Dominance (VMD) techni-vector mesons such as ρT can couple to currents of ordinary fermions [3,
p33]. Vector Meson Dominance is phenomenology that takes into account tree-level exchanges
of vector meson resonances in the Goldstone scattering amplitudes and in their coupling to
the Standard Model gauge fields [32, p8]. Chivukula and others found that a large class of
technicolor and top-color assisted technicolor models that contain colored technifermions have
already been excluded by the first set of LHC data, and only models with high technipion
masses are still viable [38, p13]. More about the vector meson ρt is discussed in chapter 6.
If the cut-off scale Λ used in the models with a four-fermion interaction using the Nambu–
Jona-Lasinio model is much higher than the electroweak scale, a perturbative renormalization
group analysis incorporating Standard Model gauge interactions yields reliable values for the
top quark and Higgs boson masses. The appearance of composite Higgs doublets at the scale
Λ is incorporated in the compositeness condition, used as a boundary condition at the Λ
63
Figure 5.2: The four-fermion interaction can be seen as an effective interaction of, for example, a
two-fermion-boson interaction.
scale in the renormalization group analysis [9, p88]. The compositeness condition means that
the couplings of the terms ∂µ φ∂ µ φ and φ4 of the Standard Model Lagrangian are zero [9,
p80]. This then becomes a boundary condition for renormalized couplings at the scale Λ [9,
p81]. The masses predicted by the theory are at a scale of Λ = 1019 GeV mt = 218 GeV and
mH = 239 GeV. These masses are only larger for larger scales of Λ; large Λ gives a fine-tuning
problem just like in the Standard Model. If these particles are very heavy, it could be that it is
an effective 4-fermion interaction of twice a 2-fermion-boson interaction, as depicted in figure
5.2. A heavy Higgs, like a composite top-Higgs, is compatible with electroweak precision data
for a large T parameter (one of the Peskin-Takeuchi parameters) [13, p7].
Neutral top pions that decay mainly to t̄c through flavor mixing could be detected at the
LHC [3, p123]. The abundance of this decay is governed by a model-dependent parameter
Utc that depends on matrices that diagonalize the up-quark mass matrix. Han and others
found that triple top production together with top charge asymmetry is a good probe for
top-color-assisted Technicolor [39, p11]. More about this is discussed in chapter 6.
Angular and radial excitations (particles) resulting from topcolor couple mostly to the topHiggs state tt̄ before the breaking of electroweak symmetry. After this symmetry is broken,
the right-handed top can mix with the right-handed charm and right-handed up quarks. The
right-handed top quark can freely mix without violating flavor bounds. Top-pions, top-Higgs,
and top-rho can contribute to tt̄ production, and can explain an observed forward-backward
asymmetry in this production [35, p3]. More on how topcolor-assisted technicolor can explain
forward-backward asymmetry in top quark pair production is discussed in the next chapter.
In appendix E a way to study the phenomenology of a general model with a scalar,
pseudoscalar, or vector Higgs (whether or not composite) is explained. This method will not
be pursued further in this thesis.
5.4.1
Top Seesaw Phenomenology
In a model that combines Minimal Walking Technicolor with top seesaw, Fukano and Tuominen found that this model is still viable when taking into account LHC data [40, p26]. A
working group at a conference in Les Houches in 2011 found that a composite Higgs could be
produced through exchange of a heavy gluon between a t̄t pair that then decays to a fermion
resonance, which then in turn decays into a Higgs and a top quark [41, p147]. Also, Chivukula
and others wrote that the top seesaw assisted technicolor theory is still a viable theory with
a large top-Higgs mass. [37, p7]. At higher energy colliders, direct pair production of the χ
states could produce six top states that can serve as a signature of the top seesaw theory [3,
p149]. The extra vector-like fermions of top seesaw allow the electroweak precision parameters
S and T to be completely consistent with data [3, p68]. The phenomenology of top seesaw
will not be further pursued in this thesis.
64
Chapter 6
Forward-Backward Asymmetry of
the Top Quark
Top condensation theories could give an explanation of the observed forward-backward asymmetry of the top quark [35]. Forward-backward asymmetry (AF B ) at the Tevatron ([42], [43],
[44]) is the difference between the number of events where a particular final-state particle
moves forward with respect to a chosen direction (NF ), and the number of events where this
final-state particle moves backward (NB ) divided by their sum:
AfF B =
NFf − NBf
NFf + NBf
,
(6.1)
where f stands for fermion. Alternatively, forward-backward asymmetry of a fermion is
defined in terms of the cross section:
AfF B
f
σFf − σB
f
σFf + σB
(6.2)
[19, p48]. It is used, for example, by LEP to measure the difference in interaction strength
between ZψL and ZψR (with ψ a fermion), which can give a precision measurement of θW
[19, p50]. Large top-up couplings would lead to a top forward-backward asymmetry through
t-channel exchange [35, p1], and top condensation models can provide particles with large topup couplings [35, p3]. After the electroweak symmetry is broken by the top quark condensate,
the Lagrangian can be written as a low-energy effective Lagrangian. A term of the form
t̄L uR πt0 arises in the effective Lagrangian of topcolor-assisted technicolor from which the
cross section of the same-sign top product can be calculated. This can be seen by looking
at the Lagrangian from equation 3.38 from section 3.4. This expression contains the term
gt0 (ψ̄L tR Ht + h.c.) that can give rise to top and up quark mixing as a result of the mass
matrix for these. Top-up mixing could possibly explain forward-backward asymmetry of the
top quark.
6.1
Effective Lagrangian
The top-Higgs doublet can be approximated just as in the Standard Model by its vacuum
expectation value (now v = fπt ) plus a fluctuation (in this case ht ), from which an effective
65
linear sigma model can be written1 . A sigma field Σ = exp
matrices, and
doublet:
πta
iπ a τ
√t a
2fπt
, with τa the Pauli
the Goldstone bosons or top-pions, can be used to represent the top-Higgs
!
a fπt + √12 ht
iπt τa
Ht = exp √
.
(6.3)
0
2fπt
This can be approximated as follows:
!
!
a fπt + √12 ht
fπt + √12 ht
iπt τa
iπta τa
+ ...
exp √
=
1+ √
0
0
2fπt
2fπt
!
fπt + √12 ht
i fπt
1 a
'
+ √ πt τa
fπt i
0
2
!
fπt + √12 ht
i fπt
1
πt3
πt1 − iπt2
=
+√
−πt3
fπt i
0
2 πt1 + iπt2

 0

!
π
−
i  fπt  √t2 πt  fπt + √12 ht
=
+
π0
fπt
i
0
πt+ − √t2
 0

!
!
iπ
iπt−
fπt + √12 ht
fπt + √12 ht
1  √2t

=
+
0
fπt iπt+ − iπ
0
0
√t
2
!
!
iπt0
fπt + √12 ht
√
ht
2
=
+
+ O(
)
+
fπt
0
iπt
!
fπt + √12 (ht + iπt0 )
+ ...
=
iπt+
√
2
where πt0 = πt3 and πt± = √12 (πt1 ± iπt2 ). It can be seen that, since Ht was complex, it is now
written as a + ib with a, b real, so ht and πt0 are real.
Let us denote the third generation doublet by T = (t, b), the second by C = (c, s), and
the first by U = (u, d). The effective (Sigma) Lagrangian contains a term, coming from the
last term in equation 3.38, that generates interactions with top-pions [35, p9]. For a coupling
λt associated with the Yukawa term of the doublet T and tR , these interaction terms are:
LΣ = gt0 (ψ̄L tR Ht + h.c.)
= λt T̄L Ht tR + h.c. + other terms
!
fπt + √1 (ht + iπt0 )
2
= λt t̄L b̄L
tR + h.c. + o. t.
iπt+
1
0
= λt t̄L fπt + √ (ht + iπt ) tR + h.c. + bL . . .
2
λt
= √ (fπt t̄L tR + t̄L iπt0 tR + t̄L ht tR ) + h.c. + other terms.
2
1
See appendix D for more on the linear sigma model.
66
(6.4)
From the first term above it can be seen that the top mass generated by topcolor is mt =
λt
√
f , so that the top quark mass is now related to the toppion decay constant. The last two
2 πt
terms result in a mix of the t and u quarks after diagonalizing the Yukawa couplings, and
subsequently contribute to forward-backward asymmetry of the top quark. This diagonalization of the Yukawa couplings, in order to obtain diagonal mass matrices, is discussed in the
next section.
6.2
Mass matrices and mass eigenstates
Through the breaking of the electroweak symmetry by both the technicolor Higgs (with a
VEV of hHi = fT ) and top-Higgs (with a VEV of hHt i = fπt ), effective mass matrices of
the quarks, the precise form of which is shown below, appear in the Lagrangian. The entries
of these mass matrices come from two types of “Yukawa” terms. The Yukawa terms of the
first type consist of thetechnicolor
Higgs
field H and a mix between the first two generations
u
c
U and C (with U =
and C =
). Those are then terms such as YuC CL HuR and
d
s
YsU UL HsR .
The Yukawa terms of the second type consist of a mix of the third with the other two
generations. These must be invariant under U (1); this is solved by containing both the
technicolor Higgs field H and the topcolor-breaking field Φ in those terms, as well as factors
of det Φ. Hypercharges then add up to zero: the left-handed third generation (tL , bL ) and
first and second generations have Y3 = 31 , Y1/2 = 13 , respectively; the right-handed generations
4
2
3 , − 3 ; then the H and Φ carry together the opposite charges under U (1)3 and U (1)1/2 , since
the third and first/second generations do not cancel each other’s hypercharge. The Yukawa
terms are furthermore constrained by the fact that the operators must be at least dimension
5 [35, p9]; as fermions are dimension 32 and Φ and H are of mass dimension 1, this comes
out correctly. The reason for higher dimensions is that higher-dimensional operators affect
oblique parameters, or the Peskin-Takeuchi parameters S and T ; these must be included by
strong dynamics models in order to avoid claims that these models are ruled out by S and T
bounds [3, p.67]. Taking the symmetry and dimensions into account, the Yukawa terms then
Φ det Φ
†
look like cT c T̄L HcR Φ
Λ for doublets of the third generation and cU b ŪL H bR Λ Λ3 for the first
and second generations, with det Φ ≡ 16 εijk εlmn Φil Φjm Φkn [3, p121].
When Φ takes a vacuum expectation value, entries in mass matrices incorporating these
Yukawa terms, where Y denotes mixing of the first two generations and c a mix of the the
third with another generation, depend on M
Λ , with M = hΦi the top-gluon mass (of the
order of 1 TeV) and Λ the energy scale of the operators of the mixing of generation 3 with
generations 1 and 2. Here the scale Λ should be larger than the vacuum expectation value
of Φ, so Λ > hΦi = M and we define M
Λ ≡ ε. Breaking of the topcolor and electroweak
symmetries (so that H → hHi = fT ) then gives mass matrices [35, p10]:

YuU YuC
cU t ε4
cCt ε4  ,
MU = fT  YcU YcC
t
cT u ε cT c ε cT t ε3 + m
fT


YdU YdC cU b ε4
MD = fT  YsU YsC cCb ε4  ,
cT d ε cT s ε cT b ε3

(6.5)
from which it can be seen that the top quark mass is partially generated by the topcolor top
quark condensate. In order to have a bottom quark mass that is not too small and a top
quark mass that is generated by topcolor, as well as to have mixing of tR with generation 1
67
and 2 of O(1), it is assumed that ε ' 0.3; furthermore, it is also assumed that O(ε3 ) can be
neglected [35, p10].
The effective mass matrices can be diagonalized just as it was done with the fermion mass
matrices for the Standard Model in section 1.2.4. If the fermion mass matrices are given by
MU and MD , then we diagonalize these with matrices V as follows:
MU = VLu MUdiag VRu† ,
diag d†
MD = VLd MD
VR ,
(6.6)
where the matrices V are constrained by 1) the fact that they are unitary, and by the form
of the CKM matrix2 K as from experiment: K = VLu VL†d . This is the same as was done in
section 1.2.4:
~ūL MU ~uR + d~¯L MD d~R = ~ūL VLu† VLu Mu VRu† VRu ~uR + d~¯L VLd† VLd MD VRd† VRd d~R
= ~ū M diag ~u
+ d~¯ M diag d~ ,
L,m
R,m
U
L,m
D
R,m
(6.7)
where m denote the mass eigenspinors. These are composites, in that for example each entry
of ~uR,m = VRu ~uR is a linear combination of uR , cR and tR . So in going back to the mass basis,
the right handed top quark is then rotated as [35, p11]:
Ru
Ru
Ru
tR → V11
tR + V21
cR + V31
uR .
(6.8)
This rotation changes the term in equation 6.4 as follows:
λ
Ru
√t iπt0 t̄L V31
uR + other terms
2
λt fπt 0
√ iπ t̄L uR + o.t.
≡ g13πt
mt 2 t
= gutπt iπt0 t̄L uR ,
λ
√t t̄L iπt0 tR =
2
f
(6.9)
λ
where the mass of the top quark is defined as mt = π√t 2 t and g13πt = gutπt . Then the other
term t̄L ht tR from 6.4 changes to gtuht ht t̄L uR . Since the Yukawa couplings were not fixed from
the beginning and are unknown, the exact diagonalization of the mass matrix and the relation
between the Yukawa couplings and the new coupling gutπt will not be calculated here. For the
purpose of studying the impact of different masses of the new particles on forward-backward
asymmetry of the top-quark (see section 6.6), the couplings will, for practical reasons, be set
33 , is taken to be m .
equal to unity. The entry for the bottom quark mass, or MD
b
6.3
Vector Resonances
In the Standard Model, chiral perturbation theory is an effective theory that is consistent
with the chiral symmetry of QCD as well as parity and charge conjugation. A model where
both condensates of quarks (such as mesons) and single quarks exist and interact at the same
time results from chiral perturbation theory. With this theory, low-energy dynamics of QCD
can be studied. In the perturbations of QCD, however, it is impossible to calculate coefficients
2
The Cabibbo-Kobayashi-Maskawa matrix is a unitary matrix which contains information on the strength
of flavor-changing weak decays. It is used to diagonalize the Yukawa couplings. This causes mixing of the
quark flavors [19, p4].
68
of higher orders in the perturbation [45, p184]. Therefore, phenomenology is used to further
describe low-energy effects of QCD. Side effects of vector mesons, in particular those of the ρ
meson, are phenomenologically the most important [45, p184]. Already before the Standard
Model was born, so-called vector meson dominance, or the idea that vector mesons play an
important dynamical role, was known [45, p185]. The linear sigma model (as described in
appendix D) does not have a ρ meson (only a scalar) and does not agree with data. Deviations
from the lowest order chiral relation must be such that low-energy effects of light resonances,
in particular the ρ meson, must be reproduced. This has broad phenomenological support
but cannot be proven [45, p185].
Since topcolor, technicolor, and topcolor-assisted technicolor are based on QCD, it is expected that again vector mesons are phenomenologically important in the low-energy effective
theory. The tt̄ condensate is thus expected to couple to excitations of itself, the lightest of
which is the top-rho ρt (analogous to the ρ-meson in QCD) [35, p11]. It can contribute to
forward-backward asymmetry of the top quark if it is less than or equal to one TeV. Its
Lagrangian can be written as [35, p12]:
Lρ = gttρ ρµ t̄γ µ t + gtcρ ρµ t̄R ϕµ cR + gtuρ ρµ t̄R γ µ uR .
(6.10)
The last term in the above can give rise to forward-backward asymmetry. We have now three
terms possibly causing AtF B :
LAtF B = gtuπt iπt0 t̄L uR + gtuht ht t̄L uR + gtuρ ρµ t̄R γ µ uR + h.c.
(6.11)
For simplicity, we set the top-Higgs and top-pion couplings to a constant gπt , and we define
gtuρ ≡ gρ . Then the interactions of the Lagrangian simplify to
LAtF B = gπt iπt0 t̄L uR + gtuht ht t̄L uR + gρ ρµ t̄R γ µ uR + h.c.
(6.12)
First some experimental results will be reviewed; then one consequence of one of the interactions above for the cross section of tt̄ will be analytically calculated. Finally, these interactions
will be used in MadGraph in addition to the Standard Model (without Higgs) to produce a
plot of the change in the tt̄ cross section σtt̄ versus the mass of each of the new particles ht , ρµ ,
and πt .
6.4
Experimental Results
The CDF collaboration at the Tevatron measured a discrepancy with the forward-backward
asymmetry predicted by the Standard Model [35]. Using Monte Carlo Simulations at next
leading order, they found for an invariant mass Mtt̄ > 450GeV an asymmetry of 0.030 ± 0.007
for tt̄ plus background, while from the real data they measured an asymmetry of 0.210 ± 0.049
[42, p17]. This is a discrepancy of more than three standard deviations from the predictions.
√
This was measured for s = 1.96 TeV [42].
The LHC is a symmetric collider under charge conjugation (since we have here pp (or lead
nuclei) collisions instead of pp̄ collisions), so forward-backward asymmetry is not the same as
at the Tevatron. A predominance of top quarks emitted in one direction (or antitop quarks in
the opposite direction) would not be found there. However, if one focuses on a specific range of
momenta, a charge asymmetry could be found; different widths of the rapidity distribution are
expected for the top and for the antitop quark. The LHC can thus be used to measure charge
69
asymmetry inconsistencies with the Standard Model [46, p3], so this would be worthwile to
explore.
If there is new physics affecting top quark pair production and that could explain the
forward-backward asymmetry measured at the Tevatron, it should be consistent with other
experimental measurements of the top quark, such as its cross section and invariant mass.
The recent LHC measurements of the total cross section σtt̄ of tt̄ could be used for this. The
invariant mass distribution could also be a good discriminator of models that can explain
AtF B [46, p8].
Han, Liu, Wu, and Yang ([39, p2]) examined the correlation between charge asymmetry
of the top, AtC , and AtF B at the Tevatron , as top charge asymmetry can be a direct test of
AtF B at the LHC. They found that the LHC constraints on the top rho mass mρt and top
pion mass mπt are already more restricted than those of the Tevatron [39, p8].
6.5
Analytical Result
Before numerical results of simulations of topcolor-assisted technicolor phenomenology are
presented, one amplitude will be computed analytically. Specifically, the amplitude of the
process uū → tt̄ when taking into account the neutral top-pion only will be calculated. The
two possibilities that result are shown in the diagram in figure 6.1.
The term that arises in the Lagrangian of equation 6.9 after the top quark is rotated
can cause top–anti-top production through processes such as uū → tt̄ as in figure 6.1. Two
diagrams are shown there, because both the QCD and the new physics from topcolor-assisted
technicolor must be taken into account. The cross section σtt̄ resulting from these diagrams
can be calculated; it depends on equal the absolute value squared of the sum of the two
diagrams. The QCD diagram (see figure 6.1(a)) is an s-channel-diagram3 , meaning the amplitude Ag ∝ s−m1gluon is proportional to the inverse of the factor s defined as s = (p1 + p2 )2
[5, p156, 157]. The diagram with pion exchange in figure 6.1(b) is a t-channel diagram, so
1
that the amplitude Aπt ∝ t−m
with t = (p1 − p3 )2 . The relevant terms from the Lagrangian
π
t
in equation 6.12 and the Standard Model QCD Lagrangian ψ̄(∂µ − igs Gaµ T a γ µ )ψ are
Lπt = gπt (iπt0 t̄L uR − iπt0 ūR tL ) =
iπt0
gπ (t̄u − ūt + t̄γ 5 u − ūγ 5 t)
2 t
(6.13)
and igs Gµa Ta (t̄γµ t + ūγµ u). With these, the amplitude of the pion exchange diagram reads:
igπt 2
1
Aπt (p1 , p2 ) = 2
2
(p1 − p3 )2 − m2π0
t
× ūt (p3 , st )γ 5 uu (p1 , su )v̄u (p2 , Su )γ 5 vt (p4 , St ).
(6.14)
Here the possibility of neither vertex being a γ 5 vertex is incorporated in the factor of 2; the
calculation for such a diagram yields the same as for one with vertices carrying a γ 5 factor.
3
When a two-particle to two-particle diagram, such as uū → tt̄, contains only one virtual particle, it is
usually said that this particle is in a certain channel, where the channel can be read from the form of the
diagram and leads to a specific angular dependence of the cross section [5, p157]. The s, t, and u are three
possible channels, and are called Mandelstam variables. If p1 and p2 are the incoming momenta, and p3 and
p4 the momenta of the outgoing particles, then s = (p1 + p2 )2 = (p3 + p4 )2 , t = (p1 − p3 )2 = (p4 − p2 )2 , and
u = (p4 − p1 )2 = (p3 − p2 )2 .
70
t
u
p1
p1
p3
p3
u
t
0
u
t
t
p2
u
p4
p4
p2
t
(a) The QCD process of an up and (b) The topcolor assisted technianti-up quark producing a gluon and color process of two quarks exchangsubsequently decaying into a top and ing a top-pion and changing flavor
antitop in the s-channel.
to two top quarks in the t-channel.
Figure 6.1: Two examples of diagrams contributing to top-antitop production. In order to calculate
the amplitude of two up quarks going to two top quarks, one needs to take into account both the QCD
process and the neutral pion exchange from topcolor-assisted technicolor.
The amplitude of the QCD diagram reads (save a factor of the gauge group, that will be dealt
with later):
Ag (p1 , p2 ) = (igs )2
η µν
ūt (p3 , st )γµ vt (p4 , St )v̄u (p2 , Su )γν uu (p1 , su ).
(p1 + p2 )2 − m2g
(6.15)
The amplitude taking into account QCD and a neutral top pion must be squared in order to
calculate a cross section [2, p132]:
|M(uū → tt̄)|2 = |Aπt (p1 , p2 ) + Ag (p1 , p2 )|2
= Aπt (p1 , p2 )|2 + |Ag (p1 , p2 )|2 + [Aπt (p1 , p2 )]∗ Ag (p1 , p2 )
+[Ag (p1 , p2 )]∗ Aπt (p1 , p2 )
= |Aπt (p1 , p2 )|2 + |Ag (p1 , p2 )|2
−2ReAπt (p1 , p2 )∗ Ag (p1 , p2 )
≡ Mπt πt + Mgg + Mgπt .
(6.16)
The πt squared term can be evaluated as follows:
!2
igπt 2 2
1
Mπt πt = |2
|
2
(p1 − p3 )2 − m2π0
t
=
=
× vt† (p4 , St )γ 5 [v̄u (p2 , Su )]† u†u (p1 , su )γ 5 [ūt (p3 , st )]†
× ūt (p3 , st )γ 5 ūu (p1 , su )v̄u (p2 , Su )γ 5 vt (p4 , St )
gπ4t
4(t − m2π0 )2
t
× v̄t (p4 , St )γ 5 vu (p2 , St )ūu (p1 , su )γ 5 ut (p3 , st )
× ūt (p3 , st )γ 5 ūu (p1 , su )v̄u (p2 , St )γ 5 vt (p4 , St )
gπ4t
4(t − m2π0 )2
t
× [v̄t (p4 , St )γ 5 vu (p2 , St )v̄u (p2 , St )γ 5 vt (p4 , St )]
× [ūt (p3 , st )γ 5 ūu (p1 , su )ūu (p1 , su )γ 5 ut (p3 , st )].
71
(6.17)
In experiments, incoming quarks are unpolarized, and the polarization of outgoing quarks
is not measured. Then we can average over initial spins and sum over final spins using the
relation [5, p132]:
X
X
v(p, s)v̄(p, s) = p
(6.18)
u(p, s)ū(p, s) = p
/ − m.
/ + m,
s
s
The averaging gives an extra factor of 14 in the final squared amplitude. The term on the last
line of equation 6.17 is summed over and a trace is taken over it. Ignoring the up quark mass,
it now becomes
X
ūt (p3 , st )γ 5 ūu (p1 , su )ūu (p1 , su )γ 5 ut (p3 , st ) =
{St ,st ,Su ,su }
1
5
5
= Tr[(p
/3 + mt )γ p
/1 γ ],
4
(6.19)
with mt the mass of the top quark. Now one can see that the squared amplitude in equation
6.17 results in:
Mπt πt
=
gπ4t
1
2
2
4(t − mπ0 ) 4
t
5
5
× Tr[(p
/3 + mt )γ p
/1 γ ]
5
5
× Tr[(p
/2 γ ]
/4 − mt )γ p
gπ4t
=
4(p3 · p1 )(p4 · p2 )
4(t − m2π0 )2
t
=
gπ4t
4(t −
(t
m2π0 )2
t
− m2t )2
(6.20)
where in the second-to-last line the trace technology Tr[γ µ γ ν ] = 4η µν was used. In the last
line we used t = (p3 −p1 )2 = −2p3 ·p1 +m2t and thus 2p3 ·p1 = −(t−m2t ); since p3 −p1 = p2 −p4 ,
the same holds for 2p2 · p4 = −(t − m2t ).
The square of the gluon diagram Mgg is calculated in a similar way. An SU (3) group
theory factor is added; the diagram in figure 6.1(a) carries (ta )i0 i (ta )j 0 j [5, p569] with i, i0
the initial and final colors of the u quark and j, j 0 the initial and finial colors of the t quark.
This factor will be squared in the term Mgg , and just like we did for spin in the previous
calculation we must now sum over final colors and average over initial colors:
2
1
1
11
2
b a
Tr[t t ] = [C(r)]2 δ ab δ ab =
·8= .
3
9
94
9
(6.21)
The factor C(r) is dependent on the representation of SU (3) and is 12 in the fundamental representation (see equation A.8 and further). Furthermore, the coupling gπt should be replaced
by the strong coupling gs , and the factors in the trace change since we are now looking at the
s-channel. The mass of the gluon is zero, so the numerator of the propagator becomes merely
72
s2 . When squaring the amplitude and averaging and summing over spins, we obtain
2 gs4 1
µ
ν
Tr[(p
/3 + mt )γ (p
/4 − mt )γ ]
9 s2 4
× Tr[p
/ 1 γµ p
/ 2 γν ]
4
2 gs 1 µ ν
=
4[p p + pν3 pµ4 − η µν (p3 · p4 + m2t )]
9 s2 4 3 4
× 4[pµ1 pν2 + pν1 pµ2 − η µν (p1 · p2 )]
2 gs4
=
8[(p3 · p1 )(p4 · p2 ) + (p3 · p2 )(p4 · p1 ) + (p1 · p2 )m2t ].
9 s2
Mgg =
(6.22)
(6.23)
In the second-to-last line the trace property Tr[γ µ γ ν γ ρ γ σ ] = 4(η µν η ρσ − η µρ η νσ + η µσ η νρ ) was
used. This can be further evaluated using
s2
+ m4t + st − 2tm2t
(6.24)
2
where the definitions of the Mandelstam variables (most importantly, s = 2p1 · p2 ) and
conservation of momentum (p1 + p2 = p3 + p4 ) were used. The amplitude Mgg now reads:
2[(p3 · p1 )(p4 · p2 ) + (p3 · p2 )(p4 · p1 ) + (p1 · p2 )m2t ] = t2 +
4 gs4 2
[2t + s2 + 2m4t + 2st − 4tm2t ].
(6.25)
9 s2
Now we need only still calculate the interference term, or Mgπt . This term amounts to
Mgg =
Mgπt
= −2
gπ2t
2 gs2 1
2
(t − mπ0 ) 9 s 4
t
X
Su ,su ,St ,st
5
× v̄t (p4 , St )γ vu (p2 , Su )ūu (p1 , su )γ 5 ut (p3 , st )
× ūt (p3 , st )γµ vt (p4 , St )v̄u (p2 , Su )γ µ uu (p1 , su )
gπ2t
2 gs2 1 X
= −2
(t − m2π0 ) 9 s 4
Su ,su ,St ,st
t
µ
× v̄u (p2 , Su )γ uu (p1 , su )ūu (p1 , su )γ 5 ut (p3 , st )
× ūt (p3 , st )γµ vt (p4 , St )v̄t (p4 , St )γ 5 vu (p2 , Su )
gπ2t
2 gs2 1
= −2
(t − m2π0 ) 9 s 4
t
5
µ
5
× Tr[(p
/3 + mt )γµ (p
/4 − mt )γ p
/2 γ p
/1 γ ].
(6.26)
The trace in the last line can be evaluated using the trace properties γµ γ ν γ µ = −2γ ν and
γ µ γ ν γ ρ γµ = 4η νρ :
5
µ
5
µ
Tr[(p
/3 + mt )γµ (p
/4 − mt )γ p
/2 γ p
/1 γ ] = −Tr[(p
/3 + mt )γµ (p
/4 − mt )p
/2 γ p
/1 ]
µ
2
µ
= −Tr[p
/ 3 γµ p
/4 p
/2 γ p
/1 ] + Tr[mt γµ p
/2 γ p
/1 ]
2
= −4p1 · p3 Tr[p
/4 p
/2 ] − 2mt Tr[p
/2 p
/1 ]
= −4p1 · p3 × 4p4 · p2 − 2m2t × 4p2 · p1
= −4[(t − m2t )2 + m2t s]
= −4[t2 + m4t − 2m2t + m2t s].
73
(6.27)
The interference term then reads:
Mgπt
= −2
gπ2t
2 gs2 1
2
(t − mπ0 ) 9 s 4
t
× − 4[t2 + m4t − 2m2t + m2t s]
4gπ2t gs2 [m2t (s − 2t) + m4t + t2 ]
=
.
9(t − m2π0 )s
(6.28)
t
The cross section of two tops σtt̄ can be calculated using the formula for the differential
cross section in the center of mass frame where one particle is massless (in this case the up
quark is, like the c, d, and s quarks, treated as massless) [5, p154]. If we take the neutral pion
as the only new particle and then include QCD and neutral pion exchange only, we obtain
the following:
dσ
M2
(6.29)
=
dΩ CM
64π 2 (E + k)2
where k is the momentum that each particle has when scattering in the center of mass frame.
See also figure 6.2. The energy E 2 = k 2 can be expressed in terms of the center of mass energy
Ecm = E +k. The four-momenta of the two top quarks that leave are p3 = (k, ~k) = (k, k cos θ),
and p4 = (k, −k cos θ). Now the differential cross section is
dσ
1
=
|M(uū → tt̄)|2 ,
(6.30)
2
2
dΩ CM
64π ECM
where the amplitude can now be obtained by combining equations 6.16, 6.20, 6.25, and 6.28:
|M(uū → tt̄)|2 =
gπ4t
1
4 gs4 2
2 2
(t
−
m
)
+
[2t + s2 + 2m4t + 2st − 4tm2t ]
t
2
9 s2
(t − mπ0 )2 4
t
4g 2 g 2 [m2 (s − 2t) + m4t + t2 ]
+ πt s t
.
9(t − m2π0 )s
(6.31)
t
For rough properties of the amplitude, we can set s = 4m2t , t = u = −m2t , and mρ mt .
Then the above expression becomes
|M(uū → tt̄)|2 =
4gπ2t gs2 m2t
gπ4t m4t
4gs4
+
−
.
9
(m2t + m2π0 )2
9(m2t + m2π0 )
t
6.6
(6.32)
t
Numerical results using MadGraph
To explore the possibility of topcolor-assisted technicolor as an explanation of measured
forward-backward asymmetry of the top quark, it would be interesting to investigate the
effect of new particles on this asymmetry as a function of their mass. To generate events and
data for this, the program MadGraph 5 [47] will be used. Feynman rules will be generated
for MadGraph using Feynrules. This is done through adding three new particles with two
new couplings to the Standard Model, as well as a new Lagrangian, namely the effective
Lagrangian with the relevant terms from equation 6.12. An exact description of how these
programs were used is given in appendix F.
74
Figure 6.2: Two particles scattering off each other. In the center of mass frame they collide with equal
momenta so that p~1 = −~
p2 , and scatter off with equal angles θ, so that p~3 = −~
p4 . Figure taken from
http://farside.ph.utexas.edu/teaching/336k/newton/node51.html.
The parton distribution function, or PDF4 , used is, as for other cuts and simplifications,
as in [35, p15] CTEQ6l. One has to be careful when choosing a PDF. If the default PDF
of MadGraph 5 is used, CTEQ6l1, the asymmetries will be lower; CTEQ6l assigns a higher
probability to up quarks, enhancing the asymmetry.
The rapidity of a particle is defined as
y=
1
E + pz
log
,
2
E − pz
(6.33)
with E the energy of the four-momentum and pz the third direction of the three-momentum.
The pseudorapidity of a particle, which is often for convenience called the rapidity of a particle,
is
θ
.
(6.34)
η = − log tan
2
This is equal to the rapidity for massless particles. Cuts are imposed on the rapidity of tops
so that |ηt,t̄ | < 2.0|. Furthermore, the invariant mass combined of the top and antitop is
Mtt̄ =
p
(pt + pt̄ )2
(6.35)
with pt NOT the transverse momentum of the particle but the four-momentum of the top.
This quantity is cut from a minimum so that Mtt̄ > 450 GeV. The couplings gπ and gρ are
set to 1 to simplify the process where we will just vary the mass of the new particles to see
the effect thereof on AtF B .
The correction order used, or the renormalization and factorization scale used, is the
top mass mt = 172.9 GeV [48]; for simplicity a mass of mt = 173 GeV is used. There is one
renormalization scale, and there are two factorization scales, which are all set to the top mass.
The two factorization scales each are for a different purpose: one is for the parton distribution
4
In experiments, it is impossible to collide individual quarks. When, for example, two protons are collided,
and one wants to study a specific process with specific initial particles, it is impossible to know whether those
particles collided. All that is known is that the two protons collided, and there is a mere probability that a
tt̄ pair was produced by a collision of a u and ū. In order to incorporate this probability in measurements,
a parton distribution is used. This function is the probability density for finding a particle with a certain
longitudinal momentum fraction at a certain momentum transfer. Because a QCD binding state has a nonperturbative effect, parton distribution functions cannot be obtained by perturbative QCD. Lattice QCD could
be a solution, but due to its current limitations the known parton distribution functions are obtained using
experimental data.
75
function, and one for the cross section. Collinear particles coming from the beams can then
be taken care of in either one of the PDF or the cross section, depending on the scales used.
The specific value of the strong coupling αs was not given in [35]. This can have an impact
on the results for forward-backward asymmetry of the top. The value of the strong coupling
at the renormalization scale mt , or αs (mt ), which is the value ultimately used, depends on its
initial value (specified by its value at MZ , or αs (MZ )), and on the beta function that describes
the renormalization. If two loops are taken into account (next to leading order), αs decreases
more slowly, and the Standard Model contribution is decreased giving a higher asymmetry. If
only one loop is taken into account, i.e. the calculation is performed at leading order, the beta
function becomes negative. In this case there is stronger running, or a faster decrease of αs ,
and an increased Standard Model contribution; then a lower asymmetry emerges. Both PDF
functions CTEQ6l and CTEQ6l1 are for leading order (one loop) calculations, but by default
the initial value for the strong coupling αs (MZ ) is given by 0.118 for the former PDF, as
calculated at next to leading order, and by 0.130 for the latter PDF, as calculated at leading
order. A difference in forward-backward asymmetry that emerges as a result of two distinct
initial values of αs is shown in figure 6.4 by an error bar.
The forward-backward asymmetry of the top quark, or [42, p5]
AtF B =
NF − NB
NF + NB
(6.36)
is calculated using the rapidity ratios of all events that pass the imposed cuts. The factorization and renormalization scale is, as mentioned above, chosen to be the top mass mt . To
simulate the Tevatron, the beam energies should be 980 GeV each (the Tevatron ran at an
energy of 1960 GeV ∼ 2 TeV). The width of all three particles is kept at 1 GeV5 . The
tt̄ cross section measured at the Tevatron with a 4.6 fb−1 luminosity was measured to be
σtSM
= 7.70 ± 0.52 pb for top mass of 172.5 GeV [49]. However, this takes next leading orders
t̄
into account and largely deviates from the Standard Model tree level cross section calculated
in MadGraph: 6.322 ± 0.0023 pb. The comparison between the Standard Model cross section
as generated by MadGraph and the cross section as a result of new effective particles added
to the Lagrangian was made after cuts were applied. The Standard Model top cross section
after cuts in MadGraph that was used for comparison was 2.22 pb.
Madevent produces both weighted and unweighted events. Unweighted events are concentrated in an area of interest: along the beam. They are processes that occur with the same
probability as in nature. This is achieved after a transformation in phase space, whereafter
the weight of a specific phase space point is compared with a random number between zero
and the maximum possible weight; if the weight is larger, then that phase-space configuration
is kept as an unweighted event [50]. Unweighted events were used in this small numerical
study.
For all three new particles included in the effective Lagrangian, MadGraph generates the
diagrams depicted in figure 6.3. In order to generate information about the forward-backward
asymmetry, the momentum in the third direction of the top quark was used. To be precise,
the forward-backward asymmetry was N (ptop
> 0) − N (ptop
< 0), normalized to the total
z
z
5
The cross section was checked for two runs of 10000 events taking into account all three new particles, one
run with a width of 1 GeV for all particles and one run with a width of 5 GeV for all particles. For the first
width, the total tt̄ cross section was 7.354 ± 0.0089 pb, for the second this was 8.61 ± 0.0078 pb. The width
thus does matter, but in order to not generate a large cross section deviating from the Standard Model cross
section the width is kept low.
76
number of events. The deviation from the cross section when taking into account various
particles is shown in figure 6.5. The results for the forward-backward asymmetry of the top
taking into account new particles are shown in figure 6.4.
Figure 6.4 consists of more fluctuations in the data points than the figure from [35, p16];
this discrepancy could be overcome by using more events so that statistical fluctuations are
decreased. The trend of the data in this plot coincide with the data in the figure from [35,
p16]. From the plot in figure 6.4 it is evident that the measured asymmetry at the Tevatron
0.210 ± 0.049 from [42, p17], which means 0.2051 ≤ AtF B ≤ 0.259, could be explained by the
addition of three particles, but quite less so by the addition of one or two particles (excluding
the ρ particle) to the Standard Model. Thus only with a ρ particle is it possible to reach such
an asymmetry. This is what we expect by looking at QCD on which top condensation theories
are modeled; in QCD a ρ meson also must be included in the effective theory [45, p184]. An
error bar is shown in figure 6.4 to differentiate between different initial values at the Z boson
mass of the strong coupling αs (MZ ). This, as well as the number of loops taken into account
when the strong coupling runs and is calculated at the scale of the top quark mass mt , can
change the asymmetry of the top quark in the simulation made with MadGraph.
In order to agree with existing measurements, the cross section including new particles
should not deviate much from the known cross section of tt̄. This happens for technicolor and
topcolor by themselves, but when they are combined into topcolor-assisted technicolor, the
predicted cross section does agree with the measured cross section. At the LHC and Tevatron,
the cross section is enhanced at next leading order. The tree level cross section as measured
after the cuts specified above was used in figure 6.5. From figure 6.5 we see that when all
three, only two, or just the ρ particle are taken into account, this holds for a particle πt of
various masses 100 GeV ≤ mπt ≤ 400 GeV depending on the data set. If only one of πt and
ht are taken into account, the constraint of small deviations from the Standard Model cross
section limits the possible masses to higher values; however, such high values for their masses
would not explain the measured asymmetry anymore. The data from an addition of all three
particles seem to fit best. The data in this plot agree with the data in [35, p18].
Deviations of the data in figures 6.4 and 6.5 from the figures in [35] could be the result
of unspecified parameter inputs or cuts for which different values may have been used. As
explained above, the value of αs (MZ ) can make a difference for the results. Another difference detected was a result of the definition of the forward-backward asymmetry of the top
quark, of which several variants exist. Also, the mass of the top quark used may differ as
new measurements could change this value. The matrix elements of several diagrams as a
result of the models from Feynrules and processes generated by MadGraph were compared
to the matrix elements as calculated from analytical expressions; these agreed to a very high
accuracy. For more on this way of verifying models, see section F.2 in the appendix.
While the forward-backward asymmetry of the top can be reached with the addition of
these three new particles or with a ρ only, it cannot be reached with just the effective topHiggs and/or top-pion that we derived from the original four-fermion interaction. This is
to be expected as the theory is modeled on QCD, where such a ρ particle is also merely
added to the effective Lagrangian, not derived. What remains unknown is the mass of this ρ
particle, both in QCD and in topcolor-assisted technicolor. It must also be mentioned that
the particles added to the Standard Model were quite general and could be particles with
similar properties from some other theory.
77
Figure 6.3: The diagrams contributing to the tt̄ cross section taking into account all three new particles
ρ, πt , and ht , as generated by MadGraph.
78
0.6
πt, ht, and ρ, with mh = mπ + 100 GeV, mρ = 500 GeV
πtt and ht t with mh = mπ + 100 GeV
t
t
ht or πt only
ρ only
ρ only with αs(MZ)=1.3 (lower AFBt) and αs(MZ)=0.118
t
Forward-backward asymmetry of top quark A FB
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
100
200
300
400
500
600
700
Mass of πt (ρ for ‘ρ only’ or ‘ht only’ data) in GeV
800
900
Figure 6.4: The forward-backward asymmetry of the top quark as function of the pion mass mπt when
taking into account a new ρ, new ht , and new πt particle (the red crosses); when including only a
πt and ht in the effective Lagrangian (green crosses); when including a single scalar ht or πt in the
effective Lagrangian (blue crosses); or when including only a vector meson ρ (pink crosses). For the
data where more than one particle is included, mρ = 500 GeV and mht = mπt + 100 GeV. When
single particles are included, the data is plotted as a function of the mass of that particle. The results
for ht are the same as for the other scalar πt . If more events had been generated and used for each
possible particle mass, the curves would be more precise and have fewer fluctuations. The error bar
shows the different results for a ρ meson of 320 GeV for different initial values of the strong coupling.
If the initial value of the strong coupling at the mass of the Z boson is αs (MZ ) = 0.130, which is the
default value for the PDF CTEQ6l1, the asymmetry is much lower than when the initial value of the
strong coupling is taken to be αs (MZ ) = 0.118, the default value for CTEQ6l.
79
40
πt, ht, and ρ, with mh = mπ + 100 GeV, mρ = 500 GeV
πtt and ht t with mh = mπ + 100 GeV
t
t
ht or πt only
ρ only
Percent change of t-tbar cross section σttbar
30
20
10
0
-10
-20
-30
-40
100
200
300
400
500
600
700
Mass of πt (ρ or ht for ‘ρ only’ or ‘ht only’ data) in GeV
800
900
Figure 6.5: The deviation from the measured tt̄ cross section σtSM
= 2.22 pb of the Standard Model
t̄
(after applied cuts specified above) in percentage, taking into account one or more new particles from
topcolor-assisted technicolor. The data represented by red crosses include all three particles, where
mρ = 500 GeV, and the green crosses include only the two scalars πt and ht . Both data sets are a
function of mπt , and mht = mπt + 100 GeV. The blue crosses represent the difference in cross section
when only a scalar ht or πt is added to the Standard Model, and the pink crosses include only an
added vector meson ρ. When single particles are included, the data is plotted as a function of the
mass of that particle. The results for ht are the same as for the other scalar πt .
80
Chapter 7
Discussion and Outlook
If the Higgs is found, several questions remain, such as whether it is composite or whether
there are more Higgs particles than just one [13, p7]. A discovery would not mean immediate
exclusion of top condensation theories like topcolor-assisted technicolor. Forward-backward
asymmetry of the top quark found at the Tevatron could even be explained by topcolorassisted technicolor.
However, one problem with forward-backward asymmetry of the top quark is that in the
investigation of the phenomenology of topcolor-assisted technicolor as a possible explanation
of the forward-backward asymmetry of tt̄ pairs, the ρ particle’s existence was assumed. Although there is phenomenological support for it, its existence cannot be proven. Without this
top-rho, the forward-backward asymmetry decreases to a value below the measured value at
the Tevatron [35, p16].
Furthermore, other explanations of forward-backward asymmetry of the top quark exist;
topcolor-assisted technicolor is not the only viable model that can yield this result. One example is supersymmetry [51]. Electroweak contributions could also diminish the discrepancies
of the measured top forward-backward asymmetry with the Standard Model predictions [52].
Furthermore, background of W + 4 jets could also explain the measured AtFt̄ B [53]. The fact
that only tree levels were taken into account in the above numerical analysis makes the claim
that electroweak contributions could account for the forward-backward asymmetry more convincing. The asymmetry was measured at an experiment, where inevitably all order beyond
tree level are taken into account. The idea that new particles account for this asymmetry at
tree level may be prone to error.
Finally, the programs and methods used to generate predictions may not be completely
accurate; the accuracy of those used at the Tevatron for Monte Carlo simulations at next
leading order was at that time still under study [42, p20]. Indeed this demands more investigation, as do the claims of other explanations for forward-backward asymmetry of the top;
however, the alternative explanations as well as the questioned accuracy of predictions indicate that more predictions of topcolor-assisted technicolor that coincide with experimental
results are needed to support the likelihood of the theory.
Another area for investigating viability of top condensation models would be triple top
production that could be measured at the LHC. For most top condensation theories such
as topcolor-assisted technicolor, the flavor-changing neutral currents in those theories induce
triple top production at tree level, which according to Han, Liu, Wu, and Yang ([39, p3])
would be a good way to see if this theory is still viable.
81
Figure 7.1: Regions excluded by Atlas and CMS upper bounds on the Ht → W W branching ratio for
different top-pion masses (from [37, p3]) as a function of sin ω. The sin ω gives the proportion of the
vacuum expectation value that is made up of the toppion decay constant: fπt = v sin ω. The region
of a top-pion of 130 GeV is colored dark wine; the region of a top-pion of 150 GeV red; the region of
a top-pion of 172 GeV orange; the region of a top-pion of 400 GeV yellow; the region of a top-pion of
masses saturating Tevatron bound is dark blue.
Currently, topcolor-assisted technicolor is tightly constrained for light top-Higgs states.
Top condensation models with top-Higgs masses larger than 300 GeV are still allowed [37, p2].
For topcolor-assisted technicolor, however, it is difficult to reach such a heavy mass. Even with
subleading corrections in the number of colors to the Nambu–Jona-Lasinio approximation it
would be difficult, because larger values of a cutoff are expected by electroweak precision
measurements and diminish the enlarging effects of these corrections [37, p7]. A top-Higgs
mass of twice the top quark mass or less, or ≤ 350 GeV, would be expected, so this would very
much constrain topcolor-assisted technicolor. A plot of exclusion regions for the top-Higgs
for various top-pion masses taken from [37] is shown in figure 7.1. Not much room is left for
topcolor-assisted technicolor, but it is not yet excluded.
82
Additional note regarding recent data Recently, the ATLAS and CMS collaborations
announced a discovery1 of a particle consistent with the Higgs boson. The access in the
HSM → γγ search channel of these collaborations could correpsond to a neutral top-pion of
mass m0πt = 125 GeV [54, p19]. This would not lead to an excess in the W W channel, for
example, because the neutral top-pion is a pseudoscalar. However, with a significance of a
few Standard deviations, this was already presented by CMS. Not much can be said yet about
higher energies where there are still possibilities for composite Higgs theories. Current limits
have now excluded the top-Higgs particle of a mass up to 400 GeV [54, p19].
1
See http://www.quantumdiaries.org/2012/07/04/so-is-higgs-finally-here/ for more on what is meant by a
discovery.
83
Appendices
84
Appendix A
Group Theory: Lie Algebras
Group theory in mathematics is the study of algebraic structures called groups. A group
consists of a set and an operation that combines any two elements of a group to form a third
element of that group. In physics, groups are important because they describe the symmetries
of the laws of physics. According to Noether’s theorem, every symmetry of a physical system
corresponds to a conservation law of the system. New groups (i.e. new symmetries) can point
to new theories in physics, for example new gauge theories in addition to or incorporating the
Standard Model gauge groups (symmetries). Many symmetries or approximate symmetries
form groups called Lie groups, the precise definition of which is given below.
Physical laws can be derived from symmetries of the Lagrangian: for example, the invariance of the Lagrangian under space translations corresponds to conservation of momentum [5,
p19]. Turning this around, a Lagrangian can be required to be invariant under certain transformations in order to conserve certain quantities. When there are several transformations of
the same kind, they can form a group, with the following properties:
1. Two consecutive transformations g and h form together an element gh of that group;
2. There is an inverse transformation so that g −1 g = 1;
3. There is an identity transformation e so that eg = g;
4. Transformations are associative, so that g(hi) = (gh)i.
Groups can be continuous and discrete; an example of a discrete symmetry group is that
of parity under which x → x0 = −x. A continuous group with an infinitesimal group element
written as [5, p495-502]
g(α) = 1 + iαa T a + O(α2 ),
(A.1)
with αa the group parameters and the Hermitian operators T a the generators of the symmetry
group, is called a Lie group [5, p495]. In order that the generators T a span the space of
infinitesimal group transformations they obey the commutation relations
[T a , T b ] = if abc T c ,
(A.2)
with f abc the so-called structure constants. The commutation relations of A.2 obey the
Jacobi identity1 . A Lie group with a finite number of generators and finite-dimensional
1
The Jacobi identity is defined as [T a , [T b , T c ]] + [T b , [T c , T a ]] + [T c , [T a , T b ]] = 0. A given set of commutation relations should satisfy this identity in order for those relations to define a Lie algebra.
85
unitary representations is called compact. A commuting generator generates an independent
continuous abelian group called U (1) with the structure of the group of phase rotations
ψ → eiα ψ.
(A.3)
If the Lie-algebra has no commuting elements and thus no U (1)-factors it is called semisimple; if it also cannot be divided into two mutually commuting sets of generators it is
called simple. Almost all compact simple Lie-algebras belong to one of three infinite families
[5, p499], which are:
1. Unitary transformations of N -dimensional vectors
If the inner product of two N -dimensional vectors ηa∗ ξa is conserved, the transformation
ηa → Uab ηb , ξa → Uab ξb is unitary. A U (1) subgroup is formed by the phase transformations ξa → eiα ξa ; leaving out this subgroup and requiring det(U )=1 we obtain
the simple Lie group SU (N ) with all N × N unitary transformations. The N 2 − 1
independent generators of SU (N ) are N × N Hermitian matrices ta with tr[ta ] = 0.
2. Orthogonal transformations of N -dimensional vectors
This is a subgroup of unitary N × N transformations preserving the inner product
ηa δab ξb ; in other words, this is the rotation group SO(N ) (with reflection it would be
O(N )) with N (N − 1)/2 independent generators (one rotation for each plane).
3. Symplectic transformations of N -dimensional vectors
This is the subgroup Sp(N ) of unitary N × N transformations
for even N , preserv
0 1
ing the antisymmetric inner product ηa Eab ξb , Eab =
. It has N (N + 1)/2
−1 0
independent generators.
Tensors supply a group with a representation. A representation can be reducible if groups
of it transform among themselves. For example, for SO(N ), a tensor T ij that transforms
0
as T ij → T ij = Oil Ojm T lm (with O an orthogonal matrix, OT O = 1 and det O = 1) can
be written as a symmetric and an antisymmetric tensor [2, p462]. The symmetric tensor
S ij ≡ 21 (T ij + T ji ) transforms into the symmetric Oil Ojm S lm and the antisymmetric Aij ≡
1
ij
ji
il jm Alm . The trace of T ij , T ≡ δ ij T ij ,
2 (T − T ) transforms into the antisymmetric O O
transforms into itself (T → T 0 = T ). This can be used to form a traceless tensor Qij ≡
T ij − N1 δ ij T (from a symmetric tensor T ij ) with 21 N (N + 1) − 1 elements. Then we see
how two vectors of dimension N can form a tensor, which can then be decomposed into a
symmetric traceless combination, a trace, and an antisymmetric tensor:
1
1
N ⊗ N = [ N (N + 1) − 1] ⊕ 1 ⊕ N (N − 1),
2
2
(A.4)
which in SO(3) amounts to 3 ⊗ 3 = 5 ⊕ 1 ⊕ 3.
These tensors have a slight extra in SU (N ) (with U † U = 1 and det U = 1): both upper
and lower indices are possible, so that the complex conjugate of a tensor with the index
reversed transforms the same as before. This works as follows:
ϕi → ϕ0 = Uji ϕ;
ϕ∗i → (Uji )∗ ϕ∗j = (U † )ji ϕ∗j ;
ϕi → ϕ0i = (U † )ji ϕj .
(A.5)
As before, symmetric, antisymmetric, and traceless representations of a tensor can be formed.
0
The representation of the traceless tensor ϕij transforming according to ϕij → ϕji = Uli (U † )nj ϕln =
86
Uli ϕln (U † )nj or ϕ → ϕ0 = U ϕU † . Now a unitary matrix can also be written as U = eiH , with
H Hermitian and traceless so that U † U = 1 and det U = 1 still hold [2, p466]. Because a
N by N Hermitian traceless matrix can be written as a linear combination of the (N 2 − 1)
linearly independent Hermitian traceless matrices T a , a = 1, 2, . . . , N 2 − 1, U can be written
a
as U = eiθT with a sum over a and θa real numbers. The ath generator in this so-called
adjoint representation then gives
ϕ → ϕ0 ∼ (1 + iθa T a )ϕ(1 + iθa T a )† ∼ ϕ + iθa T a − ϕiθa T a = ϕ + iθa [T a , ϕ].
(A.6)
An example of an SU (N ) group is the SU (3) of Gell-Mann and Neumann which transforms
the three quarks u, d, and s into linear combinations of each other. It has the isospin SU (2)
of Heidelberg as a subgroup, which transforms only u and d.
Several things can be said about the representations of Lie algebras. Let tar denote an
irreducible Hermitian d × d (with d the number of dimensions) representation matrix of a Lie
algebra. The trace of two such Hermitian generators is positive-definite [5, p498], so we can
choose a basis such that
tr[tar tbr ] = C(r)δ ab ,
(A.7)
with C(r) a constant for each irreducible representation r. Then with the commutation
relations A.2 the structure constants become
i
f abc = −
tr[tar , tbr ]tcr ,
(A.8)
C(r)
with f abc fully antisymmetric. In the fundamental representation of an SU (N ) group, which
is denoted by r = N , we have C(N ) = 12 [5, p806].
For each irreducible representation there is an associated conjugate representation r. The
representation r gives the infinitesimal transformation φ → (1 + iαa tar )φ and its complex
conjugate φ∗ → (1 − iαa (tar )∗ )φ∗ [5, p498] so that for the conjugate representation r̄ we have
tar̄ = −(tar )∗ = −(tar )T .
(A.9)
The representation r is real if r = r and tra = U tar U † ; in this case there is a matrix Gab so that
Gab ηa ξb (with η and ξ belonging to r) is invariant. If Gab is symmetric the representation
is strictly real; if Gab is antisymmetric r is pseudoreal. The N -dimensional complex vector
is the basic irreducible representation for SU (N ), which in SU (2) is the pseudoreal spinor
representation. In SO(N ) the N -dimensional vector is a strictly real representation; in Sp(N )
it is a pseudoreal representation [5, p499].
Another irreducible representation for Lie algebras, the one to which the generators of the
algebra belong, is the adjoint representation (introduced above) denoted by r = G where the
representation matrices are given by
(tbG )ac = if abc .
(A.10)
Notice that the indices a, c denote the matrix entries, whereas the index b and G denote it is
the bth generator of the adjoint representation of the group. Now the generators satisfy
([tbG , TGc ])ae = if bcd (tdG )ae .
(A.11)
The adjoint representation is always real because the structure constants are antisymmetric;
hence, taG = −(taG )∗ . The dimensions of the adjoint representation for the three groups
described above are the same as before.
87
If we now look at the covariant derivative acting on a field in the adjoint representation,
(Dµ φ)a = ∂µ φa − igAbµ (tbG )ac φc
= ∂µ φa + gf
abc
Abµ φc ,
(A.12)
(A.13)
we see that the infinitesimal transformation for a field Aaµ in a non-abelian gauge theory (in
particular, in Yang-Mills theory) can be expressed as [5, p499]
Aaµ →
1
Aaµ + ∂µ αa + f abc Abµ αc
g
1
= Aaµ + (Dµ α)a .
g
(A.14)
(A.15)
The quantity εµνλσ [Dν , [Dλ , Dσ ]] vanishes by antisymmetry and can be rewritten as the
Bianchi identity of a non-abelian gauge theory [5, p500],
εµνλσ (Dν Fλσ )a = 0,
(A.16)
with Fλσ = ∂λ Aaσ − ∂σ Aaλ + gf abc Abλ Acσ the field tensor of the non-abelian gauge theory [5,
p491].
88
Appendix B
Gap Equation in BCS Theory
The dynamical generation of mass, or the generation of mass through interactions, in particle
physcis, stems from ideas from condensed matter physics. The theory of Bardeen, Cooper, and
Schrieffer, the BCS theory of superconductivity, describes how electrons in superconductors
form condensates called Cooper pairs, and how these condensates give photons a mass. The
generation of the photon mass is a result of an energy gap, or an amount of energy needed
to break up a Cooper pair. This amount of energy can be found from the famous gap
equation which follows from BCS theory. In this appendix the physics of this condensation
and dynamical mass generation is explained, and the mathematics of BCS theory leading to
the gap equation and computations involved are shown.
B.1
Physical Motivation
An electron can be excited to a higher energy level by a photon with a sufficient high energy,
or frequency of hν. The energy of this photon must be larger than a mininum energy in order
to have the electron jump to a higher level; a so-called “energy gap” exists. Any frequency
lower than this mininum energy will have no effect. In a superconductor, absorption occurs for
frequencies greater than 1011 Hz [55, p116], so that the energy gap is of order hν = 10−4 eV.
Superconducting material can be represented by a lattice, and electrons by waves. Thermal
fluctuations cause vibrations that result in scattering of electrons with the lattice. These
interactions are mediated by quantized phonons, which are to sound waves what photons are
to light waves [55, p118].
At absolute zero, all energies
up to the Fermi energy εF are occupied and they occupy
√
a sphere of radius kF = 2mεF , also known as the Fermi sea. The Fermi energy depends
on the superconducting material. If then two electrons are added to this Fermi sea, they
must have momenta k > kF (with k 2 = 2mε2 ), or, in other words, have an energy larger
than the Fermi energy by the Pauli exclusion principle (remember that by the Pauli exclusion
principle fermions cannot be in the same state). Cooper showed that when any attraction
exists between the fermions, they form a bound state with a total energy less than 2εF
[55, p121]. This attraction results from the scattering between electrons, or their giving off
phonons.
Since the net momentum of any two electrons should be conserved, their momenta add
up to a constant. The largest number of allowed scattering processes yields the maximum
lowering of the energy of a pair; this happens when electrons with equal but opposite momenta
89
are paired up [55, p123]. Furthermore, this lowering of the energy is greatest when each
scattering has the greatest change in potential, or when the paired electrons have opposite
spins; opposite spins are also required in order to make the new “particle” antisymmetric.
Thus, these so-called Cooper pairs of electrons can be described by
φ(ki ↑, −ki ↓) = ψ↑ (ki )ψ↓ (−ki )
so that the wave function describing the Cooper pair is [55, p124]
X
Φ(~x1 , ~x2 ) =
ai φ(ki ↑, −ki ↓)
(B.1)
(B.2)
i
with |ai |2 the probability of finding electrons with momenta ki and −ki . Thus, in short, the
interaction between two electrons in a Cooper pair lowers the potential energy by more than
their kinetic energy exceeded twice the Fermi energy 2εF before. The momentum range of
the electrons is within kF ± ∆k = kF ± mhνL , with νL a phonon frequency typical of the
lattice [55, p123-125].
For many electrons in a superconductor, a many-electron wave function is simply the
product of many one-Cooper-pair wave functions as in equation B.2. This product gives for
many electrons the following combination of all individual Cooper pair wave functions, which
we will call the grand wave function:
ΨG (~x1 , ~x2 , . . . , ~xn ) = Φ(~x1 , ~x2 )Φ(~x3 , ~x4 ) . . . Φ(~xn−1 , ~xn )
(B.3)
These composite particles, then, since they are all in the same wave function, are not subject to
Pauli’s exclusion principle and all occupy the same state, thus obeying Bose-Einstein statistics
[55, p126]. The fact that all composite particles are regarded as being in the same quantum
state and having the same energy (i.e., all scattering occurs within ∆k of kF ) also follows
from the fact that they cannot be distinguished. However, there is a limit to the number of
electrons that can occupy these states, as for electrons to scatter from state (ki ↑, −ki ↓) to
(kj ↑, −kj ↓), the former must be occupied and the latter unoccupied [55, p126]. As more
Cooper pairs form, scattering becomes increasingly less probable, and eventually the decrease
in potential energy does not outweigh the increase in kinetic energy anymore [55, p131].
The probability that a Cooper pair in the grand wave function in equation B.3 occupies
state (ki ↑, −ki ↓), is not equal to ai in equation B.2, but is given by
"
#
1
εi − εF
p
hi =
,
(B.4)
2
εi − ε F + ∆ 2
where ∆ is given by
∆ = 2hνL exp[−{N (εF )V }−1 ],
(B.5)
with νL the phonon frequency depending on the material, −V the matrix element describing
the lowering of potential through the scattering interaction, and N (εF ) the density of states for
electrons at the Fermi energy of the metal. After a pair is broken up, the two particles left are
quasiparticles with momenta ki and kj ; a quasiparticle in ki ↑ means that the complementary
state −ki ↓ is empty so it cannot pair up. The amount of energy required to break up a Cooper
pair into such quasiparticle states is then [55, p130]:
q
p
(B.6)
E = Ei + Ej = (εi − εF )2 + ∆2 + (εj − εF )2 + ∆2 .
90
The minimum of this energy E occurs when εi = εj = εF . This means that the mininum
amount of energy to break up a Cooper pair is then 2∆, so that a radiation frequency is
absorbed only if it is greater than this energy gap of 2∆: hν > 2∆. This gap is a result of a
“binding energy” between the electrons, and the total energy is increased because there are
less scattering events [55, p131]: remember that the most scattering events occur with the
highest decrease in potential energy when states are of opposite momenta and spin - which
are absent here.
B.2
Mathematical Formulation
In statistical field theory, the partition function Z can be written as [24, p263]:
Z
1
∗
Z = d[φ∗ ]d[φ]e− h̄ S[φ ,φ] ,
(B.7)
R
R QM
R QM Q
dφ∗j,n,α dφj,n,α
∗
where d[φ∗ ]d[φ] ≡
j=1
n,α (2πi)(1±1)/2 [24, p134], and the action
j=1 d[φj ]d[φj ] ≡
for the interacting fermions, or interacting electrons in a superconductor, is given by [24,
p275]:
Z
X Z h̄β
h̄2 ∇2
∂
∗
∗
−
− µα φα (x, τ )
S[φ , φ] =
dτ dxφα (x, τ ) h̄
∂τ
2mα
α=↑,↓ 0
Z h̄β
Z
(B.8)
+
dτ dxV0 φ∗↑ (x, τ )φ∗↓ (x, τ )φ↓ (x, τ )φ↑ (x, τ ),
0
where µ is the chemical potential, n = (nx , ny , nz ) denotes the three quantum numbers
required to specify one-particle eigenstates in the external potential [24, p111], mα the mass of
a particle in spin state α, β = kB T , and V0 stands for a space-time independent approximation
of a short-ranged interaction V (x − x0 ) ' V0 δ(x − x0 )1 . The chemical potential can be
approximated by µ↓ = µ↑ = µ by assuming an equal amount of particles in each hyperfine
state. As the fields of this action form Cooper pairs of two fermions (with opposite spin,
because they together need to be in an antisymmetric state), they are approximated by a
pairing field ∆(x, τ ):
h∆(x, τ )i = V0 hφ↓ (x, τ )φ↑ (x, τ )i .
(B.9)
By inserting an identity into the partition function according to the method of the
Hubbard-Stratonovich transformation2 , the four-fermion interaction term can be eliminated
[24, p277]. The identity inserted is (along the lines of [24]) up to a constant:
Z
1
−1
∗
1 = d[∆ ]d[∆]exp
(∆ − V0 φ↓ φ↑ |V0 |∆ − V0 φ↓ φ↑ )
(B.10)
h̄
where the inner product stands for
Z h̄β
Z
dτ dx(∆∗ (x, τ ) − φ∗↑ (x, τ )φ∗↓ (x, τ )V0 )V0−1 (∆(x, τ ) − φ↑ (x, τ )φ↓ (x, τ )V0 ).
(B.11)
0
The approximation V (x − x0 ) ' V0 δ(x − x0 ) is valid when the interactions are short-ranged and when the
interacting particles have a large de Broglie wavelength (or a small relative momentum) [24, p227].
2
Here, the Hartree-Fock approximation is followed (see [24, p178-182]). This approximation has nonlinearities which are solved by iteration as in Dyson’s equation (see section 3.2); hence it is also called a self-consistent
field theory.
1
91
After this, the action becomes [24, p277]:
Z
Z h̄β
|∆(x, τ |2
dτ dx
S[∆∗ , ∆, φ∗ , φ] = −
V0
0
Z
Z
Z
Z
h̄β
h̄β
X
0 0
0 0
dτ 0 dx0 φ∗α (x, τ )G−1
dτ dx
−h̄
0;α (x, τ ; x , τ )φα (x , τ )
0
α=↑,↓ 0
h̄β
Z
+
Z
dx φ∗↑ (x, τ )φ∗↓ (x, τ )∆(x, τ ) + ∆∗ (x, τ )φ↓ (x, τ )φ↑ (x, τ ) ,
dτ
0
(B.12)
where the noninteracting Green’s function is given by
1
∂
h̄2 ∇2
−1
0 0
G0;α (x, τ ; x , τ ) = −
h̄
− µ δ(x − x0 )δ(τ − τ 0 ).
−
h̄
∂τ
2mα
(B.13)
This can be written in matrix notation:
Z h̄β
Z
|∆(x, τ )|2
∗
∗
S[∆ , ∆, φ , φ] = −
dτ dx
V0
0
Z h̄β
Z
Z h̄β
Z
φ↑ (x0 , τ 0 )
,
−h̄
dτ dx
dτ 0 dx0 [φ∗↑ (x, τ ), φ↓ (x, τ )] · G−1 ·
φ∗↓ (x0 , τ 0 )
0
0
(B.14)
with
0 0
0 0
G−1 (x, τ ; x0 , τ 0 ) = G−1
0 (x, τ ; x , τ ) − Σ(x, τ ; x , τ ),
where
"
0 0
G−1
0 (x, τ, x , τ ) =
and
1
Σ(x, τ ; x , τ ) =
h̄
0
0
0 0
G−1
0
0;↑ (x, τ ; x , τ )
−1
0
−G0;↓ (x, τ ; x0 , τ 0 )
0
∆(x, τ )
∆∗ (x, τ )
0
(B.15)
#
δ(x − x0 )δ(τ − τ 0 ).
(B.16)
(B.17)
The field ∆ can be approximated, as in the Higgs mechanism (see 1.1), by a fluctuation
expansion ∆(x, τ ) = ∆ + ∆0 (x, τ ), with ∆ a minimum. We demand that terms linear in the
fluctuations vanish (as terms linear in fluctuations giving zero ensure that we have expanded
around a minimum); we then obtain a self-consistent equation for ∆ . In that case, the action
92
becomes:
0∗
0
Z
∗
S[∆ , ∆ , φ , φ] = −
h̄β
Z
dτ
dx
0
|∆|2 + |∆0 (x, τ )|2
V0
0
∆0 (x, τ )∆∗ + ∆ ∗ (x, τ )∆
−
dτ dx
V0
0
Z
Z
Z
Z
h̄β
h̄β
X
0 0
0 0
dτ dx
dτ 0 dx0 φ∗α (x, τ )G−1
−h̄
0;α (x, τ ; x , τ )φα (x , τ )
Z
h̄β
Z
α=↑,↓ 0
Z
h̄β
0
Z
n
dx φ∗↑ (x, τ )φ∗↓ (x, τ )(∆ + ∆0 (x, τ )) +
0
o
0
∗
(∆ + ∆ ∗ (x, τ ))φ↓ (x, τ )φ↑ (x, τ )
Z
Z h̄β
|∆|2
dτ dx
≡ −
V0
0
Z h̄β
Z
0
∆0 (x, τ )∆∗ + ∆ ∗ (x, τ )∆
dτ dx
−
V0
0
Z h̄β
Z
Z h̄β
Z
φ↑ (x0 , τ 0 )
0
0 ∗
−1
,
−h̄
dτ dx
dτ
dx [φ↑ (x, τ ), φ↓ (x, τ )] · G ·
φ∗↓ (x0 , τ 0 )
0
0
+
dτ
(B.18)
where fluctuations are small so that |∆0 (x, τ )|2 ≈ 0, and now G−1 = G−1
∆ − Σ∆ . The latter
two are given by
#
"
0, τ 0)
0 )δ(τ − τ 0 )/h̄
(x,
τ
;
x
−∆δ(x
−
x
G−1
−1
0 0
0;↑
,
(B.19)
G∆ (x, τ ; x , τ ) =
0 0
−∆∗ δ(x − x0 )δ(τ − τ 0 )/h̄
−G−1
0;↓ (x, τ ; x , τ )
and
1
Σ∆ (x, τ ; x , τ ) =
h̄
0
0
0
∆0 (x, τ )
0∗
∆ (x, τ )
0
δ(x − x0 )δ(τ − τ 0 ).
(B.20)
Since the action in equation B.18 is a Gaussian integral, it can Rbe evaluated. For any
n × n diagonal matrix G−1 , the partition function evaluates to Z = √ dx n exp{ 12 x · G−1 ·
(2π)
R
R Qn
x} = exp{− 21 Tr[log(−G−1 ]}, with dx =
dx
[24,
p18].
This
then also holds for
j
j=1
3
−1
−1
any positive definite matrix G , since all antisymmetric parts of G
give a vanishing
contribution to x · G−1 · x, and any symmetric matrix G−1 can be written as a diagonal
matrix S · G−1 · S−1 with S orthonormal (so that det S = 1; S−1 = ST ); S indeed vanishes
when taking a trace by the trace property Tr(A−1 BA) = Tr(AA−1 B) = Tr(B) [24, p19]. For
fields φ∗ and φ, this integral amounts to ([24, p28])




Z
n
X

Y
dφ∗j dφj
 exp

0
φ∗j G−1
φ
= exp{−Tr[log(−G−1 )]}.
(B.21)
0
j
j,j
 0

2πi
j=1
3
j,j
A complex Hermitian matrix M is positive definite if z ∗ M z > 0 for all non-zero complex vectors z.
93
Analogously, for Grassmann variables4 (fermions), the integral amounts to




Z
n
X

Y

0
φ∗j G−1
= exp{Tr[log(−G−1 )]}.
dφ∗j dφj  exp
φ
0
j
j,j
 0

j=1
(B.22)
j,j
Thus, the action in equation B.18, which is part of the exponent in the partition function,
becomes
Z h̄β
Z
|∆|2
0∗
0
Seff [∆ , ∆ ] = −
dτ dx
− h̄Tr[log(−G−1 )]
V
0
0
Z
Z h̄β
0
0
∆ (x, τ )∆∗ + ∆ ∗ (x, τ )∆
dτ dx
.
(B.23)
−
V0
0
The second term in this expression can be further evaluated using (schematically)
Tr[log(−G−1 )] = Tr[log(−G−1
∆ + Σ∆ )]
= Tr[log(−G−1
∆ (1 − G∆ Σ∆ ))]
= Tr[log(−G−1
∆ )] + Tr[log((1 − G∆ Σ∆ )]
" ∞
#
X 1
= Tr[log(−G−1
(G∆ Σ∆ )m
∆ )] + Tr
m
m=1
∞
X
1
)]
+
Tr[(G∆ Σ∆ )m ].
= Tr[log(−G−1
∆
m
(B.24)
m=1
0
For m = 1 above, the trace gives rise to terms linear in the fluctuations ∆0 and ∆ ∗ . For
m = 1 we have:
Z
Z
Z
Z h̄β
h
h̄ h̄β
0
0
dx0 G∆;12 (x, τ ; x0 , τ 0 )∆ ∗ (x, τ )
h̄Tr[G∆ Σ∆ ] =
dτ dx
dτ
h̄ 0
0
×δ(x − x0 )δ(τ − τ 0 )
+ G∆;21 (x, τ ; x0 , τ 0 )∆0 (x, τ )δ(x − x0 )δ(τ − τ 0 )
Z h̄β
Z
h
0
=
dτ dx G∆;12 (x, τ ; x, τ )∆ ∗ (x, τ )
0
+ G∆;21 (x, τ ; x, τ )∆0 (x, τ ) .
(B.25)
As stated before, the terms linear in the fluctuations in the action should vanish. Therefore
we have, putting together parts of equation B.25 and equation B.23:
Z h̄β
Z
∆
0
dτ dx G∆;12 (x, τ ; x, τ ) −
∆ ∗ (x, τ ) = 0;
(B.26)
V0
0
Z h̄β
Z
∆∗
dτ dx G∆;21 (x, τ ; x, τ ) −
∆0 (x, τ ) = 0.
(B.27)
V0
0
(B.28)
4
Grassmann
variables anticommute: for Grassmann variable φ, φ2 = 0, and as a consequence
R
and dφ φ = 1.
94
R
dφ 1 = 0
Or, put differently,
∆ = V0 G∆;12 (x, τ ; x, τ ) = V0 hφ↓ (x, τ )φ↑ (x, τ )i .
This is the so-called gap equation; it can also be written in Fourier space:
V0 X
∆=
G∆;12 (k, iωn ).
h̄βV
(B.29)
(B.30)
k,n
Here ωn = π(2n + 1) are the Matsubara frequencies for fermions. The inverse of the matrix G−1
∆ can be found in Fourier space, using the expression of the bare fermion propagator in Fourier space G0,↑ (k, iωn ) = −ih̄ωn−h̄
+εk,↑ −µ . The inverse of this propagator is then
−ih̄ω +ε
−µ
2 2
n
k,↑
k
G−1
([24, p145]), where εk = h̄2m
. Assuming that in this case
0,↑ (k, iωn ) =
−h̄
−1
εk,↑ = εk,↓ = εk , we have for G∆ :
1 −ih̄ωn + εk − µ
∆
.
(B.31)
G−1
(k,
iω
)
=
−
n
∆
∆∗
−ih̄ωn − εk + µ
h̄
Then the inverse G∆ is
G∆ (k, iωn ) =
−h̄
−ih̄ωn − (εk − µ)
∆
.
∆∗
−ih̄ωn + εk − µ
−(h̄ωn )2 − [(εk − µ)2 + |∆|2 ]
(B.32)
p
From this it becomes apparent that the function G∆ (k, ω) has poles at h̄ω = (εk − µ)2 + |∆|2 ≡
h̄ωk . The latter, h̄ωk , is a fermionic single-particle excitation in the presence of Bose-Einstein
Condensation of Cooper pairs [24, p284]. The minimum energy of such an excitation occurs
at εk = µ; |∆| is then the minimum of the excitation spectrum: the excitation spectrum is
gapped. The amount of energy necessary to break up a Cooper pair is twice this minimum,
or 2∆. Using the expression for the Green’s function G∆ in B.32, we have, performing the
Matsubara sum:
V0 X
V0 X
h̄∆
∆ =
G∆;12 (k, iωn ) =
−
h̄βV
h̄βV
(h̄ωn )2 + (h̄ωk )2
k,n
k,n
1
1 X 1
h̄
h̄
→−
=
+
V0
ωk −ih̄ωn + h̄ωk ih̄ωn + h̄ωk
2h̄2 βV
k,n
=
1 X 1 − 2NFD (h̄ωk )
,
V
2h̄ωk
(B.33)
k
1
where NFD (ε) = eβ(ε−µ) +1 is the Fermi-Dirac distribution for an ideal Fermi gas [24, p77]. In
the zero-temperature limit, where NFD (ε) = 0, the gap equation results in (after rewriting
the sum over k as an integral):
8
|∆| = 2 εF e−π/2kF |a| ,
(B.34)
e
h̄2
where εF = 2m
(6π 2 n)2/3 is the Fermi energy at zero temperatures; this is the energy up to
which all states are filled in the zero-temperature limit of an ideal Fermi gas [24, p77,78]. The
h̄2 k2
constant kF is related to this quantity through εF = 2mF ; a is the s-wave scattering length
[24, p223]. The s-wave scattering length is a single parameter that characterizes two-body
interactions in an ultracold gas of atoms; it can be measured in experiments [24, p219].
95
Appendix C
The Goldberger-Treiman relation
The Goldberger-Treiman relation relates the coupling of a (pseudo-) Nambu-Goldstone boson
(such as the pion) to other (effective) particles (such as nucleons) to the pion decay constant
and the mass of those other particles. Because of the strong coupling, the calculations of some
matrix elements of a process involving such a coupling would be impossible as infinite numbers
of diagrams appear [2, p204]. By looking at relationships, however, experimentally verifiable
quantities that are otherwise theoretically uncalculable can be related. One important such
a relation is the Goldberger-Treiman relation.
Let us take a look at pion decay: π + → µ+ + νµ . Here the matrix element of interest
µ
|πb i where from Lorentz invariance we already obtain [2, p204]:
would be h0|J5a
h0|J5µ |ki = f k µ
(C.1)
with k the momentum of the pion and f the so-called pion decay constant. Then we can
see in a few steps again that the pion is massless (mπ = 0) if there were an exact symmetry
(chiral symmetry) with a conserved (axial) current (∂µ J5µ (x) = 0) from the following set of
equations [2, p205]:
kµ h0|J5µ |ki = f k 2 = f m2π
h0|J5µ (x)|ki = h0|J5µ (0)|ki e−ik·x
h0|∂µ J5µ (x)|ki = −ikµ h0|J5µ (0)|ki e−ik·x ,
where in the second line translation invariance is used. The matrix element for pion decay is
known up to the constant f [14, p185]:
µ
h0|J5a
|πb i =
if δab pµπ eipπ ·x
p
2(2π)3/2 2p0π
(C.2)
where pπ is the momentum of the pion. Even though f cannot be calculated directly, the
pion decay rate is known [14, p186]:
(GF cos θC )2 f 2 m2µ (m2π − m2µ )2
Γ(π → µ + ν) =
= (2.5033 · 10−8 s)−1
16πm3π
(C.3)
where GF is the Fermi constant and θC the Cabibbo angle; both of these and the value of the
pion decay rate above are known from experiment. Hence, the pion decay constant f can be
calculated and is f ' 184 MeV.
96
The emission of low energy pions in collisions of nucleons can be calculated with information from, for example, the matrix element between two nucleons, say, a proton and a neutron
with momenta p and n. From Lorentz invariance and parity [14, p186], [2, p203]:
hp|J5µ (x)|ni =
1 iq·x
e ūp [−iγ µ γ5 f (q 2 ) + q µ γ5 f (q 2 ) + q µ γ5 g(q 2 ) +
(2π)3
+iqν [γ µ , γ ν ]γ5 h(q 2 )]un ,
(C.4)
with q = p − n. Here f (q 2 ) and g(q 2 ) are form factors that are unknown [2, p203]. The
factor h(q 2 ) vanishes due to charge conjugation and isospin symmetries [2, p203], [14, p186].
If the SU (2) ⊗ SU (2) symmetry is exact, as mentioned above, current conservation says that
qµ hp|J5µ (x)|ni = 0. The Dirac equation
ūp (ip
n + mN )un = 0,
/ + mN ) = (i/
(C.5)
with mN the mass of a nucleon (i.e. a proton or neutron), can be used and rewritten as
follows:
ūp (ip
n + mN )un
/ + mN )γ5 un = 0 = −ūp γ5 (i/
→ ūp (ip
n)γ5 un = −2mN ūp γ5 un
/ − i/
= qµ ūp iγ µ γ5 un = −2mN ūp γ5 un
and if the last line is used in equation C.4 multiplied by qµ on both sides, the left hand
side gives zero by current conservation, and we obtain from the right hand side the relation
−2mN ūp γ5 un f (q 2 ) = −ūp q 2 γ5 g(q 2 )un , or:
2mN f (q 2 ) = q 2 g(q 2 ).
(C.6)
The factors of f (q 2 ) and g(q 2 ) cannot be calculated because they give an infinite number of
diagrams [2, p206]. If we let q 2 go to zero, it seems that either the nucleon mass is zero,
mN = 0 (which is definitely not true), or that f (0) = 0 which has been shown in experiment
not to be true. The latter was shown by measuring the matrix element in equation C.4 to be
f (0) ≡ gA = 1.2573 6= 0 [14, p187]. The infinite number of diagrams in fact have poles due
to the massless pion and give rise to the term f q µ q12 gπN N ūp γ 5 un [2, p206], with gπN N the
pion-nucleon-nucleon coupling. By comparing to equation C.6 and its preceding sentence, it
can be seen that in the limit that q 2 goes to zero the form factor qµ g(0)γ 5 ∼ f qµ q12 gπN N γ 5
so that the relation in equation C.6 gives [2, p206], [14, p187]:
2mN f (q 2 ) = q 2 g(q 2 ) −→ 2mN gA = f gπN N .
q 2 →0
This is the famous Goldberger Treiman relation.
97
(C.7)
Appendix D
(Non-)Linear Sigma Model
Quantum field theory seemed useless for doing calculations involving the strong interaction
because the coupling there is too strong for perturbation theory. In computing, for example,
the Goldberger-Treiman relation, field theory also seemed unnecessary [2, p318]. However, if
results could be obtained from general properties such as spontaneous symmetry breaking;
hence, if a Lagrangian incorpates such properties, it could give results as well [2, p318]. Low
energy effective field theory is an approach of composing such a Lagrangian for lower energies
while having it obey the desired properties. One example of a low energy effective field theory
is the σ model, which has been used in topcolor and technicolor effective theories. In this
appendix, a short description of the (non)linear σ model is given.
In the linear σ model, an additional meson field σ is added to the Lagrangian (originally
by Gell-Mann and Lévy) in order to keep the larger SU (2)L ⊗ SU (2)R symmetry [2, p319].
The term ψ̄L (σ + i~τ · ~π )ψR + h.c. = ψ̄(σ + i~τ · ~π γ5 )ψ is then invariant, making the Lagrangian
invariant under SU (2)R ⊗ SU (2)R .
The kinetic energy term ψ̄iγ∂ψ = ψ̄L iγ∂ψL + ψ̄R iγ∂ψR is invariant under SU (2)L ⊗
SU (2)R , as ψL/R transforms as a doublet under SU (2)L/R , or ψL ∼ ( 21 , 0) and ψR ∼ (0, 12 ); the
SU (2)I isospin symmetry is the diagonal subgroup of the larger symmetry SU (2)L ⊗ SU (2)R .
When a mass term comes in, however, the larger symmetry no longer holds: mψ̄ψ = mψ̄L ψR +
mψ̄R ψL is a 4-dimensional representation of SU (2)L ⊗ SU (2)R , since ψ̄L ψR ∼ ( 12 , 12 ) [2, p319].
Only the isospin symmetry SU (2)I seems to hold. In order to keep the original symmetry,
four meson fields are needed that couple to the four bilinears that can be constructed from
ψ̄L and ψR : the singlet ψ̄ψ and the triplet ψ̄iγ 5 τ a ψ. Three of these fields are known: they
are the pion fields. The fourth is the σ field postulated by Gell-Mann and Lévy.
Gell-Mann and Lévy insisted that the symmetry SU (2)L ⊗SU (2)R holds in the Lagrangian
even though it was broken by mass terms. To obtain a four-dimensional representation, the σ
field is added to the pion fields: (σ, ~π ). Now ψ̄L (σ + i~τ · ~π )ψR = ψ̄(σ + i~τ · ~π γ 5 )ψ is invariant.
The invariant Lagrangian now reads [2, p319]:
L = ψ̄[iγµ ∂ µ + g(σ + i~τ · ~π γ5 )]ψ + L(σ, ~π ),
(D.1)
where g is the coupling strength and the last part is derived from a Lagrangian of 4 (or N , if
we allow N − 1 pions) scalar fields φi (x) [5, p349]:
1
1
λ
L = (∂µ ϕi )2 + µ2 (ϕi )2 − [(ϕi )2 ]2
2
2
4
98
(D.2)
so that for φi = (σ, ~π ) the last part of the Lagrangian in equation D.1 reads:
1
µ2
λ
L(σ, ~π ) = (∂σ)2 + (∂~π )2 + (σ 2 + ~π 2 ) − (σ 2 + ~π 2 )2 .
2
2
4
(D.3)
This is known as the linear sigma model, where our old field ϕ = (ϕ1 , ϕ2 , ϕ3 , ϕ4 ) is now
written as (σ, ~π ). The mass term of the fermion ψ is still missing. However, the
p vacuum
expectation value of ϕ can be chosen to be in the first direction so that h0|σ|0i = µ2 /λ ≡ v
is a nonzero vacuum expectation value, and h0|~π |0i = 0. When, as with the Higgs mechanism,
we then expand around v such that σ = v + σ 0 , the fermion ψ obtains a mass M = gv [2,
p319]. The pion indeed has no mass, but the σ meson does.
After σ obtains a vacuum expectation value, it is possible to relate the axial current to
this value and obtain the Goldberger-Treiman relation again in the form of M = gv (see [2,
p319-321, p. 498]).
D.1
The linear σ model in the Higgs Lagrangian
Without gauge couplings, the Higgs Lagrangian is invariant under SU (2)L ⊗ SU (2)R , which
can be seen if the Higgs doublet is written as follows [13, p5]:
π2 + iπ1
ϕ1 + iϕ2
(D.4)
=
H=
σ − iπ3
ϕ3 + iϕ4
As described in section 2.2, this is the right column of a matrix M :
1
M = √ (σ1 + i~τ · ~π ) = (iτ2 H ∗ , H).
2
(D.5)
The Higgs fields can be recovered from this matrix as follows:
M
1 − τ3
= (0, H),
2
M
1 + τ3
= (iτ2 H ∗ , 0).
2
(D.6)
Weak gauge bosons Wµa , for a ∈ 1, 2, 3, gauge the SU (2)L symmetry; the third generator of
SU (2)R is taken to be the hypercharge generator [13, p5]. Then the covariant derivative is
now
τa
τ3
Dµ M = ∂µ M + igWµa + ig 0 Bµ ,
(D.7)
2
2
and the Higgs Lagrangian is
1
LHiggs = Tr[(Dµ M )† Dµ M ] + µ2 Tr[M † M ] − λTr[M † M M † M ].
2
(D.8)
This Lagrangian is has a form similar to the one of the linear σ Lagrangian that was introduced for describing chiral symmetry breaking. Under a transformation of a matrix
UL/R ∈ SU (2)L/R , M transforms as:
M → UL M UR† .
(D.9)
In other words, SU (2)L ⊗SU (2)R → SU (2)V , a diagonal subgroup [13, p6], which is equivalent
to our earlier SU (2)L ⊗ U (1)Y → U (1)EM because of our explicit choice of the hypercharge
generator in this case.
99
D.2
The nonlinear σ model
Instead of treating the Lagrangian L(σ, ~π ) as one with the constraint σ 2 + ~π 2 = v 2 , which is
where
√ the vacuum expectation value is assumed, one could solve for σ and plug the solution
σ = v 2 + ~π 2 into the Lagrangian. This gives the so-called nonlinear sigma model [2, p320]:
(~π · ∂~π )2
1
1
1
2
(∂~π ) + 2
= (∂~π )2 + 2 (~π · ∂~π )2 + . . .
L=
(D.10)
2
2
f − ~π
2
2f
Any Lagrangian with the right symmetry properties should describe the same low energy
physics; the linear and nonlinear sigma Lagrangians are attempts at this. The nonlinear
sigma Lagrangian can take many forms. A general definition given by Zee in his book on
quantum field theory is a Lagrangian with a simple form but with the fields in it subject to
some nontrivial constraint. One example is given by the model used by Hill and Simmons in
their review on strong electroweak symmetry breaking:
L(U ) =
f2
Tr(∂µ U † · ∂ µ U ),
4
(D.11)
where U (x) ∈ SU (2) is a matrix. This takes the form of equation D.10 if U = e(i/f )~π·~τ so
that L(U ) = 12 (∂~π )2 + 2f1 2 (~π · ∂~π )2 + . . . as above [2, p320].
In the non-linear sigma model, a field U appears that transforms under UL ⊗ UR so that
U → LU R† , with
U = exp(iπ a τ a /f )
L=
f2
Tr[∂ µ U † ∂µ U ].
4
(D.12)
This can be used in chiral symmetry as well as with the Standard Model Higgs mechanism.
In the latter case, σ is the Higgs field.
100
Appendix E
Phenomenology of a General Model
This thesis focuses on specific phenomenology of topcolor-assisted technicolor, and in turn
specifically the top forward-backward asymmetry that could arise from this theory. It is,
indeed, also possible to study the phenomenology of a more general model, not necessarily
with a top-Higgs but with any scalar, pseudoscalar, or vector. A short description of how to
approach this is given in this appendix. For more details, see for example [56]. This method
of studying physics beyond the Standard Model will, apart from this concise appendix, not
be pursued further in this thesis.
One possibility to explore the phenomenology of a general model when specific parameters
cannot be measured is to make comparisons of parameters. When ratios of cross sections are
calculated, it is possible that only two parameters, of which one known, are calculated, and
these can be checked with the actual cross sections and any deviations thereof from the
standard model. Take for example the standard model with Higgs Lagrangian, but leave
the Higgs a general spin-0 state instead of specifically the Higgs boson; that is, let it be
Φ ∈ {h, S, P } where h is the Standard Model Higgs, S is a (composite) scalar, and P is a
(composite) pseudoscalar (for example, the top-pion) [56]. Now various Yukawa terms are
possible. Ignoring whether the theory is renormalizable or not, this could be for example
t̄tΦ.In the Standard Model, below electroweak symmetry breaking this
t̄th
wouldbe
with
0 1
tL
φ1
t=
just a Dirac spinor; above it, it would be ij qi qj tR = tL bL
t =
tR
−1 0
φ2 R
tL φ2 tR − bL φ1 tR ., since above the VEV of electroweak symmetry breaking L and R must
be written explicitly. Now Yukawa terms are possible like tt̄h, tt̄S, tγ 5 t̄P , where the last
term is parity-invariant because both P and tγ 5 t̄ are parity-odd (tt̄P breaks both P and
CP symmetry). One would typically expect couplings of CP-violating terms to be small
from experiment. The first Yukawa term appears in the Standard Model with Higgs. The
Wh
term m2v
Wµ+ W −µ appears in the Standard Model, but for a pseudoscalar we would have
something of the form Λ1 P µνρσ W µν W ρσ with W µν = ∂ µ W ν − ∂ ν W µ − ig[W µ , W ν ], where
µνρσ appears to make the term be dimension five. The term must be dimension five to
prevent a model from being ruled out by bounds on the Peskin-Takeuchi parameters S and
T.
The LHC looks at cross sections of processes of proton collisions (here SM − h means the
Standard Model minus Higgs theory):
σ(pp → final state) = σSM −h (pp → final state) + σΦ (pp → Φ → final state)
(E.1)
This then looks the same as the Standard Model if the SM Higgs and Φ couplings are the
101
g
t
t
g
t
t
Figure E.1: Two gluons decaying into the general scalar Φ, which subsequently produces two photons.
q
W/Z
W/Z
t
q
g
q
t
(a) Two quarks producing a Z or
W boson, which then produces a
Z or W and Φ; the latter decays
into two photons.
t
t
W
q
t, b
W
t
g
t, b
t
(b) Two quarks producing W (c) Two gluons producing a Φ
bosons, which then produces a Φ and two top or bottom quarks;
particle. The latter decays into the Φ decays into two photons.
two photons.
Figure E.2: Different production modes leading to photon production.
same. An example of this is γγ production, as shown in figure E.1. Let Br denote the
branching ratio. A cross section of two protons decaying into a general spin-0 state Φ and
then producing a final state x can be written as [56, p8]:
σ(pp → Φ → x) = σ(pp → Φ) · Br(Φ → x)
Γ(Φ → x)
= σ(pp → Φ)
.
Γ(Φ → all possible final states)
(E.2)
Γ denotes the decay rate. Examples of decays into photons are shown in figure E.2. Let us
now focus on Φ = S for simplicity.
A cross section is the amplitude squared times phase space, or the diagram squared, which
2 , with Y
gives in the case of figure E.3 the factor gs4 YStt
Stt the Yukawa coupling of Stt̄. We
would then obtain (defining hereby a production constant κprod
gg,S ) [56, p5]:
κprod
gg,S ≡
Y2
σ(gg → S)
= Stt
2 ,
σ(gg → h)
Yhtt
(E.3)
since all other couplings are equal and thus cancel. Define then also a decay constant:
κdec
S,γγ ≡
Y2
σ(S → γγ)
= Stt
2 .
σ(h → γγ)
Yhtt
(E.4)
Note that S → γγ is composed of different diagrams, like the ones in figure E.4. It would then
be interesting to know the ratio of cross sections of photon production involving a general
102
g
t
g
t
S
Figure E.3: The square of two gluons going to a scalar spin-0 state S, which would give the cross
section.
W
t
S
S
+
t
W
Figure E.4: Different possibilities of S → γγ.
scalar S versus h. It can be calculated using equation E.2 as follows:
σ(gg → S → γγ)
σ(gg → h → γγ)
σ(gg → S)Γ(S → γγ)Γ(h)
σ(gg → h)Γ(s)Γ(h → γγ)
2
YStt
Γ(h) dec
=
κ
2
Yhtt Γ(S) S,γγ
Γ(h) dec
= κprod
κ
.
gg,S
Γ(S) S,γγ
=
(E.5)
Note that Γ(S), which is the total decay (i.e. all possible decays) of S, depends on definitions
of couplings. A process as h → tt̄ would be left unchanged under a change of coupling
because it would not be allowed under a different coupling; this is because experiments put
limits on cross sections and thus processes contributing to those. Then in order to study the
compatibility of a theory with experiment, studying coupling ratios would be sufficient.
The pseudoscalar coupling is suppressed by a factor of the inverse cut-off if the pseuW
doscalar is light. This is because W µν has derivatives in it, giving a factor of Λ1 (pW
µ + pν ) for
P → W W . These momenta are on shell proportional to the mass of the pseudoscalar mP ;
m2P
→W W )
the ratio Γ(P
Γ(S→W W ) ∼ Λ2 is then only large if mP Λ, meaning only in that case are its
couplings large. Calculating ratios as these can help decide which models can be ruled out
and which remain.
103
Appendix F
Feynrules and MadGraph
In section 6.6, numerical results of a Monte Carlo simulation of an effective topcolor-assisted
technicolor were studied. For this study, the programs MadGraph and Feynrules were used.
In this appendix, the settings and parameters that were used in these programs are given.
Note that a difference in parameters and cuts that are not specified could have an impact
on the results of for example the forward-backward asymmetry of the top-quark or the cross
section.
F.1
Feynrules
For Feynrules, a copy of the existing file “SM.fr” containing all Standard Model parameters
was edited. Three new particles were added in the section “Particle Classes”. One vector
particle was added to the subsection “Gauge bosons: physical vector fields”, this is the rho
particle; two particles, the top-Higgs and the top-pion, were added to the subsection “Higgs:
physical scalars”. The parts added looked as follows:
( ∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ )
( ∗ ∗∗∗∗ P a r t i c l e c l a s s e s ∗∗∗∗ ∗ )
( ∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ )
M$ClassesDescription = {
( ∗ Gauge b o s o n s : p h y s i c a l v e c t o r f i e l d s ∗ )
...
V [ 5 ] &=& {
ClassName
−> rho ,
SelfConjugate
−> True ,
Mass
−> mrho ,
Width
−> 1 ,
P a r t icleName
−> ‘ ‘ rho ’ ’ ,
PDG
−> 9 9 9 9 ,
P r o p a g a t o r L a b e l −> ‘ ‘ rho ’ ’ ,
PropagatorType
−> C,
PropagatorArrow −> None ,
FullName
−> ‘ ‘ rho ’ ’
},
...
( ∗ Higgs : p h y s i c a l s c a l a r s
∗)
104
...
S [ 4 ] &=& {
ClassName
SelfConjugate
Mass
Width
PropagatorLabel
PropagatorType
PropagatorArrow
PDG
P a r t icleName
FullName
},
S [ 5 ] &=& {
ClassName
SelfConjugate
Mass
Width
PropagatorLabel
PropagatorType
PropagatorArrow
PDG
P a r t icleName
FullName
},
−>
−>
−>
−>
−>
−>
−>
−>
−>
−>
ht ,
True ,
{mht ,
wht ,
‘ ‘ ht ’
D,
None ,
9998 ,
‘ ‘ ht ’
‘ ‘ ht ’
−>
−>
−>
−>
−>
−>
−>
−>
−>
−>
pit ,
True ,
{ mpit ,
wpit ,
‘ ‘ pit ’
D,
None ,
9997 ,
‘ ‘ pit ’
‘ ‘ pit ’
200} ,
’,
’,
’
150} ,
’,
’,
’
Three couplings for each of the three particleswere also added in the “Parameters” section.
The reason that, even though all couplings were later set to 1, three couplings were added
was so that vertices could be easily turned on or off later in MadGraph to study effects of
individual particles. Notice that in the code below the couplings are already given the value
of 1.
( ∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ )
( ∗ ∗∗∗∗∗
Parameters
∗∗∗∗∗ ∗ )
( ∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ )
M$Parameters = {
(∗ External parameters
...
g e f f r h o &=& {
ParameterType
−>
BlockName
−>
OrderBlock
−>
Value
−>
I n t e r a c t i o n O r d e r −>
TeX
−>
Description
−>
},
g e f f p i &=& {
ParameterType
−>
BlockName
−>
OrderBlock
−>
Value
−>
I n t e r a c t i o n O r d e r −>
∗)
External ,
EFFECTIVERHO,
1,
1.0 ,
{EFTRHO, 1 } ,
S u b s c r i p t [ g , rho ] ,
‘ ‘ E f f e c t i v e c o u p l i n g rho ’ ’
External ,
EFFECTIVEPIT,
1,
1.0 ,
{EFTPIT, 1 } ,
105
TeX
Description
−> S u b s c r i p t [ g , p i ] ,
−> ‘ ‘ E f f e c t i v e c o u p l i n g p i ’ ’
},
g e f f h &=& {
ParameterType
BlockName
OrderBlock
Value
InteractionOrder
TeX
Description
},
...
−>
−>
−>
−>
−>
−>
−>
External ,
EFFECTIVEHT,
1,
1.0 ,
{EFTHT, 1 } ,
Subscript [ g , h ] ,
‘ ‘ Effective coupling h ’ ’
To the “Lagrangian” section, the following effective Lagrangians were added:
( ∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ )
( ∗ ∗∗∗∗∗
Lagrangian
∗∗∗∗∗ ∗ )
( ∗ ∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ )
...
FSG [ \ [Mu] , \ [ Nu ] ] = d e l [ rho [ \ [ Nu ] ] , \ [Mu ] ] − d e l [ rho [ \ [Mu ] ] , \ [ Nu ] ]
L E f f e c t i v e r h o := g e f f r h o t b a r . ProjM . Ga [ mu ] . ProjP . u rho [ mu ] + g e f f r h o ubar .
ProjM . Ga [ mu ] . ProjP . t rho [ mu ] −1/4 FSG [ mu, nu ] FSG [ mu, nu ] ;
L E f f e c t i v e p i := g e f f p i t b a r . ProjP . ProjP . u p i t + g e f f p i ubar . ProjM . ProjM . t p i t ;
L E f f e c t i v e h := g e f f h ubar . ProjM . ProjM . t ht + g e f f h t b a r . ProjP . ProjP . u ht ;
and were added to the Standard Model Lagrangian:
LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost + L E f f e c t i v e r h o +
LEffectivepi + LEffectiveh ;
This model was loaded in the “SM.nb” Mathematica notebook that is included in Feynrules. In this notebook, the masses of the lighter quarks were set to zero with the following
code:
(∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗)
(∗
R e s t r i c t i o n f i l e f o r SM. f r
∗)
(∗
∗)
(∗
Only t h e t and b quarks , and t h e tau l e p t o n a r e m a s s i v e
∗)
(∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗)
M$Restrictions = {
Me −> 0 ,
yme−>0,
ye −>0,
MM −> 0 ,
ymm−>0,
106
ym−>0,
MU −>
ymup−>0,
yup−>0,
MD −>
ymdo−>0,
ydo−>0,
MS −>
yms−>0,
ys −>0,
MC −>
ymc−>
yc −>
0,
0,
0,
0,
0,
0
}
Massless.rst
A directory named “Standard Model UFO” is then generated by Mathematica; this contains
the Feynman rules for MadGraph 5.
F.2
Checking the model
In order to check the validity of a model, the squared amplitude of a diagram can be calculated and checked against analytical calculations of single diagrams. An example is the pion
exchange diagram Mπt πt as calculated in section 6.5. Its expression given in equation 6.20
was:
gπ4t
Mπ t π t =
(t − m2t )2 .
4(t − m2π0 )2
t
Using MadGraph, one can generate a process of uū → tt̄ and create a small directory for
checking results using a standalone option:
mg5> import model s m p l u s p i t
mg5> g e n e r a t e u u˜ > t t ˜
mg5> output s t a n d a l o n e c h e c k p i t
A file called “check sa.f” under the directory “SubProcesses” can then be altered to display
a calculation of the squared amplitude from the analytical expression. Variables shat, that,
and Msq are declared at the beginning of the file:
PROGRAM DRIVER
C∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
C
THIS IS THE DRIVER FOR CHECKING THE STANDALONE MATRIX ELEMENT.
C
IT USES A SIMPLE PHASE \ p a r t {CE GENERATOR
C
Fabio M al t on i − 3 rd Febraury 2007
C∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
C
C
LOCAL
C
INTEGER I , J ,K
REAL∗8 P ( 0 : 3 ,NEXTERNAL)
! f o u r momenta . Energy i s t h e z e r o t h component .
REAL∗8 SQRTS,MATELEM, shat , that , Msq
! s q r t ( s )= c e n t e r o f mass
energy
107
REAL∗8 PIN ( 0 : 3 ) , POUT( 0 : 3 )
CHARACTER∗120 BUFF(NEXTERNAL)
}
These variables are then defined just before the section where the matrix elements are called:
s h a t=2d0 ∗ dot ( p ( 0 , 1 ) , p ( 0 , 2 ) )
t h a t=pmass ( 3 ) ∗∗2−2d0 ∗ dot ( p ( 0 , 1 ) , p ( 0 , 3 ) )
write (∗ ,∗) ” shat ” , shat
write (∗ ,∗) ” that ” , that
Msq=(MT∗∗2 − t h a t ) ∗∗2
Msq=Msq/ ( 4 d0 ∗ ( that −MPIT∗ ∗ 2 ) ∗ ∗ 2 )
w r i t e ( ∗ , ∗ ) ” |M| ˆ 2 from t h e a n a l y t i c a l e x p r e s s i o n ”
w r i t e ( ∗ , ∗ ) Msq
c
c
c
Now we can c a l l t h e matrix e l e m e n t !
In addition, the matrix element of the QCD process is turned off by setting it to zero, since
we only look at the analytical expression of a pion exchange diagram. This is done in the file
“matrix.f” under the directory “SubProcesses/P0 uux ttx”:
C
Amplitude ( s ) f o r diagram number 1
CALL FFV1 0 (W( 1 , 4 ) ,W( 1 , 3 ) ,W( 1 , 5 ) , GC 11 ,AMP( 1 ) )
AMP( 1 ) =(0d0 , 0 d0 )
Now the makefile has to be invoked by running “make”. After that, by running “./check” the
amplitude as calculated from the analytical expression can be compared to the amplitude as
calculated by MadGraph. The output then should show the matrix elements are the same:
% . / ch e c k
...
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
shat
1000000.0000000000
that
−657790.14438888500
|M| ˆ 2 from t h e a n a l y t i c a l e x p r e s s i o n
0.24259214276568336
Matrix e l e m e n t =
0.24259214276568336
GeVˆ
0
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
F.3
MadGraph and Madevent
The effective topcolor-assisted technicolor model as generated by Feynrules and saved as
Standard Model UFO output for MadGraph 5 is used to generate a process in MadGraph.
The effective model is imported in the MadGraph program and a process is defined and
generated as follows:
108
mg5>
mg5>
mg5>
mg5>
import model TC2
d e f i n e p = g u u˜ d d˜ s s ˜ c c ˜
generate p p > t t˜
output pp ttTC2
It was verified that the bottom quark did not contribute to the forward-backward asymmetry
of the top, so this particle is ignored in this process.
The run card (“run card.dat”) is modified in order to satisfy the Tevatron settings and
the choices of the CDF collaboration. In particular, the Tevatron has incoming beams of 980
GeV, and both a proton (1) and antiproton (-1) beam. The renormalization and factorization
scales are chosen to be fixed and of the order of the top quark mass, and the parton density
function used is CTEQ6l. The lines altered were the following:
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Tag name f o r t h e run ( one word )
∗
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
pptttc2
= r u n t a g ! name o f t h e run
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Number o f e v e n t s and rnd s e e d
∗
# Warning : Do not g e n e r a t e more than 1M e v e n t s i n a s i n g l e run
∗
# I f you want t o run Pythia , a v o i d more than 50 k e v e n t s i n a run .
∗
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
50000 = n e v e n t s ! Number o f unweighted e v e n t s r e q u e s t e d
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# C o l l i d e r type and e n e r g y
∗
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
1
= lpp1
! beam 1 type (0=NO PDF)
−1
= lpp2
! beam 2 type (0=NO PDF)
980
= ebeam1 ! beam 1 e n e r g y i n GeV
980
= ebeam2 ! beam 2 e n e r g y i n GeV
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# PDF CHOICE : t h i s a u t o m a t i c a l l y f i x e s a l s o a l p h a s and i t s e v o l .
∗
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
’ cteq6 l ’
= pdlabel
! PDF s e t
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# R e n o r m a l i z a t i o n and f a c t o r i z a t i o n s c a l e s
∗
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
T
= fixed ren scale
! i f . true . use f i x e d ren s c a l e
T
= fixed fac scale
! i f . true . use f i x e d f a c s c a l e
173.0 = s c a l e
! f i x e d ren s c a l e
173.0 = dsqrt q2fact1
! f i x e d f a c t s c a l e f o r pdf1
109
173.0
1
= dsqrt q2fact2
= scalefact
! f i x e d f a c t s c a l e f o r pdf2
! s c a l e f a c t o r f o r event−by−e v e n t s c a l e s
Using another PDF, such as CTEQ6l1, can change the results for the forward-backward
asymmetry of the top quark. For example, the probability of the up quark differs per PDF.
A different value and different running for the strong coupling αs could also change results,
as discussed in section 6.6.
The parameter card of the process is changed in such a way that the masses of the new
effective particles are as desired; for example, if all three are taken into account, the mass of
the top-pion should be varied (changed for each run), the mass of the top-Higgs should be
100 GeV more than the top-pion mass for each run, and the mass of the top-rho should be
500 GeV. An example of the changed parameter card for a process including all three new
particles:
###################################
## INFORMATION FOR MASS
###################################
Block mass
5 4 . 7 0 0 0 0 0 e+00 # MB
6 1 . 7 2 0 0 0 0 e+02 # MT
15 1 . 7 7 7 0 0 0 e+00 # MTA
23 9 . 1 1 8 7 6 0 e+01 # MZ
25 1 . 2 0 0 0 0 0 e+02 # MH
9997 m a s s o f p i t # mpit
9998 m a s s o f h t # mht
9999 5 . 0 0 0 0 0 0 e+02 # Mrho
In this card, “massofpit” and “massofht” are replaced each run with a different mass. With
this, 800 runs of 50000 events are made, where each run has a different mass input for the
top-pion and the top-Higgs.
F.4
MadAnalysis
With MadAnalysis, the unweighted events of the runs are analyzed. For all events, several
cutoffs are specified. The pseudorapidity is cut off to be |ηt,t̄ | < 2.0|, and the invariant mass
of the top and antitop is chosen to be Mtt̄ > 450 GeV. These cuts are specified in the file
“ma card.dat”. The top and antitop were added to the list of classes:
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Put h e r e your l i s t o f c l a s s e s
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Do NOT put s p a c e s b e f o r e c l a s s names !
# Begin C l a s s e s # This i s TAG. Do not modify t h i s l i n e
j e t 1 −1 2 −2 3 −3 4 −4 21
# C l a s s number 1
b
5 −5
# C l a s s number 2
t
6
# C l a s s number 3
t b a r −6
# C l a s s number 4
mET 12 −12 14 −14 16 −16 1000022
# M i s s i n g ET c l a s s , name i s r e s e r v e d
110
The histograms are normalized:
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
o r d e r i n g f u n c t i o n pt # o r d e r s p a r t i c l e s i n c l a s s e s a c c o r d i n g t o t h e i r pt
normalization 1
# h i s t o g r a m n o r m a l i z a t i o n , x s e c o r number ( e . g . 1 )
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
and rapidity and invariant mass cuts were implemented:
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Cuts on p l o t t e d e v e n t s
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Modify t h e c u t s and remove t h e pounds / h a s h e s t o apply t h o s e c u t s
# Do NOT put s p a c e s a t t h e b e g i n n i n g o f t h e f o l l o w i n g l i n e s !
# Begin Cuts # This i s TAG. Do not modify t h i s l i n e
etamax
3 1 2d0
etamax
4 1 2d0
mijmin
3 1 4 1 450 d0
# End Cuts # This i s TAG. Do not modify t h i s l i n e
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
The function for the rapidity “eta” and the function for the invariant mass “mij” are defined in
the file “kin fun.f” written in Fortran. The variable “pseudo-rapidity” is defined as follows:
DOUBLE PRECISION FUNCTION e t a ( p )
c ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
c
Returns pseudo−r a p i d i t y o f p a r t i c l e i n t h e l a b frame
c ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
IMPLICIT NONE
double p r e c i s i o n p ( 0 : 3 )
double p r e c i s i o n costh
i f ( abs ( c o s t h ( p ) ) . l t . 0 . 9 9 9 9 9 9 9 9 9 9 d0 ) then
e t a = −d l o g ( dtan ( 0 . 5 d0 ∗ d a c o s ( c o s t h ( p ) ) ) )
e l s e i f ( c o s t h ( p ) . ge . 0 . 9 9 9 9 9 9 9 9 9 9 d0 ) then
e t a = 10 d0
else
e t a = −10d0
endif
end
and the invariant mass, named “Invarient mass” [sic], is defined as:
d o u b l e p r e c i s i o n f u n c t i o n mij ( P1 , P2 )
c ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
c
I n v a r i e n t mass o f 2 p a r t i c l e s
c ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
IMPLICIT NONE
d o u b l e p r e c i s i o n p1 ( 0 : 3 ) , p2 ( 0 : 3 )
d o u b l e p r e c i s i o n sumdot
e x t e r n a l sumdot
111
mij = d s q r t (max( Sumdot ( p1 , p2 , 1 d0 ) , 0 d0 ) )
RETURN
END
An own function was defined that yields a negative number when an event is backward and
a positive number when an event is forward was written in this same file:
d o u b l e p r e c i s i o n f u n c t i o n XY2( pa , pb )
C∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
C
User f u n c t i o n f o r p l o t t i n g and c u t s
C∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
i m p l i c i t none
d o u b l e p r e c i s i o n pa ( 0 : 3 ) , pb ( 0 : 3 )
double p r e c i s i o n y
XY2=y ( pa )−y ( pb )
end
With this number, the forward-backward asymmetry can be calculated. This was done here
by dividing the negative and positive values of this function in two bins, and then applying the
formula for calculating the forward-backward-asymmetry. This is specified in “ma card.dat”
(note the line with “XY2”):
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Put h e r e t h e p l o t r a n g e s
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Do NOT put s p a c e s a t t h e b e g i n n i n g o f t h e f o l l o w i n g l i n e s !
# Begin PlotRange # This i s TAG. Do not modify t h i s l i n e
pt
10
0 500
# b i n s i z e , min v a l u e , max v a l u e
et
10
0 500
# b i n s i z e , min v a l u e , max v a l u e
etmiss
10
0 500
# b i n s i z e , min v a l u e , max v a l u e
ht
20
0 1500
y
0 . 2 −5 5
# etc .
mij
20
0 1500
dRij
0.1 0 5
X1
1 0 0 0 0 . 0 −10000.0 +10000.0
XY2
1 0 0 0 0 . 0 −10000.0 +10000.0
#d e l t a p h i 0 . 1 0 3 . 1
#X1
1
0 100
#XYZA1
1
0 100
# End PlotRange # This i s TAG. Do not modify t h i s l i n e
In the same file it was specified that the results for the forward-backward asymmetry must
be included in the outputfile “plots.top” generated when running “./plot events”:
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Put h e r e t h e r e q u i r e d p l o t s
#∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
# Do NOT put s p a c e s a t t h e b e g i n n i n g o f t h e f o l l o w i n g l i n e s !
112
# Begin P l o t D e f s # This i s TAG. Do not modify t h i s l i n e
...
eta
eta
3 1
4 1
# p l o t pseudo−r a p i d i t y f o r t h e f i r s t two p a r t i n t h e 2nd c l a s s
# p l o t pseudo−r a p i d i t y f o r t h e f i r s t two p a r t i n t h e 2nd c l a s s
...
#X1
XY1
XY1
XY2
XY2
3
4
3
4
3 1
1
1
1
1
# p l o t X1 ( d e f i n e d i n k i n f u n c . f )
...
# End P l o t D e f s # This i s TAG. Do not modify t h i s l i n e
Using these settings, the forward-backward asymmetry of and the cross section of the top
and antitop are then extracted from the data of events that passed the imposed cuts. The
forward-backward asymmetry is extracted from the file “plots.top”, and the cross section is
extracted from the standard output of the program “plot events”. MadAnalysis was used for
the extraction of data, and Gnuplot was used to plot the data.
113
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