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A TOPCOLOR MODEL WITH
FLAVOR-UNIVERSAL HYPERCHARGE
SECTOR

By

Felix Braam

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER of Science

Department of Physics and Astronomy

2007

ABSTRACT

A TOPCOLOR MODEL WITH FLAVOR—UNIVERSAL
HYPERCHARGE SECTOR

By

Felix Braam

The standard model of the electroweak interactions, including electroweak symmetry
breaking and the fermion masses, is in good agreement with most of the current data.
Nevertheless its key ingredient, the “Higgs”-particle, is not completely satisfactory.
Technicolor models have been able to give an alternative explanation of the symmetry
breaking pattern. In order to explain the origin of fermion masses technicolor has
been embedded in a larger model, called “extended” technicolor. The large mass
of the top quark further suggests the existence of new dynamics called “topcolor”
associated with the third generation of fermions.

The subject of this thesis is a topcolor model that by assuming an additional strong
gauge interaction coupling only to the third quark generation explains the large top
mass. Unlike previously discussed topcolor models, this model does not distinguish
between the third and the first, and second generation of fermions in the hypercharge

sector. Implications from current experimental data are discussed.

ACKNOWLEDGMENTS

I especially would like to thank my advisors Prof. R. Sekhar Chivukula, and Prof.
Elizabeth H. Simmons and my fellow student Michael FloBdorf. Further, I would
like to mention Dr. Kazuhiro Tobe and Baradhwaj Coleppa who helped me during

various long discussions.

iii

TABLE OF CONTENTS

LIST OF TABLES .............................. vii
LIST OF FIGURES ............................. viii
1 Introduction ................................ 1
1.1 The Goldstone Theorem ......................... 2

1.2 Chiral Symmetry Breaking in massless QCD .............. 5
1.3 The Standard Model ........................... 8

1.4 Technicolor ................................ 12
1.5 Extended Technicolor ........................... 15
1.6 Topcolor .................................. 19

2 Topcolor with a Flavor-Universal Hypercharge Sector ...... 20
2.1 Symmetry Breaking at the Scale A ................... 21
2.2 Symmetry Breaking in the Electroweak Sector ............. 26
2.3 The Nambu-Jona-Lasinio model ..................... 31

3 Theoretical Constraints ......................... 35
3.1 Constraints From the Gap Equation ................... 35
3.2 The Landau-Pole ............................. 37

4 First Constraints from Existing Data ................. 40
4.1 Flavor Changing Neutral Currents (FCNC) ............... 40
4.1.1 FCNC in R0 — K0 systems ................... 41

4.1.2 FCNC in BO — B0 systems .................... 43

4.2 Constraints on effective four-fermion couplings ............. 44

5 Constraints from electroweak precision data ............. 46
5.1 A Fit to LEPl data ............................ 46
5.2 A fit to a larger set of observables .................... 59
5.2.1 Definition of electroweak parameters .............. 59

5.2.2 Constraints from the global fit .................. 63

6 Summary and Conclusions ....................... 67
6.1 Summary of the Results ......................... 67
6.2 Expectations on Future Experiments .................. 68
APPENDICES ................................ 70

iv

A.1 Constraints from Electroweak Precision Data on Models with a Flavor
Non-Universal Hypercharge Sector ....................

BIBLIOGRAPHY ..............................

70

2.1

5.1

5.2

5.3

5.4

Al

A2

LIST OF TABLES

The transformation behavior of the fermion generations and the com-
posite scalars under the gauge group from eq.(2.1). “SM” means that
the fermions transform under the respective group as they would under
the corresponding standard model gauge group. ............

The input parameters we used in ZFITTER. .............

shows the best-fit one-loop predictions of the standard model for the

set of observables that we used to perform our fit to calculated by
ZFITTER for different Higgs masses. .................

Correlation matrices from [7] for the experimental values in Figure 5.1 .

The values for the electroweak parameters obtained from a global fit
to LEPl and LEP2 data, according to [2]. ...............

The correlation matrix of the experimental values for the flavor non-
universal case. ..............................

The submatrix for the correlations of the lepton left-right asymmetries.

vi

2O

48

57

58

65

73

74

1.1

1.2

2.1
2.2
2.3
3.1

3.2

3.3
4.1
4.2

4.3

4.4

5.1

5.2

5.3

LIST OF FIGURES

Coupling of the Wi bosons to the pions. ...............

The generation of the fermion masses in extended technicolor. In the
last step, the technicolor interaction becomes strong and closes the
external technifermion lines to a loop ...................

Interaction of fermions via Topgluon and Z ' -boson exchange ......
Topgluon and Z' exchange as effective four fermion interaction.

The process of dynamical mass creation as viewed in the NJ L-model.
The triangle of allowed values for ml and n3. .............

The curves (1) and (ii) show the allowed parameter space in the K3 — n1
plane for a scale A = 500 GeV, 1000 GeV respectively. The solid lines
represent the tip of the gap triangle in Figure 3.1. ..........

1n(AH/A) as a function of 51. ......................
Box diagrams of W bosons causing FCNC in the R0 — K 0 system.

Lower bound for (n3, MC) from FCNC in R0 — K 0 systems .....

Lower bound for MC as a function in $3 from FCNC in Bg-Meson
systems ..................................

Lower limits for Al and M Z’ from comparison with LEP2 data.

The first column shows experimental values from [7] for the set of
observables we used to fit the predictions of our model to. In the second
column the best-fit results for our model are listed and the graph on
the right-hand side shows the pulls for each observable. In the last line

the xgm-n-value and the best fit values for our fit-parameters are given.

The plot shows the allowed region on 95% confidence-level after fitting
to the data listed in Figure 5.1. Curve (i) and (ii) enclose the allowed
parameter space for m H = 800 GeV and m H = 1.5 TeV respectively.

The allowed parameter space for M Z’ and n1. .............

vii

14

18
25
26
32

36

37
39
41

43

44

45

52

53

54

5.4

5.5

5.6

A.1

A2

The dashed curve encloses the allowed parameter space in the n3-n1
plane for different values of p. The bold lines represent the tip of the
gap triangle (see Fig.3.1). ........................

The ratio of the two contributions to the p—parameter arising from the
top gluon and the electroweak sector. .................

The constraints on the parameter space of our model from a fit to LEPl
and LEP2 data [2] under the assumption of a Higgs mass of 800 GeV.
The dashed line shows the result from our fit in the previous section
for comparison. ..............................

The first column shows experimental values from [7] for the set of
observables we used to fit the predictions of our model to. In the second
column the best-fit results for our model are listed and the graph on
the right—hand side shows the pulls for each observable. In the last line

the xgm-n-value and the best fit values for our fit-parameters are given.

The plot shows the allowed parameter space on 95% confidence-level
for a flavor non-universal model after fitting to electroweak precision

data [7]. ..................................

viii

55

63

66

71

73

Chapter 1
Introduction

The standard model provides a theory which is able to explain the currently ob-
served physics, including the heavy gauge boson and fermion masses, very accurately.
Nevertheless, there are several indications (“fine-tuning”, etc.), associated with the
Higgs-particle which drives the electroweak symmetry breaking, that suggest that the
standard model should be regarded as effective theory up to the TeV-scale. Techni-
color models [10] can give an alternative, dynamical, explanation of the symmetry
breaking pattern, yielding correct predictions for the heavy gauge boson masses. In
order to explain the origin of the fermion masses technicolor has been embedded
into a larger model, called “extended” technicolor [10]. The discovery of the large
top mass led to complications that could not be compensated within extended tech—
nicolor. Topcolor models [11] associate new dynamics with the third generation of
fermions in order to create the large top mass.

The subject of this thesis is a topcolor model with a flavor universal hypercharge
sector.

In order to give a picture of the framework in which this model has to be seen and
to establish the necessary terminology, we shall briefly review the relevant parts of

modern field theory:

Goldstone’s theorem and chiral symmetry breaking. Then we review the standard
model and the dynamical alternatives, technicolor, extended techninicolor, and top—

color.

1.1 The Goldstone Theorem
In a classical field theory [9] the action of a field (bl-(:13), is defined as

Il¢i(x)l = / d4z£ei<x>a¢i<x>> (1.1)

The fields that extrernize the action, obey the equation of motion

8E 85 -
u a O a l‘z, 1.2
an “mama "11 ( )

where in the last step the canonical conjugate momentum H’“ has been introduced.
A continuous transformation a : ¢i(:r) +——+ ¢i(z, a), where ¢i(:z:, 0) E gb‘fr) is called
a symmetry, if the variation of the Lagrangian can be written as a four divergence of

an arbitrary function F(:1:):

(SE = 8"Ffl, where (if E % . (1.3)

(1:0

It can be straightforwardly shown that, if the above holds, the action is invariant
under the (infinitesimal) transformation (1.

According to Noether’s Theorem, the four-divergence of the current
j“ 2 Hyatt — F" (1.4)

vanishes, which yields the conserved charge Q E f d3cr 30.

In a quantum field theory, following the approach through canonical quantization, gbi

and II,- E 1'1? are promoted to operators which obey

Mil-’13., t), Hj(37, t)] = 26;: 63(53’ -—- 37),
[¢"(a‘=‘.t).¢"(at)] = 0

and [n,-(a,t),r1,(g,t)] = o (1.5)

Starting with a symmetry group G, with elements exp(z’aaTa), where Ta are the
generators of G', we can write an infinitesimal transformation of the field (bi under G
as

¢i —> ((527 + zang) (bl, so that (Saabi = zTéijj. (1.6)
In analogy to the case of a classical field theory one can define a current and its

corresponding charge (whether conserved or not) as
at = .nmw => Qa(t) = . f c132: me, t)T§j¢j(f. t). (1.7)

Using the equal time commutation relations from eq.(1.5), one can show that any

local operator 0(t) built out of the fields (iii and the momenta II,- obeys
[Qa(t), C(15)] = —z 6a0(t). (1.8)

In particular, if the Hamiltonian of the theory commutes with the charge associated
with the generator of the transformation, Q, is conserved (time-independent) and
thus generates a symmetry according to eq.(1.8).

In a quantum field theory, one postulates a state of lowest energy, the vacuum [0), as

a normalizable eigenstate of the momentum operator with

13,, p) = 0. (1.9)

A field theory with a symmetry, in the sense of the above definition, is said to expe-

rience a spontaneous symmetry break down1 if and only if

(0| 6a My) I0) 7f 0 (1.10)

for any field ’l/J(y). This means that the vacuum expectation value of the field is not
invariant under the symmetry transformation, even though the Hamiltonian is.

The Goldstone Theorem says that if a quantum field theory experiences a sponta-
neous symmetry break down, there is massless particle associated with each broken
generator Ta in the spectrum.

Following [9] closely, we show that if for a quantum field theory with a conserved cur-
rent ja every state but the vacuum has p“p,,> e for some 6 > 0, then (0|61/1(y)|0) = 0
for all fields w(y).

Proof:

(Oljalx =2 (OIja.(x )In) (WW )|0>

Z< <0|eszJa< '0“ )e—szIn>(nlethw(O)C-2Pylo)
=2 (OIJa(O())In (nIt/J(0 )I0)61Pn( y—z)
= /d4kZ 54(k—pn)(0|jg(o)ln)<nl¢(0 0)I0)e zk( (—y x)

2 /d4k kupa(k2)ezk(y—x) (1.11)

In the second last line, the sum over n does not give a contribution for n = 0, since
(Oljg(0)I0) must be zero. By assumption, there is some 6 > 0 such that for k2 < 6,

there is no other state with 13,2, < c. This yields for k2 < e : p(k2) = 0.

 

1For a more intuitive introduction to spontaneous symmetry breaking in quantum field theories
see for example [18].

Since the four divergence of j: vanishes, we can see from the last line k2p(k2) = 0,

which yields p(k2) for k2 > 0. Thus
(0|jé‘(:r)2/)(y)l0) = 0 (It analogously follows that (0|w(y)jg($)l0) = 0) (1.12)
This yields

/ c132: (Olljt‘(w),w(y)ll0> = amen) = o. qed (1.13)

Inverting the statement of the proof tells us that if we have a spontaneous break
down of a symmetry (0] 60 7r0,(p) [0) 7f 0, there has to be a massless state in the
spectrum. This state is called Goldstone boson and will be denoted as 7ra. It has to
couple to the conserved current, since if it did not, the above steps of the proof of
the Goldstone theorem would yield eq.(1.13) in contradiction to the assumption of a
symmetry break down. From the Lorenz structure of the current and the fact that

its four divergence vanishes, we obtain

(Olaf (113)7Tb(P)|0> = zf1rID“<5ab 62px, (1-14)
where the so—called pton decay constant f1; has been introduced as the strength of the
coupling of the Goldstone boson to the conserved current.
1.2 Chiral Symmetry Breaking in massless QCD

In massless quantum chromodynamics2 with the two quark-flavors u and d, if one

neglects the electroweak interactions, the Lagrangian reads [9]

1
L = (702prqu — EtrIF’wa/I, (1.15)

 

2The typical energy scale of phenomena determined by QCD, AQCD is of order of a few hundred
MeV. It is therefore a reasonable approximation to assume mu and md to be zero.

where q = (u, d) and foI denotes the S U (3)-color gauge-covariant derivative

cc’
(1

In terms of the matrix valued fields A” E Afig‘QE, the field strength tensor reads
Flt” = 6,114,, -— 8W4” + zg3IAfl, AV]. (1.17)

The Lagrangian from eq.(1.15) is invariant under local 5' U (3) transformations by
construction3. Additionally, there is a global symmetry of the theory. In order to
demonstrate this, we introduce the chiral projectors PR / L = % (1 i 75) which satisfy
PR+ PL =1 and PR PL = 0. Defining qL/R = (“L/RadL/R) E PL/R(u,d) and using

{’75, 7,1} = O, we can rewrite eq.(1.15) as

L = C-I-LZIIDCIL ‘I‘ (IRZEQR — cgauge- (1-18)

This Lagrangian is manifestly invariant under global unitary transformations of the

fermion fields UL x U3 6 U (2) L X U (2) R independently:

qL —* ULqL, (IR “t URCIR- (1-19)

It is generally believed that QCD dynamics induce a non-zero vacuum expectation
value for the operator ciq = q’LqL + fiRqR. Since the formation of this condensate is
assumed to occur because the non-Abelian color interactions become strong at AQCD,
there is no mathematical proof, because at this point the perturbative calculational
tools break down. But assuming that this is what happens leads to impressive pre-

dictions, as we shall see in the following.

From

(qLQL + clean) at 0, (1.20)

3For a detailed discussion of gauge theories, see e.g. chapter 15 in [15].

 

we see that the vacuum expectation value is no longer invariant under separate U (2) L
and U (2) R transformations. However, it remains invariant under U (2)V (“vector”-)
transformations:

a

UV = UL = UR = exp[zflv] eXpIzaIl/T7], (1.21)

where a = 1, 2, 3 and r“ are the usual Pauli-matrices.

We write the original symmetry group as a direct product according to
U(2)L X {1(2)}? = U(1)V X U(1)A X SU(2)V X “Sf/(2)14”, (1.22)

where the subscript A stands for axial vector transformations, which are generated

by

eprzHAry5] = eprzBAIPR + epr—zOAIPL for U(1)A

Ta Ta Ta.
exp[zt9A—2-ry5] = exp[zaf}4?]PR + epr—zafi—Z—IPL for “SU(2)A”. (1.23)

Note that S U (2) A does actually not form a proper subgroup, because the algebra of
its generators €175 does not close. But since we only need the infinitesimal transfor-
mations, regardless of whether they form a group or not, this is not important.
Even though eq.(1.22) is the symmetry group of the Lagrangian from eq.(1.18), quan-
tum effects actually spoil the invariance of the theory under U (1) A right away.4 So
only “SU (2) A” is broken by the appearance of the non-trivial vacuum expectation
value in eq.(1.20). Therefore we expect three massless Goldstone bosons in the spec-
trum which couple to the currents associated with the spontaneously broken genera-
tors of the symmetry “5 U (2) A”

. _ T“
J?“ = ““5aq = q7“75-2—q (1-24)

 

4For details see chapter 19 in [15].

according to

(Oljé‘“(0)lrb(q)> = 2f1rCI“<5“b- (1.25)

Since the current 3'?“ transforms as axial—vector under parity transformations and on
the right-hand side of the equation stands a Lorenz vector, we can conclude that the
Goldstone bosons have to transform as pseudo-scalars under parity.

The behavior of the Goldstone bosons under the (unbroken) symmetry transforma-
tions can be determined by looking at the commutator of the unbroken charges with
jg“. One obtains that they form a triplet under S U (2)V (“isospin”) and have charge
zero under U (1)V (“baryon number”). This isotriplet of Goldstone bosons is taken to
be the triplet of the lightest experimentally observed hadrons, the pions whose masses
are not exactly zero, albeit small compared to AQCD- The pion decay constant is
found to be f1r m 93 MeV.

In reality, the quarks have masses, which explicitly breaks the axial symmetries and
if mu and md are not degenerate, also breaks SU(2)V. Since the masses and their
difference are small, one assumes the symmetry group from eq.(1.22) to be an “ap-
proximate symmetry” of the Lagrangian in eq.(1.18). This leads to a pion mass

proportional to its constituent masses (see [9]).

1.3 The Standard Model

The Glashow-Weinberg-Salam theory [18] of the weak interactions, which is generally
referred to as the “standard model” starts from a Lagrangian that is gauge-invariant
under the gauge group S'U (2) L x U (1)y, where the subscript Y denotes the so-

called hypercharge. For simplicity we shall sketch the model only for one family5

 

5In reality there have been observed three generations of fermions. The other two generations
can be seen, within the standard model, as replicates of the first generations with heavier masses.

of fermions e, V, a, d,. The left handed fermions ( f L E PL f , where PL denotes
the chiral projector as defined in the previous section) transform as doublets un-
der SU(2)L, EL = (VL,€L)Y=_1/2 for leptons and QL = (“LadL)Y=1/6- The right
handed fermions are singlets with respect to SU (2) L and their hypercharge assign-
ments are chosen to be equal to their electric charge, for reasons, that will become
obvious later.

The Lagrangian describing the interactions of fermions with the gauge fields and

among the gauge bosons themselves reads

— 1 1
£fermions = Z zlflfDH-f _ ZF/WF/JV - 106”“!sz (126)
f
with
D” = 8,, + zg'BfiY + ngawg. (1.27)

The sum runs over all left— and right-handed fermions.6 B" denotes the gauge field
associated with U (1)y and WI} represents the triplet of the S U (2) L gauge fields. The
r“ E a“ / 2, where 0“ denote the Pauli matrices, are the generators of SU (2) L- The
last two terms in eq.(1.26) represent the kinetic Lagrangian for the gauge bosons.

A key ingredient to the standard model is a complex doublet scalar field, qt E
(¢+’¢0)Y=1/2- As we shall see in the following it drives the symmetry breaking
creating the masses of the heavy gauge bosons. It is also responsible for the genera-

tion of fermion masses in this model. Its dynamics are described by
‘Cscalar = (DM¢)T(D#¢) — V(¢1l¢)? (128)
with the covariant derivative from eq.(1.27) and a potential of the form

WW») 2 #2 (chic/6) + IAI ((12%)? . (1.29)

6Note, that since the right handed fermions are singlets under S U (2) L they are eigenvectors of
the S U (2) L generators 7“ corresponding to the eigenvalue zero.

 

Additionally, we are free to add an interaction term via Yukawa couplings of the

scalar doublet to the fermions
LYukawa = —Ge EL¢6R - Gu QLg/MtR - Gd CabQLa (15;; (13 + h.c. (1.30)

This term is gauge-invariant under S U (2) L x U (1)y. The mass of the neutrinos in this
simple version of the model is assumed to be zero, which is equivalent to excluding
right- handed neutrinos from the beginning.

If we let the parameter p in eq.(1.29) be negative, the potential has a continuous set
of non-trivial degenerate minima, implicitly given by I¢minl2 = (—,u2/2|A|) E v2 / 2.
We choose the vacuum expectation value of the scalar field to be

<asmmm=(i). am)
V2

which obviously breaks the generators of the gauge groups SU (2)]; x U (1)y, except

from the linear combination

(T3 + Y) (<15) 5 Q ((1)). (132)

which is identified with the electric charge Q. Unlike in QCD where a global symmetry
broke down, no Goldstone bosons appear in the spectrum, but they become the
longitudinal polarizations of the formerly massless gauge bosons associated with the

broken generators. One can see this by expanding the scalar field around its vacuum

a a 0
¢ = exp 2“ “”7 (that . (1.33)
21) V2

which is gauge—equivalent to

expectation value:

 

0
(15 —’ I = (IhIzI+vI)- (1'34)
«2

10

Inserting this into eq.( 1.28), we find

(w w + 1— (92 (wt)? (wi)2+ (—gw2+g'2u)2)

+ “interaction terms” . (1.35)

The term proportional to v2 has the form of a mass term for three massive vector

bosons. Defining the weak mixing angle via

9

c036 E —— (1.36)
92 + 9/2
allows us to denote them as follows:7
W; = 53 (W; IF 2W3) with mass mW = g; ; (1.37)
Z2 = (cos 6W3 — sin BB”) with mass m2 = g 92 + 9’2 . (1.38)

The fourth vector field, orthogonal to 20, remains massless and is identified with the
photon

A], = (sin 6W3 + cos OB”) . (1.39)
In order to obtain the couplings of the mass eigenstates it is useful to replace the
canonical gauge fields in the covariant derivative from eq.(1.27) by the appropriate

linear combinations of the mass eigenstates:

DM = Bfl—i—g— (Wlfr+ + WII'F) — 2le (g cosO’r3 - g’sinGY)

fl
—i-——g—’g——Ap (7'3 + Y) , (1.40)

where Ti = 1/2(o1 :l: 202). In order to complete the association of the residual U ( 1)
gauge symmetry with electrodynamics, we define

I
e = —9—’1-—. (1.41)

1/92_I_912

7Comparison to low-energy measurements where the weak interactions appear as effective four
fermion interactions, allows us to relate the vacuum expectation value of the scalar condensate to
the Fermi-constant according to v = (flGp)(‘1/2) z 264 GeV.

 

11

The first two terms in eq.(1.35) describe the physical “Higgs”-particle with mass
m H = —\/§/J. > 0.
The Yukawa-coupling of the fermions to the scalar field in eq.(1.30) gives rise to

fermion masses and fermion-Higgs interactions:

+12 -— - v
£Yukawa = _ 2 Gf ’0 (foR + fRfL) z) mf = G'f_‘- (1-42)
f fl \/§

One can picture the fermions acquiring mass as an effect of their motion through an

 

“ether” consisting of the condensates of the scalar field:

(425} I

fL
fR

fL

1.4 Technicolor

The standard model as described in the previous section has proven to be extremely
successful describing what has been measured in experiments so far. Nevertheless,
the existence of a fundamental scalar field gives rise to conclusions of unsatisfactory

character:

0 The bare existence of a fundamental scalar field has no observed analogy in

12

experiments so far.

0 Loop corrections lead to a quadratically divergent mass correction to the tree-
level mass of the physical Higgs-field h.(:r:) as derived in the previous section. If
one wants the theory to be valid up to high energy scales, this leads to a severe

fine-tuning problem [19].

0 Beyond the requirement of gauge invariance, there are no further constraints on
the self-coupling of the Higgs field or on its Yukawa—couplings to the fermions.

Hence, we are left with a large number of free parameters.

Technicolor addresses mainly the first two statements. In this theory there is no
fundamental scalar particle. The electroweak symmetry breaking is achieved by em-
ploying an analogy to QCD [10] as described in section 1.2.

In a massless two—flavor version of QCD the Lagrangian is invariant under global
SU(2)L x SU(2)R x U(1)V, (1.43)
transformations. Further we assume the existence of a gauged subgroup of eq.(1.43)
SU(2)L x U(1)y. (1.44)

So we have four massless gauge fields of the unbroken standard model Wi, W2, Bl“
in addition to the discussion in section 1.2. At AQC D the chiral symmetry breaks
down to S U (2)V x U (1)V, which would make three massless Goldstone bosons appear
in the spectrum, if not for the presence of the gauge fields in this case. The interaction
of the massless gauge bosons with the pions shifts the propagator of the W: according

to
giw—q"<1"/q2 gull-(1"QV/ (:2

M" = _)
(12 6120+ I1012))

 

 

(1.45)

13

where II(q2) denotes the vacuum polarization. The process is schematically shown in

the diagram below.

 

Figure 1.1. Coupling of the Wi bosons to the pions.

Since the pion is massless, its propagator has a pole at q2 —-> 0. The current 3'15

couples to 7r; with strength f”, which yields a contribution to II(q2) according to

 

(1.46)

This shifts the pole in the propagator in eq.(1.45) to 21-ng1%, so the W; gauge field
has acquired a mass of mW = %gf7r. An analogous, albeit more complicated, calcu-
lation yields a massless photon— and a ZB-analog with the mass from eq.(1.38), with
f7, instead of 12.

So the chiral symmetry breaking in QCD breaks the standard model gauge group
S U (2) L x U (1)y down to U (1)Q and the pions become the longitudinal components
of the electroweak gauge bosons. Unfortunately, the masses of the gauge bosons are
too light (by a factor of z 2600).

The idea of Technicolor is now to introduce a new type of fermion, the technifermions,
which have a new strong gauge interaction, but where the chiral symmetry breaking
occurs at a higher scale8 ATC >> AQC’D: such that the technipion decay constant
F7r can be numerically equal to the vacuum expectation value of the standard model

scalar field 11. From the above calculation we see that this yields the correct masses

 

8This suggests that the technifermions have an S U (N )Tc gauge interaction whose coupling be-
comes strong at a higher scale Arc.

14

for the weak bosons.
Various models involving different sets of technifermions have been studied (see
e.g.[10],[12]). The simplest model, providing the correct symmetry breaking pattern,

consists of only one 5 U (2) L doublet of technifermions

TL = (2L) , (AR)Y=1/22 (BR)Y=—1/2 - (1-47)
L Y=0

The technifermions are color-singlets, but transform as a non-trivial representation of
technicolor S U (N )T0- In this model there are only three exact Goldstone bosons, the
technipions.9 So the techni-hadronic spectrum consists of the analogs to the hadrons
in massless two-flavor QCD. Their masses are estimated by scaling up the masses
of their QCD-analogs by ATC/AQCD (see again the references given above). The
existence of the technihadrons also plays an important role in the unitarization of
W—scattering.

More realistic scenarios provide, besides the three technipions as exact Goldstone-
bosons, a rich spectrum of “Pseudo-Goldstone-Bosons” which acquire mass due to

their non-trivial transformation behavior under the standard model gauge groups

[10].

1.5 Extended Technicolor

Although technicolor models can explain the mechanism of electroweak symmetry
breaking, they do not provide a complete theory that could replace the standard

model, since they cannot give an explanation for the generation of fermion masses.

 

9The “eaten” and the physical pions are actually linear combinations of the technipions and the
QCD-pions, but since the decay constants appear as weights in the linear combination and F,r >> f,”
the physical pions consist in good approximation only of the QCD-pions and the eaten pions are
mainly technipions.

15

Extended technicolor (ETC) ([12] and references therein), as the name indicates, is a
model which preserves the technicolor-scheme for the electroweak symmetry breaking
and adds a mechanism creating the masses of the fermions.

The idea is to embed the technicolor gauge group SU (N )TC’ and possibly parts of

the full gauge group
SU(N)TC X SU(3)C X SU(2)L X U(1)y (1.48)

into a larger gauge group gETC- At a high scale A ETC >> ATC the ETC gauge
group g breaks down, giving masses to the ETC gauge bosons which mediate interac-
tions among technifermions and ordinary fermions. This can be seen in the following

example [10]:

Suppose having a technicolor gauge group SU (N )TC and some tech-
nifermions in its fundamental representation (N -dim). Now embed
the technifermions in S U (N + 1) ETC, which somehow breaks down to
SU (N )TC- This leaves us with an N -tuple of technifermions and a sin-
glet under technicolor, an ordinary fermion. The broken generators are
exactly those 2N + 1 generators of S U (N + 1) ETC which had. “mixed”
the N technifermions with the additional component of the N + 1-dim

fundamental representation of S U (N + 1) ETC-

The fermion masses are assumed to be generated by the following mechanism, which
is sketched in Figure 1.2:
The interactions of massive ETC gauge bosons (m ETC 3 A ETC) with the fermions

can be, at energies well below A ETC: described as an effective four fermion interaction

 

2
L oc (Xi/‘2‘; ) (107127“an fryflTaF) (1.49)

16

where F denotes the technifermions, f the ordinary fermions and T“ are the broken
generators of gETC- A Fierz-rearrangement of this Lagrangian contains a term of

the form

2
_ (7’92) [(FLFR)(foR) + h.c. +...] (1.50)
ETC

such that at energies below ATC B 0.5 — 1.0 TeV, Where the technifermions condense
(FF)

(FF) 75 0, this takes the form of a mass term with mf a: gETC’ A .
ETC

17

QETC QETC

AETC >> E 2 Arc

 

AT0>E

 

V
V

f f

Figure 1.2. The generation of the fermion masses in extended technicolor. In the last
step, the technicolor interaction becomes strong and closes the external technifermion lines
to a loop.

The extended technicolor gauge bosons cause flavor changing neutral currents, which
forces the scale A ETC to be very large, in order not to disagree with the experimental
results from for e.g. the measurement of the keen mass-splitting. At this very high

scale, the coupling constant of the ETC-group is very small. This makes it difficult

18

to generate fermion masses of the appropriate size.10 A possible solution may be
a so—called “walking” of the technicolor coupling constant [13] up to A ETC- It is

generally believed that this can explain masses up to the bottom quark mass.

1.6 Topcolor

The extremely high mass of the top-quark suggests that there are extra dynamics
associated with the top-sector that single out the top-quark. The idea of topcolor
[11] is to introduce a strong interaction, that makes the top-quark form a condensate
and thus dynamically creates the top-mass via the coupling of the top—quark to its
condensate, in analogy to the mass generation for the light fermions in extended
technicolor models.

Various ansatze have been discussed in the past [12]. The subject of our work is
a topcolor model with a flavor universal hypercharge sector and how its parameter

space is constrained by existing data from experiments.

 

1The ratios of the different fermion masses can be adjusted, but the problem is to create the right
absolute size. Even the charm quark mass can barely be realized.

19

Chapter 2

Topcolor with a Flavor-Universal

Hypercharge Sector

We start from the gauge group

SU(3)1 X SU(3)2 X SU(2) X U(1)1 X U(1)2, (2.1)

where the corresponding couplings are chosen such that 93(1) > 93(2). Quarks and
leptons transform only under the stronger U (1) group and in the color sector the
third generation of quarks transforms under the stronger S U (3), whereas the light

quarks transform under the weaker S U (3) group.

 

[gen [I SU(3)1 I SW33 | SUQII U(1)1 ITJUIzl

 

 

 

I’ll 1 “SM” “SM” “SM” 0
111 ‘(SM” 1 “SM” “SM” 0

- _ L
(DA 3 3 5% 21/6
451-20 1 2 0 %
(pt 1 1 2 —% 0

 

 

 

 

 

 

 

 

 

Table 2.1. The transformation behavior of the fermion generations and the composite
scalars under the gauge group from eq.(2.1). “SM” means that the fermions transform
under the respective group as they would under the corresponding standard model gauge

group.

20

At the scale A

AETC > A > Aweak: (2-2)

a condensate ((1) A) = FA (Sufi, that transforms under the initial gauge group in eq.(2. 1)
according to Table 2.1 forms, which breaks the color sector and the hypercharge

groups into their diagonal subgroupslz
SU(3)1 x SU(3)2 —» SU(3)C. U(1)1 x U(1)2 —» U(1),» (2.3)

This symmetry breaking also triggers the condensation of top quarks (got) at a much
lower scale. The discussion of this mechanism shall be postponed to section 2.3.
The remaining standard model electroweak gauge group is then broken by the tech-

nifermion and the top condensate

0 0
((151m) = (gm). tot) = ( f ) (2.4)
«5 7'5
according to
SU(3)C X SU(2)L X U(1)Y —> U(1)Q. (2.5)

2.1 Symmetry Breaking at the Scale A

The relevant part of the Lagrangian (i.e. omitting the S U (2) gauge fields for this

discussion) is given by
£=11121D11+tr(0<1>)l0#<1> +£ 26
It A A Gauge: ( - )
with

Du = 3p + wan/111,17"? + mum/13,17"? + z91(1)Bl)uY1 + Z91(2)132;1Y2 (2-7)

 

1Within this thesis we will not specify the nature of this particle any further. We will just
investigate the implications of this model for low-energy physics.

21

where T“ = A; are the generators of S'U (3). All,“ and A23,“ are the SU (3) gauge fields
and B}, and 3,2,, the fields associated with the respective U (1) gauge groups. £Gauge
denotes the gauge kinetic terms for all four gauge fields.

According to the Goldstone theorem (see section 1.1), there appears one massless
particle associated with each generator of a broken global symmetry. In this case, a
local symmetry is broken, which triggers the Higgs-mechanism and makes the Gold-
stone bosons become the longitudinal components of the gauge fields, belonging to
the broken generators. In other words, these gauge bosons acquire mass as the mass-
less Goldstone bosons vanish from the spectrum.

To demonstrate this, we look at the zeroth order term of the expansion of the field
(I) A around its vacuum expectation value ((1) A). The covariant derivative from eq.(2.7)

acting on ((13 A) gives2

0,, (m) = 3p ((1)11) +ZQ3(1)Afp (3;) (“I’Al — z93(2)A(2zu (CPA) (A2?)
+zgl(1)Ble1 ((DA) + 291(2)82#Y2 ((1)11)
= 2A (2‘2") (93(1)A‘1‘u ‘ gum/‘5?)

+ZA§‘% (91(1)Blp ‘ 91(2)82p)

2The minus sign in the first line of eq.(2.8) accounts for the condensate transforming, according
to Table 2.1, under the conjugate representation of S U (3)2.

 

22

Inserting this in the kinetic term of (I) A in eq.(2.6) yields

91091. <9) W02 <9) )1

M M b ”
=~n<-.>(91<)<>

 

 

h—NP—d
:%5ab
2
p 1
+ $91-21 (*291(1)31p + z91(2)}5'2p) (291(1)Bll — 291(2)B’2)I
=3
A?“ I 93(1) ‘93(1)93(2) 0 0 ) Al“
:93 g“ -93(1)93(2) 93(2) 20 2 0 A21“
2 81,. 0 0 (If; gfu) (5’) 921(1)91(2) BI:
3211 I 0 0 -(E) 91(1)91(2) (5) 9f(2) / 32

(2.9)

The upper block of the matrix appearing in the last step can be diagonalized, by

rotating to the basis of the mass eigenstates:

 

 

 

 

0““ = 93(2)”? + 93(1)A(2m, (2.10)
793(1) + 93(2)
a. . 93(1)Ai“ — gun/“21“
0’: 2 2 , an)
793(1) + 93(2)

where the first octet of fields belongs to the eigenvalue zero and is thus identified with

the massless gluon and the second octet is the so called top gluon, with mass

An analogous diagonalization can be performed with the lower block of the matrix in

eq.(2.9), yielding the massless BI, and the massive field Z’

 

 

 

 

B“ + B“
szmm;-gf)K an)
\/91(1) + 91(2)
B“ —- B”
791(1) + 91(2)

23

The mass of the Z’ boson is then given by

2

M2" = 24' A2 (gin) + giml (2°15)

The gauge covariant derivative from eq.(2.7) can then be expressed in terms of the

mass eigenstates:

D), = (9,) + 293(1) (cosw Cf] + sinw 0):) T1“ + zg3(2) (— sinw Cf: + cosw Gfi) T;

+ 291(1) (cos (j) Z], + sin (b BL)Y1+ 291(2) (— sin (I) Z], + cos d) BL) Y2, (2.16)

where the notation

93(1) 93(2)

 

 

 

 

 

 

 

 

cosw = 2 2 , sinw = 2 2 (2.17)
793(1) + 93(2) 793(1) +9242)
_ 91(1) - _ 91(2)
cos (b — 2 + 2 , Sln¢ — 2 + 2 (2.18)
\/91(1) 91(2) \/91(1) 91(2)

has been used.
The interactions of fermions with the massive vector bosons, arising from the fermionic

kinetic term in the Lagrangian in eq.(2.6), then have the form

_ ,\a . _ A“
LintC = 93(1)COSW WII ’7” (“2'“) 10111 - 93(2)smw $1,117“ (7) z/JIJII Cf],
(2.19)
LintZ’ = 91(1)COS¢ U)“ f) Yf 2,,“ (2.20)

where W,- denotes the quark fields of the i—th generation and f any fermion field. This
suggests that we define the coupling of the Z ' to the fermions and the coupling of the
top gluon to the third quark generation as

gin) 93(1)

E , [£3 E ,
7%) + 9i<2> 793(1) + 93(2)

 

 

 

 

“1

(2.21)

24

respectively. The analogous expressions to eq.(2.19) and (2.20) describing the inter-

actions of the massless vector bosons are given by

A“ A“
£intG= [93(1) Sinw 117111 '1“ (— ) 19111 + 93(2) 0089! 1/11, II 7“ (— 2 ) 1121,11] Gil

= 930939; [11; u (A2“) 11:] of], (2.22)
93(1)+93(2)

£9992 =91(1) sincb (W f) Y’ BL =

 

 

91(1)91(2)
79in) + gitz)

The respective couplings of the bosons associated with the unbroken gauge groups

 

 

(Hi f) Yf B,’,. (2.23)

are defined as

 

 

 

 

— g3(l)g3(2) and a =93
_\/92 +92 S_47r
3(1) 3(2)
’= 91(1)91(2) and a —g,2 (224)
_ 2 2 3“: 47r '
\/91(1) + 91(2)

Rewriting eqns.(2.19) and (2.20) in terms of 193, a S and n1, ay yields

A“ A“
5mm: m [flit/2111 7" (— ) 111111 - 7:11-21, II 7“ (— 2 ) 191,11] 03 (225)
£int Z’ = Isl (Ir-’7’“ f) Yf Z; (2.26)

These Lagrangians give rise to processes as sketched below.

C 2’

Figure 2.1. Interaction of fermions via Topgluon and Z’ -boson exchange.

Since the gauge boson masses are presumably at least of order of a few TeV, one
can, for processes with a momentum transfer well below this energy, integrate out the

intermediate vector bosons. The effective four fermion interactions

25

1 1
cc —2' oc —-s-.
MC MZ,

Figure 2.2. Topgluon and Z’ exchange as effective four fermion interaction.

can be described by effective Lagrangians

a a.

2 - A .. ,\ 2
£0 = _A_4% [fl #2111 7;» (7) 30m - 3% WU 7,2 (7) Wu] (2-27)

 

27r - 2
£21: — 2 In (f'y“Yf) (2.28)
M Z’
Note, that with the couplings as defined in eqns.(2.21) and (2.24), we can express the

masses of the heavy gauge bosons in terms of p, A, (:1 and 53:

(K3 + as)2
53

N1 + ay)2

ME. = 471'A2 , Mg, = 7r (131))2 ( (2.29)

“1

2.2 Symmetry Breaking in the Electroweak Sector

In the previous section, we analyzed the symmetry breaking at the scale A and ob-
tained in the U (1)-sector a massless field B1, coupling via the hypercharge Y 2 Y1 +Y2
as in the case of the unbroken standard model gauge group U (1y). Additionally there
is a massive vector field ZL. Electroweak symmetry breaking is driven by two com-
posite scalar fields ¢TC and (pt, with the non-trivial vacuum expectation values of
eq.(2.4).The effects of the extra ingredients on the breaking of the standard model
gauge symmetry will be subject to the following analysis.

The Lagrangian describing the electroweak sector is given by
L = @1sz + tr [(DpaTCfl Wm] + tr [(Dmotfl Dim] + Lemme, (2.30)

26

with
. 2'
D” = 8,) ‘l' nglzl'gQ—l + ZQIYBL + 2 (91(1) COS ¢ Y1 — 91(2) Sin ¢ Y2) Z11. (2.31)
Rewriting this in terms of the familiar fields, the photon An, coupling via the con-
served electric charge Q = T3 + Y and the Z boson, by rotating the W3 and the BL

fields by the weak mixing angle 6, gives

 

+ ._
DH: ,2 _g_w+0__ .2. ~i_ Au LT3_ '29le
('3 +z\/.2_ 2 +z\/.2_W 2 +zeQ +zsin6c030( Qsm )
e - 2 I
Y —Y Z” 2. 2
+zcos05in¢cos¢( 1 sm (1)) ’ ( 3 )

where we have expressed the couplings in terms of the electric charge (2 and the
rotation angles:

9’ e I 9, e

__=_..____ =———=————— 2.
cosqb cosOcosgb’ 91 sinqb cosGsingb (33)

g=-.——, gé=

Evaluating the kinetic terms of the condensates in eq.(2.30), we find that the Z and
Z’ are no longer the appropriate superpositions of canonical gauge fields to represent
mass eigenstates.

In order to have a more convenient notation at hand, we define Y’ E Y1 — sin2 (bY

and introduce the short handed notation
<0a0b> s Z<¢i>léiéb<¢a (2.34)
i

where the sum runs over all condensates and O can denote any operator.

The proper mass eigenstates are obtained by diagonalizing the following mass matrix

[4]
' 6
Z _ SiDQCOSG . sinB (T3Y’) sin2 6 (ylyl> ' ( ' )
sm $ cos 3 sin ¢ cos ¢

27

Evaluating the entries of the mass matrix gives for the upper diagonal entry:

<T3T3> = <¢Tc>*T§T3<¢TC> + <¢t>TT§T3<¢t> + can"; T3<<I>>

=0
_ Err: -1 __1_ £613.11 1 .1._ft_
= %(F%C+f?) E v; (236)

ft denotes the decay constant of the top-condensate (in analogy to f,“ the pion decay

constant in QCD). For the off-diagonal entries, we obtain
(T3Y'> = (T3Y1) — sin2 ¢<T3Y) = (T3Y1) + sin2 ¢<T3T3), (2-37)

where in the last step the conservation of electric charge has been employed: Y =

Q - T3 with Q = 0 => Y = —T3. Then we can calculate

2
T Y = TTIY TTJY ft YtL— YltR _f__t _ _.fL
< 3 1) (¢TC) 3 1(2‘0) +(¢t> 1(¢t)= “242“ 1 )fl_ 8
- —3
(2.38)
and conclude
I 1 - 2 2 2
(T3Y ) = g (sm 45v — ft). (2.39)
Finally, the lower diagonal entry gives
<Y’Y’> -—- <(Y1 — sin2 «bl/)2) = «Y1 + sin2 323)?)
= (Ylyl) + 2 Sin2 $(T3Y1) + Sin4 ¢(T3T3)
2 2
= ((1))lY1lY1(<I>) + (affix/1(a)) — 2sin2 (3%— + sin4 ((38—
1
-p§ (p 2A2 + ft2 + sin4 (151)2— —2sin2 (bf?) . (2.40)
Factoring out
6 2 2 _ 2
(2sin0cosé’) v _ MZISM’ (2'41)

28

the mass matrix (eq.2.35) then takes the form

1a
2_ 2
MZ—MZ|SM( )

023

where a and :c are given by

 

 

 

= sin 0 (T3Y’)_ sin 0 __ IL
a _ sin (b cos ()5 (T3T3)— sin 45 cos (b '02
a: = sin2 0 (Y' Y’ ) sin2 0 Sinz¢pv( A)2
_ sin2 45 cos2 d) (T373) zsinz (1) cos2 (b

From diagonalizing the mass matrix, we obtain the eigenvalues

1 40:2
M2 _ M2 _

a2

2 X 1 — 7c-
ZISM ,
IE

 

 

to first order in 1/23.

 

2
2 2 2 2
e v v f
4:) M 2 = l — —— sin2 — i
Z 4 sin2 0 cos2 6 p2A2 ( ()5 122)
M 2 — €2P2A2 .
Z 4 cos2 6 sin2 (b cos2 d)

 

The corresponding eigenvectors take the form

2;: w=Z“——Z’“
1:

I
nléw— — —Zu + 2’”.

(2.42)

(2.43)

(2.44)

(2.45)

(2.46)

(2.47)

(2.48)

Horn here on, we shall denote the new mass eigenstates simply as Z l‘ and Z’ I‘. The

couplings of the new states to fermions can be read off directly from eq.(2.47) and

29

eq.(2.48)

 

 

 

 

 

2 2
__€__ _ . 2 _ ”U - 2 _ _f_t_ _ - 2
92 _sin 6cos 6 T3 Qsm 6 p2A2 (Sln 6) v2) (Y1 Ysm (b)
_ =Ycos2¢
’ 2 2
_ 6 ’U 2 , 2 - .fL
—sin6cos6 T3 (1+ 122A2 COS d) (mm o 122))
v2 f2
— Q sin2 6 + p2A2 cos2 (2 sin2 (b — i (2.49)
6 2 8
= = —Y t 2.50
gZ’ cos6sin¢cos¢ OS ()5 cos6 C0 ¢’ ( )

where we have used the fact, that all fermions transform only under U (1)1. 3

One should notice, that the mixing angle 6 in the gauge couplings deviates from the
weak mixing angle that appears in the standard model couplings. There are now
several ways to define a weak mixing angle at tree-level. One of them is to define the
mixing angle in terms of the three most precisely measured electroweak quantities a,

CF and MZ
7m

—, (2.51)
fiGFMg

sin2 62 cos2 62 E

corrections in sin 6 arise from the shift in the Z-mass (since the other constants remain

the same as in the standard model). Plugging in the results from eq.(2.45) yields

 

 

2 ° 2 2 2
. 2 _ , 2 1) sm 6003 6 FTC 2
sm 6g — sm 6+ p2A2 cosz6—sin26 ( v2 — cos qfi , (2.52)

’U

to second order in K' In this model the above definition deviates from the weak

mixing angle defined by the ratio of the masses of the W and the Z bosons:

2 2 2 2
2 _ MW 2 v FTC 2
6 = = 6 1 — —— -— . 2.53
cos W ( M2) cos + (73A) ( 02 cos (b) ( )

3Working to this order, we obtain exactly the same result for the Z ' mass and its coupling to the
fermions as in the previous section.

 

3O

Therefore, we have to be very careful, when calculating observables by shifting the
couplings of the Z boson and the weak mixing angle in the standard model predictions,

about the definition of the weak mixing angle.

2.3 The Nambu—Jona—Lasinio model

Let us now address the question how in the topcolor model the large top mass is
created. The idea can be understood in terms of a useful toy—model, the NJ L—model.
We have a strong interaction SU(3)1 which couples more strongly to the third gener-
ation of quarks than to the others. When SU(3)1 x SU(3)2 breaks at the scale A, the
top gluons acquire mass, according to eq.(2.12). At lower energies their interaction
with fermions can be viewed as effective four fermion interaction (see Figure 2.1). As
in the extended technicolor model in section 1.5, we notice that the Lagrangian from

eq.(2.27) contains after a Fierz—rearrangement [3] terms of the form

8WK3
2
MC

 

£NJL = [(1% Hz) (t-L 11112) + h-C-l - (2-54)

We will refer to this as the NJL—Lagrangian. The NJL—model now assumes that the
self-interaction of the top quarks is strong enough for a top-condensation, which leads
to a mass creation as sketched in Figure 2.3. Self-consistently, one finds that the mass
has to obey the following relation4

__ 87rn3 d4]: 2 1 5
—zm—‘M—(2} Wtr[(m)§(l+7)]ch

(2.55)

 

4Here one makes the rather strong dynamical assumption that the terms in eq.(2.54) are the only
ones that contribute to the mass creation. We will therefore use the results that we obtain from the
NJ L-model only use for a qualitative analysis.

31

The trace gives a contribution of 4m, since the trace of an odd number of "y-matrices

vanishes. To carry out the integration, we rotate into Euclidean space

 

2m

2
_ 167rNcn3m /M0 1 2422;, (mg.
0

 

Mg (2704 4,2,: — m2
= 2 MB m (1 —- fl710g <—q—22—)) . (2.56)
7r MC m

From here we can see that we get constraints on K3 and A in order to make eq.(2.56)
give us a finite positive solution for a top—mass (which will be treated as being equiv-

alent to the formation of a top—condensate) according to

1r m2 Mg
= ln —§',
m

_ _ __ 2.57

where we have used that MC >> m. Eq.(2.57) is the so—called gap-equation for a

dynamical fermion mass.

 

317(1)) m) : ppm—m

Figure 2.3. The process of dynamical mass creation as viewed in the NJ L-model.

The above calculation lacks an explanation for the non-appearance of a bottom-quark
condensation, since the top-gluons do not distinguish between top and bottom quarks.
But we did not take into account the four fermion interaction due to exchange of a
heavy Z’ boson yet. Taking this interaction into account, we expect a constraint on
141, the effective coupling of the Z’ to fermions. The process can then be understood

as follows:

32

The coupling of the top-gluon 143 to t and b quarks has to be strong and very close
to its critical value at which a top- and bottom-quark condensation occurs. Then
the self-interaction due to Z’ -exchange, which depends on the hyperpercharge Y and
hence distinguishes between t0p- and bottom-quarks, tilts the total self-interaction
to be either strong enough to create a condensate or not.

The calculation taking the 2’ into account is analogous to the steps shown above,
albeit more complicated, and we modified the results from [17] to our model. The

resulting gap-equation for our model is:

2 2 2
_ mez' mf Mz’
mf — 01 871'2 [1 (HZ—I) 1n ( ]

 

 

 

 

 

2
mi
3me3. mf 2 Mg.
G’ 1— — l —— 2.
+ 3 842 l (MC) n m} l ( 58)
where
87f f f -
G1 = 2 [ill/L YR for all fermions, (2.59)
. M Z! '
G3 = 0 for leptons, (2.60)
4 2
= ”203 for quark gen. 1,11, (2.61)
M06313
47r
= —2—K.3 for quark gen. III. (2.62)
MC

This model does not take into account the unbroken gauge interactions, which can
contribute to the condensation as well. Including the appropriate correction terms

yields according to [17] the gauged gap equation

2 6 mM2 m 2 M2
mf(1‘;aS‘;Y5YfaY)=GI gn22'[1-(.M_;)1n( le

3meg. mf 2
+ G3 87:2 [1- (M5) 1n

 

 

 

33

A positive solution for a fermion mass to the gap—equation is taken to be equivalent
to the occurrence of a condensation of the respective fermion. We will choose the
couplings such that we obtain a top-, but no bottom-condensate.

The top-condensate (ft) 74 O breaks chiral symmetry which makes three top-pions
appear in the spectrum.5 Their decay constant can be estimated using the Pagels-

Stokar relation [12] to be

 

f3 2 NC 42

 

5Since there is no reliable estimation of their masses, we just have to assume that they are heavy
enough not to give rise to potentially dangerous effects [12].

34

Chapter 3
Theoretical Constraints

In this chapter we shall investigate how the parameters have to be chosen, so that the
model can provide the required generation of the top mass. We will also briefly discuss
the Landau-pole of the U (1)1 group and how the parameter space gets constrained

by the requirement that we want the theory to be valid up to high scales.

3.1 Constraints From the Gap Equation

Inserting the masses of the heavy gauge bosons from eq.(2.29) into the gap equation

and assuming A > m f, yields
2 6 f f _ mi 2 2
In order to obtain a non trivial, positive solution for m f in eq.(2.63), i.e. a fermion

condensate ( f f ) # O,

1 2 6

must hold.

Since we only want top-condensation to occur, we have to make sure that there is a

35

solution for the top quark-mass, but none for the other fermions.1 The three strongest
bounds come from the following set of inequalitieszz

2 27r 4 4

(it) 75 0, if K33 + 2—'7K.1 Z —3— — gas — gay
— 1 2 4 2
<bb> = 0, If K3 — 577-51 S —37: — 5045' + 'Q—CIY
(fr) =2 0, if rel g 27r — Gay, (3.3)

which are plotted in Figure 3.1.

 

 

 

 

 

Figure 3.1. The triangle of allowed values for ml and 53.

Above the line (i) is the area, where (it) 74 0; (iii) is the upper limit, for non-
appearance of T condensation and (ii) the lower bound from (5b) = 0.

Note, that curves (ii) and (iii) remain the valid limits for the (6b) = ($7) = 0 re-
gardless of the size of the scale A. Since we do not want to have any mass being

created by condensation of b-quarks or T’s, the approximation that leads to eq.(3.1)

 

1We do not know the exact value of the part of the top mass that has to be generated via this
mechanism, since we also expect a small contribution of a few GeV from the Extended Technicolor
sector. But fortunately our analysis turns out not to be sensitive to small deviations in the top mass
that comes from top—condensation.

2Even though the last inequality follows from eq.(3.2), the formation of a T-condensate does not
fit into our picture of the mass generation, since the leptons do not couple to the strongly interacting
sector. During our following analysis it will turn out that this region of the parameter space will
not be of interst.

36

is always valid for m f —-> 0 + c. On the other hand the limit for top-condensation
becomes a single line in the K3 — 191 plane, if we assume the condensation to happen
at a particular scale. In Figure 3.2 the lines that represent the allowed parameter
space for top-condensation at a particular choice for A are plotted for different values
of A, assuming the hypercharge p of the condensate that forms at that scale to be

one.

 

 

 

 

 

1.85

K3

Figure 3.2. The curves (i) and (ii) show the allowed parameter space in the K3 — 521 plane
for a scale A = 500 GeV, 1000 GeV respectively. The solid lines represent the tip of the gap
triangle in Figure 3.1.

Of course the values for A are always within a certain interval, but on can see that if

the scale is forced to be higher than 1 TeV, the parameter space will be very narrow.3

3.2 The Landau-Pole

In loop calculations in gauge theories one generally encounters infinities that force us
to think about the meaning of the coupling constants and masses appearing in the

Lagrangian. The procedure of redefining the constants in a sensible way, such that

 

3The line that would belong to A = 10 TeV is already so close to the line from top condensation
at an infinite scale that the resolution of the plots was not high enough to separate them.

37

those infinities cancel, is called renormalization. According to the standard literature
(e. g. [15]), the coupling constants become dependent on the momentum transfer in the
loops (“running couplings”). For Abelian gauge theories like QED, the strength of
the coupling increases with energy, whereas in non-Abelian gauge theories like QCD
the coupling constant is strong at low energies and converges to a small value at high
energies (“asymptotic freedom”). The Landau-Pole denotes the scale at which the
running coupling constant of an Abelian theory becomes infinitely strong, i.e. the
physics at this energy cannot be described by that theory any more.

In the model, that we are investigating, we expect the coupling of the U (1)1 group
4

to exhibit such a running behavior.

In analogy to QED, the running of the coupling constant

 

 

 

gin)
(11/1 5 —47r— = Ozy + [$1 (3.4)
is given by
ayl
AH 1 — ayl 3% ln (#)
A

 

where C denotes the sum over the squared U (1)1-charges of all particles contributing
to the self-energy of the U (1)1 gauge boson. Taking all standard model fermions in a
chiral representation into account, we obtain 0' = 5. Inserting eq.(3.4) into (3.5) and
assuming that ay(A) it: ay(MZ) = a(MZ)/cos 6, we obtain the ratio of AH/A as a

function of R1.

 

4Since there are no fermions transforming under the second U (1) group, there are no interactions
that could cause a running behavior.

38

 

 

 

 

 

0 0.5 1 1.5 2
K1

Figure 3.3. ln(AH/A) as a function of K4.

As one can see from Figure 3.3, ml has to be very small, if we want our theory to be

valid up to a very high scale.

39

Chapter 4
First Constraints from Existing Data

Our model shifts the predictions of the standard model for well measured observables.
We shall investigate in this and the following chapter how the parameters of our model
are constraint by the requirement that our predictions for observables are in agreement

with current experimental data.

4.1 Flavor Changing Neutral Currents (FCNC)

The fermion ficlds appearing in the effective Lagrangian eq.(2.27) are in their gauge
eigenstates. As the top-gluons single out the third quark generation, there can occur
a further unknown mixing among the mass eigenstates of up- and down-type quarks
respectively.

For a first estimation of how this mixing limits the free parameters of our model,
we assume that only the left handed down-type quarks mix, according to the CKM—

mechanism from the electroweak theory. This yields FCN C at tree-level according

4O

to:

 

27r T — A“ 2 0132 f — A“ 2
= — M02 [‘3((V3ivj3) Bi'YME'Bj) + — (VjiVik) Bflfl—Z-Bk
t - A“ t -. A“ .

where the primed fields denote the gauge eigenstates and the unprimed, the mass
eigenstates, which are obtained by rotating the gauge eigenstates with V. Further, I
used the abbreviations D = (d, s)T and B = (d, s, b)T.

In order to find the constraints on the free parameters of our model, we compare the

FCNC arising from top gluon exchange to the standard model prediction.

4.1.1 FCNC in R0 — K0 systems

In the standard model, F CNC arise at loop-level from interactions as displayed in the

box diagrams in Figure 4.1.

d W 8 d U,C,t s

6H m §I
E
E

cm
on
F

m A
@4-
9—:

W d
Figure 4.1. Box diagrams of W bosons causing FCNC in the K0 — K 0 system.

This interaction can be described in terms of an effective Lagrangian of the form [3]

Leff SM = —95M(JL7"SL)2, (42)

41

where

 

_G’F a
:= V V-* . 4.3
sst —T 167T(MWISII1 0w)2 2 22217717, ( )

The corresponding physical observable would be the K 0 — K 0 mass splitting, but for
our purposes it is sufficient to just compare the factors (I in the Lagrangian. The

Lagrangian yielding the AS = 2 neutral currents at tree-level in our model reads:1

 

£0 = — 5:2(n3(V113V32)+a—';22-l-[(V111V12) (mgr/”)2 + 2 (V1I1V12V112V22)]
—20zs [(V1I1V12V113V32 + V1:2V22V1I3V32)]) (JL7/‘-/\2—GSL)2 .
as

In order to compare the factors in the Lagrangians, we have to bring the currents in
eq.(4.4) to the form of the current of the effective standard model Lagrangian eq.(4.2).

We sum over the indices of the Cell-Mann matrices A“ and employ the relation

A“ [\a 1 1
— — =— 5 a —-6 6 4.
2(2)afl(2)76 2(01613’7 30376) (5)

a

This yields a term of the shape

— - — 2
(dLa’7“SL{3) (dLfi'l’uSLa) = (-)(—)(dL7”8L) , (4-6)
where the first minus sign comes from a Fierz—rearrangement and the second one

from interchanging Grassmann valued fermion fields. Plugging this in the Lagrangian

 

1Note, that for gauge bosons that couple equally to all fermion generations, the couplings could
be factored out and the sum over the matrix elements would give 1, since the mixing matrix has to
be unitary. In that case there is no residual flavor changing neutral current.

42

eq.(4.4) and comparing the factors, demanding
2“ ’r 2 es2 1‘ 2 t 2 ’r 1‘
“SM > — W (K3 (V13V32) + 7c? [(V11V12) + (V12V22) + 2 (V11V12V12V22)

(4.7)

yields the constraints for 523 and MC shown in Figure (4.2).

 

1200 . . . -
1000 »//"‘ .
800 . «
600 >
400 -
200 »

MC [GeV]

 

 

 

1.6 1.8 2 2.2 2.4
K3

Figure 4.2. Lower bound for (n3, MC) from FCNC in R0 — K 0 systems

4.1.2 FCNC in BO — BO systems

The same procedure can be repeated for BO -— B0 systems, where we obtain similar
constraints from the mass shift in B3 or B2 mesons. The corresponding graph is

shown in Figure(4.3).

43

 

8000 . - . . .
7000l__ifl__flflg_gfl,,_....—————-""”’”'F
6000» .
5000»
__. 4000 '
U 3000 ’

2000*

1000L

GeV]

M

 

 

1.6 1.8 2 2.2 2.4
K3

 

Figure 4.3. Lower bound for MC as a function in 53 from FCNC in BEE-Meson systems

One can see directly from Figure 4.3 that this constraint is stronger than the one
from the kaon sector. From the gap triangle in Figure 3.1, we have 533 z 2, which

yields a lower bound on the top gluon mass of approximately 6 TeV.

4.2 Constraints on effective four-fermion couplings

Data from LEP2 puts upper limits on the coupling in effective four fermion contact
interactions. The data is presented in the following form:

If the Lagrangian has the form

2

 

ceff = i (TL’YMPL) (15137“ 0;.) , (4.8)

:l: 2
ALL

2
with £7 E 1, then LEP2 data gives lower limits for Af L or upper limits for the overall

coefficient in the Lagrangian.
This becomes interesting for us, if we look at the effective four-fermion interaction,

due to Z’ exchange:

44

LIZ; = _M27r2m (ffWYfi) (37“ij) . (4.9)
2!

 

Even though one might expect the most restrictive constraint for K1 and M Z’ to
follow from leptonic contact interactions (eeee) or (eepp), it arises from the contact
interaction (eeuu), for which the factor A: L = 23.3 TeV is the largest. 2

The graph in Figure 4.4 shows the lower limit for the allowed region for (al, M Z’)'

 

2 S T v v v *1 fi—‘r

Mz [TCV]

 

 

 

 

Figure 4.4. Lower limits for 101 and M z' from comparison with LEP2 data.

In order to keep M Z’ on the order of a few TeV, K1 is restricted to be smaller than 1.

 

2Notice, that, even though our Lagrangian had initially an overall minus sign, we have to use the
value for A according to an overall positive Lagrangian, as the hypercharge eigenvalues for electrons

and quarks have opposite signs.

45

Chapter 5

Constraints from electroweak precision

data

In this chapter we will investigate the bounds on our parameter space from elec-
troweak precision measurements. At first we present our fit to the LEPl data given
in [7]. In the second section we compare our results to the fit by Barbieri et al [2] to

data from LEP1-, LEP2-, low energy-experiments.

5.1 A Fit to LEPl data

In order to find constraints on the parameter space of the model, we have to derive
expressions for the observables. These can be obtained at tree level by shifting the
coupling of the Z boson and the weak mixing angle in the standard model observ-
ables according to eqns.(2.49) and (2.52). The observables are then functions of the
very well measured quantities M Z, a and G’ F as well as the free parameters in the
electroweak sector cos gb and pA. Note that ft and hence FTC are already determined
in terms of these parameters according to eqns.(2.64) and (2.36).

‘U

Since we expect pA >> 11 z 246 GeV, we expand to second order in 57? Every

46

observable then takes the form
v2

0' = 0.19M +60z(cos¢) x W,

(5.1)

where 60i(cos 06) is a polynomial in cos 05. From this we see, that in the limit A -—> 00
and cosgb ——> O, we approach the standard model. 03.9.1” represents the tree-level
standard model value for the observable written in terms of sin 6Z-

In order to get bounds on the free parameters, we have to compare our expression for

of to the experimental value Gimp :

2
o o c . v
0:31:19 i 0023:}, 2 0.23M + 602((303 Q6) X W

(5.2)
We expect that the new effects coming from our model are of the same order as
one-loop corrections to the standard model. To make sure that we do not mistake
standard model one-loop corrections for new physics, we replace 0% M in eq.(5.2)
by the best-fit standard model one-loop value for the corresponding observable. We
obtain those values using the program ZFITTER [1]. We used the sample program
given in [1] and updated the subroutine which is in charge of the initialization. The
values we used as input parameters are listed in Table 5.1.

The one—loop correction to the standard model involves a Higgs particle loop, which
strongly depends on the Higgs mass. The particle in our model corresponding to
the Higgs would be the particle that unitarizes the W scattering, which is in this
class of models assumed to be the Technicolor-analog to the QCD-p. Its mass can be
estimated by upscaling the QCD-p mass to be of the order of a TeV.

In order to see how the mass of the particle that plays the role of the Higgs in

the loop-calculations affects our results, we performed the fit for two different input

parameters m H = 800 geV and m H = 1.5 TeV.

47

 

 

MZ [GeV] a_1(Mz) 015 mt [GeV] mH [Gav]

 

 

91.1876 128.887 0.117 174.2 800/1500

 

 

 

 

 

 

 

 

 

 

Table 5.1. The input parameters we used in ZFITTER.

To combine the constraints from measurements of several observables, one has to
perform a least-squares fit, i.e. the best fit values are obtained by minimizing
2 . - o v 2
x (cos ¢,pA) = Z 023,, — 0.29M — 60'(cos 03) x (17/?)
233'

. . , v 2
x (apafijl (Gimp — OJSM -— (503 (cos 05) X (1]) ) , (5.3)

where ,0 denotes the correlation matrix of the observables and a is the diagonal matrix

containing their standard deviations. The allowed parameter space is obtained, by

2

requiring that Ax2 = X2 — Xmin does not exceed a certain value, associated with the

confidence level and the number of fit parameters. We will work at a 20 confidence
level and have two fit parameters pA and cos (b, which yields according to [8] 6.17 2
AXZ.

Since we obtained the first bounds on our parameter space from the gap equation
1

(3.2), we want in the end to translate our constraints back into the 103-101 plane.

Comparing eq.(2.18) and eq.(2.21) yields

’"31

cos2 90 = —.
K1 +ay

(5.4)

In order to impose limits on :03, we use the gauged gap equation for the desired top

 

1The results of this part of the analysis have a rather qualitative character, since one should not
assume that the self-consistent ansatz of the NJL—approximation holds to such high accuracy.

48

2 2 1
_ _ _ _ = __Yt yt
1 97raS 37raY 471‘ L Rm

 

 

2 M2
110104 0]
le mt
3 mt 2 Mg;
+ 57; n3 [1 (V0) log (m? (5.5)

Inserting the relation for the heavy gauge boson masses from eq.(2.29), one can see

 

that eq.(5.5) depends only on the free parameters A, K1, R3 and p

2 2 1 m2 r01 47r(pA)2(n1 + Oty)2
1— — — — = —— 1 - t 1
97raS 37raY 36W"$1 ] (47r(pA)2(n1 + (ll/)2) 0g ( mg [$1

+ 3 1 _ p2 m? K3 1 47r(p/\)2(K3 + as)2
2— "3 4 A 2 2 03 2 2 °
W 7T0) ) (“33+OS) P mm

(5.0)

 

 

 

 

In order to eliminate pA we solve eq.(5.2) for pA and obtain pA as a function of K1.
Inserting this into eq.(5.6) enables us to translate the constraint into the 53 — n1
plane, with a dependency on the parameter p.

Recall that p denotes the absolute value of hypercharge assigned to the condensate
that drives the first symmetry breaking. It is assumed that p is of the order of one,

but we shall demonstrate how different choices of p affect our results.

According to eq.(2.53), the shift in the W mass is given by

'U 2 F2 2
M3], zit/1% cos2 0 1 + (—A) TC — cos2 0)
p

 

v2

 

M2 20 +M2 ( v )2 (203462 ch 20 2 (57)
= cos — —— — cos , .
HLPZ/ Z 19A 00326 Z — sin2 9Z 112
MeVISM * v 4
v 2 2
=(5r) WW

where in the last step we have used eq.(2.52) to express cos2 0 in terms of cos2 0 Z and

 

expanded to second order in v/pA. As outlined in the previous section, we replace

49

M2

W] S M by the best-fit one-loop value, obtained by running ZFITTER (see Table 5.2).

The partial decay width of the Z boson into a fermion-antifermion pair (ff)

is given by (see e.g. [3]):

NZ ff)= gig—20;, +9”) (5.8)

Using the expression for the new Z-coupling from eq.(2.49), one obtains

62

2477 sin2 6 cos2 6

2 2 2
2 f - 2 v 2 . 2 f
x (Tchos 6—YL (SID 6+ W608 (b (sm gb— v—tz)))
’02 f2 2
+(Yf (sin2 6 + p2A2 cos2 ()5 (sin2 (b — 0%))) (5.9)

Rewriting this in terms of 6 Z using the result from eq.(2.52) and expanding to second

 

(Z->ff)=

 

order in p7”; yields an expression of the form of eq.(5.1)

212

r(z —+ ff) = NZ —> fflsmtree level + mg —5 ff) x r727 (5.10)

Neglecting the small contribution from the Z decaying into other gauge bosons and
hadronic states we can write the decay width into hadrons as
PZ—5had = Z NcI‘(Z —> if), (5.11)
f=u,d,s,c,b
where NC = 3 is the color factor.2 Fzfiinv is given by summing eq.(5.10) over all
neutrinos. The total Z—boson decay width was calculated by summing all partial

decay-widths into fermion-antifermion pairs (excluding the top quark ).

 

2The decay into a if pair is kinematically impossible at the Z pole, since the top mass exceeds
the center of mass energy by far.

50

The experimental results in [7] further include values for the pseudo-observables as

defined in the following: The hadronic cross section of the Z boson ”had is given by

 

a = (5.12)
had Mg I‘Z tot
The branching ratios into quarks and leptons are defined as
I‘ _, — I‘
I‘Z-ahad I‘z—ui'

Futher, the left-right and forward-backward asymmetry parameters are introduced as

 

f f f f
Af a 0L _UR 1 , Af E (11711, (5.14)
LR 0£+01f2 (lPel) FB 0£+UB

where in the first expression (lPel) denotes the polarization of the initial state elec-
trons. Important for us is that in the standard model they can be conveniently
rewritten in terms of another asymmetry parameter, which is only a function of the

coupling of the Z boson to fermions

2 2
9 L ‘ 9 R
Af E H. (5.15)
gfL + ng
With the above definition we obtain
f _ f _ 3

In order to calculate the predictions for the pseudo-observables in our model, we
again replace the Z-coupling by the shifted expression from eq.(2.49) and further
express sin 6 by sin 6 Z according to eq.(2.52). We expand to second order in a”; and
replace the zeroth order term by the standard model prediction at one—loop level
from ZFITTER[1] listed in Table 5.2. In order to obtain values for the asymmetry

parameters, the ZFITTER-based program Smatasi [14] was used.3

 

3 At this point, the author would like to thank his fellow student Michael F loBdorf who was
responsible for the complicated but in the end successful installation of this program, which enabled
us to perform this improved fit.

51

Since in our model the hypercharge sector is flavor-universal, we used the values from
[7] which were calculated under the assumption of lepton-universality. The ZFITTER
values for the flavor-universal observables 0’ were calculated by averaging over the
values for 06, O“, and 0". In the fitting procedure we minimized the Xz—function in
eq.(5.3) under the conditions pA, rel > 0. The experimental values and their errors
are displayed together with the results of our fit (for m H = 800 GeV) in Figure 5.1.
The correlation matrices for the observables can be found in Table 5.3 at the end of
this section. In the “Fit”-column the predictions of our model for the observables
using the best-fit values for the fit parameters pA and 1621 are listed. The “Pulls”
quantify how much the prediction of our model deviates from the experimental value

in units of its standard deviation.

 

Experimental Value Fit Pull

Fl: (2.4952i0.0023) GeV 2.4978 GeV
chad: (41.54lzt0.037) nb 41.485 nb ‘

 

R1: 2076740025 20.733
A'FB= 00171100010 0.0161
P.= 014653600033 0.1465

AlLR(SLD)= 0.1513i0.0021 0.1465
Rb: O.21629i0.00066 0.2160

 

RC: 01721400030 0.1723
Aha: 00992300016 0.1027
14°53: 00707¢00035 0.0733
Ab“: 0923:0020 0.935
Ac”: 067010.027 0.668

MW: (8040310029) GeV 80.402 GeV

 

 

 

 

0 1 2 3
26266515772 at <pA>=4038 rev,(x, )=0

 

 

Figure 5.1. The first column shows experimental values from [7] for the set of observables
we used to fit the predictions of our model to. In the second column the best-fit results
for our model are listed and the graph on the right-hand side shows the pulls for each
observable. In the last line the XEm-n-value and the best fit values for our fit-parameters are
g1ven.

52

For the fit where we assume a heavier Higgs-like particle, the Xw2m'n value improves

slightly compared to the one in Figure 5.1 to
x3... = 14.1726 at (pA) = 3.785 TeV, and (151) = 0 (5.17)

The allowed parameter space on 95% confidence-level for the fits with m H = 800

GeV and m H = 1.5 TeV is shown in Figure 5.2.

 

1.24 ' ' r *
1 l
0.8 7
0.6- ' (i)
0.4- . 1
(ID
0.2»

0 1

107316

 

 

 

3.25 3.5 3.75 4 4.25 4.5 4.75
pAlTeV]

Figure 5.2. The plot shows the allowed region on 95% confidence-level after fitting to
the data listed in Figure 5.1. Curve (i) and (ii) enclose the allowed parameter space for
my = 800 GeV and mg 2 1.5 TeV respectively.

From Figure 5.2 we can see that the results do not depend significantly 011 the mass of
the Higgs-like particle. Therefore we will assume in the following analysis the Higgs-
like particle to have a mass of 800 GeV.

The value for x7271," being only slightly larger than the number of degrees of freedom

(= 13 — 2) tells us that the predictions of our model fit the experimental data well4:
anm/dor. = 1.434.

According to Pearson’s XZ-statistic the probability to find x2-values greater than this

is approximately 15%.

 

4For the standard model with a Higgs mass larger than 800GeV, we estimated sznin z 100.

53

The fit gives a lower bound on pA of about 3.5 TeV and restricts 181 to be less
than 0.0012. Using the relation from eq.(2.29), we can translate the bounds on the

parameter space from Figure 5.2 into the M Z’ — 141 plane.

 

1.2 1'

1 .
0.8 -
0.6 .
0.4 *
0.2 -

10-3K1

 

 

2 4 6 8 l 0
M2 [TeV]

 

Figure 5.3. The allowed parameter space for M z' and 161.

The plot tells us that we can exclude the Z’ to be lighter than 2 TeV on 95%
confidence-level. In fact, since the best fit value for 151 is zero, it is likely that the Z’
is very heavy.

In order to find an estimate for the top-gluon mass, we translate this constraint back
into the K3 — 1521 plane, where we encounter a dependence on the hypercharge p of the
condensate driving the first symmetry breaking. Since we expect p to be of order one
(by comparison with the hypercharges of the familiar particles), the constraints are
plotted for p = %, 1, 2 below, in order to demonstrate how the choice of p affects the

evaluation.

54

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2 ' fi' pzb's
1‘ s:
_ 0-8’ :‘i
>< ::
'1‘ 0.61 ::
S SE
0.41 55
0.2 . §§
o '.' . . . . . -
1.957 1.958 1.959 1.96 1.961 1.962 1.963
K3
LZ’ fl Pél
1 - 3
_ 0-8~ :' i
x : :
"r 0.6 5 :
2 : 5
0.4» 5 5
0-2’ i .5 1
o . . i .1‘ . . . 1 .
1.957 1.958 1.959 1.96 1.961 1.962 1.963
K3
1.2l . ' ‘ p=2 f/x’ 2‘“
1 . ,’ 1:
_ 0.8 ’ II” '1'
¥ III '1'
f? 0.6 i ,’ '1
S. I
0.4 ' l’ '1'
0.2» 1
0+ " '

 

 

 

1.957 1.958 1.959 1.96 1.961 1.962 1.963
K3

Figure 5.4. The dashed curve encloses the allowed parameter space in the 193-161 plane for
different values of p. The bold lines represent the tip of the gap triangle (see Fig.3.1).

For small values of p the parameter space becomes extremely narrow and approaches

55

the line that represents the limit of top-condensation occuring at an infinite scale A.
This is because small p corresponds to a large top gluon mass. On the other end,
large values for p drive the allowed parameter space into a region where, according to
the NJ L—approximation, b-condensation can occur. Regardless of the exact choice of
p, as long as it is of the order of one, we obtain [63 z 1.96. Inserting this in eq.(2.29),

we find, that the top gluon mass is limited by
15 TeV 3 MC 3 26 TeV. (5.18)

Even though the estimation of 163 requires the NJL-approximation to hold to a
(perhaps unreasonably) high accuracy, the general statement that 153 (a) has to be
large compared to a S in order to insure top—condensation and is (b) on the other
hand restricted not to exceed a certain value to avoid (50) 75 0 has to hold, in order
to satisfy the general ansatz of our model. Together with the claim of p z 1, this

always yields upper and lower bounds on MC in analogy to eq.(5.18).

56

Table 5.2. shows the best-fit one-loop predictions of the standard model for the set of
observables that we used to perform our fit to calculated by ZFITTER for different Higgs

masses.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

oi Zfit (m H = 800GeV) Zfit (m H = 1500GeV)
rZ [GeV] 2.488069 2.485958
chad [nb] 41.4881 41.4887
Re 20.715 20.7118
13,, 20.715 20.7121
3,, 20.715 20.7589
in, 0.01444 0.01411
4%,; 0.01444 0.0144
g3 0.01444 0.0144
ER 0.1388 0.1372
AgR 0.1388 0.1372
212 0.1388 0.1372
Rb 0.2158 0.2159
1?.C 0.1722 0.1722
A13 0.9340 0.9340
1R 0.6642 0.6635
MW [GeV] 80.23222 80.20047

 

 

 

 

57

 

 

 

 

I‘Z chad R) A323

 

 

PZ 1.000
“had -0.297 1.000

R, 0.004 0.183 1.000

I
AFB 0.003 0.006 -0.056 1.000

 

 

 

 

 

 

 

 

 

 

b b
Rb RC AFB 153 ALR A212

 

 

Rb 1.00

RC 018 1.00

b

AFB -010 0.04 1.00
A353 0.07 —0.06 0.15 1.00

b
ALR -0.08 0.04 0.06 —0.02 1.00

AER 0.04 -0.06 0.01 0.04 0.11 1.00

 

 

 

 

 

 

 

 

Table 5.3. Correlation matrices from [7] for the experimental values in Figure 5.1 .

58

5.2 A fit to a larger set of observables

Barbieri et al. [2] have performed a fit to high energy precision data from the LEPl
and LEP2 experiments as well as low energy precision data, like atomic parity viola-
tion in Cs atoms.

In the following we use their results to impose bounds on our parameter space and
compare it to our results from the previous section. In order to do so, we first have

to find a translation for our parametrization into their language.

5.2.1 Definition of electroweak parameters

The standard model predictions at tree—level fit the electroweak precision measure-
ments already very well, so that one expects the effects from beyond the standard
model physics to be maximally of the order of the standard model one-loop predic-
tions.

Both the corrections to the standard model one-loop contributions due to new physics
(new fermions loops, for example), as well as the deviations from the standard model
predictions arising from a whole class of “universal” models at tree-level, can be
parametrized in terms of four parameters. Universal, in this context, denotes theories
in which the corrections to the standard model predictions can be expressed solely by
modifications to the two-point correlation functions of electroweak gauge currents of

fermions. Using the parametrization given in [5], the matrix element for the neutral

59

weak current is given by

(T 3 - 82G?) (T :3 - 8262’)
2 2 S 1
(87:76— _ T577) + 472016 (1— aT+ 43%?)

+ x/EGF-S—Z‘gfrflg + 4¢20F (Ap — 6:!) (Q — T3) (62’ — Té) (5.19)

2Q_Qf
p2

 

"MNC = e +

and the matrix element of the charged current reads

(T+TL + T_Tjr) /2

 

 

(16 (T+TL + T.T_,+_)

 

_ M = 2G
CC (52_S)P2+ 1 (1+ a6)+\/_ Fszcz 2
22 T67? 4&0 F 43252
(5.20)
P2 denotes the Euclidean four-momentum transfer, P2 = —q2, where q is the usual

Minkowski momentum. Ap corresponds to the deviation from unity of the ratio of the
strengths of isotriplet weak neutral current and charged current scattering at zero mo-
mentum transfer (P = 0). The factors a6 and (Ap — aT) correspond to the strength
of effective contact interactions via extra heavy gauge bosons (from additional S U (2)
or U (1) gauge groups). 3 and T are the parameters from the STU-formalism [16]
introduced by Peskin and Takeuchi to characterize the so called “oblige” corrections,
i.e. the corrections at one-loop level to the self energies of the standard model gauge
bosons due to new physics. 32 denotes the weak mixing angle as defined by the on-
shell Z coupling to the fermions.5
The parametrization is chosen such that for the standard model all parameters are
zero. Since the standard model predictions are in good agreement with current ex-
periments, one can assume the values of the parameters to be very small compared

to one.

In order to find the expressions for the electroweak parameters in our model, we com-

 

5It is, besides eq.(2.51) and(2.53) a fourth possibility to define a weak mixing angle, such that in
the standard model at tree level: .92 = sin 6w = sin 62 = sin 6.

60

pare eq.(5.19) and(5.20) to the respective matrix elements, that we obtain from our

model at tree level

2
_ 82_Q___Q' 92 92’
"MN0 — p2 + 52274} + 17;

2
2 v2 2
__ e2—Q——Q, e - 2 ft
_ P_2— +_sin26cos26 [1+pv 2—A—2 cos 205 (Sln ¢_fi>]
2

2
(T3 — Q [sin2 6 — 5% cos2 6cos2 4) (£3 - sin2 ¢)])

 

 

 

 

 

x
2 2 2 . f2
2 e v __ v 2 _
P + 4cosg6sin26 (1 122112 (8m ¢ 5%))
82 2
—'2'5C0t d)
+ “’3 .212 (Q — T3) (62’ — T5)
4cos2 6sin2 (boos2 d)
2 f2 2
QQ’ (T3 — Q [sin2 6 — fix? cos2 600$2 05 (35 — sin2 (0])
:82 2 + 2 '
P sin26c0526 2 112 2 f ' 2 132 M* 2
T 1+ Wcosqfi fi—smqfi +( Z)
4
+ p2A2 “034 ¢ (Q — T3) (Q, — T8), (5'21)

where the contribution of the heavy Z’ appears as a contact interaction, assuming
(P2 < M Z’)' The factor in the second line has been pulled out in order to give the
numerator the shape of the corresponding term in eq.(5.19). In the next step, its

inverse appears to second order in v / A in the denominator. (M2)2 is given by

2 2 F4
:1: 2 _ ’U 2) TC 4
(M2) — I |:].— W ('72— — COS (6)] . (5.22)

Analogously, we obtain for the charged current

1 (T+T’_ T4117.)
sin2 6 P2 _ e2122
82 4 sin2 6

 

— MCC = (5.23)

From here, we can straightforwardly derive the electroweak parameters in terms of

the free parameters of our model [6], by comparing the coefficients in eq.(5.19) and

61

 

eq.(5.20) with those from eq.(5.21) and eq.(5.23). At first we express .52 in terms of
our parametrization
v 2 f2
.52 = sin2 6 + — sin2 (I) — —t2 cos2 05 cos2 6. (5.24)
[311 ’0
Then, we find
016 _
m — 0 , (5.25)

 

 

 

since there is no contact interaction term in eq.(5.21) proportional to T3714. Further

1 v2 02 ’02 Ffo 4
fill—0T): “—0570: [1"p2A2(v2 —COS¢

 

 

 

 

¢=> aT = fi’%2(féQ-cos4¢). (5.26)

 

Here we have used 1/2GF % l/vz. Comparing the term in the propagators of the

standard model gauge bosons proportional to P2 yields

 

2
023 = 4% cos2 6 cos2 qb (sin2 45 — 155) . (5.27)

 

 

 

From the Z’-term, we obtain

 

2
(Ap — aT) = EEK? cos4 (1) . (5.28)

 

 

 

Ap is defined by the ratio of the couplings of the T3Té terms in eq.(5.19) to that
of the (T +T'_ + TLT_) / 2 term in eq.(5.20). Of course it can also be obtained, by

combining eqns.(5.26) and (5.28). Either way yields

 

v2

F4
Ap = W‘zfic' (5.29)

 

 

 

The top-gluon sector also shifts the p parameter. The main contribution arises from

single top gluon exchange across the top and bottom quark loops of W and Z vacuum

62

polarization diagrams. According to [17] this contribution is given by6

2
167rza f2

A (01% Y t , 5.30

p 3sin2 6w ( MCMZ n3 ( )

 

where MC denotes the top gluon mass and sin 6w denotes some weak mixing angle,

which we set equal to sin 6 Z7. In Figure 5.5, the ratio of the two contributions to Ap

from eq.(5.29) and (5.30) are plotted over the relevant energy regime.

 

0.06 ' ' 7
0.05 .
0.04 5
0.03 -
0.02 r
0.01 -
O u 1 - . .

0 5 10 15 20

pA[TeV]

U

32

f7’
N

 

 

 

Figure 5.5. The ratio of the two contributions to the p-parameter arising from the top
gluon and the electroweak sector.

Evidently, the contributions from the top—gluon sector are very small and can be

safely neglected in the following.

5.2.2 Constraints from the global fit

At first, I shall briefly outline what the results from [2] that we used to constrain our
parameter Space correspond to.

The first step of the general fitting procedure closely follows the steps outlined in the

 

6The model [17] has a flavor-universal top gluon-sector, but the couplings of the top gluon to the
third generation of fermions are the same in both models.

7Ap is already at the order of (v/pA)2. Since we are working only to that order in general, the
different expressions for the weak mixing angle are equal to each other.

63

beginning of Chapter 5: For a fit with 1‘ parameters to 'n observables, one first ex-
presses all observables in terms of the chosen parametrization 6 = (61, ..., 6r), yielding
terms of the form of eq.(5.1). Since all the parameters are expected to be close to
zero, the expressions for the observables are expanded to linear order in 6. Then, the
standard model tree-level predictions are replaced by the best fit one-loop prediction
of the standard model, which depends on the Higgs mass. This ensures that our
parametrization just describes the effect of the new physics.

In the next step, a least-square fit is performed, by minimizing

x209) = Z (02.... — 0:06)) OH)” (0%.... — 0:},(0) . (5.31)

2,]:
where Vij = oikpklolj, with the correlation matrix p of the observables and their
error matrix 0. Since all 01(6) are linear in 6, Ax2(6) = X2(6) — anin is a quadratic
form in 6 which can be rewritten as

r ..
2138(9) = Z ((9‘) — 9") (11—1)” ((93) — 91'), (5.32)
i,j=1
with U = opo denoting the covariance matrix of the best-fit values. This is the
desired expression, containing the best-fit value (6), the standard deviation 0 and
correlation p for the fit parameters.8

In order to use the constraints on the electroweak parameters from the global fit

performed by Barbieri et al [2], we have to express the parameters they used in terms

 

8The division of U = opo is unique, since the correlation matrix p is defined to have diagonal
entries of 1.

64

of our parametrization. According to [5], the Barbieri parameters read in our basis

 

 

 

 

. _ 1 2 016 v2 02 F7230 2
S : E (015 '1' 4C (Ap — OT)) ‘1' 27 = p2A2 ? 7 COS (25 (5.33)
2 F2

. v

5 5p = p2A2 {2Q (5.34)

2 2 2
_ C _ 'U C 4
025

W E 452c2 = 0 (5.36)

The constraints 011 those parameters obtained from the fit are listed in Table 5.4.

 

 

mH[G'eV] 103 (5) 103 (2“) 103 (Y) 103 (W)

 

 

800 —0.9 :1: 1.3 2.0 :l: 1.0 0.0 :1: 1.2 —0.2 :l: 0.8

 

 

 

 

 

 

 

 

 

 

 

Table 5.4. The values for the electroweak parameters obtained from a global fit to LEPl
and LEP2 data, according to [2].

The corresponding correlation matrix, regardless of the Higgs mass, given in [2] is

1 0.68 0.65 —0.12

0.68 1 0.11 0.19
p = (5.37)
0.65 0.11 1 —0.59

-—0.12 0.19 —0.59 1
Since we only have two free parameters in our model Ax2 is not zero for the best-fit

values of our parameters. The allowed parameter space at 95% (20)-Confidence level

is obtained by requring

<9) _ 9 T (5") _ 9
2 (T) - T —1 (T) — T

6.17 + Ax ((1019451)) .>. (y) _ Y (Up 0) (y) _ Y 1 (538)
(W) — W (W) — W

with o = diag (05, 0T, 01’, 0W). Inserting eqns.(5.33)-(5.36) and the values from

Table 5.4, yields the bounds on the parameter space shown in Figure 5.6:

65

 

1.75 - '
1.5»
1.25 .

10%,

0.75 .
0.5 ’
0.25 -

 

 

 

 

3.5 4 4.5 5 5.5 6
pA [TeV]

Figure 5.6. The constraints on the parameter space of our model from a fit to LEPl and
LEP2 data [2] under the assumption of a Higgs mass of 800 GeV. The dashed line shows
the result from our fit in the previous section for comparison.

Surprisingly the parameter space becomes wider if one takes more data into account.
In [2] there is unfortunately neither a xgm.” value nor the pulls for the observables
given which limits our possibilities to understand this. There has to be a tendency
among the extra observables to pull the best fit value for pA toward larger values.
The lower limit on pA remains the same, so that the lower bound on M Z’ does not

shift.

66

Chapter 6

Summary and Conclusions

6.1 Summary of the Results

Based on the fits to electroweak precision data, we can exclude 011 95% confidence-
level Z’ masses below 2 TeV. With the upper bound on its coupling to fermions
(r91 _<_ 0.0015), this tells us that our model does not get in conflict with the bounds
on effective four-fermion interactions from the LEP2 measurements (see section 4.2).
The strong upper bound on 151 also ensures that, in the sense of our results from the
calculation of the Landau-pole (section 3.2), our theory can (technically) be valid up
to scales beyond the Planck-scale.

Given the analysis in section 4.1 and the estimation of the topgluon mass in eq.(5.18),
we do not expect our model to give rise to flavor changing neutral currents in a

detectable range.

For further investigation of this model it may be useful to express our results

in terms of the universal parameters S, T, and Ap from eq.(5.26) through (5.29):

Ap = air = 2.9 x 10-3 and as = 0. (6.1)

67

From here we see that we are basically left with one free parameter, because K1 —+ 0.
Looking back at eq.(3.3), we find that the “tilting” that makes the top quark form
a condensate but not the bottom quark is dominated by the unbroken electroweak
gauge interactions. The exchange of a heavy Z ’ boson hardly affects the tilting process
(5:1 << ay). This forces [$3 to be “fine-tuned” in the sense that it has to be very close
to the critical coupling at which a condensation of third generation quarks occurs.
This corresponds to the very narrow parameter space in the 753 — 751 plane as shown
in Figure 5.4.1

This leaves us with a “sterile” Z' boson which in a good approximation does not
couple to fermions at all. Nonetheless it is a key ingredient in the model, since the
goodness of our fit relies on its contribution to the shift in the coupling of the ordinary

Z boson to fermions:

2 2 2 ° 2
, f _ _ . 2 12 _IL cos 6Zs1n 6Z
l1mOgZ — (T3 Sln 6g) + (p—A) (1 ) Q (6.2)

 

51—» 122 cos2 6g — sin2 62 .
6.2 Expectations on Future Experiments

The most exciting experiment (not only regarding our model) will evidently be the
search for the Higgs-boson at the LHC. If a Higgs-particle with standard model-like
couplings to the other particles will be detected, our model, and nearly the entire
class of Technicolor models, becomes obsolete.

Apart from this, it is hard to make predictions from our model for possible observa-
tions at LHC or ILC, since the masses of the heavy gauge bosons are already, by the
current electroweak precision data, forced beyond the reach of those colliders. The Z’

is not necessarily very heavy but its coupling to fermions as well as its mixing with

 

1The graphs were obtained using the NJ L-approximation, but the statement about the “fine-
tuned” parameter 153 holds for any possible model describing this pattern of symmetry breaking.

68

the standard model Z is very small which makes direct searches unrealistic.

From ILC one can expect upper bounds on effective four-quark interactions2 similar
to the LEP2 data that we used in section 4.2. In contrast to the analysis of the 2'-
exchange, where 151 —> 0 can compensate for arbitrarily small masses of the Z ’ boson,
163 has to be large compared to a S in order to provide the desired top-condensation.
A strong lower bound on the mass of the top-gluon MC may, according to eq.(2.29),
drive the scale A out of the region allowed by electroweak precision data.3 The general
ansatz of explaining the large top-mass via extra dynamics singling out the third gen-
erations of fermions, leads to potentially detectable decays of for example top-pions

into 05, but predictions for such processes exceed, unfortunately, the subject of this

thesis.

 

2The author would like to thank his advisor Prof. Elizabeth Simmons for pointing this out.
3At least under the assumption that the hypercharge of the condensate driving the first symmetry
breaking is of order 1.

69

APPENDIX

70

APPENDIX A

A.1 Constraints from Electroweak Precision Data on Models

with a Flavor Non-Universal Hypercharge Sector

Flavor non-universal topcolor models (as in [17]) usually assume that the third gener-
ation of fermions transforms under the U (1)1 group and first and second generations
under the U (1)2 group (see Table 2.1). Therefore, the Z boson couples differently to
the third than to the other fermion generations. This can be seen by looking back at

eq.(2.49), where we have used that, in the flavor-universal model, the operator
Y1 — (Y1 + Y2) sin26 = Y c0826, (Al)

for all fermions. This relation has to be modified in the flavor non-universal case. For
the third generation of fermions the Z boson coupling stays the same as in eq.(2.49),

but for the first two generations the coupling takes the form

2 2

12 e ’U . 2 .2 ft

1 =__ T 1___ __
gZ sin6cos6 [ 3( 132A2 sm (“(8111 lb 12 ))

2
2 2
— Q (sin2 6 —— gag/l2 sin2 (25 (sin2 05 -— i—g)>[ (A2)

The fitting procedure from the last section can be easily applied to this class of models

 

as well. While calculating the values for observables as predicted by these models,

one has to keep track of the couplings of the Z boson. As set of experimental values

71

we used the same as in the previous section, with the exception that the values for
observables depending on the coupling of the Z boson t0 leptons have to be replaced
by those which are given in [7] for the flavor non—universal case.

We have only performed this fit assuming a Higgs-like particle of mass mH = 800
GeV. The result of our fit and the experimental values are given in Figure A1 The

additional correlation matrices to Table 5.3 are given in Table Al and A2.

 

Experimental Value Fit Pull
F2: (2.4952100023) GeV 2.4892 GeV
0m: (41.54110037) nb 41.4598 nb

 

Re: 20.80410050 20.7428
Ry: 20.78510033 20.7431
RT: 2076410045 20.7515

A°FB= 0.0145100025 0.01682
A”FB= 00169100013 0.01682
Arm: 0.0188100017 0.0160
ATLR= 01465100033 0.1417
AeLR(SLD)= 01516100021 0.1493
A”LR(SLD)= 014210.015 0.1493
ATLR(SLD)= 0.13610015 0.1417
R1,: 0.216291000066 0.2160
= 01721100030 0.1722
A pg: 00992100016 0.1046
A°FB= 00707100035 0.07454

 

 

AbLR= 0.92310020 0.9342
ACLR= 0.67010027 0.6688
Mw= (8040310029) GeV 80.2970 GeV

0 1 2 3
sz,,,=49.9627 at (pA)=6.629TeV,(K1)=0

 

 

Figure A.1. The first column shows experimental values from [7] for the set of observables
we used to fit the predictions of our model to. In the second column the best—fit results
for our model are listed and the graph on the right-hand side shows the pulls for each
observable. In the last line the Xgnin-value and the best fit values for our fit—parameters are
given.

The Xgm-n-value in this case is large which makes the it unlikely that a model like this
fits the data well. Comparing the pulls in this case to Figure 5.1, one can see that

the agreement of the predicted values with the experimental data is not as good as

72

for the flavor-universal model.

From the analysis of the flavor universal case, we have seen that 191 = ay cot2 05
has to be very small in order to fit the data well. In the flavor non-universal model
this corresponds to make the shift compared to the standard model coupling in the
Z coupling to third generation small. At the same time this shift becomes large in
the coupling to the first two fermion generations which spoils the fit. Therefore one
might expect the best fit value for 191 to be positive which is not the case. One has
to take into account that the coupling of the Z boson contains a flavor-independent
dependency on [$1 due to the shift in the weak mixing angle in eq.(2.52). Obviously,
this contribution primarily has to be zero in order to fit the data.1 The “allowed”
parameter space is shown in Figure A.2, but since the xgm-n-value is so large the

meaning of this becomes ambiguous.

 

Swr—V' 4

DJ

10_3K1

N

 

 

 

4 5 6 7 8
9A [TeV]

Figure A.2. The plot shows the allowed parameter space on 95% confidence-level for a
flavor non-universal model after fitting to electroweak precision data [7].

Generally speaking, one can conclude that model with a flavor non-universal hyper-

charge sector are disfavored by the current electroweak precision data.

 

1I double-checked this by interchanging the couplings of the third generation with the couplings
of the first two fermion generations to the Z boson.

73

Table A.1. The correlation matrix of the experimental values for the flavor non-universal
case.

Table A.2. The submatrix for the correlations of the lepton left-right asymmetries.

 

 

 

 

 

 

I“z Uhad Re Ru 37 A333 Alia 199
1‘2 1.000
”had -0.297 1.000
Re 0.011 0.105 1.000
H), 0.008 0.131 0.069 1.000
R, 0.006 0.092 0.046 0.069 1.000
35.3 0.003 0.001 -0371 0.001 0.003 1.000
AgB 0.002 0.003 0.020 0.012 0.001 -0024 1.000
;.3 0.001 0.002 0.013 -0.003 0.009 -0.020 0.046 1.000

 

 

 

 

 

 

6
LR

)2
ALB

T
ALR

 

 

 

 

8
ALB

p
ALR

e
AFB

 

 

1.000

0.038

0.033

1.000

0.007 1.000

 

 

 

 

74

 

 

BIBLIOGRAPHY

75

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77

 
 

 
 

 
 
 
   
   
   
 
   
   
 
   
   

 

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