# Electroweak Chiral Lagrangian from TC2 Model with nontrivial TC fermion
condensation and walking

Feng-Jun Ge1, Shao-Zhou Jiang2111Email address: jsz@gxu.edu.cn(S.Z.Jiang).,
Qing Wang1,3222Corresponding author at: Department of Physics, Tsinghua
University, Beijing 100084, P.R.China
Email address: wangq@mail.tsinghua.edu.cn(Q.Wang). 1Department of Physics,
Tsinghua University, Beijing 100084, P.R.China
2College of Physics Science and Technology, Guangxi University, Nanning,
Guangxi 530004, P.R.China
3Center for High Energy Physics, Tsinghua University, Beijing 100084,
P.R.china

(April 29, 2011)

###### Abstract

The electroweak chiral Lagrangian for the topcolor assisted technicolor model
proposed by K. Lane, which uses nontrivial patterns of techniquark
condensation and walking, was investigated in this study. We found that the
features of the model are qualitatively similar to those of Lane’s previous
natural TC2 prototype model, but there is no limit on the upper bound of the
$Z^{\prime}$ mass. We discuss the phase structure and possible walking
behavior of the model. We obtained the values of all coefficients of the
electroweak chiral Lagrangian up to an order of $p^{4}$. We show that although
the walking effect reduces the S parameter to half its original value, it
maintains an order of $2$. Moreover, a special hyper-charge arrangement is
needed to achieve further reductions in its value.

PACS numbers: 12.60.Nz; 11.10.Lm, 11.30.Rd, 12.10.Dm

††preprint: TUHEP-TH-11175

##  I Introduction

Modern technicolor (TC) models of dynamical electroweak symmetry breaking
require assistance for top-color interactions that are strong in the TeV
energy region to provide the large mass of the top quark, and a walking
technicolor (WTC) gauge coupling to aid in the avoidance of large flavor-
changing neutral current (FCNC) effects. The first addition consists of a
class of topcolor-assisted technicolor (TC2) models made through the careful
arrangement of TC, topcolor, extended hypercharge groups, and relevant
techniquark and Standard Model (SM) fermion representations. With the help of
extended technicolor (ETC), we expect that technicolor condensates will form
and provide the mass for the weak vector bosons. ETC provides the mass for the
light quarks and leptons and a bottom-quark-sized mass to the top. The largest
contribution to the top-quark mass is from the formation of a top-quark
condensate through the dynamics of the topcolor gauge sector. The second
addition is based on the phase diagram of strongly coupled TC gauge theories
involving fermions in arbitrary representations of the gauge group. With
suitable choices for the TC group and techniquark representations, WTC is a
natural option for situations with asymptotic freedom that are nearly
conformal. In this case, the TC gauge coupling has an approximate infrared-
stable fixed point (the zero of the beta function) $\alpha_{*}$ which is
slightly larger than the critical value $\alpha_{c}$ necessary for techniquark
condensate formation. In such a theory, for values of $\alpha$ above
$\alpha_{*}$, as the energy scale decreases $\alpha$ increases. However, its
rate of increase decreases to zero as $\alpha$ approaches $\alpha_{*}$. Hence,
over an extended energy interval, $\alpha$ is order O(1), and it is slowly
varying which leads a large anomalous dimension $\gamma\simeq 1$ for the
bilinear local techniquark operator. This results in the enhancement of the SM
fermion and those undiscovered pseudo goldstone boson masses, which achieve
realistic scales while maintaining sufficient suppression of FCNC effects.

The typical gauge group of the TC2 models is

$\displaystyle SU(N)_{\mathrm{TC}}\otimes SU(3)_{1}\otimes SU(3)_{2}\otimes
SU(2)_{L}\otimes U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}$ (1)

in which the topcolor and extended hypercharge groups $SU(3)_{1}\otimes
SU(3)_{2}\otimes U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}$ spontaneously break into
their diagonal subgroups $SU(3)_{C}\otimes U(1)_{Y}$ at an energy of a few
TeV. The remaining electroweak groups $SU(2)_{L}\otimes U(1)_{Y}$
spontaneously break into their electromagnetic subgroup $U(1)_{\mathrm{em}}$
at electroweak scale because of a combination of a top-quark condensate and
techniquark condensate. In the simplest example of Hill’s TC2 model Hill95 ,
there are separate color and weak hypercharge gauge groups for the heavy third
generation quarks and leptons and for the two lighter generations. The third
generation transforms under a strongly coupled $SU(3)_{1}\otimes U(1)_{1}$ and
maintains its usual charges. However, the light generations transform
conventionally under a weakly coupled $SU(3)_{2}\otimes U(1)_{2}$. Near 1 TeV,
these four groups break into a diagonal subgroup of ordinary color and
hypercharge, $SU(3)_{C}\otimes U(1)_{Y}$. The desired condensation pattern
occurs because the $U(1)_{1}$ couplings are such that the spontaneously broken
$SU(3)_{1}\otimes U(1)_{1}$ interactions are supercritical only for the top
quark.

After Hill’s proposal was made, Chivukula, Dobrescu, and Terning CDT  claimed
that the techniquarks required to break the top and bottom quark chiral
symmetries are likely to have custodial-isospin violating couplings to the
strong $U(1)_{1}$. To maintain a $\rho\simeq 1$, the $U(1)_{1}$ interaction
must be so weak that it is necessary to fine-tune the $SU(3)_{1}$ coupling.
This results in the implementation of the theory being unnatural. To remedy
this isospin violation and improve the suitability of the model, K. Lane
proposed a natural prototype TC2 model in Ref.Lane95 . In that model, the
different techniquark isodoublets, $T^{t}$ and $T^{b}$, provide ETC mass to
the top and bottom quarks. These doublets then could have different $U(1)_{1}$
charges, which are isospin conserving for the right and left handed parts of
each doublet. The $U(1)$ symmetries presented in the model automatically avoid
the problem of $B_{d}-\bar{B}_{d}$ mixing raised by KominisKominis . To
achieve the mixing of the magnitude observed between the heavy and light
generations while breaking the strong top-color interactions near 1 TeV, K.
Lane also proposed an alternative model based on the nontrivial patterns of
techniquark condensation and discussed its phenomenologyLane96 . In this new
model, to break the extended hypercharge groups into $U(1)_{Y}$, a set of
electrically neutral $SU(2)$ singlet techniquarks belonging to the
antisymmetric tensor representation of the TC group were added into the model.
This, in combination with other techniquarks, further ensures the technicolor
coupling walks. With so many techniquarks, one may wonder whether the $S$
parameter of the model can be small. Although qualitatively the large number
of techniquarks will increase the value of $S$, walking effects and certain
arrangements of the hypercharges of the techniquarks may compensate for this
increase, and result in a small overall $S$ parameter. One aim of this paper
is to examine this possibility.

In fact, our interests are not limited to the S parameter, which is one of the
low energy constants (LECs) of the bosonic part of the standard electroweak
chiral Lagrangian (EWCL)EWCL . Rather, our interests include all EWCL LECs. In
our previous studies, we compiled a formulation for computing the bosonic part
of the EWCL LECs for orders up to $p^{4}$ for the one-doublet TC model
discussed in Ref.1D , Hill’s schematic TC2 model Hill95  in Ref.HongHao08 , K.
Lane’s natural prototype TC2 model Lane95  in Ref.JunYi09  and a hypercharge-
universal TC2 model Sekhar  in Ref.LangPLB . Here, the bosonic part of the
EWCL is the part that only involves SM electroweak gauge fields and
corresponding Goldstone fields. This part describes the electroweak symmetry
breaking effects on the electroweak gauge fields, but the parts of the EWCL
dealing with matter also include SM fermions which describe the electroweak
symmetry breaking effects on the SM fermion fields. In the literature, these
two parts are proposed in Refs.EWCL  and EWCLfermion  separately because they
have independent characteristics. The reason that we choose to compute the
bosonic part of the EWCL in isolation is that the matter part is more complex
than the bosonic part. Moreover, some of the three-dimensional fermion mass
terms and six-dimensional FCNC terms were already discussed in Lane’s original
paper Lane96 . In this paper, we only discuss the bosonic part of the EWCL for
the first stage of computing the LECs that are generalized from the $S$
parameter, and leave the part dealing with matter for future discussion. The
EWCL is an universal platform which enables us to compare different underlying
models with experimental data and extract the true physical theory that guides
our world. To achieve this comparison, we compute the EWCL coefficients model
by model. This study is the fourth paper in a series, starting with
Ref.HongHao08 , in which we compute these strongly coupled physics models.
Here, we focus on K. Lane’s alternative TC2 model with nontrivial TC fermion
condensation and walkingLane96 , which was mentioned previously. Corresponding
to recent advances in the understanding of the phase diagram of the $SU(N)$
gauge theories and the new possibilities for model buildingNewWalking , this
work offers a modern way to investigate walking effects in a realistic
strongly-coupled theory with complex structures.

In this paper, except for some conventional calculations that are similar to
those in our previous papers, we focus on the effects of walking that have not
been discussed before. We will compare the different situations of walking,
ideal walking, and running; and examine their effects on the $S$ parameter. In
the next section, we first review K. Lane’s alternative TC2 model with
nontrivial condensation and walkingLane96  and discuss its phase structure. In
section III, we apply our formulation developed in Ref.HongHao08  to Lane’s
model Lane96 . We perform these dynamical calculations through several steps:
first we integrate in the Goldstone field, U. Then, we integrate out the
technigluons and techniquarks by solving the Schwinger-Dyson equation (SDE)
for techniquarks. Next, we integrate out the colorons and $Z^{\prime}$,
perform a low energy expansion, and compute the effective action. Finally, we
obtain the EWCL coefficients. For simplicity, some details of the derivation
and computation in this section are placed in the appendices. Section IV.
contains numerical results and discussions. Section V. is a short summary and
discussion.

##  II Review of the Model and its phase structure

Consider K. Lane’s TC2 model Lane96  with nontrivial TC fermion condensation
and walking, in which the group is given by (1). Because we are only
interested in the bosonic part of EWCL, which is independent of the SM
fermions, we do not list their representations and $U(1)$ charge arrangements
here. The left gauge charges for the techniquarks are shown in Table I. There
are three sets of techniquarks. The first set includes $T^{1}$ and $T^{2}$.
These are the specific techniquarks of the model and are expected to have
twisted condensates that generate $SU(3)_{1}\otimes SU(3)_{2}\rightarrow
SU(3)_{c}$ and electroweak breaking, and a sufficient level of generation
mixing. The second set includes $T^{l}$, $T^{t}$ and $T^{b}$, which are the
standard TC2 techniquarks from Lane’s first natural prototype TC2 model Lane95
. They supply the ETC mass to the SM fermions, including the top and bottom.
The third set consists of the high-dimensional representation field $\psi$,
which is responsible for generating $U(1)_{1}\otimes U(1)_{2}\rightarrow
U(1)_{Y}$ and ensuring theory walking.

TABLE I. Gauge charge assignments of the techniquarks in Lane’s TC2 model.

field$\setminus$group | $SU(N)_{\mathrm{TC}}$ | $SU(3)_{1}$ | $SU(3)_{2}$ | $SU(2)_{L}$ | $U(1)_{1}$ | $U(1)_{2}$
---|---|---|---|---|---|---
field,coupling | $G^{\alpha}_{\mu},g_{\mathrm{TC}}$ | $A^{A}_{1\mu},h_{1}$ | $A^{A}_{2\mu},h_{2}$ | $W_{\mu}^{a},g_{2}$ | $B_{1\mu},q_{1}$ |  $B_{2\mu},q_{2}$
$T_{L}^{1}$ | N | 3 | 1 | 2 | $u_{1}$ | $u_{2}$
$U_{R}^{1}$ | N | 3 | 1 | 1 | $v_{1}$ | $v_{2}+\frac{1}{2}$
$D_{R}^{1}$ | N | 3 | 1 | 1 | $v_{1}$ | $v_{2}-\frac{1}{2}$
$T_{L}^{2}$ | N | 1 | 3 | 2 | $v_{1}$ | $v_{2}$
$U_{R}^{2}$ | N | 1 | 3 | 1 | $u_{1}$ | $u_{2}+\frac{1}{2}$
$D_{R}^{2}$ | N | 1 | 3 | 1 | $u_{1}$ | $u_{2}-\frac{1}{2}$
$T_{L}^{l}$ | N | 1 | 1 | 2 | $x_{1}$ | $x_{2}$
$U_{R}^{l}$ | N | 1 | 1 | 1 | $x_{1}^{\prime}$ | $x_{2}^{\prime}+\frac{1}{2}$
$D_{R}^{l}$ | N | 1 | 1 | 1 | $x_{1}^{\prime}$ | $x_{2}^{\prime}-\frac{1}{2}$
$T_{L}^{t}$ | N | 1 | 1 | 2 | $y_{1}$ | $y_{2}$
$U_{R}^{t}$ | N | 1 | 1 | 1 | $y_{1}^{\prime}$ | $y_{2}^{\prime}+\frac{1}{2}$
$D_{R}^{t}$ | N | 1 | 1 | 1 | $y_{1}^{\prime}$ | $y_{2}^{\prime}-\frac{1}{2}$
$T_{L}^{b}$ | N | 1 | 1 | 2 | $z_{1}$ | $z_{2}$
$U_{R}^{b}$ | N | 1 | 1 | 1 | $z_{1}^{\prime}$ | $z_{2}^{\prime}+\frac{1}{2}$
$D_{R}^{b}$ | N | 1 | 1 | 1 | $z_{1}^{\prime}$ | $z_{2}^{\prime}-\frac{1}{2}$
$\psi_{L}$ | $\frac{1}{2}N(N-1)$ | 1 | 1 | 1 | $\xi$ | $-\xi$
$\psi_{R}$ | $\frac{1}{2}N(N-1)$ | 1 | 1 | 1 | $\xi^{\prime}$ | $-\xi^{\prime}$

The details of the ETC interaction are not specified in Lane’s original
paperLane96 ; this prohibits quantitative computations. The effects on the
EWCL LECs from these ETC operators can be ignored in our calculation because
the relevant operators are small. Unfortunately, although we know from Ref.
JunYi09  that its contribution to the EWCL LECs is small, the effective four-
fermion coupling may become strong enough to change the results of the current
walking theoryETC . When the effective four-fermion coupling exceeds its
critical value, the position of the infrared fixed point changes
significantly. For the first step of the investigation, we ignore this case by
assuming that the four-fermion coupling does not exceed the critical value and
leave discussion of more general effects for future studies.

A number of constraints were given in Lane’s original paperLane96  to limit
and simplify the charges:

  * •

To ensure that the techniquark condensates conserve electric charge,
$u_{1}+u_{2}=v_{1}+v_{2}$, $x_{1}+x_{2}=x_{1}^{\prime}+x_{2}^{\prime}$,
$y_{1}+y_{2}=y_{1}^{\prime}+y_{2}^{\prime}$, and
$z_{1}+z_{2}=z_{1}^{\prime}+z_{2}^{\prime}$.

  * •

The $U(1)_{1}$ charges of the techniquarks respect custodial isospin.

  * •

For the $U(1)_{1}$ charges of $T^{1}$ and $T^{2}$: while $u_{1}\neq v_{1}$,
the broken $U(1)_{1}$ interactions favor the condensation of $T^{1}$ with
$T^{2}$. If this interaction is stronger than the $SU(3)_{1}$ attraction of
$T^{1}$ to itself and we neglect the other vacuum-aligning ETC interactions,
then $\langle\bar{T}^{i}_{L}T^{j}_{R}\rangle\propto(i\tau^{2})_{ij}$ in each
charge sector.

  * •

$u_{1}\neq v_{1}$ implies $Y_{1i}\neq Y_{1i}^{\prime}$ for the fermions.

  * •

For the $SU(N)_{\mathrm{TC}}$ antisymmetric tensor $\psi$,
$\xi^{\prime}\neq\xi$ guarantees $U(1)_{1}\otimes U(1)_{2}\rightarrow
U(1)_{Y}$ when $\langle\overline{\psi_{L}}\psi_{R}\rangle$ forms.

The Lagrangian of the model is

$S[G,A_{1},A_{2},W,B_{1},B_{2},\bar{T},T,\bar{\psi},\psi]=\int
d^{4}x[\mathcal{L}_{\mathrm{gauge~{}kinetic}}+\mathcal{L}_{\mathrm{techniquark}}+\mathcal{L}_{\mathrm{SM~{}fermion}}]\;,$
(2)

with

$\displaystyle\mathcal{L}_{\mathrm{gauge~{}kinetic}}=-\frac{1}{4}\bigg{[}G_{\mu\nu}^{\alpha}G^{\alpha,\mu\nu}+A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1\mu\nu}B^{1,\mu\nu}+B_{2\mu\nu}B^{2,\mu\nu}\bigg{]}$

and

$\displaystyle\mathcal{L}_{\mathrm{techniquark}}=$
$\displaystyle+\bar{T}^{1}[i\not{\partial}\\!-\\!g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!\\!-\\!h_{1}\frac{\lambda^{A}}{2}\not{A}_{1}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1}P_{L}\\!\\!-\\!q_{2}u_{2}\not{B}_{2}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1}P_{R}\\!\\!-\\!q_{2}(v_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{1}$
$\displaystyle+\bar{T}^{2}[i\not{\partial}\\!-\\!g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!\\!-\\!h_{2}\frac{\lambda^{A}}{2}\not{A}_{2}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1}P_{L}\\!\\!-\\!q_{2}v_{2}\not{B}_{2}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1}P_{R}\\!\\!-\\!q_{2}(u_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{2}$
$\displaystyle+\bar{T}^{l}[i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!-q_{1}x_{1}\not{B}_{1}P_{L}\\!-q_{2}x_{2}\not{B}_{2}P_{L}\\!-q_{1}x_{1}^{\prime}\not{B}_{1}P_{R}\\!-q_{2}(x_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{l}$
$\displaystyle+\bar{T}^{t}[i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!-q_{1}y_{1}\not{B}_{1}P_{L}\\!-q_{2}y_{2}\not{B}_{2}P_{L}\\!-q_{1}y_{1}^{\prime}\not{B}_{1}P_{R}\\!-q_{2}(y_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{t}$
$\displaystyle+\bar{T}^{b}[i\not{\partial}-g_{\rm
TC}t^{\alpha}\not{G}^{\alpha}\\!-g_{2}\frac{\tau^{a}}{2}\not{W}^{a}P_{L}\\!-q_{1}z_{1}\not{B}_{1}P_{L}\\!-q_{2}z_{2}\not{B}_{2}P_{L}\\!-q_{1}z_{1}^{\prime}\not{B}_{1}P_{R}\\!-q_{2}(z_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2}P_{R}]T^{b}$
$\displaystyle+\bar{\psi}[i\not{\partial}-g_{\rm
TC}\tilde{t}^{\alpha}\not{G}^{\alpha}\\!-q_{1}\xi\not{B}_{1}P_{L}\\!+q_{2}\xi\not{B}_{2}P_{L}-q_{1}\xi_{1}^{\prime}\not{B}_{1}P_{R}\\!+q_{2}\xi^{\prime}\not{B}_{2}P_{R}]\psi\;.$
(3)

Where $\lambda^{A}$ is the three-dimensional Gellman matrix for topcolor
interaction, $\tau^{a}$ is the Pauli matrix for the electroweak interaction,
$t^{\alpha}$ is the $SU(N)_{\mathrm{TC}}$ fundamental representation matrix,
$\tilde{t}^{\alpha}$ is the $SU(N)_{\mathrm{TC}}$ antisymmetric tensor
representation matrix. We do not specify $\mathcal{L}_{\mathrm{SM~{}fermion}}$
which is not relevant to our discussions for the present approximation.

Now we will discuss the phase structure of the model. The two-loop $\beta$
function of the $SU(N)_{\mathrm{TC}}$ coupling, $g_{\mathrm{TC}}$, is333The
reason that we chose the two-loop $\beta$ function instead of the one-loop
version is that it can generate the walking effects needed for the model.
Otherwise, the model setting must be rearranged. Physically, we expect that
the most significant contribution should come from the TC interaction. The SM
particle mass does not reach the TC scale, and the masses of the colorons and
$Z^{\prime}$ slightly exceed this scale, all of their contributions are
expected to be smaller than those of the TC interactions. For simplicity in
the first stage approximation, we ignore the possible effects from SM
particles, colorons, and $Z^{\prime}$. We also ignore the high-dimension ETC
interactions. We will investigate the accuracy of this approximation in a
future study of all of these effects.

$\displaystyle\beta(\alpha)$ $\displaystyle=$
$\displaystyle-\beta_{0}\frac{g_{\mathrm{TC}}^{3}}{(4\pi)^{2}}-\beta_{1}\frac{g_{\mathrm{TC}}^{5}}{(4\pi)^{4}}\hskip
56.9055pt\alpha\equiv\frac{g_{\mathrm{TC}}^{2}}{4\pi}\;.$ (4)

In this case, the two coefficients $\beta_{0}$ and $\beta_{1}$444Here we apply
the convention of Ref.ConformalWindow . are

$\displaystyle 2N\beta_{0}$ $\displaystyle=$
$\displaystyle\frac{11}{3}C_{2}(SU(N)_{\mathrm{TC}})-\frac{4}{3}[T(R_{1})+T(R_{2})+T(R_{3})]$
(5) $\displaystyle(2N)^{2}\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{34}{3}C_{2}^{2}(SU(N)_{\mathrm{TC}})-{\displaystyle\sum_{i=1}^{3}}[\frac{20}{3}C_{2}(SU(N)_{\mathrm{TC}})T(R_{i})+4C_{2}(R_{i})T(R_{i})]\;.$
(6)

The representations of the three sets of techniquarks mentioned above are
labeled $R_{1}$, $R_{2}$ and $R_{3}$. Their corresponding parameters are given
in Table II.

TABLE II. The representation parameters of this model. $d(R)$ is the dimension
of the representation, and $d(SU(N)_{\mathrm{TC}})$ is the number of group
generators. $C_{2}(R_{i})$ and $C_{2}(SU(N)_{\mathrm{TC}})$ are the quadratic
Casimir operators of the representation $R_{i}$ and the adjoint
representation, respectively. $N_{f}$ is the number of techniquarks in the
same representation, $N_{f}C_{2}(R)d(R)=T(R)d(G)$

$i$ | $d(R_{i})$ | $C_{2}(R_{i})$ | $C_{2}(SU(N)_{\mathrm{TC}})$ | $T(R_{i})$ | $d(SU(N)_{\mathrm{TC}})$ | $N_{f}$
---|---|---|---|---|---|---
1 | $N$ | $N^{2}-1$ | $2N^{2}$ | $N_{f}N$ | $N^{2}-1$ | 12
2 | $N$ | $N^{2}-1$ | $2N^{2}$ | $N_{f}N$ | $N^{2}-1$ | 6
3 | $N(N-1)/2$ | $2(N+1)(N-2)$ | $2N^{2}$ | $N_{f}N(N-2)$ | $N^{2}-1$ | 1

The reason that we only use the two-loop $\beta$ function is that the three-
loop term of the $\beta$ function is scheme dependent. Usually, it is only
used for error estimates. The behavior of the TC coupling, $\alpha$, is guided
by the renormalization group equation
$\mu\frac{\partial\alpha}{\partial\mu}=\beta$. From the equation, we know that
$\beta_{0}>0$ corresponds to the case in which the TC interaction allows
asymptotic freedom. However, $\beta_{0}<0$ corresponds to the loss of
asymptotic freedom, or non-asymptotic freedom. From (5) and Table II, we find
that the critical value dividing asymptotic freedom and non-asymptotic freedom
is determined by $\beta_{0}=0$ and leads $N=32/9$. If further ($\beta_{0}>0$
and $\beta_{1}<0$), TC interaction creates a Banks-Zaks infrared fixed point
$\alpha_{*}=-\frac{4\pi\beta_{0}}{\beta_{1}}$ BZ , which corresponds to the
zero of the $\beta$ function. In the more general case, an infrared fixed
point may not exist , which often happens in the situation in which the number
of fermions is small. This is the case for QCD. In this model, because there
are already too many technifermions, we have checked that the infrared fixed
point always exists. The existence of an infrared fixed point requires that
the coupling remains nearly constant over a given range of infrared energy
scales, i.e., it walks. This is the modern realization of the walking
mechanism. When an infrared fixed point exists, the two-loop $\beta$ function
dictates the following energy scale dependence of the TC coupling:

$\displaystyle\frac{1}{\alpha(x)}=\frac{\beta_{0}}{2\pi}\ln
x+\frac{1}{\alpha_{*}}\ln\frac{\alpha(x)}{\alpha_{*}-\alpha(x)}\hskip
85.35826ptx=\frac{q^{2}}{\Lambda^{2}_{w}}\;.$ (7)

Where the parameter $\Lambda_{w}$ is roughly the length of the interval of
constant coupling in the infrared region. At this scale, the coupling constant
completes the walk and begins a fast run in which it exhibits typical
asymptotic freedom behavior. In Section IV, we show that in the ideal walking
situation, $\Lambda_{w}$ can be interpreted as the ETC scale. It is often
referred to as $\Lambda_{\mathrm{ETC}}$ in the literatureYamawaki . Moreover,
in the standard running situation, $\Lambda_{w}$ can be treated as the TC
scale (or $\Lambda_{\mathrm{TC}}$). Realistically, in our model, the system is
somewhere between the cases of running and ideal walking, which suggests that
$\Lambda_{\mathrm{TC}}<\Lambda_{w}<\Lambda_{\mathrm{ETC}}$. This change from
$\Lambda_{\mathrm{ETC}}$ to $\Lambda_{w}$ also reflects the fact that
$\alpha(x)$ in the presence of some walking effects does not depend on the
value of $\Lambda_{\mathrm{ETC}}$ too much. However, in the ideal walking
theory they are very much correlated. Furthermore, the existence of both
asymptotic freedom and an infrared fixed point will divide the theory into two
different phases. One phase is the asymptotic freedom phase in which
$\alpha\leq\alpha_{*}$. In this case, the coupling $\alpha$ increases from
zero to $\alpha_{*}$ monotonically while the energy scale decreases from the
ultraviolet region to the infrared region. The other phase is the non-
asymptotic freedom phase, where $\alpha\geq\alpha_{*}$. In this case, the
coupling $\alpha$ decreases from infinity to $\alpha_{*}$ monotonically while
the energy scale decreases from the ultraviolet region to the infrared region.
Furthermore, the ladder approximation Schwinger-Dyson equation (SDE) for
techniquark self-energy predicts a critical coupling:

$\displaystyle\alpha_{c}=\frac{2\pi N}{3C_{2}(R)}$ (8)

for techniquarks that belong to the techni-gauge group representation, $R$.
While the infrared fixed point $\alpha_{*}$ exceeds its critical coupling
$\alpha_{c}$, spontaneous chiral symmetry breaking occurs, and the SDE
automatically develops nonzero techniquark self-energies and condensates.
However, when $\alpha_{*}$ is less than $\alpha_{c}$, there is no spontaneous
chiral symmetry breaking, and the techniquark self-energy vanishes. Later, we
will see that to ensure the correctness of our $\beta$ function, the nonzero
values of the techniquark self-energy and condensate must be small enough
compare to $\Lambda_{w}$. This dictates that $\alpha_{*}$ can only be larger
than $\alpha_{c}$ by a small amount. In practice, $\alpha_{*}$ may not be so
close in value to $\alpha_{c}$, this will cause inaccuracy in our
computations. We will estimate this error in later calculations. For the cases
discussed above for different values of TC coupling and different choices of
$N$, our model may exhibit different behaviors and then form different phases.
We present555Because $N_{f}$ is fixed in the model, we depict the phase
diagram in terms of $N$ and $\alpha$, instead of $N$ and $N_{f}$, which is
more commonly done in the literature. Comparing our Fig.1 to the phase diagram
depicted by Fig.1 in Ref.ConformalWindow , our phase diagram corresponds to a
horizontal line with a fixed $N_{f}$ in their diagram. Their phase diagram
only provides information about $N_{f}$ and $N$. Our phase diagram does not
provide information about $N_{f}$ , but does provide more information about
the running coupling constant. a phase diagram of our model in Fig.1.

Figure 1: Phases of Lane’s alternative TC2 model with nontrivial TC fermion
condensation and walking. The blue solid line represents the infrared fixed
point $\alpha_{*}$. The red dashed line denotes the critical coupling of the
first and second techniquark sets(fundamental representation of
$SU(N)_{\mathrm{TC}})$). The black dashed-dotted line denotes the critical
coupling of the third techniquark set(antisymmetric representation of
$SU(N)_{\mathrm{TC}})$). The magenta dotted line shows the value $N=32/9$ from
$\beta_{0}=0$.

From Fig.1, we can see that the blue line (infrared fixed point) divides the
phase space into two parts: the region above the blue line represents the non-
asymptotic freedom phase and that below the blue line represents the
asymptotic freedom phase.

In the asymptotic freedom phase, $\alpha$ runs from $\alpha_{*}$ (blue line)
to zero, as the energy scale increases. The blue line crosses the red dashed
line (critical coupling of the first and second techniquark sets) and the
black dashed-dotted line (critical coupling of the third techniquark set) at
two points, which divide the blue line into three segments. The trapezoids
(and triangle) under these segments form the three sub-regions of the
asymptotic freedom phase. From left to right, the blue region is the conformal
region, where $\alpha$ is always below its critical value and no techniquark
condensation forms. Therefore, there is no spontaneous chiral symmetry
breaking. The second red region is the intermediate mixture region, where
$\alpha$ is always below the critical value $\alpha_{c,1}=\alpha_{c,2}$, but
will cross $\alpha_{c,3}$ as the energy scale decreases. This means the third
set of techniquarks forms condensates, but the first and second sets do not.
The yellow and green regions are the ones that we mainly focus on in this
paper. In these regions, $\alpha$ will cross all its critical values as the
energy scale decreases. Thus, all techniquarks have nonzero self-energies and
condensates. Therefore, this is the model required for spontaneous chiral
symmetry breaking.

In the yellow region, the unique TC coupling in the infrared energy region
approaches that of the infrared fixed point, critical values $\alpha_{c}$ of
the first and second techniquark sets (within a magnitude of 0.2 ), and that
of the third techniquark set (within a magnitude of 0.4 ), as the energy scale
decreases. This causes a near conformal behavior in which the value of the
techniquark self-energy is very small (corresponding to a tiny mass). For at
least two reasons, this region is the most important to the investigation of
the walking effect. First, the lower the techniquark self-energy, the more
accurate and reliable our estimate of the $\beta$ function over the energy
region will be. This is because we have used the $\overline{\mathrm{MS}}$
scheme, which assumes massless techniquarks, to obtain the coefficients of the
$\beta$ function in (5) and (6). Second, if a techniquark has a significant
mass, it will decouple and not contribute to the $\beta$ function in the low
energy region. Therefore, in the extreme infrared region, because of
spontaneous chiral symmetry breaking, we cannot treat techniquarks as
massless. Therefore, we need to ignore techniquark contributions if they have
mass. The coupling without these techniquark contributions will run (rather
than walk) to a very large value and will not reach its original infrared
fixed point. We show this special running behavior in the infrared energy
region for $N=6$ using a dashed magenta line near the vertical axis in Fig.2.
A techniquark self-energy on the order of $F_{\mathrm{TC}}$ leads to an
infrared interval of the same order size, which is small in comparison to the
typical scale for $\Lambda_{w}$. The smaller the $F_{\mathrm{TC}}$ is, the
more accurately (7) describes the coupling walking behavior. Therefore, we
expect that replacing the running behavior in this region with an infrared
fixed point will only cause errors of order $F_{\mathrm{TC}}/\Lambda_{w}$ in
the solution of the SDE for the techniquark self-energy. In this model,
because our techniquarks belong to different representations of the TC group,
which leads to different critical couplings, there is not a unique point where
the $\alpha_{*}$ is equal to all the critical coupling values. Usually this is
a necessary component of modern walking theory.

Furthermore, the minimum integer $N$ closest to the conformal region is $N=6$,
but the value $N=4$ was chosen in Lane’s original paperLane96  and does not
satisfy the walking requirements of this study. Although we do not have an
unique $\alpha_{*}$ that is equal to all the critical coupling values and
$N=6$ is perhaps too far from the conformal region, our numerical results
given in section IV show that walking effects are present. Therefore, we do
achieve the situation where the infrared fixed point is not enough but
sufficiently close to the critical coupling. In fact, even if we found a
unique infrared fixed point $\alpha_{*}$ meets all the critical couplings and
an integer $N$ very near the conformal region, the walking results would not
be significantly more reliable. This is because of the large number of
assumptions made in our calculations. These assumptions include: ignoring
higher-order loops (error of $1/16\pi^{2}$), SM particles of mass $m$ (error
of $m^{2}/F^{2}_{\mathrm{TC}}$), and gauge fields such as coloron and
$Z^{\prime}$ (error of $F^{2}_{\mathrm{TC}}/M^{2}_{\mathrm{coloron}}$ and
$F^{2}_{\mathrm{TC}}/M^{2}_{Z^{\prime}}$ in the $\beta$ function). The
precision in the critical value is now only at the two-loop level. As we
mentioned before, the ETC effects may also play a role. One known effect from
the ETC interactionETC  is that while the coupling of the ETC-induced
effective four-fermion interaction exceeds its critical value, the area of the
conformal window will be substantially reduced. In this sense, we must include
all the above-mentioned corrections before we can quantitatively improve the
precision of the present calculation of the possible walking effects of the
model. In the asymptotic freedom phase, we show the scale dependence of the TC
coupling according to formula (7) for different values of $N$ in Fig.2.

Figure 2: Energy scale dependence of the TC coupling, $\alpha$, determined
using (7).

From Fig.2, it can be seen that in the asymptotic freedom phase, the smaller
the value of $N$, the flatter the curve. In other words, the smaller the slope
of the curve or corresponding value of the $\beta$ , the larger the impact on
the walking effect. From Fig.1, we know that when $N\leq 5$, there is no
overall spontaneous chiral symmetry breaking. Therefore, the minimum value of
$N$ at which spontaneous chiral symmetry breaking occurs and results in the
largest walking effect is $N=6$. Throughout this paper, we will use $N=6$ in
our quantitative computations.

##  III Derivation of the EWCL from Lane’s Model

Our goal is to obtain

$\displaystyle\exp\bigg{(}iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]\bigg{)}$
$\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}\mathcal{D}T^{1}\mathcal{D}\bar{T}^{2}\mathcal{D}T^{2}\mathcal{D}\bar{T}^{l}\mathcal{D}T^{l}\mathcal{D}\bar{T}^{t}\mathcal{D}T^{t}\mathcal{D}\bar{T}^{b}\mathcal{D}T^{b}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(9)
$\displaystyle\times\exp\bigg{(}iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\mu}^{a},B_{1\mu},B_{2\mu},\bar{T},T,\bar{\psi},\psi]\bigg{)}\bigg{|}_{A^{A}_{\mu}=0}$
$\displaystyle=$
$\displaystyle\mathcal{N}[W_{\mu}^{a},B_{\mu}]\int\mathcal{D}\mu(U)\exp\bigg{(}iS_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]\bigg{)}\;,$
(10)

where $S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]\equiv\int
d^{4}x{\displaystyle\sum_{i}}\mathcal{L}_{i}$ is the action of the EWCL.
$B_{\mu}$ is the gauge field of $U(1)_{Y}$ and $Z^{\prime}_{\mu}$ is the gauge
field of $U(1)^{\prime}\equiv U(1)_{Y_{1}}\otimes U(1)_{Y_{2}}/U(1)_{Y}$. They
are related to $B_{1\mu}$ and $B_{2\mu}$ through the mixing angle $\theta$ by

$\displaystyle\begin{pmatrix}B_{1\mu}&B_{2\mu}\end{pmatrix}=\begin{pmatrix}Z_{\mu}^{\prime}&B_{\mu}\end{pmatrix}\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\hskip 56.9055ptg_{1}\equiv
q_{1}\sin\theta=q_{2}\cos\theta\;.$ (11)

In (9) $A^{A}_{\mu}$ is the gluon field of $SU(3)_{c}$ and $B^{A}_{\mu}$ is
the gauge field of $SU(3)_{1}\otimes SU(3)_{2}/SU(3)_{c}$. They are related to
$A^{A}_{1\mu}$ and $A^{A}_{2\mu}$ through the mixing angle $\theta^{\prime}$
by

$\displaystyle\begin{pmatrix}A_{1\mu}^{A}&A_{2\mu}^{A}\end{pmatrix}=\begin{pmatrix}B_{\mu}^{A}&A_{\mu}^{A}\end{pmatrix}\begin{pmatrix}\cos\theta^{\prime}&-\sin\theta^{\prime}\\\
\sin\theta^{\prime}&\cos\theta^{\prime}\end{pmatrix}\hskip
56.9055ptg_{3}\equiv h_{1}\sin\theta^{\prime}=h_{2}\cos\theta^{\prime}\;.$
(12)

In the next section, we will use Schwinger-Dyson analysis that the $SU(N)_{\rm
TC}$ interaction induces techniquark condensates
$\langle\overline{\psi_{L}}\psi_{R}\rangle\neq 0$ and
$\langle\overline{T_{L}}^{i}T^{j}_{R}\rangle\neq 0$ for $i,j=1,2$. They
trigger the extended hypercharge symmetry breaking, $U(1)_{Y_{1}}\otimes
U(1)_{Y_{2}}\rightarrow U(1)_{Y}$, and the topcolor symmetry breaking,
$SU(3)_{1}\otimes SU(3)_{2}\rightarrow SU(3)_{c}$, at a TeV energy scale.
These processes leave a singlet heavy state $Z_{\mu}^{\prime}$ in broken
$U(1)^{\prime}$ and colorons $B^{A}_{\mu}$ in the broken $SU(3)_{1}\otimes
SU(3)_{2}/SU(3)_{c}$, respectively. Because this work is only concerned with
the EWCL, we ignored the gluon field by taking $A^{A}_{\mu}=0$.

In (10), $U$ is the standard electroweak Goldstone boson, which can be
expressed in terms of a dimensionless unitary unimodular $2\times 2$ matrix
field, $\mathcal{D}\mu$ denotes the normalized functional integration measure
on $U$. The normalization factor $\mathcal{N}[W_{\mu}^{a},B_{\mu}]$ is
determined through the requirement that when the TC interaction is switched
off, $S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]$ must vanish. This fixes it at:

$\displaystyle\mathcal{N}[W_{\mu}^{a},B_{\mu}]$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}\mathcal{D}T^{1}\mathcal{D}\bar{T}^{2}\mathcal{D}T^{2}\mathcal{D}\bar{T}^{l}\mathcal{D}T^{l}\mathcal{D}\bar{T}^{t}\mathcal{D}T^{t}\mathcal{D}\bar{T}^{b}\mathcal{D}T^{b}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(13)
$\displaystyle\times\exp\bigg{(}iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\mu}^{a},B_{1\mu},B_{2\mu},\bar{T},T,\bar{\psi},\psi]\bigg{)}\bigg{|}_{A^{A}_{\mu}=0,\mbox{\tiny
ignore TC interation}}\;.~{}~{}~{}$

In Ref.EWCL , the EWCL was constructed with building blocks which are
$SU(2)_{L}$ covariant and $U(1)_{Y}$ invariant as $T\equiv
U\tau^{3}U^{\dagger}$, $V_{\mu}\equiv(D_{\mu}U)U^{\dagger}$,
$g_{1}B_{\mu\nu}$, $g_{2}W_{\mu\nu}\equiv
g_{2}\frac{\tau^{a}}{2}W_{\mu\nu}^{a}$. Where $B_{\mu\nu}$ and $W_{\mu\nu}$
are the field strengths of the $U(1)_{Y}$ and $SU(2)_{L}$ gauge fields,
respectively. Alternatively, in Ref.HongHao08 , we reformulated the EWCL
equivalently using $SU(2)_{L}$ invariant and $U(1)_{Y}$ covariant building
blocks as $\tau^{3}$, $X_{\mu}\equiv U^{\dagger}(D_{\mu}U)$,
$g_{1}B_{\mu\nu}$, $\overline{W}_{\mu\nu}\equiv U^{\dagger}g_{2}W_{\mu\nu}U$.
In which, $\tau^{3}$ and $g_{1}B_{\mu\nu}$ are both $SU(2)_{L}$ and $U(1)_{Y}$
invariant, but $X_{\mu}$ and $\overline{W}_{\mu\nu}$ are bilinearly $U(1)_{Y}$
covariant. The second formulation was used throughout this paper. In Table
III, we detail the relationship between the two formalisms.

TABLE III. Symmetry breaking sector of the EWCL
$S_{\mathrm{eff}}[U,W_{\mu}^{a},B_{\mu}]=\int
d^{4}x{\displaystyle\sum_{i}}\mathcal{L}_{i}$
Formulation in Ref.EWCL  Formulation in Ref.HongHao08  ${\cal L}^{(2)}$
$\frac{1}{4}f^{2}{\rm
tr}[(D_{\mu}U^{\dagger})(D^{\mu}U)]=-\frac{1}{4}f^{2}{\rm tr}(V_{\mu}V^{\mu})$
$-\frac{1}{4}f^{2}{\rm tr}(X_{\mu}X^{\mu})$ ${\cal L}^{(2)\prime}$
$\frac{1}{4}\beta_{1}f^{2}[{\rm tr}(TV_{\mu})]^{2}$
$\frac{1}{4}\beta_{1}f^{2}[{\rm tr}(\tau^{3}X_{\mu})]^{2}$ ${\cal L}_{1}$
$\frac{1}{2}\alpha_{1}g_{2}g_{1}B_{\mu\nu}{\rm tr}(TW^{\mu\nu})$
$\frac{1}{2}\alpha_{1}g_{1}B_{\mu\nu}{\rm tr}(\tau^{3}\overline{W}^{\mu\nu})$
${\cal L}_{2}$ $\frac{1}{2}i\alpha_{2}g_{1}B_{\mu\nu}{\rm
tr}(T[V^{\mu},V^{\nu}])$ $i\alpha_{2}g_{1}B_{\mu\nu}{\rm
tr}(\tau^{3}X^{\mu}X^{\nu})$ ${\cal L}_{3}$ $i\alpha_{3}g_{2}{\rm
tr}(W_{\mu\nu}[V^{\mu},V^{\nu}])$ $2i\alpha_{3}{\rm
tr}(\overline{W}_{\mu\nu}X^{\mu}X^{\nu})$ ${\cal L}_{4}$ $\alpha_{4}[{\rm
tr}(V_{\mu}V_{\nu})]^{2}$ $\alpha_{4}[{\rm tr}(X_{\mu}X_{\nu})]^{2}$ ${\cal
L}_{5}$ $\alpha_{5}[{\rm tr}(V_{\mu}V^{\mu})]^{2}$ $\alpha_{5}[{\rm
tr}(X_{\mu}X^{\mu})]^{2}$ ${\cal L}_{6}$ $\alpha_{6}{\rm
tr}(V_{\mu}V_{\nu}){\rm tr}(TV^{\mu}){\rm tr}(TV^{\nu})$ $\alpha_{6}{\rm
tr}(X_{\mu}X_{\nu}){\rm tr}(\tau^{3}X^{\mu}){\rm tr}(\tau^{3}X^{\nu})$ ${\cal
L}_{7}$ $\alpha_{7}{\rm tr}(V_{\mu}V^{\mu}){\rm tr}(TV_{\nu}){\rm
tr}(TV^{\nu})$ $\alpha_{7}{\rm tr}(X_{\mu}X^{\mu}){\rm
tr}(\tau^{3}X_{\nu}){\rm tr}(\tau^{3}X^{\nu})$ ${\cal L}_{8}$
$\frac{1}{4}\alpha_{8}g_{2}^{2}[{\rm tr}(TW_{\mu\nu})]^{2}$
$\frac{1}{4}\alpha_{8}[{\rm tr}(\tau^{3}\overline{W}_{\mu\nu})]^{2}$ ${\cal
L}_{9}$ $\frac{1}{2}i\alpha_{9}g_{2}{\rm tr}(TW_{\mu\nu}){\rm
tr}(T[V^{\mu},V^{\nu}])$ $i\alpha_{9}{\rm
tr}(\tau^{3}\overline{W}_{\mu\nu}){\rm tr}(\tau^{3}X^{\mu}X^{\nu})$ ${\cal
L}_{10}$ $\frac{1}{2}\alpha_{10}[{\rm tr}(TV_{\mu}){\rm tr}(TV_{\nu})]^{2}$
$\frac{1}{2}\alpha_{10}[{\rm tr}(\tau^{3}X_{\mu}){\rm
tr}(\tau^{3}X_{\nu})]^{2}$ ${\cal L}_{11}$
$\alpha_{11}g_{2}\epsilon^{\mu\nu\rho\lambda}{\rm tr}(TV_{\mu}){\rm
tr}(V_{\nu}W_{\rho\lambda})$ $\alpha_{11}\epsilon^{\mu\nu\rho\lambda}{\rm
tr}(\tau^{3}X_{\mu}){\rm tr}(X_{\nu}\overline{W}_{\rho\lambda})$ ${\cal
L}_{12}$ $\alpha_{12}g_{2}{\rm tr}(TV_{\mu}){\rm tr}(V_{\nu}W^{\mu\nu})$
$\alpha_{12}{\rm tr}(\tau^{3}X_{\mu}){\rm tr}(X_{\nu}\overline{W}^{\mu\nu})$
${\cal L}_{13}$
$\alpha_{13}g_{2}g_{1}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}{\rm
tr}(TW_{\rho\sigma})$
$\alpha_{13}\epsilon^{\mu\nu\rho\sigma}g_{1}B_{\mu\nu}{\rm
tr}(\tau^{3}\overline{W}_{\rho\sigma})$ ${\cal L}_{14}$
$\alpha_{14}g_{2}^{2}\epsilon^{\mu\nu\rho\sigma}{\rm tr}(TW_{\mu\nu}){\rm
tr}(TW_{\rho\sigma})$ $\alpha_{14}\epsilon^{\mu\nu\rho\sigma}{\rm
tr}(\tau^{3}\overline{W}_{\mu\nu}){\rm tr}(\tau^{3}\overline{W}_{\rho\sigma})$

From (9) and (10), it can be seen that to obtain the EWCL, we must integrate
in the electroweak Goldstone boson field, $U$. We also need to integrate out
the series of fields which include the three sets of techniquarks, $\psi$,
$T^{1}$, $T^{2}$, $T^{l}$, $T^{t}$, $T^{b}$ and the technigluon
$G_{\mu}^{\alpha}$, and the colorons $B^{A}_{\mu}$ and $Z_{\mu}^{\prime}$. In
the following subsections, we divide this work into five steps.

###  III.1 Integrating in the electroweak Goldstone boson field $U$

We introduce a local $2\times 2$ operator

$O(x)\equiv\mathrm{tr}[T^{1}_{L}\bar{T}^{1}_{R}+T^{2}_{L}\bar{T}^{2}_{R}+T^{l}_{L}\bar{T}^{l}_{R}+T^{t}_{L}\bar{T}^{t}_{R}+T^{b}_{L}\bar{T}^{b}_{R}](x)$
(14)

In this case, $\mathrm{tr}$ are the traces with respect to the Lorentz,
$SU(N)_{\mathrm{TC}}$, $SU(3)_{1}$ and $SU(3)_{2}$ indices. The transformation
of $O(x)$ under $SU(2)_{L}\times U(1)_{Y}$ is

$O(x)\rightarrow V_{L}(x)O(x)V_{R}^{\dagger}(x)\hskip
56.9055ptV_{L}(x)=e^{i\frac{\tau^{a}}{2}\theta^{a}(x)}\qquad
V_{R}(x)=e^{-i\frac{\tau^{3}}{2}\theta^{0}(x)}\;.$ (15)

Then we decompose $O(x)$ as

$O(x)=\xi^{\dagger}_{L}(x)\sigma(x)\xi_{R}(x)$ (16)

Where $\sigma(x)$ which is represented using a Hermitian matrix, describes the
modular degree of freedom; and $\xi_{L}(x)$ and $\xi_{R}(x)$, which are
represented using unitary matrices, describe the phase degrees of freedom of
$SU(2)_{L}$ and $U(1)_{Y}$ respectively. Their transformations under
$SU(2)_{L}\otimes U(1)_{Y}$ are

$\displaystyle\sigma(x)\rightarrow h(x)\sigma(x)h^{\dagger}(x)\hskip
28.45274pt\xi_{L}(x)\rightarrow h(x)\xi_{L}(x)V^{\dagger}_{L}(x)\hskip
28.45274pt\xi_{R}(x)\rightarrow
h(x)\xi_{R}(x)V^{\dagger}_{R}(x)~{}~{}~{}~{}~{}~{}$ (17)

where

$h(x)=e^{i\theta_{h}(x)\frac{\tau^{3}}{2}}$ (18)

belongs to an induced hidden local $U(1)$ symmetry group. Next, we define a
new field

$U(x)\equiv\xi_{L}^{\dagger}(x)\xi_{R}(x)\;,$ (19)

which is the nonlinear realization of the Goldstone boson field in the EWCL.
Subtracting the $\sigma(x)$ field, we find that the present decomposition
results in a constraint
$\xi_{L}(x)O(x)\xi_{R}^{\dagger}(x)-\xi_{R}(x)O^{\dagger}(x)\xi^{\dagger}_{L}(x)=0$
and its functional expression is

$\int\mathcal{D}_{\mu}(U)\mathcal{F}[O]\delta(\xi_{L}O\xi^{\dagger}_{R}-\xi_{R}O^{\dagger}\xi^{\dagger}_{L})=\mathrm{const}\;,$
(20)

where $\mathcal{D}_{\mu}(U)$ is an effective invariant integration measure;
and $\mathcal{F}[O]$ only depends on $O$ and is invariant under
$SU(2)_{L}\otimes U(1)_{Y}$ transformations. This causes the value of the
integrated quantity to be a constant. Inserting the above identity into (9),
we have

$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}\mathcal{D}T^{1}\mathcal{D}\bar{T}^{2}\mathcal{D}T^{2}\mathcal{D}\bar{T}^{l}\mathcal{D}T^{l}\mathcal{D}\bar{T}^{t}\mathcal{D}T^{t}\mathcal{D}\bar{T}^{b}\mathcal{D}T^{b}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(21)
$\displaystyle\times\int\mathcal{D}_{\mu}(U)\mathcal{F}[O]\delta(\xi_{L}O\xi^{\dagger}_{R}-\xi_{R}O^{\dagger}\xi^{\dagger}_{L})e^{iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\mu}^{a},B_{1\mu},B_{2\mu},\bar{T},T,\bar{\psi},\psi]}\bigg{|}_{A^{A}_{\mu}=0}\\!.~{}~{}~{}$

Using a special $SU(2)_{L}\otimes U(1)_{Y}$ rotation for $V_{L}(x)=\xi_{L}(x)$
and $V_{R}(x)=\xi_{R}(x)$ and labeling the fields after rotation with the
subscript, ξ, the above path integral becomes:

$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}_{\xi}\mathcal{D}T^{1}_{\xi}\mathcal{D}\bar{T}^{2}_{\xi}\mathcal{D}T^{2}_{\xi}\mathcal{D}\bar{T}^{l}_{\xi}\mathcal{D}T^{l}_{\xi}\mathcal{D}\bar{T}^{t}_{\xi}\mathcal{D}T^{t}_{\xi}\mathcal{D}\bar{T}^{b}_{\xi}\mathcal{D}T^{b}_{\xi}\mathcal{D}G_{\mu}^{\alpha}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
(22)
$\displaystyle\times\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})e^{iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}\bigg{|}_{A^{A}_{\mu}=0}\\!.~{}~{}~{}$

where we have used the result that the functional integration measure,
$\mathcal{F}[O]$ and the action on the exponential of the integrand are
invariant under $SU(2)_{L}\otimes U(1)_{Y}$ transformations. From Table I, it
can be seen that:

$\displaystyle T^{1}_{\xi
L}=e^{-i(u_{1}+u_{2})\theta_{0}}P_{L}\xi_{L}T^{1}_{L}\hskip
56.9055ptT^{1}_{\xi R}=e^{-i(v_{1}+v_{2})\theta_{0}}P_{R}\xi_{R}T^{1}_{R}$
$\displaystyle T^{2}_{\xi
L}=e^{-i(v_{1}+v_{2})\theta_{0}}P_{L}\xi_{L}T^{2}_{L}\hskip
56.9055ptT^{2}_{\xi R}=e^{-i(u_{1}+u_{2})\theta_{0}}P_{R}\xi_{R}T^{2}_{R}$
$\displaystyle T^{l}_{\xi
L}=e^{-i(x_{1}+x_{2})\theta_{0}}P_{L}\xi_{L}T^{l}_{L}\hskip
56.9055ptT^{l}_{\xi
R}=e^{-i(x^{\prime}_{1}+x^{\prime}_{2})\theta_{0}}P_{R}\xi_{R}T^{l}_{R}$ (23)
$\displaystyle T^{t}_{\xi
L}=e^{-i(y_{1}+y_{2})\theta_{0}}P_{L}\xi_{L}T^{t}_{L}\hskip
56.9055ptT^{t}_{\xi
R}=e^{-i(y^{\prime}_{1}+y^{\prime}_{2})\theta_{0}}P_{R}\xi_{R}T^{t}_{R}$
$\displaystyle T^{b}_{\xi
L}=e^{-i(z_{1}+z_{2})\theta_{0}}P_{L}\xi_{L}T^{b}_{L}\hskip
56.9055ptT^{b}_{\xi
R}=e^{-i(z^{\prime}_{1}+z^{\prime}_{2})\theta_{0}}P_{R}\xi_{R}T^{b}_{R}\;,$

Furthermore,

$\displaystyle
g_{2}\frac{\tau^{a}}{2}W^{a}_{\xi,\mu}=\xi_{L}[g_{2}\frac{\tau^{a}}{2}W^{a}_{\mu}-i\partial_{\mu}]\xi_{L}^{\dagger}$
(24) $\displaystyle
g_{1}\frac{\tau^{3}}{2}B_{\xi,\mu}=\xi_{R}[g_{1}\frac{\tau^{3}}{2}B_{\mu}-i\partial_{\mu}]\xi_{R}^{\dagger}\hskip
28.45274pt\begin{pmatrix}B_{1\xi,\mu}&B_{2\xi,\mu}\end{pmatrix}=\begin{pmatrix}Z_{\mu}^{\prime}&B_{\xi,\mu}\end{pmatrix}\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\;.~{}~{}~{}~{}$ (25)

Note the fields without the subscript ξ in (22) are the fields that are
invariant under $SU(2)_{L}\otimes U(1)_{Y}$ rotation.

###  III.2 Integrating out the technigluons

As a second step,we integrate out the technigluon in (22) using:

$\displaystyle\int\mathcal{D}G_{\mu}^{\alpha}e^{iS[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}=e^{iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]+iS_{\mathrm{TC1}}[A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}\;,~{}~{}~{}$
(26)

where we choose

$\displaystyle
e^{iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}=\int\mathcal{D}G_{\mu}^{\alpha}~{}e^{i\int
d^{4}x(-\frac{1}{4}G_{\mu\nu}^{\alpha}G^{\alpha,\mu\nu}-g_{TC}G_{\mu}^{\alpha}J^{\mu\alpha})}$
(27) $\displaystyle
S_{\mathrm{TC1}}[A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]=S[G_{\mu}^{\alpha},A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]\bigg{|}_{G_{\mu}^{\alpha}=0}~{}~{}$
(28)

and

$\displaystyle J^{\mu\alpha}$ $\displaystyle=$
$\displaystyle\bar{\psi}\tilde{t}^{\alpha}\gamma^{\mu}\psi+\tilde{J}^{\mu\alpha}$
(29) $\displaystyle\tilde{J}^{\mu\alpha}$ $\displaystyle=$
$\displaystyle\bar{T}^{1}_{\xi}t^{\alpha}\gamma^{\mu}T^{1}_{\xi}+\bar{T}^{2}_{\xi}t^{\alpha}\gamma^{\mu}T^{2}_{\xi}+\bar{T}^{l}_{\xi}t^{\alpha}\gamma^{\mu}T^{l}_{\xi}+\bar{T}^{t}_{\xi}t^{\alpha}\gamma^{\mu}T^{t}_{\xi}+\bar{T}^{b}_{\xi}t^{\alpha}\gamma^{\mu}T^{b}_{\xi}\;.$
(30)

Integrating out the technigluon fields in (27), we get

$\displaystyle
iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]=\sum_{n=2}^{\infty}\int
d^{4}x_{1}\ldots
d^{4}x_{n}\frac{(-ig_{\mathrm{TC}})^{n}}{n!}G_{\mu_{1}\ldots\mu_{n}}^{\alpha_{1}\ldots\alpha_{n}}(x_{1},\ldots,x_{n})J_{\alpha_{1}}^{\mu_{1}}(x_{1})\ldots
J_{\alpha_{n}}^{\mu_{n}}(x_{n})\;,~{}~{}~{}$ (31)

where
$G_{\mu_{1}\ldots\mu_{n}}^{\alpha_{1}\ldots\alpha_{n}}(x_{1},\ldots,x_{n})$ is
a n-point Green’s function for the technigluons.

###  III.3 Integrating out the techniquarks

Combining (22) and (26), our starting $S_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]$,
after integrating in the electroweak Goldstone boson field $U$ and integrating
out the technigluons, becomes

$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}_{\xi}\mathcal{D}T^{1}_{\xi}\mathcal{D}\bar{T}^{2}_{\xi}\mathcal{D}T^{2}_{\xi}\mathcal{D}\bar{T}^{l}_{\xi}\mathcal{D}T^{l}_{\xi}\mathcal{D}\bar{T}^{t}_{\xi}\mathcal{D}T^{t}_{\xi}\mathcal{D}\bar{T}^{b}_{\xi}\mathcal{D}T^{b}_{\xi}\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}$
$\displaystyle\times\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})e^{iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]+iS_{\mathrm{TC1}}[A_{1\mu}^{A},A_{2\mu}^{A},W_{\xi,\mu}^{a},B_{1\xi,\mu},B_{2\xi,\mu},\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]}\bigg{|}_{A^{A}_{\mu}=0}\\!.$

After some detailed derivations and approximations which can be found in
Appendix A, we get:

$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})\int\mathcal{D}B_{\mu}^{A}\mathcal{D}Z_{\mu}^{\prime}~{}\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}$ (33)
$\displaystyle+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1,\mu\nu}B^{1,\mu\nu}+B_{2,\mu\nu}B^{2,\mu\nu})]$
$\displaystyle+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{]}_{A^{A}_{\mu}=0}\;,$

The various quantities appearing in (33) are defined at the end of Appendix A.
Furthermore, in Appendix B, we have shown that the techniquark self energies
$\tilde{\Sigma}$, $\hat{\Sigma}$, $\bar{\Sigma}$ and $\bar{\Sigma}_{5}$
satisfy the following SDEs,

$\displaystyle\tilde{\Sigma}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N+1)(N-2)}{4\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\tilde{\Sigma}(q_{E}^{2})}{q_{E}^{2}+\tilde{\Sigma}^{2}(q_{E}^{2})}$
(34) $\displaystyle\hat{\Sigma}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N^{2}-1)}{8\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\hat{\Sigma}(q_{E}^{2})}{q_{E}^{2}+\hat{\Sigma}^{2}(q_{E}^{2})}$
(35) $\displaystyle\bar{\Sigma}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N^{2}-1)}{8\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\bar{\Sigma}(q_{E}^{2})}{q_{E}^{2}+\bar{\Sigma}^{2}(q_{E}^{2})+\bar{\Sigma}_{5}^{2}(q_{E}^{2})}$
(36) $\displaystyle\bar{\Sigma}_{5}(p_{E}^{2})$ $\displaystyle=$
$\displaystyle\frac{3(N^{2}-1)}{8\pi^{3}N}\int{d^{4}q_{E}}\frac{\alpha[(p_{E}-q_{E})^{2}]}{(p_{E}-q_{E})^{2}}\frac{\bar{\Sigma}_{5}(q_{E}^{2})}{q_{E}^{2}+\bar{\Sigma}^{2}(q_{E}^{2})+\bar{\Sigma}_{5}^{2}(q_{E}^{2})}\;,$
(37)

where the technigluon propagator is parameterized though the TC running
coupling constant $\alpha$ as

$\displaystyle
G_{\mu\nu}^{\alpha\beta}(x,y)=\\!\int\\!\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\frac{-i\delta^{\alpha\beta}}{p^{2}[1\\!+\\!\Pi(-p^{2})]}\bigg{(}g_{\mu\nu}\\!-\frac{p_{\mu}p_{\nu}}{p^{2}}\bigg{)}\hskip
28.45274pt\alpha(p_{E}^{2})\equiv\frac{g^{2}_{\mathrm{TC}}}{4\pi[1\\!+\\!\Pi(p_{E}^{2})]}\;.~{}~{}~{}$
(38)

###  III.4 Integrating out the colorons and the low energy expansion

Before integrating out the coloron field, we first discuss its mass which is
determined by the kinetic and mass terms. From the exponential of the
integrand in (33), it can be seen that there is already a standard coloron
kinetic term from
$-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu})$. The
first set of techniquarks contributes to the quantum loop corrections to the
coloron kinetic and mass terms through the term
$\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]$
in (33). Through detailed computations, we find that these corrections are

$\displaystyle\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{|}_{\mbox{\tiny
coloron kinetic and mass terms}}$ $\displaystyle=\frac{i}{4}\int
d^{4}x\bigg{[}Cg_{3}^{2}(\tan\theta^{\prime}\\!+\\!\tan\theta^{\prime})^{2}B^{A}_{\mu}B_{A}^{\mu}-(\partial^{\mu}B^{A}_{\nu}-\partial^{\nu}B^{A}_{\mu})^{2}[\mathcal{K}g_{3}^{2}(\cot^{2}\theta^{\prime}\\!+\\!\tan^{2}\theta^{\prime})$
$\displaystyle+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}g_{3}^{2}(\tan\theta^{\prime}\\!-\\!\cot\theta^{\prime})^{2}+\frac{1}{2}\hat{E}g_{3}^{2}(\tan\theta^{\prime}+\cot\theta^{\prime})^{2}]\bigg{]}\;,$
(39)

In this case, the coefficients are given at the beginning of Appendix C.
Combining the standard coloron kinetic term in (33) and the techniquark
quantum loop correction given by (39), we find the formula for the coloron
mass to be:

$\displaystyle
M_{\mathrm{coloron}}^{2}=\frac{C}{\hat{E}+2(\mathcal{K}+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})+(2/g_{3}^{2}-8\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})/(\cot\theta^{\prime}+\tan\theta^{\prime})^{2}}\;.$ (40)

In Appendix C, we integrate out the coloron fields and perform the low energy
expansion. Finally we obtain,

$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle e^{i\int
d^{4}x[-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}]}\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})\int\mathcal{D}Z_{\mu}^{\prime}~{}e^{iS_{0}+iS_{Z^{\prime}}}\;.~{}~{}~{}~{}~{}$
(41)

Where detailed expressions of $S_{0}$ and $S_{Z^{\prime}}$ are given in (112)
and (116) respectively in Appendix C.

###  III.5 Integrating out $Z^{\prime}$

We denote the resulting action after the integration over $Z^{\prime}$ as

$\displaystyle\int\mathcal{D}Z^{\prime}_{\mu}~{}e^{iS_{Z^{\prime}}}=e^{i\bar{S}_{Z^{\prime}}}\;.$
(42)

We can use the loop expansion to calculate the above integral:

$\displaystyle\bar{S}_{Z^{\prime}}=S_{Z^{\prime}}\bigg{|}_{Z^{\prime}=Z^{\prime}_{c}}+\mbox{loop
corrections}$ (43)

where the classical field $Z^{\prime}_{c}$ satisfies:

$\displaystyle\frac{\partial}{\partial
Z^{\prime}_{c,\mu}(x)}\bigg{[}S_{Z^{\prime}}+\mbox{loop
corrections}\bigg{]}=0\;.$ (44)

Using this method, we integrate out the $Z^{\prime}$ field in Appendix D and
simplify the result $\bar{S}_{Z^{\prime}}$ given in (140) into the form of
EWCL. Furthermore, combining (42) and (41) together, we find

$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle e^{i\int
d^{4}x[-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}]}\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})~{}e^{iS_{0}+i\bar{S}_{Z^{\prime}}}\;.~{}~{}~{}~{}~{}$
(45)

Comparing this with (10) and Table.III, we obtain all the EWCL LECs. Our final
analytical results for the EWCL LECs (up to an order of $p^{4}$) are

$\displaystyle f^{2}=5\hat{F}_{0}^{2}\hskip
28.45274pt\beta_{1}=\frac{10a_{3}^{2}\hat{F}_{0}^{2}}{\bar{M}_{Z^{\prime}}^{2}}\hskip
28.45274pt\alpha_{1}=\frac{5}{2}(1-2\beta_{1})(\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})+\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\gamma\beta_{1}}{2a_{3}}$
$\displaystyle\alpha_{2}=(\beta_{1}-\frac{1}{2})(\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})+\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\gamma\beta_{1}}{2a_{3}}\hskip
28.45274pt\alpha_{3}=(\beta_{1}-\frac{1}{2})(\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0})~{}~{}~{}~{}~{}~{}$
$\displaystyle\alpha_{4}=(2\beta_{1}+\frac{1}{4})(\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})+(\frac{5}{16}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-\frac{5}{32}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})+\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}$
$\displaystyle\alpha_{5}=-\frac{5}{2}(4\beta_{1}+\frac{1}{4})\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{4}(3\beta_{1}+\frac{1}{4})\hat{\mathcal{K}}_{14}^{\Sigma\neq
0}+\frac{5}{32}(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\hat{\mathcal{K}}_{4}^{\Sigma\neq
0})-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}$
$\displaystyle\alpha_{6}=-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\beta_{1}^{2}}{4a_{3}^{2}}[-(2a_{0}^{2}+\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-10a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+5a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}$ $\displaystyle\hskip
14.22636pt+2a_{0}^{2}\hat{D}_{4}]-\frac{\beta_{1}}{2}(\frac{5}{2}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+15\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}-5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})$
$\displaystyle\alpha_{7}=\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}-\frac{\beta_{1}^{2}}{4a_{3}^{2}}[(\frac{5}{2}a_{3}^{2}+a_{0}^{2}+\frac{1}{2}\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+(a_{0}^{2}+\frac{1}{2}\hat{a}_{0}^{2}-\frac{5}{2}a^{2}_{3})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-10a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+5a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}+a_{0}^{2}\hat{D}_{3}]$
$\displaystyle\hskip
14.22636pt-\frac{\beta_{1}}{2}(\frac{5}{4}\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\frac{5}{4}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-15\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}+5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})$
$\displaystyle\alpha_{8}=-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}+10\beta_{1}(\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}\\!-\hat{\mathcal{K}}_{13}^{\Sigma\neq 0})\hskip
28.45274pt\alpha_{9}=-\frac{\beta_{1}f^{2}}{2M_{Z^{\prime}}^{2}}\\!+\beta_{1}(5\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}\\!-10\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}\\!+\frac{5}{4}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0})~{}~{}~{}~{}$
$\displaystyle\alpha_{10}=\frac{5\beta_{1}^{2}}{4}(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0})+\frac{\beta_{1}^{4}}{8a_{3}^{4}}g_{4Z}-\frac{\beta_{1}^{3}}{2a_{3}^{3}}[(2a_{3}^{3}+6a_{0}^{2}a_{3}+3\hat{a}_{0}^{2}a_{3})(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq 0})+2a_{0}^{2}a_{3}\hat{D}_{2}]$
$\displaystyle\alpha_{11}=\alpha_{12}=\alpha_{13}=\alpha_{14}=0\;.$ (46)

##  IV Numerical results and discussion

We first analyze the general features of the EWCL LECs obtained in the
previous section, which are similar to those in Lane’s first natural prototype
TC2 modelJunYi09 :

  * •

The contributions of the $p^{4}$-order coefficients are divided into two
parts: the contribution from the three sets of techniquarks and the
$Z^{\prime}$ contribution

  * •

All correction terms from the $Z^{\prime}$ particle to the EWCL LECs are
proportional to powers of $\beta_{1}$ which vanish if the mixing disappear
($\theta=0$). This can be seen from (46) and (128) which show that:
$\beta_{1}=\frac{10g_{1}^{2}\hat{F}_{0}^{2}\tan^{2}\theta}{16\bar{M}_{Z^{\prime}}^{2}}$.
By using the relation $\alpha_{\mathrm{em}}T=2\beta_{1}$, we can express all
LECs in terms of the $T$ parameter. Later in the paper, we show the $T$
dependence of the LECs.

  * •

From (46) (for $f^{2}$ and $\beta_{1}$), combined with (128), (121) , the
relation $\alpha_{\mathrm{em}}T=2\beta_{1}$ and the relationships of the
hyper-charges from Ref.Lane96 , we have

$\displaystyle\alpha_{\mathrm{em}}T$ $\displaystyle=$
$\displaystyle\bigg{[}1+\frac{2}{5}[\frac{81\tilde{F}_{0}^{2}}{4\hat{F}_{0}^{2}}+716+4(1-\frac{F_{0}^{\prime
2}}{\hat{F}_{0}^{2}})](u_{1}-v_{1})^{2}(1+\cot^{2}\theta)^{2}\bigg{]}^{-1}\;.~{}~{}~{}$

If we include the numerical result that $F^{\prime 2}_{0}<\hat{F_{0}^{2}}$,
the above result implies that $T$ must be positive and has an upper bound. The
upper bound is:

$\displaystyle\alpha_{\mathrm{em}}T_{\mathrm{Max}}=\frac{1}{1+\frac{2}{5}[\frac{81\tilde{F}_{0}^{2}}{4\hat{F}_{0}^{2}}+716+4(1-\frac{F_{0}^{\prime
2}}{\hat{F}_{0}^{2}})](u_{1}-v_{1})^{2}}\;.~{}~{}$ (47)

  * •

Because numerical calculation shows that $\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}\\!-\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}<0$ and $\beta_{1}$ is positive,
$\alpha_{8}$ is negative based on (46). Then $U=-16\pi\alpha_{8}$ which is a
coefficient given in Ref.EWCL  , is always positive in the present model.

Combining (121), (122) and (136), we find,

$\displaystyle
2\frac{\tilde{F}_{0}^{2}}{M_{Z^{\prime}}^{2}}g_{1}^{2}(\cot\theta+\tan\theta)^{2}\xi^{2}+4\frac{\hat{F}_{0}^{2}}{M_{Z^{\prime}}^{2}}(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})-8\frac{F^{\prime
2}_{0}}{M_{Z^{\prime}}^{2}}a_{0}^{2}$ (48)
$\displaystyle=1+[4(\cot\theta+\tan\theta)^{2}\xi^{2}+2\tan^{2}\theta+8\hat{v}+3\tan^{2}\theta+\hat{y}]\mathcal{K}g_{1}^{2}+4(\cot\theta+\tan\theta)^{2}\xi^{2}\tilde{\mathcal{K}}_{2}^{\Sigma\neq
0}g_{1}^{2}$
$\displaystyle+8(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+[40a_{3}^{2}+2(\hat{t}+\hat{s})g_{1}^{2}]\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-15\hat{D}_{0}a_{0}^{2}\;.$

We treat the above equation as a constraint on $\mathcal{K}$. This is done as
following: A suitable choice is made for the hypercharges (this will be
discussed later), electroweak gauge coupling, $T$ and $M_{Z^{\prime}}$. We
already know most of the parameters in (48), except $\tilde{F}_{0}$,
$\hat{F}_{0}$, $F^{\prime 2}_{0}$, $\tilde{\mathcal{K}}_{2}^{\Sigma\neq 0}$,
$\hat{\mathcal{K}}_{2}^{\Sigma\neq 0}$, $\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}$ and $\hat{D}_{0}$. By solving the SDEs, (34), (35), (36), (37), we can
obtain the techniquark self-energies, $\tilde{\Sigma}$, $\hat{\Sigma}$,
$\bar{\Sigma}$, $\bar{\Sigma}_{5}$. Furthermore, substituting the resulting
techniquark self-energies into the formulae given in Appendix E and (107), we
can obtain $\tilde{F}_{0}$, $\hat{F}_{0}$, $F^{\prime 2}_{0}$,
$\tilde{\mathcal{K}}_{2}^{\Sigma\neq 0}$, $\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}$, $\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}$ and $\hat{D}_{0}$ from (48). Now,
aside from $\mathcal{K}$ all the parameters in (48) are known. Then we can use
(48) to fix the value of $\mathcal{K}$. Once $\mathcal{K}$ is fixed, with the
help of (96), we can determine the ratio of the infrared cutoff $\kappa$ and
ultraviolet cutoff $\Lambda$. Numerical calculations show that this is unlike
the results in Refs.JunYi09 ; LangPLB , where the condition $\Lambda>\kappa$
occurs through the definitions used for the calculations and offers stringent
constraints on the allowed region for $T$ and the upper bound for
$M_{Z^{\prime}}$. In our model, $\Lambda>\kappa$ is naturally satisfied for
real values of $M_{Z^{\prime}}$. For example, we find that $\ln\kappa/\Lambda$
is about $-7.6$ and $-9.0$ for $M_{Z^{\prime}}$ values of 0.5TeV and 1TeV,
respectively.

With the above qualitative features, we now can generate numerical results.
First, we take $N=6$ which yields an infrared fixed point of
$\alpha_{w}=88\pi/523$. Then, we take $f=250$GeV. This completely fixes the
two-loop value at $\Lambda_{w}=5.5$TeV through the running behavior of (7),
SDE (35), $f^{2}=5\hat{F}_{0}^{2}$ and (142) which sets up the relationship
between $\hat{F}_{0}^{2}$ techniquark self-energy. This value of $\Lambda_{w}$
is smaller than the expected conventional ETC scale. Therefore, we cannot
interpret it as $\Lambda_{\mathrm{ETC}}$. Later, we will see that this is
because the walking effect is not large enough, and more ideal walking can
lead to a larger $\Lambda_{w}$. The current result with
$\Lambda_{w}\ll\Lambda_{\mathrm{ETC}}$ shows that our running coupling
constant cannot always walk from extreme infrared energy regions to the ETC
scale, $\Lambda_{\mathrm{ETC}}$. Instead, it can only walk a shorter distance
to the scale, $\Lambda_{w}$. Beyond $\Lambda_{w}$, it will run and fall
quickly exhibiting conventional asymptotic freedom behavior. Another
theoretical parameter is the coloron mass given by (40), which theoretically
depends on the values $\theta^{\prime}$, introduced in (12) and $\Theta$,
introduced in (91). We find the largest coloron mass occurs for
$\Theta=\pi/2$, i.e., the self-energies for the first set of techniquarks are
completely contributed by the twisted part of the set,
$\bar{\Sigma}_{5}=\hat{\Sigma}\sin\Theta$ and $\bar{\Sigma}=0$. Using this
value of $\Theta=\pi/2$, in Fig.3, we plot the coloron mass in terms of the
$T$ parameter. We used four values of $M_{Z^{\prime}}=0.5,1,2,5$TeV
(corresponding to $\ln\kappa/\Lambda\sim$ -7.6,-9.0,-9.4 and -9.5). We found
that that the coloron mass is not sensitive to $\theta^{\prime}$.

Figure 3: Coloron mass for Lane’s model.

From Fig.3, it can be seen that the coloron mass is roughly half the 1 TeV
expected in Lane’s original paperLane96 . The reason is that we included a
techniquark loop correction in the coloron kinetic term, which appeared in
(40) with the coefficients $\hat{E}$, $\mathcal{K}$ and
$\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}$. If we denote the coloron mass without
this correction as $M_{\mathrm{bare~{}coloron}}$ which was the notation used
in Lane’s original workLane96 , then our numerical calculation shows that:
$M_{\mathrm{bare~{}coloron}}/M_{\mathrm{coloron}}\sim\frac{2}{3}(\tan\theta^{\prime}+\cot\theta^{\prime})$.
This leads to a larger value for $M_{\mathrm{bare~{}coloron}}$. In fact, if we
carefully examine the denominator of (40), the structure of this kinetic term
correction can be divided into three parts: the tree order term
$2/g_{3}^{2}(\cot\theta^{\prime}+\tan\theta^{\prime})^{2}$, the techniquark
self-energy dependent part $\hat{E}+2\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-8\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}/(\cot\theta^{\prime}+\tan\theta^{\prime})^{2}$, and the techniquark self-
energy independent part $2\mathcal{K}$. The numerical calculation shows that
the main contribution comes from the techniquark self-energy dependent part,
which is an order of magnitude larger than the contributions from the other
two parts. Because the coloron mass is small666The small coloron mass forces
us to switch the order of integration over the coloron and Z’, i.e., instead
of integrating out the coloron before the Z’ boson, we need to integrate out
Z’ and then the coloron. We have performed the computation using this new
procedure and found the same result as that of the present paper, i.e.
switching the order of integration yields no correction. We found that the
possible correction from switching this order of integration depends on the
classical field $B_{A,c}^{\mu}$ caused by the coloron integration. These
classical coloron fields are determined by stationary equations. In both
cases, the stationary equations offer the null solution, $B_{A,c}^{\mu}=0$ ,
which was used in our results., we will use $\Theta=\pi/2$ to give the largest
coloron mass for all the following computations.

To provide numerical values for all the EWCL LECs, we need to choose the
various hyper-charges for the model. Note that the arrangement of the hyper-
charges given in Lane’s original paperLane96  is not suitable here because
that result used $N=4$. We showed in Section II that for the modern
interpretation of our two-loop based phase structure model, we use $N=6$, and
recalculate the hyper-charges. According to a series of relations among
different hyper-charges given by K. Lane in Ref.Lane96 , we need to use three
hyper-charges $x_{1}$, $y_{1}$ and $y_{1}+y_{2}$. We use a treatment similar
to the one used by K. Lane in Ref.Lane96 . Namely, we use $x_{1}=y_{1}$,
$y_{1}+y_{2}=0$. Furthermore, this requires that $u=(u_{1}-v_{1})/2\sim 1$.
These fully fix the typical values of all the hyper-charges. By ”typical” we
mean that the value of the hyper-charges must satisfy all 23 constraint
equations given in Ref. Lane96  and two more constraints: $x_{1}=y_{1}$,
$y_{1}+y_{2}=0$. The last two constraints were not explicitly mentioned in
Ref.Lane96 , but the detailed example used them. These typical hyper-charges
are: $a=-39$, $a^{\prime}=-46$, $b=14$, $b^{\prime}=8.2$, $c=-39$,
$c^{\prime}=-46$, $d=-12$, $d^{\prime}=-14$, $\xi=4.6$, $\xi^{\prime}=-4.6$,
$x_{1}=25$, $x^{\prime}_{1}=19$, $x_{2}=-26$, $x^{\prime}_{2}=-19$,
$y_{1}=25$, $y^{\prime}_{1}=23$, $y_{2}=-25$, $y^{\prime}_{2}=-23$,
$z_{1}=-7.7$, $z^{\prime}_{1}=19$, $z_{2}=7.7$, $z^{\prime}_{2}=-19$,
$u_{1}=-4.1$, $v_{1}=-6.1$, $u_{2}=4.2$, $v_{2}=6.2$.Using this set of typical
hyper-charges, combined with the other necessary inputs for the model, which
were discussed in the previous paragraph, (47) yields an upper bound,
$T_{\mathrm{max}}=0.035$. We show $S=-16\pi\alpha_{1}$ in Fig.4, and
$U=-16\pi\alpha_{8}$ in Fig.5.

Figure 4: $S$ parameter for Lane’s model.

Figure 5: $U$ parameter for Lane’s model.

From Fig.4, it can be seen that the value of $S$ is generally larger than 2,
which is not in agreement with experimental data. This value of the $S$
parameter already includes the walking effects in the model, which we will
discuss later. To examine the possibility of reducing the value of the $S$
parameter through the choice of hyper-charges, we found that when the input
hyper-charges $x_{1},y_{1}$ are not constrained by the requirement
$x_{1}=y_{1}$ and are much larger than 1, $S$ may achieve small values. Fig.6
shows the case with: $x_{1}=-50,y_{1}=36,y_{2}=-12$ which leads $a=-19$,
$a^{\prime}=-22$, $b=7$, $b^{\prime}=4$, $c=-19$, $c^{\prime}=-22$, $d=-6$,
$d^{\prime}=-7$, $\xi=2.3$, $\xi^{\prime}=-2.3$, $x_{1}=-50$,
$x^{\prime}_{1}=-53$, $x_{2}=2.7$,
$x^{\prime}_{2}=5.7$,$y_{1}=36$,$y^{\prime}_{1}=35$,$y_{2}=-12$,
$y^{\prime}_{2}=-11$, $z_{1}=20$, $z^{\prime}_{1}=33$, $z_{2}=3.6$,
$z^{\prime}_{2}=-9.4$, $u_{1}=0.41$, $v_{1}=-0.59$, $u_{2}=-0.41$,
$v_{2}=0.59$. The $S$ parameter can achieve negative values with larger values
of $T$. There may be other sets of hyper-charges which can also yield small or
even negative values of $S$, but typically these hyper-charges have large
values.

Figure 6: $S$ parameter for various choices of the hyper-charges:
$x_{1}=-50,y_{1}=36,y_{2}=-12$.

Excluding the $S$ and $U$ parameters, the leftmost eight non-zero parameters
$\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6},\alpha_{7},\alpha_{9},\alpha_{10}$
are shown in Fig.7 to Fig.13. $\alpha_{3}$ and $\alpha_{10}$ are independent
of $M_{Z^{\prime}}$ and are shown in the same figure.

Figure 7: $\alpha_{2}$ parameter for Lane’s model.

Figure 8: $\alpha_{3}$ and $\alpha_{10}$ parameters for Lane’s model.

Figure 9: $\alpha_{4}$ parameter for Lane’s model.

Figure 10: $\alpha_{5}$ parameter for Lane’s model.

Figure 11: $\alpha_{6}$ parameter for Lane’s model.

Figure 12: $\alpha_{7}$ parameter for Lane’s model.

Figure 13: $\alpha_{9}$ parameter for Lane’s model.

We found that $\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},$ are on the order
of $10^{-2}$, $\alpha_{6},\alpha_{7},\alpha_{9}$ are on the order of $10^{-5}$
and $\alpha_{10}$ is on the order of $10^{-10}$.

Previously, we discussed the three other TC2 modelsHill95 ; Lane95 ; Sekhar .
In Table IV., we list the different features and the orders of magnitude for
all the LECs of these TC2 models. In Fig.14, Fig.15, Fig.16,Fig.17 and Fig.18,
we show the ten nonzero LECs from these four TC2 models for comparison. This
comparison may be useful to other researchers as they consider the needs of
future models.

TABLE IV. Features and LECs of the TC2 models Hill95 , Lane95 , Sekhar  and
Lane96

Property or LEC |  Schematic TC2${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Hill95}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ |  Natural TC2${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Lane95}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ |  Hypercharge Universal${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Sekhar}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ |  Present${}^{\mbox{\tiny\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{Lane96}{\@@citephrase{(}}{\@@citephrase{)}}}}}$
---|---|---|---|---
Upper bound of $M_{Z^{\prime}}$ | $\surd$ | $\surd$ | $\surd$ | $\times$
Negative $S$ |  $M_{Z^{\prime}}\\!<\\!0.44$TeV or $T\\!>\\!0.17$ | $\times$ | $T\geq 10^{-1}$ | choose hypercharges
Typical $S\\!=\\!-16\pi\alpha_{1}$ | $\sim 0.3$ | $\sim 0.8$ | $\sim 1$ | $\sim 2$
$\alpha_{2}$ | $-10^{-3}$ | $-10^{-3}$ | $-10^{-3}$ | $-10^{-2}$
$\alpha_{3}$ | $-10^{-3}$ |  $3\times$ result of Hill95  | $-10^{-3}$ | $-10^{-2}$
$\alpha_{4}$ | $10^{-3}$ |  $3\times$ result of Hill95  | $10^{-3}$ | $10^{-2}$
$\alpha_{5}$ | $-10^{-3}$ |  $3\times$ result of Hill95  | $-10^{-3}$ | $-10^{-2}$
$\alpha_{6}$ | $\sim-10^{-4}$ | $\sim-10^{-3}$ | $\sim-10^{-4}$ | $\sim-10^{-5}$
$\alpha_{7}$ | $\sim 10^{-4}$ | $\sim 10^{-3}$ | $\sim 10^{-4}$ | $\sim 10^{-5}$
$\alpha_{8}=-\frac{U}{16\pi}$ | $\sim-10^{-4}$ |  $3\times$ result of Hill95  | $\sim-10^{-4}$ | $\sim-10^{-5}$
$\alpha_{9}$ | $\sim-10^{-4}$ |  $3\times$ result of Hill95  | $\sim-10^{-4}$ | $\sim-10^{-5}$
$\alpha_{10}$ | $\sim-10^{-8}$ | $\sim-10^{-8}$ | $\sim-10^{-7}$ | $\sim 10^{-10}$

Figure 14: $\alpha_{1}$ and $\alpha_{2}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.

Figure 15: $\alpha_{3}$ and $\alpha_{4}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.

Figure 16: $\alpha_{5}$ and $\alpha_{6}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.

Figure 17: $\alpha_{7}$ and $\alpha_{8}$ of the TC2 model Hill95 -Hill, Lane95
-Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve are the
masses of the $Z^{\prime}$ boson in TeV.

Figure 18: $\alpha_{9}$ and $\alpha_{10}$ of the TC2 model Hill95 -Hill,
Lane95 -Lane(I), Sekhar -Chiv and Lane96 -Lane(II). The numbers on each curve
are the masses of the $Z^{\prime}$ boson in TeV.

Finally, we estimate the magnitude of the walking effect in the present model.
Because the primary contribution to the walking effect is from the running
coupling constant, which appears in the kernel of the SDE, we can measure the
walking effect by comparing two other running behaviors:

  * •

Running $\alpha$: Rather than using a two-loop running coupling constant (7)
which exhibits an approximation of walking behavior in $N=6$ and spontaneous
chiral symmetry breaking, we used the one-loop running coupling constant used
in our previous workHongHao08 ; JunYi09 ; LangPLB  as

$\displaystyle\alpha(x)=\frac{4\pi}{\beta_{0}}\times\left\\{\begin{array}[]{lll}7&&\ln
x\leq-2\\\ 7-\frac{4}{5}(2+\ln x)^{2}&&-2\leq\ln x\leq 0.5\\\ \frac{1}{\ln
x}&&\ln x\geq 0.5\end{array}\right.\hskip
56.9055ptx=\frac{p^{2}}{\Lambda^{2}_{\mathrm{TC}}}\;.~{}~{}~{}~{}$ (52)

Eq.(52) was originally introduced in Ref.Runalpha . The general principle of
the technique is to use a plateau in the low energy region to normalize the
possibly infinite value in the infrared region that is predicted using the
perturbative result and smoothly connect this infrared plateau with the
ultraviolet asymptotic freedom running behavior. Note that if we ignore the
two-loop term in the $\beta$ function in this model and normalize the infrared
coupling constant such that it has a finite value, we can qualitatively obtain
the above form of the running coupling constant. Furthermore, this
approximation at the one-loop level suggests that $\Lambda_{w}$ must be
treated as $\Lambda_{\mathrm{TC}}$ in this running situation. The change from
one-loop running to two-loop walking reflects the evolution of our
understanding of the gauge-coupling running behavior in non-abelian gauge
theory. In addition, the decision to use the latter model in this study is
important because it confirms the existence of the infrared fixed pointLattice
which qualitatively supports the modern two-loop-based explanation of walking.

  * •

Ideal walking $\alpha$: Rather than using a two-loop running coupling constant
(7) and a value of $\alpha_{*}=88\pi/523$ that is not close in value to the
critical coupling $\alpha_{c}=4\pi/35$ for the first and second set of
techniquarks, we use the same running coupling constant but change the value
of $\alpha_{*}$ in (7) by artificially requiring that
$\alpha_{*}=1.02\alpha_{c}=1.02*4\pi/35$. Although this is not a realistic
case for the model, it is closer to the conformal situation, and therefore,
ideal walking.

The reason we must consider the above cases is because our analytical
estimation using the $\beta$ may cause some error. Therefore, we can use these
two extremes to investigate the effect of changes in the situation on our
results. We show three different behaviors of $\alpha$ in Fig.20. It can be
seen that $\alpha_{r}$ is much bigger than $\alpha_{w}$ only in the extreme
infrared region, and that the running behavior corresponding to
$1.02\alpha_{c}$ is smaller than that corresponding to $\alpha_{w}$ over most
of the energy region. From a comparison of Fig.20 with Fig.2, it can be seen
that the running effect increases the height of the infrared plateau and
narrows its length. To contrast other differences resulting from these
different couplings, in Fig.21, we show the techniquark self-energies,
$\tilde{\Sigma}$ and $\hat{\Sigma}$, which are determined by the SDEs (34) and
(35). We found that the closer the system came to walking, the lower and wider
the techniquark self-energy plateau was. By contrast, during running, the
plateau was higher and narrower. For fixed $f=250$GeV, we found that the
running situation produces a value of $\Lambda_{\mathrm{TC}}=0.21$TeV
($\Lambda_{\mathrm{ETC}}$ in the running case cannot be determined solely by
the running behavior and requires some other physical parameters to be known).
This result is consistent with the estimate of $\Lambda_{\mathrm{TC}}\simeq
2f\sqrt{3/N}$ given in Ref.LambdaTC . Our walking and ideal walking situations
yield:

$\displaystyle\Lambda_{w}=\left\\{\begin{array}[]{lll}5.5\mathrm{TeV}&&\mbox{walking}\\\
958\mathrm{TeV}&&\mbox{ideal walking}\end{array}\right.$ (55)

From this, it can be seen that $\Lambda_{w}$ is very sensitive to the walking
effect. The closer the system is to ideal walking, the bigger the value of
$\Lambda_{w}$. This was further checked by calculating $\Lambda_{w}$ for
several values of
$\alpha_{*}/\alpha_{c}=1.04,1.06,1.08,1.1,1.12,1.14,1.16,1.18,1.2$. These
points were then plotted as a curve in Fig.19 to quantitatively show the
sensitivity of $\Lambda_{w}$ to the degree of walking.

Figure 19: Dependence of the $\Lambda_{w}$ (TeV) on the degree of walking.

The small value of $\Lambda_{w}$ in our walking situation suggests that the
walking effect in the present model is not large enough. In an ideal walking
situation, $\Lambda_{w}$ is large and can be treated as
$\Lambda_{\mathrm{ETC}}$.

Figure 20: Three different couplings. $\alpha_{w}$ is the coupling used in our
calculation. $\alpha_{r}$ is the running coupling, which is given in (52).
Here, we show $\alpha_{r}/5$ to facilitate comparison between the couplings.
$1.02\alpha_{c}$ is the ideal walking coupling, where
$\alpha_{*}=1.02\alpha_{c}$.

Figure 21: Techniquark self-energies for three different couplings:
$\hat{\Sigma}_{w}$ and $\tilde{\Sigma}_{w}$ the self-energies for the second
and third sets of techniquarks for the coupling that we used in our
calculation. $\hat{\Sigma}_{r}$ and $\tilde{\Sigma}_{r}$ are the self-energies
for the second and third sets of techniquarks for the running coupling, which
is given in (52). Here, we show $\hat{\Sigma}_{r}/5$ and
$\tilde{\Sigma}_{r}/5$ to facilitate comparison between the self-energies.
$\hat{\Sigma}_{1.02\alpha_{c}}$ and $\tilde{\Sigma}_{1.02\alpha_{c}}$ are the
self-energies for the second and third sets of techniquarks for the ideal
walking coupling, where $\alpha_{*}=1.02\alpha_{c}$.

To show the effect of walking on the $S$ parameter, in Fig.22, we show the
value of $S$ for couplings corresponding to running and ideal walking. It can
be seen that for ideal walking (the upper bound on $T$ is reduced to 0.012 in
this case), $S$ is only slightly smaller than 2. Therefore, our prediction
that $S$ is about 2 is not significantly altered, even as one approaches the
walking region. However, Fig.22 shows that for running, $S$ is doubled by
reaching a value of 4. This implies that because of the existence of the
infrared fixed point, the walking only reduces the $S$ parameter by a factor
of 2. Furthermore, comparing the values of the $S$ parameters at different
couplings with their perturbative values
$S_{\mathrm{pert}}=N_{D}*N/6\pi=9/\pi$, we found that the perturbative value
of $S$ lies just between our realistic value and that of the running case.

Figure 22: $S$ parameters for the running and walking cases.

For the effect of walking on the other EWCL LECs, our numerical calculation
shows that for $\alpha_{2},\alpha_{3},\alpha_{4}$ walking reduces these LECs
to roughly $65\%$ of their original values in the running case. $\alpha_{5}$,
similar to the $S$ parameter, is reduced by the walking effect to half of its
original value in the running case. $\alpha_{6},\alpha_{7},\alpha_{9}$ are
reduced by one order of magnitude by the walking effect, but their signs are
preserved. $\alpha_{10}$ is reduced by two orders of magnitude and changes in
sign. Using the expression for $\alpha_{10}$ given by (46), the numerical
computation shows that some cancellations occur here. It is these
cancellations that result in $\alpha_{10}$being the smallest among the EWCL
LECs. Because of this cancellation, if the techniquark self-energy is changed,
more sign changes may occur. This cancellation may reduce the reliability of
our estimate of $\alpha_{10}$ and $\alpha_{10}$ may be seen as one of the
limitations of the calculation for the approximations used. We found that not
all LECs are sensitive to how close to ideal walking the theory is. The only
major exception is $\alpha_{10}$. Finally, we found that walking has almost no
effect on the coloron mass. We interpret this to mean that the techniquark
self-energy will change the value of the coloron mass significantly, but
walking, which changes the form of the techniquark self-energy, does not have
a large effect on the coloron mass. In fact, some quantities, such as
$\Lambda_{w}$ are sensitive to this detailed form of the techniquark self-
energy, but some other quantities, such as the coloron mass, are not.

##  V Summary

In this paper, we discuss K. Lane’s TC2 Model in the presence of nontrivial TC
fermion condensation and walking. We focus on the walking effects in the
model, which has not been discussed before. We also discuss the phase
structure of the model in terms of the two-loop $\beta$ function of the TC
coupling of the model. We found that to have both an infrared fixed point and
spontaneous chiral symmetry breaking, the minimum $N$ for the TC group $SU(N)$
is $N=6$. This is the optimal choice because it is the value that is the most
conformal that can be used in our model. Although this choice differs from the
critical values, $N^{c}_{1,2}=5.42$ for the first and second sets of
techniquarks and $N^{c}_{3}=4.93$ for the third set of techniquarks (Fig.1),
walking effects occur in the computed EWCL LECs. We can understand this
explicit walking effect qualitatively through the relation, $N-N^{c}_{i}\ll
N^{c}_{i}$ for $i=1,2,3$. For $N=6$, using the technique used in our previous
studiesHongHao08 ; JunYi09 ; LangPLB ,we derive the EWCL from Lane’s model and
calculate the EWCL LECs up to an order of $p^{4}$. We found that the primary
contributions to the $p^{4}$ order coefficients arise from the three sets of
techniquarks and $Z^{\prime}$. There is no limit on the upper bound of the
$Z^{\prime}$ mass which differs from the TC2 modelsHill95 ; Lane95 ; Sekhar
that we discussed previously. Moreover, all corrections from the $Z^{\prime}$
particle are at least proportional to $\beta_{1}$ and vanish for a mixing of
$\theta=0$. It is especially important that the scale parameter,
$\Lambda_{w}$, appears in the solution of the two-loop $\beta$. This signifies
that the scale of walking cannot be assumed to be $\Lambda_{\mathrm{ETC}}$ in
this model because, generally,
$\Lambda_{\mathrm{TC}}\leq\Lambda_{w}\leq\Lambda_{\mathrm{ETC}}$. We found
that $\Lambda_{w}=5.5$TeV. The value of $\Lambda_{w}$ is small because it is
sensitive to the walking effect. However, our choice of $N$ differs from its
critical value, and does not exhibit a sufficient walking effect. We verified
that in a more ideal walking case, $\Lambda_{w}$ can be increased by at least
two orders of magnitude. The ratio
$(\Lambda_{\mathrm{ETC}}-\Lambda_{w})/\Lambda_{\mathrm{ETC}}$ can be used as a
measurement of the deviation of our theory from ideal walking. We also found
that the coloron mass is roughly half of its expected value of 1 TeV and is
independent of the walking effect. The small coloron mass occurs as the result
of including a correction from the coloron kinetic term for which the main
contribution is from the techniquark self-energy. The $T$ and $U$ parameters
are positive, and there is an upper bound for the $T$ parameter. For our
choice of typical hyper-charges, the upper bound of the $T$ parameter is
0.035, which is well below the experimentally measured bound from PDG. The $S$
parameter is about 2 for our choice of typical hyper-charges, which already
exceeds the experimentally verified constraint that it be half of the value
from the running case, but similar to that of the ideal walking case. To
reduce the value of the $S$ parameter, one can change hyper-charges. This can
result in $S$ being negative for slightly larger values of $T$. This allows
for a case in which both $S$ and $T$ are within the bound from PDG. The
leftmost nine nonzero LECs, $\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5}$ are
on the order of $10^{-2}$ which matches the estimate obtained from naive
dimensional analysis. $\alpha_{6},\alpha_{7},\alpha_{9}$ are on the order of
$10^{-5}$ and $\alpha_{10}$ is on order of $10^{-10}$. This is because
$\alpha_{6},\alpha_{7},\alpha_{9}$,and especially $\alpha_{10}$, are sensitive
to walking effects. Comparing these results with the constraints imposed by
the precision dataprecision , we find that the results are consistent with the
constraints from the precision data. However, $\alpha_{3}$ has the correct
order of magnitude, but the wrong sign.

Previously, we investigated bosonic contribution to the EWCL LECs for most of
the TC2 models. In the future, we will focus on calculating the EWCL LECs in
four areas: The first will be to explore new physics models, including the
non-TC2-type models. The second will be to investigate the part of the EWCL
dealing with matter. In particular, we will focus on the top quark. The third
will be to deepen our understanding of the structure of the model we are
currently discussing in areas such as phase diagrams and the infrared behavior
of the gauge coupling constant. The fourth will be to improve the precision of
the computation and reduce the number of approximations necessary. With an
increasing number of models in our EWCL platform, it will be effective for
future investigations of the electroweak symmetry breaking mechanisms.

## Acknowledgments

This work was supported by the National Science Foundation of China (NSFC)
under Grants No. 10875065 and 11075085.

##  Appendix A Process of integrating out the techniquarks

To integrate out the techniquarks, which we have done in previous
studiesHongHao08 ; JunYi09 ; LangPLB , we assume only four fermion
interactions in (31), because a naive dimensional analysis indicates that the
contributions from higher dimensional operators are usually suppressed in the
low energy region. Also, this approximation leads to the conventional ladder
approximation, which is often used in discussions of the SDE. This yields:

$\displaystyle
iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]\approx\int
d^{4}x_{1}d^{4}x_{2}\frac{(-ig_{\mathrm{TC}})^{2}}{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})J_{\alpha_{1}}^{\mu_{1}}(x_{1})J_{\alpha_{2}}^{\mu_{2}}(x_{2})$
$\displaystyle=-\frac{g^{2}_{\mathrm{TC}}}{2}\int
d^{4}x_{1}d^{4}x_{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})\bigg{[}\bar{\psi}(x_{1})\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}\psi(x_{1})\bar{\psi}(x_{2})\tilde{t}^{\alpha_{2}}\gamma^{\mu_{2}}\psi(x_{2})$
$\displaystyle+{\displaystyle\sum_{i,j=1,2,l,t,b}}\bar{T}^{i}_{\xi}(x_{1})t^{\alpha_{1}}\gamma^{\mu_{1}}T^{i}_{\xi}(x_{1})\bar{T}^{j}_{\xi}(x_{2})t^{\alpha_{2}}\gamma^{\mu_{2}}T^{j}_{\xi}(x_{2})+2{\displaystyle\sum_{i=1,2,l,t,b}}\bar{\psi}(x_{1})\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}\psi(x_{1})\bar{T}^{i}_{\xi}(x_{2})t^{\alpha_{2}}\gamma^{\mu_{2}}T^{i}_{\xi}(x_{2})\bigg{]}$
$\displaystyle\approx\int
d^{4}x_{1}d^{4}x_{2}\bigg{[}\bar{\psi}^{\sigma}(x_{1})\tilde{\Pi}_{\sigma\rho}(x_{1},x_{2})\psi^{\rho}(x_{2})+{\displaystyle\sum_{i,j=1,2,l,t,b}}\bar{T}^{i\sigma}_{\xi}(x_{1})\Pi^{ij}_{\sigma\rho}(x_{1},x_{2})\bar{T}^{j\rho}_{\xi}(x_{2})\bigg{]}\;,$
(56)

where we have used (29) and (30). And

$\displaystyle\tilde{\Pi}_{\sigma\rho}(x_{1},x_{2})$ $\displaystyle\equiv$
$\displaystyle-g^{2}_{\mathrm{TC}}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}_{\sigma\sigma_{1}}\langle\psi^{\sigma_{1}}(x_{1})\bar{\psi}^{\rho_{2}}(x_{2})\rangle\tilde{t}^{\alpha_{2}}\gamma^{\mu_{2}}_{\rho_{2}\rho}$
(57) $\displaystyle\Pi^{ij}_{\sigma\rho}(x_{1},x_{2})$ $\displaystyle\equiv$
$\displaystyle-g^{2}_{\mathrm{TC}}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x_{1},x_{2})t^{\alpha_{1}}\gamma^{\mu_{1}}_{\sigma\sigma_{1}}\langle
T^{i\sigma_{1}}(x_{1})\bar{T}^{j\rho_{2}}(x_{2}){\rangle}t^{\alpha_{2}}\gamma^{\mu_{2}}_{\rho_{2}\rho}\;.$
(58)

To obtain (56), we have used the average field approximation and approximated
the four-fermion interactions using their vacuum expectation values (VEVs).
Furthermore, we used the result:
$\langle\bar{\psi}(x)\gamma^{\mu}\psi(x)\rangle=\langle\bar{T}^{i}(x)\gamma^{\mu}T^{j}(x)\rangle=0$,
which can be obtained from the Lorentz invariance;
$\langle\bar{\psi}(x)T^{i}(x)\rangle=\langle\bar{T}^{i}(x)\psi(x)\rangle=0$,
which was assumed in Lane’s original paper Lane96  and can be verified as a
solution to the SDE. In fact, one can confirm that the VEVs between the
different sets of techniquarks vanish and VEVs among the different
techniquarks of the second set also vanish. For (56), this yields:

$\displaystyle iS_{\mathrm{TC}}[\bar{T}_{\xi},T_{\xi},\bar{\psi},\psi]$
$\displaystyle\approx$ $\displaystyle\int
d^{4}x_{1}d^{4}x_{2}\bigg{[}\bar{\psi}^{\sigma}(x_{1})\tilde{\Pi}_{\sigma\rho}(x_{1},x_{2})\psi^{\rho}(x_{2})+{\displaystyle\sum_{i,j=1,2}}\bar{T}^{i\sigma}_{\xi}(x_{1})\bar{\Pi}^{ij}_{\sigma\rho}(x_{1},x_{2})T^{j\rho}_{\xi}(x_{2})$
(59)
$\displaystyle+{\displaystyle\sum_{i=l,t,b}}\bar{T}^{i\sigma}_{\xi}(x_{1})\hat{\Pi}_{\sigma\rho}(x_{1},x_{2})T^{i\rho}_{\xi}(x_{2})\bigg{]}$

with

$\displaystyle\Pi^{ij}_{\sigma\rho}(x_{1},x_{2})=\left\\{\begin{array}[]{lll}\bar{\Pi}^{ij}_{\sigma\rho}(x_{1},x_{2})&{}{}{}&i,j=1,2\\\
&&\\\ \hat{\Pi}_{\sigma\rho}(x_{1},x_{2})&&i,j=l,t,b\end{array}\right.\;.$
(63)

Therefore $\bar{\Pi}$, $\hat{\Pi}$ and $\tilde{\Pi}$ represent the fermion
self-energies for the first, second, and third sets of techniquarks,
respectively. Following the treatment in our previous studiesHongHao08 ;
JunYi09 ; LangPLB , these techniquark self-energies can be approximated as:

$\displaystyle\hat{\Pi}^{ij}_{\sigma\rho}(x,y)\approx-\delta_{\sigma\rho}[\hat{\Sigma}(\overline{\nabla}_{x}^{2})\delta^{4}(x\\!-\\!y)]_{ij}\hskip
14.22636pt\tilde{\Pi}_{\sigma\rho}(x,y)\approx-\delta_{\sigma\rho}\tilde{\Sigma}(\partial_{x}^{2})\delta^{4}(x\\!-\\!y)\hskip
14.22636pt\overline{\nabla}^{\mu}\\!=\\!\partial^{\mu}\\!-\\!iV_{2\xi}^{\mu}~{}~{}~{}$
(64)
$\displaystyle\bar{\Pi}^{ij}_{\sigma\rho}(x,y)\approx-[\delta_{\sigma\rho}\bar{\Sigma}(\hat{\nabla}_{x}^{2})+i\gamma^{5}_{\sigma\rho}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}_{x}^{2})]_{ij}\delta^{4}(x-y)\hskip
71.13188pt\hat{\nabla}^{\mu}=\partial^{\mu}-iV_{1\xi}^{\mu}\bigg{|}_{v_{1}=0}\;,~{}~{}~{}~{}~{}$
(65)

where $V_{2\xi}^{\mu}$, $V_{1\xi}^{\mu}$ and $v_{1}^{\mu}$ will be discussed
later in the appendices. The above approximation is the lowest order of a
dynamical perturbation originally proposed by Pagels and Stokar in Ref.DPT .
In this perturbation, all source dependent parts are expressed in terms of the
techniquark self-energy and the detailed dependence is determined by including
the minimal contribution that is covariant with the local chiral symmetry. An
important result of this dynamical perturbation is that the lowest order,
which includes the fermion loop terms, yields spontaneous chiral symmetry
breaking and is dominated by the fermion self-energy. In our previous
studiesHongHao08 ; JunYi09 ; LangPLB , the $\Pi$ functions are diagonal in the
spinor space, but in this model, $\bar{\Pi}_{\sigma\rho}(x,y)$ in (65) differs
from the conventional expression. In this case, there is an extra term
($\bar{\Sigma}_{5}$) that is proportional to $\gamma^{5}$ and $\tau^{2}$ (in
isospin space) because of the special model arrangement that generates
nontrivial twisted TC fermion condensation. This condensation will stimulate
topcolor symmetry breaking: $SU(3)_{1}\otimes SU(3)_{2}\rightarrow SU(3)_{c}$
and generate the coloron mass. Later, we will discuss the appearance of this
term and determine the functions corresponding to $\hat{\Sigma}$,
$\tilde{\Sigma}$, $\bar{\Sigma}$ and $\bar{\Sigma}_{5}$ .

With the results from (59)-(65), the techniquark interactions in (III.3)
become bilinear, and we can complete the integration over the techniquarks and
obtain (33), which is given in the text. Where:

$\displaystyle
V_{1\xi}=\begin{pmatrix}v_{1}\\!+v_{2}-g_{3}\frac{\lambda^{A}}{2}B^{A}\cot\theta^{\prime}&0\\\
0&v_{1}\\!+v_{2}+g_{3}\frac{\lambda^{A}}{2}B^{A}\tan\theta^{\prime}\end{pmatrix}\hskip
19.91684ptA_{1\xi}=\begin{pmatrix}a_{1}\\!+a_{2}&\\\
&a_{1}\\!-a_{2}\end{pmatrix}~{}~{}~{}~{}~{}$ (66) $\displaystyle
V_{2\xi}=\begin{pmatrix}v_{l}&0&0\\\ 0&v_{t}&0\\\ 0&0&v_{b}\end{pmatrix}\hskip
219.08612ptA_{2\xi}=\begin{pmatrix}a_{l}&0&0\\\ 0&a_{t}&0\\\
0&0&a_{b}\end{pmatrix}\;.$ (67)

The prime in $\mathrm{Tr^{\prime}}$ denotes the trace of the extra $2\times 2$
space for the first two sets of techniquarks, and the double prime in
$\mathrm{Tr"}$ denotes the trace of the extra $3\times 3$ space for the third
set of techniquarks with:

$\displaystyle
v_{1}=-\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}(B_{\xi}-Z^{\prime}\tan\theta)$
$\displaystyle
v_{2}=-\frac{1}{2}g_{1}(u_{2}+v_{2})(B_{\xi}-Z^{\prime}\tan\theta)-\frac{1}{2}g_{1}(u_{1}+v_{1})(B_{\xi}+Z^{\prime}\cot\theta)$
(68) $\displaystyle
a_{1}=\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}(B_{\xi}-Z^{\prime}\tan\theta)\hskip
56.9055pta_{2}=\frac{1}{2}g_{1}(u_{1}-v_{1})(\cot\theta+\tan\theta)Z^{\prime}~{}~{}~{}~{}$
(69) $\displaystyle
v_{i}=-\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{g_{1}}{2}\frac{\tau^{3}}{2}(B_{\xi}\\!-\\!Z^{\prime}\tan\theta)-\frac{g_{1}}{2}(x^{i}_{2}\\!+x^{i\prime}_{2})(B_{\xi}\\!-\\!Z^{\prime}\tan\theta)-\frac{g_{1}}{2}(x^{i}_{1}\\!+x^{i\prime}_{1})(B_{\xi}\\!+\\!Z^{\prime}\cot\theta)$
$\displaystyle
a_{i}=\frac{1}{2}g_{2}\frac{\tau^{a}}{2}W_{\xi}^{a}-\frac{1}{2}g_{1}\frac{\tau^{3}}{2}(B_{\xi}-Z^{\prime}\tan\theta)+\frac{1}{2}g_{1}(x^{i}_{1}-x^{i\prime}_{1})(\cot\theta+\tan\theta)Z^{\prime}\hskip
42.67912pti=l,t,b\;.$ (70)

We have used the relation

$\displaystyle
iq_{1}\xi{B}_{1\xi,\mu}P_{L}-iq_{2}\xi{B}_{2\xi,\mu}P_{L}+iq_{1}\xi_{1}^{\prime}{B}_{1\xi,\mu}P_{R}-iq_{2}\xi^{\prime}{B}_{2\xi,\mu}P_{R}=-ig_{1}(\cot\theta+\tan\theta)\xi
Z^{\prime}_{\mu}\gamma^{5}$ (71)
$\displaystyle-\\!h_{1}\frac{\lambda^{A}}{2}\not{A}_{1}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1\xi}P_{L}\\!\\!-\\!q_{2}u_{2}\not{B}_{2\xi}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1\xi}P_{R}\\!\\!-\\!q_{2}(v_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}$
$\displaystyle=\not{v}_{1}\\!+\not{v}_{2}-g_{3}\frac{\lambda^{A}}{2}\not{B}^{A}\cot\theta^{\prime}+(\not{a}_{1}+\not{a}_{2})\gamma^{5}$
(72)
$\displaystyle-\\!h_{2}\frac{\lambda^{A}}{2}\not{A}_{2}^{A}\\!\\!-\\!g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!\\!-\\!q_{1}v_{1}\not{B}_{1\xi}P_{L}\\!\\!-\\!q_{2}v_{2}\not{B}_{2\xi}P_{L}\\!\\!-\\!q_{1}u_{1}\not{B}_{1\xi}P_{R}\\!\\!-\\!q_{2}(u_{2}\\!\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}$
$\displaystyle=\not{v}_{1}\\!+\not{v}_{2}+g_{3}\frac{\lambda^{A}}{2}\not{B}^{A}\tan\theta^{\prime}+(\not{a}_{1}-\not{a}_{2})\gamma^{5}$
(73) $\displaystyle-
g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!-q_{1}x_{1}\not{B}_{1\xi}P_{L}\\!-q_{2}x_{2}\not{B}_{2\xi}P_{L}\\!-q_{1}x_{1}^{\prime}\not{B}_{1\xi}P_{R}\\!-q_{2}(x_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}=\not{v}_{l}+\not{a}_{l}\gamma^{5}$
(74) $\displaystyle-
g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!-q_{1}y_{1}\not{B}_{1\xi}P_{L}\\!-q_{2}y_{2}\not{B}_{2\xi}P_{L}\\!-q_{1}y_{1}^{\prime}\not{B}_{1\xi}P_{R}\\!-q_{2}(y_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}=\not{v}_{t}+\not{a}_{t}\gamma^{5}$
(75) $\displaystyle-
g_{2}\frac{\tau^{a}}{2}\not{W}_{\xi}^{a}P_{L}\\!-q_{1}z_{1}\not{B}_{1\xi}P_{L}\\!-q_{2}z_{2}\not{B}_{2\xi}P_{L}\\!-q_{1}z_{1}^{\prime}\not{B}_{1\xi}P_{R}\\!-q_{2}(z_{2}^{\prime}\\!+\\!\frac{\tau^{3}}{2})\not{B}_{2\xi}P_{R}=\not{v}_{t}+\not{a}_{t}\gamma^{5}\;.$
(76)

##  Appendix B Derivation of the Schwinger-Dyson equations for the techniquark
self-energies

In this appendix, we derive the SDE for the techniquark self-energies. We
start from the path integral given in (III.3), and fix the functional
integration over the $U$, $B^{A}_{\mu}$ and $Z^{\prime}_{\mu}$ fields. The
total functional derivative of the integrand with respect to $\bar{\psi}$ and
$\bar{T}_{\xi}^{i}$ is zero, which yields:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\mu(\psi,T)~{}\frac{\delta}{\delta\bar{\psi}^{\sigma}(x)}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}+iS_{\mathrm{source}}}\bigg{|}_{A^{A}_{\mu}=0}$
(77) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\mu(\psi,T)\frac{\delta}{\delta\bar{T}^{i,\sigma}_{\xi}(x)}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}+iS_{\mathrm{source}}}\bigg{|}_{A^{A}_{\mu}=0}$
(78) $\displaystyle\mathcal{D}\mu(\psi,T)$ $\displaystyle\equiv$
$\displaystyle\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}\bar{T}^{1}_{\xi}\mathcal{D}T^{1}_{\xi}\mathcal{D}\bar{T}^{2}_{\xi}\mathcal{D}T^{2}_{\xi}\mathcal{D}\bar{T}^{l}_{\xi}\mathcal{D}T^{l}_{\xi}\mathcal{D}\bar{T}^{t}_{\xi}\mathcal{D}T^{t}_{\xi}\mathcal{D}\bar{T}^{b}_{\xi}\mathcal{D}T^{b}_{\xi}\;,$
(79)

In this case, we have introduced source terms with external sources $\bar{I}$
and $\bar{J}$ to help to derive the SDEs:

$\displaystyle iS_{\mathrm{source}}=\int
d^{4}x\bigg{[}\bar{\psi}(x)I(x)+{\displaystyle\sum_{i=1,2,l,t,b}}\bar{T}^{i}(x)J^{i}(x)\bigg{]}\;.$
(80)

We derive $I^{\rho}(y)$ for both sides of (77) and remove all external
sources. We obtain:

$\displaystyle
0=S_{\psi\sigma\rho}^{-1}(x,y)+i[i\not{\partial}_{x}\\!+g_{1}(\cot\theta\\!+\\!\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}]_{\sigma\rho}\delta(x\\!-\\!y)-g_{\mathrm{TC}}^{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x,y)[\tilde{t}^{\alpha_{1}}\gamma^{\mu_{1}}S(x,y)\tilde{t}^{\alpha_{2}}\gamma^{\mu_{2}}]_{\sigma\rho}$
(81) $\displaystyle
S_{\psi\sigma\rho}(x,y)\equiv\langle\psi^{\sigma}(x)\bar{\psi}^{\rho}(y)\rangle=\frac{\int\mathcal{D}\mu(\psi,T)~{}\psi^{\sigma}(x)\bar{\psi}^{\rho}(y)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}{\int\mathcal{D}\mu(\psi,T)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}\bigg{|}_{A^{A}_{\mu}=0}\;.$
(82)

(81) is the SDE in coordinate space for the third set of techniquarks.
Combining (57) and (81), we find that $S_{\psi\sigma\rho}(x,y)$, which is
determined by the SDE, relates to $\tilde{\Pi}_{\sigma\rho}(x,y)$, introduced
in (57), through:

$\displaystyle
0=S_{\psi\sigma\rho}^{-1}(x,y)+i[i\not{\partial}_{x}\\!+g_{1}(\cot\theta\\!+\\!\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}]_{\sigma\rho}\delta(x\\!-\\!y)+\tilde{\Pi}_{\sigma\rho}(x,y)=0\;.$
(83)

Similarly we derive $J^{j\rho}(y)$ for both sides of (78), and remove all
external sources, We obtain:

$\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{1\xi}\\!+\\!\not{A}_{1\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)-g_{\mathrm{TC}}^{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x,y)[t^{\alpha_{1}}\gamma^{\mu_{1}}S(x,y)t^{\alpha_{2}}\gamma^{\mu_{2}}]^{ij}_{\sigma\rho}$
$\displaystyle\hskip 312.9803pti,j=1,2$ (84) $\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{2\xi}\\!+\\!\not{A}_{2\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)-g_{\mathrm{TC}}^{2}G_{\mu_{1}\mu_{2}}^{\alpha_{1}\alpha_{2}}(x,y)[t^{\alpha_{1}}\gamma^{\mu_{1}}S(x,y)t^{\alpha_{2}}\gamma^{\mu_{2}}]^{ij}_{\sigma\rho}$
$\displaystyle\hskip 312.9803pti,j=l,t,b\;,$ (85)

where

$\displaystyle S^{ij}_{T\sigma\rho}(x,y)\equiv\langle
T^{i\sigma}(x)\bar{T}^{j\rho}(y)\rangle=\frac{\int\mathcal{D}\mu(\psi,T)~{}T^{i\sigma}(x)\bar{T}^{j\rho}(y)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}{\int\mathcal{D}\mu(\psi,T)~{}e^{iS_{\mathrm{TC}}+iS_{\mathrm{TC1}}}}\bigg{|}_{A^{A}_{\mu}=0}\;.$
(86)

(84) and (85) are the SDEs in the coordinate space of the first and second
sets of techniquarks. Combining (58), (63), (84) and (85), we find that
$S^{ij}_{T\sigma\rho}(x,y)$ which is determined by the SDE, relates to
$\bar{\Pi}^{ij}_{\sigma\rho}(x,y)$ and $\hat{\Pi}^{ij}_{\sigma\rho}(x,y)$,
introduced in (58) and (63), through:

$\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{1\xi}\\!+\\!\not{A}_{1\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)+\bar{\Pi}^{ij}_{\sigma\rho}(x,y)\hskip
28.45274pti,j=1,2$ (87) $\displaystyle
0=S_{T\sigma\rho}^{ij,-1}(x,y)+i[i\not{\partial}_{x}\\!+\\!\not{V}_{2\xi}\\!+\\!\not{A}_{2\xi}\gamma^{5}]^{ij}_{\sigma\rho}\delta(x\\!-\\!y)+\hat{\Pi}^{ij}_{\sigma\rho}(x,y)\hskip
28.45274pti,j=l,t,b\;.$ (88)

Following the treatment in our previous works HongHao08 ; JunYi09 ; LangPLB ,
the techniquark self-energies $\hat{\Sigma}$ and $\tilde{\Sigma}$ in (64) and
$\bar{\Sigma}$, $\bar{\Sigma}_{5}$ in (65) are determined by removing the
gauge fields in the SDEs. Using this approximation, we find the three sets of
techniquarks:

$\displaystyle
S_{\psi\sigma\rho}(x,y)=\\!\int\\!\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\bigg{[}\frac{i}{\not{p}\\!-\\!\tilde{\Sigma}(-p^{2})}\bigg{]}_{\sigma\rho}\hskip
28.45274ptS_{T\sigma\rho}^{ij}(x,y)=\\!\int\\!\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\bigg{[}\frac{i\delta_{ij}}{\not{p}\\!-\\!\hat{\Sigma}(-p^{2})}\bigg{]}_{\sigma\rho}$
$\displaystyle\hskip 275.99164pti,j=l,t,b$ (89) $\displaystyle
S_{T\sigma\rho}^{ij}(x,y)=\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ip(x-y)}\bigg{[}\frac{i}{\not{p}-\bar{\Sigma}(-p^{2})-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(-p^{2})}\bigg{]}^{ij}_{\sigma\rho}\hskip
56.9055pti,j=1,2\;,$ (90)

In Euclidean space, we obtain(34), (35), (36) and (37)in the main text.

In terms of $\hat{\Sigma}$, comparing(35) with (36) and (37),we can construct
$\bar{\Sigma}$ and $\bar{\Sigma}_{5}$ as follows:

$\displaystyle\bar{\Sigma}(p_{E}^{2})=\hat{\Sigma}(p_{E}^{2})\cos\Theta\hskip
56.9055pt\bar{\Sigma}_{5}(p_{E}^{2})=\hat{\Sigma}(p_{E}^{2})\sin\Theta\;.$
(91)

$\Theta$ at the present stage in the computation is an arbitrary constant, and
we have verified that the vacuum energy generated by $\bar{\Sigma}$ and
$\bar{\Sigma}_{5}$ only depends on
$\bar{\Sigma}^{2}+\bar{\Sigma}^{2}_{5}=\hat{\Sigma}^{2}$, which is independent
of $\Theta$. Later we show that the coloron mass is dependent on $\Theta$ and
the present model gives a relatively small coloron mass (several hundred GeV).
In practice, we use the value of $\Theta$ which offers the largest coloron
mass. Once nonzero techniquark self-energies are present, we will have nonzero
techniquark condensates:

$\displaystyle\langle\bar{T}^{i}_{L}(x)T^{j}_{R}(x)\rangle=-2N\\!\int\\!\frac{d^{4}p_{E}}{(2\pi)^{4}}\bigg{[}\frac{\delta_{ij}\bar{\Sigma}(p_{E}^{2})}{p_{E}^{2}\\!+\\!\bar{\Sigma}^{2}(p_{E}^{2})\\!+\\!\bar{\Sigma}_{5}^{2}(p_{E}^{2})}-\frac{i\tau^{2}_{ij}\bar{\Sigma}_{5}(p_{E}^{2})}{p_{E}^{2}\\!+\\!\bar{\Sigma}^{2}(p_{E}^{2})\\!+\\!\bar{\Sigma}_{5}^{2}(p_{E}^{2})}\frac{p_{E}^{2}\\!-\\!\bar{\Sigma}^{2}(p_{E}^{2})}{p_{E}^{2}\\!+\\!\bar{\Sigma}^{2}(p_{E}^{2})}\bigg{]}$
$\displaystyle\hskip 361.3499pti,j=1,2,$ (92)
$\displaystyle\langle\bar{T}^{i}_{L}(x)T^{j}_{R}(x)\rangle=-2N\delta_{ij}\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\frac{\hat{\Sigma}(p_{E}^{2})}{p_{E}^{2}+\hat{\Sigma}^{2}(p_{E}^{2})}\hskip
142.26378pti,j=l,t,b,$ (93)
$\displaystyle\langle\bar{\psi}_{L}(x)\psi_{R}(x)\rangle=-N(N-1)\int\frac{d^{4}p_{E}}{(2\pi)^{4}}\frac{\tilde{\Sigma}(p_{E}^{2})}{p_{E}^{2}+\tilde{\Sigma}^{2}(p_{E}^{2})}\;.$
(94)

Note that the first techniquark set has a nontrivial twisted condensation:
$\langle\bar{T}^{1}_{L}(x)T^{2}_{R}(x)\rangle=-\langle\bar{T}^{2}_{L}(x)T^{1}_{R}(x)\rangle\neq
0$ resulting from the nonzero self-energies.

##  Appendix C  Integrating out the colorons and the low energy expansion

The coefficients in (39) are,

$\displaystyle C=\int
d^{4}\tilde{k}[-2\tau+\tau^{2}k_{E}^{2}+16\tau^{2}\bar{\Sigma}^{2}_{5}]$ (95)
$\displaystyle\mathcal{K}=-\frac{1}{48\pi^{2}}[\ln\frac{\kappa^{2}}{\Lambda^{2}}+\gamma]\hskip
56.9055pt\kappa,\Lambda~{}\mbox{: infrared and ultraviolet cutoffs}$ (96)
$\displaystyle\hat{E}=\int
d^{4}\tilde{k}[\tau^{2}+16\tau^{2}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}+4\tau^{2}k_{E}^{2}\bar{\Sigma}^{\prime
2}+8\tau^{2}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime\prime}+4\tau^{2}k_{E}^{2}\bar{\Sigma}_{5}^{\prime
2}-\frac{1}{3}\tau^{3}k_{E}^{2}-\frac{16}{3}\tau^{3}\bar{\Sigma}_{5}^{2}$
$\displaystyle\hskip
28.45274pt-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}-6\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}-\frac{32}{3}\tau^{3}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}-\frac{32}{3}\tau^{3}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}\bar{\Sigma}^{\prime\prime}-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}^{\prime
2}$ $\displaystyle\hskip
28.45274pt-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime\prime}-\frac{2}{9}\tau^{3}k_{E}^{4}\bar{\Sigma}_{5}^{\prime
2}-\frac{32}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}-\frac{16}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime\prime}\bar{\Sigma}_{5}^{2}-\frac{16}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}$ $\displaystyle\hskip
28.45274pt-16\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}^{2}\bar{\Sigma}_{5}^{\prime
2}+\frac{1}{18}\tau^{4}k_{E}^{4}+\frac{4}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}+\frac{2}{9}\tau^{4}k_{E}^{4}\bar{\Sigma}\bar{\Sigma}^{\prime}+\frac{2}{9}\tau^{4}k_{E}^{4}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}$
$\displaystyle\hskip
28.45274pt+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{3}\bar{\Sigma}^{\prime}+\frac{2}{9}\tau^{4}k_{E}^{4}\bar{\Sigma}^{2}\bar{\Sigma}^{\prime
2}+\tau^{4}k_{E}^{4}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}+\tau^{4}k_{E}^{4}\bar{\Sigma}_{5}^{2}\bar{\Sigma}_{5}^{\prime
2}$ $\displaystyle\hskip
28.45274pt+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}+\frac{32}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}+\frac{16}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{4}\bar{\Sigma}_{5}^{\prime
2}]$ (97) $\displaystyle\int
d^{4}\tilde{k}=N\int^{\infty}_{\frac{1}{\Lambda^{2}}}\frac{d\tau}{\tau}\int\frac{d^{4}k_{E}}{(2\pi)^{4}}e^{-\tau[k_{E}^{2}+\bar{\Sigma}^{2}(k_{E}^{2})]},\hskip
28.45274pt\bar{\Sigma}=\bar{\Sigma}(k_{E}^{2}),\hskip
28.45274pt\bar{\Sigma}_{5}=\bar{\Sigma}_{5}(k_{E}^{2})\;,$ (98)

Where $\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}$ are the coefficients that are
introduced later in (102), $\Lambda$ is a cutoff that is not sensitive to
changes for values between 10 TeV and 100 TeV for our walking theory. In our
practical calculation, we set it to 40 TeV. Combining the standard coloron
kinetic term in (33) and the techniquark quantum loop correction given by
(39), we obtain the formula for the coloron mass (40) given in the text. With
the coloron mass from (40), we can discuss coloron field integration in (40),
we then discuss coloron field integration in (33). This can be achieved using
the standard loop expansion:

$\displaystyle\int\mathcal{D}B_{\mu}^{A}~{}\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1,\mu\nu}B^{1,\mu\nu}+B_{2,\mu\nu}B^{2,\mu\nu})]$
$\displaystyle+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{]}_{A^{A}_{\mu}=0}$
$\displaystyle=\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}(A_{1\mu\nu}^{A}A^{A,1\mu\nu}+A_{2\mu\nu}^{A}A^{A,2\mu\nu}+W_{\mu\nu}^{a}W^{a,\mu\nu}+B_{1,\mu\nu}B^{1,\mu\nu}+B_{2,\mu\nu}B^{2,\mu\nu})]$
$\displaystyle+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]+\mbox{loop
corrections}\bigg{]}_{A^{A}_{\mu}=0,B^{A}_{\mu}=B^{A}_{\mu,c}}\;.$ (99)

And $B^{A}_{\mu,c}$ is determined by requiring that the result reach its
extremum at $B^{A}_{\mu}=B^{A}_{\mu,c}$. One can show that $B^{A}_{\mu,c}=0$
is one solution. Consequently, (33) becomes:

$\displaystyle e^{iS_{\mathrm{EW}}[W_{\mu}^{a},B_{\mu}]}$ $\displaystyle=$
$\displaystyle e^{i\int
d^{4}x[-\frac{1}{4}W_{\mu\nu}^{a}W^{a,\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}]}\int\mathcal{D}_{\mu}(U)\mathcal{F}[O_{\xi}]\delta(O_{\xi}-O^{\dagger}_{\xi})\int\mathcal{D}Z_{\mu}^{\prime}$
(100) $\displaystyle\exp\bigg{[}i\int
d^{4}x[-\frac{1}{4}Z^{\prime}_{\mu\nu}Z^{\prime\mu\nu}]+\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]$
$\displaystyle+\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]$
$\displaystyle+\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]+\mbox{loop
corrections}\bigg{]}_{A^{A}_{\mu}=B^{A}_{\mu}=0}\;.$

Note that we are interested in the bosonic part of the EWCL, those operators
involve explicit top quark fields, which belong to the part of the EWCL
dealing with matter, are beyond the scope of this paper. The top quark loop
term (especially the top quark condensate) is expected to essentially
contribute only to the top quark mass and not to the W and Z masses in TC2
models. This suggests that the contribution from top quark condensation to the
bosonic part of the EWCL may also be small (we will show this in the future in
a separate paper). Consequently, colorons, which are important in the
formation of top-quark condensates and contribute the majority of the top-
quark mass, only play a passive role in our present calculations. From
(12),the requirement, $A^{A}_{\mu}=B^{A}_{\mu}=0$ in (100) is equivalent to
the requirement, $A^{A}_{1\mu}=A^{A}_{2\mu}=0$.

Now, with the help of a technique used in our previous studiesHongHao08 ;
JunYi09 ; LangPLB , we take low energy expansion for the three TrLn terms in
(100):

$\displaystyle\mathrm{Trln}[i\not{\partial}+g_{1}(\cot\theta\\!+\tan\theta)\xi\not{Z}^{\prime}\gamma^{5}-\tilde{\Sigma}(\partial^{2})]\bigg{|}_{\mathrm{normal~{}part}}$
(101) $\displaystyle=i\int
d^{4}x(\cot\theta\\!+\\!\tan\theta)^{2}\bigg{[}\tilde{F}_{0}^{2}g_{1}^{2}\xi^{2}Z^{\prime
2}-(\mathcal{K}+\tilde{\mathcal{K}}_{2}^{\Sigma\neq
0})g_{1}^{2}\xi^{2}{Z}^{\prime}_{\mu\nu}{Z}^{\prime\mu\nu}-\tilde{\mathcal{K}}_{1}^{\Sigma\neq
0}g_{1}^{2}\xi^{2}(\partial^{\mu}Z_{\mu}^{\prime})^{2}$
$\displaystyle+(\tilde{\mathcal{K}}_{3}^{\Sigma\neq
0}+\tilde{\mathcal{K}}_{4}^{\Sigma\neq
0})g_{1}^{4}(\cot\theta+\tan\theta)^{2}\xi^{4}Z^{\prime 4}\bigg{]}+O(p^{6})$
$\displaystyle\mathrm{Tr^{\prime}ln}[i\not{\partial}+\\!\not{V}_{1\xi}\\!+\not{A}_{1\xi}\gamma^{5}\\!-\bar{\Sigma}(\hat{\nabla}^{2})\\!-i\gamma_{5}\tau^{2}\bar{\Sigma}_{5}(\hat{\nabla}^{2})]\bigg{|}_{\mathrm{normal~{}part}}$
(102) $\displaystyle=i\int
d^{4}x\bigg{\\{}\hat{F}_{0}^{2}A_{1\xi}^{2}-8F^{\prime
2}_{0}g_{1}^{2}u^{2}(\cot\theta+\tan\theta)^{2}Z^{\prime
2}-\frac{1}{2}\mathcal{K}\bigg{[}g_{2}^{2}W^{a}_{\mu\nu}W^{a\mu\nu}+g_{1}^{2}[1+4(u_{1}+u_{2})^{2}$
$\displaystyle+4(v_{1}+v_{2})^{2}]B_{\mu\nu}B^{\mu\nu}+g_{1}^{2}[4(u_{2}\tan\theta-
u_{1}\cot\theta)^{2}+4(v_{2}\tan\theta-v_{1}\cot\theta)^{2}+\tan^{2}\theta$
$\displaystyle+4\hat{D}_{0}u^{2}(\cot\theta+\tan\theta)^{2}]Z^{\prime}_{\mu\nu}{Z^{\prime}}^{\mu\nu}-2g_{1}^{2}[4(u_{1}+u_{2})(u_{2}\tan\theta-
u_{1}\cot\theta)$ $\displaystyle+4(v_{1}+v_{2})(v_{2}\tan\theta-
v_{1}\cot\theta)+\tan\theta]B_{\mu\nu}{Z^{\prime}}^{\mu\nu}\bigg{]}+\mathrm{tr}\bigg{[}-\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}(d_{\mu}A_{1\xi}^{\mu})^{2}+\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}(A_{1\xi}^{2})^{2}$ $\displaystyle-\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}(d_{\mu}A_{1\xi\nu}-d_{\nu}A_{1\xi\mu})^{2}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}(A_{1\xi\mu}A_{1\xi\nu})^{2}-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}V_{1\xi\mu\nu}V^{\mu\nu}_{1\xi}+i\hat{\mathcal{K}}_{14}^{\Sigma\neq
0}V_{1\xi\mu\nu}A_{1\xi}^{\mu}A_{1\xi}^{\nu}\bigg{]}$
$\displaystyle-8[\hat{D}_{1}a_{0}^{4}+\hat{D}_{2}a_{0}^{2}a_{3}^{2}]Z^{\prime
4}+\hat{D}_{3}a_{0}^{2}Z^{\prime
2}\mathrm{tr}(X^{\mu}X_{\mu})+2\hat{D}_{4}a_{0}^{2}Z^{\prime}_{\mu}Z^{\prime}_{\nu}\mathrm{tr}(X^{\mu}X^{\nu})$
$\displaystyle+4i\hat{D}_{2}a_{0}^{2}a_{3}Z^{\prime
2}Z^{\prime}_{\mu}\mathrm{tr}(X^{\mu}\tau^{3})\bigg{\\}}+O(p^{6})$
$\displaystyle\mathrm{Tr"ln}[i\not{\partial}+\not{V}_{2\xi}\\!+\not{A}_{2\xi}\gamma^{5}\\!-\hat{\Sigma}(\overline{\nabla}^{2})]\bigg{|}_{\mathrm{normal~{}part}}$
(103) $\displaystyle=i\int
d^{4}x\sum_{\eta=l,t,b}\mathrm{tr}_{f}\bigg{[}\hat{F}_{0}^{2}a^{\eta
2}-\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}(d_{\mu}a^{\eta\mu})^{2}-\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}(d_{\mu}a_{\nu}^{\eta}-d_{\nu}a_{\mu}^{\eta})^{2}+\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}(a^{\eta 2})^{2}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}(a_{\mu}^{\eta}a_{\nu}^{\eta})^{2}$
$\displaystyle-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}v_{\mu\nu}^{\eta}v^{\eta\mu\nu}+i\hat{\mathcal{K}}_{14}^{\Sigma\neq
0}a_{\mu}^{\eta}a_{\nu}^{\eta}v^{\eta\mu\nu}\bigg{]}+O(p^{6})\;,$

where

$\displaystyle
d_{\mu}A_{1\xi\nu}=\partial_{\mu}A_{1\xi\nu}-i[V_{1\xi\mu},A_{1\xi\nu}]\hskip
56.9055ptV_{1\xi\mu\nu}=\partial_{\mu}V_{1\xi\nu}-\partial_{\nu}V_{1\xi\mu}-i[V_{1\xi\mu},V_{1\xi\nu}]$
(104) $\displaystyle
d_{\mu}a^{\eta}_{\nu}=\partial_{\mu}a^{\eta}_{\nu}-i[v^{\eta}_{\mu},a^{\eta}_{\nu}]\hskip
108.12054ptv^{\eta}_{\mu\nu}=\partial_{\mu}v^{\eta}_{\nu}-\partial_{\nu}v^{\eta}_{\mu}-i[v^{\eta}_{\mu},v^{\eta}_{\nu}]$
(105) $\displaystyle F^{\prime 2}_{0}=\int
d^{4}\tilde{k}~{}2\tau\bar{\Sigma}^{2}_{5}$ (106)
$\displaystyle\hat{D}_{0}=\int
d^{4}\tilde{k}~{}[2\tau^{2}\bar{\Sigma}_{5}\bar{\Sigma}^{\prime}_{5}+\tau^{2}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}^{\prime\prime}_{5}-\frac{2}{3}\tau^{3}\bar{\Sigma}^{2}_{5}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}\bar{\Sigma}^{\prime}_{5}-\frac{4}{3}\tau^{3}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}-\frac{4}{3}\tau^{3}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}$
$\displaystyle\hskip
14.22636pt-\frac{4}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}\bar{\Sigma}_{5}^{\prime}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime\prime}\bar{\Sigma}_{5}^{2}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}-\frac{2}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime\prime}-\frac{10}{3}\tau^{3}k_{E}^{2}\bar{\Sigma}_{5}^{2}\bar{\Sigma}_{5}^{\prime
2}+\frac{1}{6}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}$ $\displaystyle\hskip
14.22636pt+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{2}+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}^{2}\bar{\Sigma}^{\prime
2}\bar{\Sigma}_{5}^{2}+\frac{4}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}\bar{\Sigma}^{\prime}\bar{\Sigma}_{5}^{3}\bar{\Sigma}_{5}^{\prime}+\frac{2}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{4}\bar{\Sigma}_{5}^{\prime
2}]$ (107) $\displaystyle\hat{D}_{1}=\int
d^{4}\tilde{k}~{}[2\tau^{3}\bar{\Sigma}_{5}^{2}-\frac{1}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}-\frac{4}{3}\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}-\frac{2}{3}\tau^{4}\bar{\Sigma}_{5}^{4}]$
(108) $\displaystyle\hat{D}_{2}=\int
d^{4}\tilde{k}~{}[2\tau^{3}\bar{\Sigma}_{5}^{2}+\frac{1}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}-4\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}]$
(109) $\displaystyle\hat{D}_{3}=\int
d^{4}\tilde{k}~{}[\frac{1}{3}\tau^{4}k_{E}^{2}\bar{\Sigma}_{5}^{2}-\frac{4}{3}\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}]$
(110) $\displaystyle\hat{D}_{4}=\int
d^{4}\tilde{k}~{}[\tau^{3}\bar{\Sigma}_{5}^{2}-\tau^{4}\bar{\Sigma}^{2}\bar{\Sigma}_{5}^{2}-\frac{1}{3}\tau^{4}\bar{\Sigma}_{5}^{4}]\;.$
(111)

$\hat{F}_{0}^{2}$ and $\hat{\mathcal{K}}_{i}^{\Sigma\neq 0}$ are functions of
the techniquark self-energy $\hat{\Sigma}(p_{E}^{2})$ which is determined by
(35). Detailed expressions for these quantities are given in (142) and (143)
of Appendix.E. Similarly, $\tilde{F}_{0}^{2}$ and
$\tilde{\mathcal{K}}_{i}^{\Sigma\neq 0}$ are functions of the techniquark
self-energy $\tilde{\Sigma}(p_{E}^{2})$ , which is determined by (34).
Detailed expressions for these quantities are given in (142) and (143) of
Appendix.E. In this case, the substitution,
$\hat{\Sigma}\rightarrow\tilde{\Sigma}$ is used.

With expansions (101),(102) and (103) and (66)-(70), and by ignoring loop
corrections, we can express (100) as (41) in the text. In this case, $S_{0}$
and $S_{Z^{\prime}}$ are $Z^{\prime}$ independent and dependent parts of the
actions:

$\displaystyle S_{0}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\bigg{\\{}-(\frac{5}{4}\mathcal{K}+\frac{1}{4}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+\frac{5}{8}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+\frac{3}{8}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})g_{2}^{2}W_{\mu\nu}^{a}W^{a,\mu\nu}-[(\frac{5}{4}+2\hat{u}+2\hat{x})\mathcal{K}+\frac{5}{8}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}$ (112)
$\displaystyle+(\frac{5}{8}+2\hat{u}+2\hat{x})\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}]g_{1}^{2}B_{\mu\nu}B^{\mu\nu}+(\frac{5}{8}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}+\frac{5}{32}\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\frac{5}{32}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-\frac{5}{8}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{16}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})(\mathrm{tr}[X_{\mu}X^{\mu}])^{2}$
$\displaystyle+(\frac{5}{16}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+\frac{5}{8}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{5}{16}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})\mathrm{tr}[X^{\mu}X_{\nu}]\mathrm{tr}[X_{\mu}X^{\nu}]+(\frac{5}{4}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}-\frac{5}{4}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})g_{1}\mathrm{tr}[\overline{W}^{\mu\nu}\tau^{3}]B_{\mu\nu}$
$\displaystyle+(-\frac{5}{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{8}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})i\mathrm{tr}[\overline{W}_{\mu\nu}X^{\mu}X^{\nu}]+(-\frac{5}{4}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{16}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ig_{1}B_{\mu\nu}\mathrm{tr}[\tau^{3}X^{\mu}X^{\nu}]$
$\displaystyle+\frac{1}{2}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}\mathrm{tr}[U^{{\dagger}}(D^{\mu}D_{\mu}U)U^{{\dagger}}(D^{\nu}D_{\nu}U)+2U^{{\dagger}}(D^{\mu}D_{\mu}U)(D^{\nu}U^{{\dagger}})(D_{\nu}U)]$
$\displaystyle+\frac{3}{4}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}\mathrm{tr}[U^{{\dagger}}(D^{\mu}D_{\mu}U)U^{{\dagger}}(D^{\nu}D_{\nu}U)+2U^{{\dagger}}(D^{\mu}D_{\mu}U)(D^{\nu}U^{{\dagger}})(D_{\nu}U)]\bigg{\\}}\;,$

where

$\displaystyle U(x)=\xi_{L}^{{\dagger}}(x)\xi_{R}(x)\hskip
71.13188ptX_{\mu}=U^{{\dagger}}(D_{\mu}U)\hskip
71.13188pt\overline{W}_{\mu\nu}=U^{{\dagger}}g_{2}\frac{\tau^{a}}{2}W^{a}_{\mu\nu}U~{}~{}~{}~{}~{}$
(113) $\displaystyle
D_{\mu}U=\partial_{\mu}U+ig_{2}\frac{\tau^{a}}{2}W_{\mu}^{a}U-ig_{1}U\frac{\tau^{3}}{2}B_{\mu}\hskip
28.45274ptD_{\mu}U^{{\dagger}}=\partial_{\mu}U^{{\dagger}}-ig_{2}U^{{\dagger}}\frac{\tau^{a}}{2}W_{\mu}^{a}+ig_{1}\frac{\tau^{3}}{2}B_{\mu}U^{{\dagger}}~{}~{}~{}~{}~{}~{}~{}$
(114)
$\displaystyle\hat{x}=(x_{1}+x_{2})^{2}+(y_{1}+y_{2})^{2}+(z_{1}+z_{2})^{2}\hskip
85.35826pt\hat{u}=(u_{1}+u_{2})^{2}+(v_{1}+v_{2})^{2}\;.$ (115)

While

$\displaystyle S_{Z^{\prime}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x~{}\bigg{[}\frac{1}{2}Z^{\prime}_{\mu}D_{Z}^{-1,\mu\nu}Z^{\prime}_{\nu}+Z^{\prime,\mu}J_{Z,\mu}+Z^{\prime
2}Z_{\mu}^{\prime}J^{\mu}_{3Z}+g_{4Z}Z^{\prime 4}\bigg{]}\;,~{}~{}~{}~{}$
(116)

with

$\displaystyle
D_{Z}^{-1,\mu\nu}=g^{\mu\nu}(c_{Z^{\prime}}^{2}\partial^{2}+\bar{M}^{2}_{Z^{\prime}})-(1+\lambda_{Z})\partial^{\mu}\partial^{\nu}+\Delta^{\mu\nu}_{Z}(X)$
(117) $\displaystyle
J_{Z}^{\mu}=J_{Z0}^{\mu}+g_{1}\gamma\partial^{\nu}B_{\mu\nu}+\tilde{J}_{Z}^{\mu}$
(118) $\displaystyle
g_{4Z}=[10a_{3}^{4}+12a_{3}^{2}(2a_{0}^{2}+\hat{a}_{0}^{2})+4a_{0}^{4}+2\hat{a}_{0}^{4}](\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq 0})$ $\displaystyle\hskip
17.07182pt+g_{1}^{4}(\tan\theta+\cot\theta)^{4}\xi^{4}(\tilde{\mathcal{K}}_{3}^{\Sigma\neq
0}\\!+\\!\tilde{\mathcal{K}}_{4}^{\Sigma\neq
0})-8\hat{D}_{1}a_{0}^{4}-8\hat{D}_{2}a_{0}^{2}a_{3}^{2}~{}~{}~{}$ (119)
$\displaystyle
J_{3Z}^{\mu}=-i[(10a_{3}^{3}+12a_{0}^{2}a_{3}^{2}+6\hat{a}_{0}^{2}a_{3})(\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{4}^{\Sigma\neq
0})+4a_{0}^{2}a_{3}\hat{D}_{2}]\mathrm{tr}[X^{\mu}\tau^{3}]\;,$ (120)

where

$\displaystyle\bar{M}^{2}_{Z^{\prime}}=2\tilde{F}_{0}^{2}g_{1}^{2}(\cot\theta+\tan\theta)^{2}\xi^{2}+4\hat{F}_{0}^{2}(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})-8F^{\prime
2}_{0}a_{0}^{2}$ (121) $\displaystyle
c^{2}_{Z^{\prime}}=1+[4(\cot\theta+\tan\theta)^{2}\xi^{2}+2\tan^{2}\theta+8\hat{v}+3\tan^{2}\theta+\hat{y}]\mathcal{K}g_{1}^{2}+4(\cot\theta+\tan\theta)^{2}\xi^{2}\tilde{\mathcal{K}}_{2}^{\Sigma\neq
0}g_{1}^{2}$ $\displaystyle\hskip
17.07182pt+8(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+[40a_{3}^{2}+2(\hat{t}+\hat{s})g_{1}^{2}]\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-16\hat{D}_{0}a_{0}^{2}$ (122)
$\displaystyle\lambda_{Z}=-2g_{1}^{2}(\tan\theta+\cot\theta)^{2}\tilde{\mathcal{K}}_{1}^{\Sigma\neq
0}-4(2a_{0}^{2}+\hat{a}_{0}^{2}+5a_{3}^{2})\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}$ (123)
$\displaystyle\Delta^{\mu\nu}_{Z}(X)=[40a_{3}^{2}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}-(4a_{0}^{2}+2\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-(4a_{0}^{2}+2\hat{a}_{0}^{2}+10a_{3}^{2})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-20a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+10a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}$ $\displaystyle\hskip
42.67912pt+2a_{0}^{2}\hat{D}_{4}]\mathrm{tr}[X^{\mu}X^{\nu}]-(20\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}+5\hat{\mathcal{K}}_{3}^{\Sigma\neq 0}-10\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})a_{3}^{2}\mathrm{tr}[X^{\mu}\tau^{3}]\mathrm{tr}[X^{\nu}\tau^{3}]$
$\displaystyle\hskip
42.67912pt+g^{\mu\nu}[(5a_{3}^{2}+2a_{0}^{2}+\hat{a}_{0}^{2})\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}+(2a_{0}^{2}+2\hat{a}_{0}^{2}-5a_{3}^{2})\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-20a_{3}^{2}\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+10a_{3}^{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq 0}$ $\displaystyle\hskip
42.67912pt+a_{0}^{2}\hat{D}_{3}]\mathrm{tr}[X^{\lambda}X_{\lambda}]-g^{\mu\nu}(5\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+10\hat{\mathcal{K}}_{13}^{\Sigma\neq 0}-5\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})a_{3}^{2}tr[X_{\lambda}\tau^{3}]\mathrm{tr}[X^{\lambda}\tau^{3}]$ (124)
$\displaystyle
J_{Z0}^{\mu}=-5ia_{3}\hat{F}_{0}^{2}\mathrm{tr}[X^{\mu}\tau^{3}]$ (125)
$\displaystyle\gamma=2[5a_{3}\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+(5a_{3}+4g_{1}\hat{w}+2g_{1}\hat{z})\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+(4\hat{w}+\frac{5}{2}\tan\theta+2\hat{z})g_{1}\mathcal{K}]$ (126)
$\displaystyle\tilde{J}_{Z}^{\mu}=10(-\hat{\mathcal{K}}_{2}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0})a_{3}\partial_{\nu}\mathrm{tr}[\overline{W}^{\mu\nu}\tau^{3}]+10(\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{1}{4}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ia_{3}\partial_{\nu}\mathrm{tr}[X^{\mu}X^{\nu}\tau^{3}]$
$\displaystyle\hskip 14.22636pt+5(\frac{1}{4}\hat{\mathcal{K}}_{3}^{\Sigma\neq
0}-\frac{1}{4}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}-\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{1}{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ia_{3}\mathrm{tr}[X^{\nu}X_{\nu}]tr[X^{\mu}\tau^{3}]$ $\displaystyle\hskip
14.22636pt+5(\frac{1}{2}\hat{\mathcal{K}}_{4}^{\Sigma\neq
0}+\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}-\frac{1}{2}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})ia_{3}\mathrm{tr}[X^{\mu}X_{\nu}]\mathrm{tr}[X^{\nu}\tau^{3}]$
$\displaystyle\hskip 14.22636pt+(-5\hat{\mathcal{K}}_{13}^{\Sigma\neq
0}+\frac{5}{4}\hat{\mathcal{K}}_{14}^{\Sigma\neq
0})a_{3}\mathrm{tr}[\overline{W}^{\mu\nu}(X_{\nu}\tau^{3}-\tau^{3}X_{\nu})]$
$\displaystyle\hskip 14.22636pt+5ia_{3}\hat{\mathcal{K}}_{1}^{\Sigma\neq
0}\mathrm{tr}\bigg{[}U^{{\dagger}}(D^{\nu}D_{\nu}U)U^{{\dagger}}D^{\mu}U\tau^{3}-U^{{\dagger}}(D^{\nu}D_{\nu}U)\tau^{3}U^{{\dagger}}D^{\mu}U-\partial^{\mu}[U^{{\dagger}}(D^{\nu}D_{\nu}U)\tau^{3}]\bigg{]}$
$\displaystyle\hskip 14.22636pt+i\hat{a}_{0}{\mathcal{K}}_{1}^{\Sigma\neq
0}\partial^{\mu}\mathrm{tr}[X^{\nu}X_{\nu}-U^{\dagger}(D^{\nu}D_{\nu}U)]$
(127)

in which

$\displaystyle
a_{0}=\frac{1}{2}g_{1}(u_{1}-v_{1})(\cot\theta-\tan\theta)\hskip
85.35826pta_{3}=\frac{1}{4}g_{1}\tan\theta$ (128)
$\displaystyle\hat{a}^{2}_{0}=\frac{1}{4}g^{2}_{1}(\tan\theta+\cot\theta)^{2}[(x_{1}-x_{1}^{\prime})^{2}+(y_{1}-y_{1}^{\prime})^{2}+(z_{1}-z_{1}^{\prime})^{2}]$
(129)
$\displaystyle\hat{a}^{4}_{0}=\frac{1}{16}g_{1}^{4}(\tan\theta+\cot\theta)^{4}[(x_{1}-x_{1}^{\prime})^{4}+(y_{1}-y_{1}^{\prime})^{4}+(z_{1}-z_{1}^{\prime})^{4}]$
(130) $\displaystyle\hat{v}=(u_{2}\tan\theta-
u_{1}\cot\theta)^{2}+(v_{2}\tan\theta-v_{1}\cot\theta)^{2}$
$\displaystyle\hat{w}=(u_{1}+u_{2})(u_{2}\tan\theta-
u_{1}\cot\theta)+(v_{1}+v_{2})(v_{2}\tan\theta-v_{1}\cot\theta)$ (131)