#  $Z^{\prime}$ Boson Mixings with $Z\\!-\\!\gamma$ and Charge Assignments

Ying Zhang1, Qing Wang2,3111Corresponding author at: Department of Physics,
Tsinghua University, Beijing 100084, P.R.China
Email address<EMAIL_ADDRESS>(Q.Wang). 1School of Science, Xi’an
Jiaotong University, Xi’an, 710049, P.R.China
2Center for High Energy Physics, Tsinghua University, Beijing 100084,
P.R.china
3Department of Physics,Tsinghua University,Beijing 100084,P.R.China

(May 17, 2009)

###### Abstract

Based on the general description for $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixing
as derived from the electroweak chiral Lagrangian, we characterize and
classify the various new physics models involving the $Z^{\prime}$ boson that
have appeared in the literature into five classes: 1. Models with minimal
$Z^{\prime}\\!-\\!Z$ mass mixing; 2. Models with minimal $Z^{\prime}\\!-\\!Z$
kinetic mixing; 3\. Models with general $Z^{\prime}\\!-\\!Z$ mixing; 4. Models
with $Z^{\prime}\\!-\\!\gamma$ kinetic and $Z^{\prime}\\!-\\!Z$ mixing; and 5.
Models with Stueckelberg-type mixing. The corresponding mixing matrices are
explicitly evaluated for each of these classes. We constrain and classify the
$Z^{\prime}$ boson charges with respect to quark-leptons by anomaly
cancellation conditions.

PACS numbers: 12.60.Cn; 14.70.Pw; 11.30.Ly; 12.39.Fe

##  I Introduction

With the running of the LHC at CERN Geneva, a TeV energy era begins and
researchers are anxiously expecting a possible new revolution in particle
physics. There are various predictions from both the Standard Model (SM) and
new physics models. Among these the appearance of possible new underlying
interactions beyond conventional strong/weak/electromagnetic gauge
interactions is of special interest. From knowledge accumulated in resent
years in particle physics, we know that the expected new interactions at least
must govern the electroweak symmetry breaking that result in the massive
$W^{\pm}$ and $Z^{0}$ bosons and may further be responsible for the origin of
masses for ordinary quarks and leptons. Theoreticians have also touted various
ambitious alternative sources of these new interactions, such as unifications,
supersymmetries, and extra dimensions. With the exception of the well-known
scalar-type interactions which suffer unnaturalness, triviality and hierarchy
problems, the typical new interaction that avoids the shortcomings of
elementary scalar fields is a gauge interaction and minimal such kind of
interaction involves an additional so-called $U(1)^{\prime}$ gauge
interaction. In most instances this extra $U(1)^{\prime}$ gauge force is a
”relic” of some larger underlying new physics gauge interactions such as those
occurring in GUT models, string theories, left-right symmetric models and
models deconstructed from extra space dimensions. Alternatively, in some
special models, the $U(1)^{\prime}$ gauge force takes on a special role: for
example 1) in little Higgs type models, it can partially remove the quadratic
divergence from the SM Higgs mass at the one loop levelRizzoARXIV2006 ; 2) in
topcolor-assisted technicolor (TC2) models, it ensures top quark condensation
while not for the bottom quark Hill ; Lane ; Chiv ; 3) in SUSY models, it can
mediate SUSY breakingZ'SUSY ; and 4) in models based on string theory, it
mediates particles communicating between the hidden and visible sectors
CasselARXIV2009 . This represents but a sampling of new physics models
involving additional $U(1)^{\prime}$ factors: a recent review of others can be
found in Ref.Langacker .

Phenomenologically, we are interested in the possibility of experimentally
finding the carrier, an electrically-neutral color singlet spin-one boson
$Z^{\prime}$, of this additional gauge force especially at the LHC. As a
detection has not been made so far, this boson has to be massive and the
corresponding $U(1)^{\prime}$ gauge symmetry must be violated. The more
preferred and exciting experimental finding would be that the $Z^{\prime}$
mass is relatively light compared with the other new physics particles, for
then it might arise as a first signature of the new physics beyond SM at the
LHC. This prospect heightens the need for theoretical studies of such a light
$Z^{\prime}$ boson and its interactions with known particles would also be of
the special importance in new physics research.

Physically, one main effect of the $Z^{\prime}$ boson derives from its mixings
with conventional $Z$ boson and $\gamma$ photon; another stems from its gauge
couplings to ordinary quarks and leptons, which leads to various charge
assignments. There exist a diversity of new physics models involving the
$Z^{\prime}$ boson, each model has its own arrangement of
$Z^{\prime}-Z-\gamma$ mixings and $Z^{\prime}$ coupling to ordinary quarks and
leptons. To compare models, a model independent investigation is needed of
these Z’ boson interactions with known particles, particularly in classifying
and comparing the role of the Z’ boson within each model. The electroweak
chiral Lagrangian (EWCL) method provides such a platform to perform model
independent research. In our previous paper Z'our , we have written down the
bosonic part up to order $p^{4}$ of the most genral EWCL involving the
$Z^{\prime}$ boson222In the Lagrangian, terms involving a neutral Higgs boson
that only plays a passive role are also included to help in matching unitarity
requirements within the theory. and known particles. This EWCL alos describes
the most general $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings, and with it we can
further classify the various $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings that
appear in each model enabling us to compare and discriminate between the
different new physics models333It should be emphasized that a $p^{4}$ order
EWCL provides some special degrees of freedom for the
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings. For example, all kinetic mixings
are from $p^{4}$ order terms in EWCL (see Eq.(II)), as a $p^{2}$ order EWCL
only cannot offer the most general $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings.
. Here the classification categorizes the general
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings into several simplifying cases that
appear in the new physics models in the literature. The reason in doing this
is because the general $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings is too
complex to be discussed analytically, while we will show that for all
simplifying cases presented in this paper, mixings can be diagonalized
exactly. This improves on the approximate diagonalization result usually used
in the literature and we can exhibit explicitly the relationship between the
various simplifying cases. The main purpose of this paper is to present these
finding s and moreover to generalize the EWCL given in Ref.Z'our  to include
the $Z^{\prime}$ boson couplings to ordinary quarks and leptons for the most
general charge assignments. In terms of these charges, new physics models
involving the $Z^{\prime}$ boson can also be classified. Because most of the
experimental searches for the $Z^{\prime}$ boson depend heavily on these
charge assignments and on how $Z^{\prime}$ mixes with $Z$ and $\gamma$, we
combine a discussions on these two issues in present paper.

This paper is organized as follows. In Sec.II, we first give a short review of
the bosonic part of the EWCL involving the $Z^{\prime}$ boson and general
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings. In Sec.III, we classify the various
models involving the $Z^{\prime}$ boson that have appear in the literatures
according to their arrangements of the $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$
mixings. In Sec.IV, we set up the general $Z^{\prime}$ boson charge
assignments to the ordinary quarks and leptons in terms of the anomaly
cancellation conditions. Sec.V provides a summary of the paper.

##  II The Bosonic part of the EWCL involving the $Z^{\prime}$ boson and
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings

As given in Ref.Z'our , the covariant derivative in the EWCL including the
$Z^{\prime}$ boson is

$\displaystyle
D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+igW_{\mu}\hat{U}-i\hat{U}\frac{\tau_{3}}{2}g^{\prime}B_{\mu}-i\hat{U}(\tilde{g}^{\prime}B_{\mu}+g^{\prime\prime}X_{\mu})I\;,$
(1)

where the two by two unitary field $\hat{U}$ represents four Goldstone boson
degrees of freedom resulting from spontaneous symmetry breaking of
$SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)^{\prime}\rightarrow U(1)_{em}$, and
$\tilde{g}^{\prime}$ is a Stueckelberg-type coupling constant associated with
which is a special kind of $U(1)$. To help in understanding this choice of
covariant derivative, we denote $SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)^{\prime}$ group elements as
$(e^{i\theta^{a}t^{a}_{L}+i\theta^{\prime}t^{\prime}},e^{i\theta t})$ for
which the Hermitian matrices $t^{a}_{L}$ ($\theta^{a}$) with $a=1,2,3$, $t$
($\theta$) and $t^{\prime}$ ($\theta^{\prime}$) are generators (group
parameters) of $SU(2)_{L}$, $U(1)_{Y}$ and an extra $U(1)^{\prime}$
respectively. The electromagnetic $U(1)_{\mathrm{em}}$ group generator has now
been generalized from its traditional expression to $t_{\mathrm{em}}\equiv
t_{L}^{3}+t+ct^{\prime}$ depending on an additional arbitrary parameter $c$.
This generator results in the $U(1)_{\mathrm{em}}$ group element
$(e^{i\theta_{\mathrm{em}}(t^{3}_{L}+ct^{\prime})},e^{i\theta_{\mathrm{em}}t})$
and we can label the representative element for the corresponding coset by
$(\hat{U},1)$. Group theory tells us that each symmetry breaking generator
corresponds to a coset which can be represented by introducing a
representative element for each coset. Denoting the representative element by
$n$, its transformation rule to $n^{\prime}$ under the action of an arbitrary
group element $g$ is then $gn=n^{\prime}h$ where $h$ is an element belonging
to the un-broken subgroup. Specifically for the above gauge group, this
transformation rule then stipulates that

$\displaystyle(e^{i\theta^{a}t_{L}^{a}+i\theta^{\prime}t^{\prime}},~{}e^{i\theta
t})(\hat{U},1)\stackrel{{\scriptstyle
gn=n^{\prime}h}}{{=====}}(\underbrace{e^{i\theta^{a}t_{L}^{a}+i\theta^{\prime}t^{\prime}}\hat{U}e^{-i\theta(t_{L}^{3}+ct^{\prime})}}_{\hat{U}^{\prime}},~{}1)\underbrace{(e^{i\theta(t_{L}^{3}+ct^{\prime})},~{}e^{i\theta
t})}_{U(1)_{\mathrm{em}}}$ (2)

which yields the following transformation rule for the Goldstone field
$\hat{U}$ under $SU(2)_{L}\otimes U(1)\otimes U(1)^{\prime}$

$\displaystyle\hat{U}^{\prime}=e^{i\theta^{a}t_{L}^{a}+i\theta^{\prime}t^{\prime}}~{}\hat{U}~{}e^{-i\theta(t_{L}^{3}+ct^{\prime})}\;.$
(3)

The choice of the Goldstone field in the two dimensional internal space
corresponds in taking the generator $t_{L}^{a}=\tau^{a}/2$, $t=t^{\prime}=1$
(Note, according to our arrangement of group elements, $t$ and $t^{\prime}$
act on different spaces, so $t=t^{\prime}=1$ will not cause confusion). With
(3) and the standard $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)^{\prime}$
transformation rule for electroweak gauge fields $W_{\mu},B_{\mu}$ and the
extra $U(1)^{\prime}$ gauge field $X_{\mu}$, we derive the action of the
covariant derivative on the Goldstone field $\hat{U}$ as:
$D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+i(gW_{\mu}+g_{X}X_{\mu})\hat{U}-i\hat{U}(\frac{\tau^{3}}{2}g^{\prime}+cg^{\prime})B_{\mu}$.
Further identifying $g_{X}\equiv-g"$ and
$cg^{\prime}\equiv\tilde{g}^{\prime}$, we obtain the result given in Eq.(1).
With symmetry breaking pattern $SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)^{\prime}\rightarrow U(1)_{em}$, the Higgs mechanism ensures that the
Goldstone bosons represented by the $\hat{U}$ field will be eaten out by the
electroweak gauge bosons $W^{\pm},Z^{0}$ and $Z^{\prime}$ which then acquire
mass. Here $W_{\mu}$, $B_{\mu}$ and $X_{\mu}$ are respectively the gauge
fields of $SU(2)_{L}$, $U(1)_{Y}$ and $U(1)^{\prime}$ before mixing.

The full bosonic part of the Lagrangian up to order $p^{4}$ is

$\displaystyle\mathcal{L}_{Stueck-SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)^{\prime}\rightarrow
U(1)_{em}}=\mathcal{L}_{0}+\mathcal{L}_{2}+\mathcal{L}_{4}\;,$ (4)

with each term in the Lagrangian defined as

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$ $\displaystyle-V(h)\;,$ (5)
$\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}h)^{2}-\frac{1}{4}f^{2}\mathrm{tr}[\hat{V}_{\mu}\hat{V}^{\mu}]+\frac{1}{4}\beta_{1}f^{2}\mathrm{tr}[T\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]+\frac{1}{4}\beta_{2}f^{2}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]$
(6)
$\displaystyle+\frac{1}{4}\beta_{3}f^{2}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}^{\mu}]+\beta_{4}f(\partial^{\mu}h)\mathrm{tr}[\hat{V}_{\mu}]\;,$
$\displaystyle\mathcal{L}_{4}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{K}+\mathcal{L}_{B}+\mathcal{L}_{H}+\mathcal{L}_{A}\;,$
(7)

where $T\equiv\hat{U}\tau_{3}\hat{U}^{\dagger}$ and
$\hat{V}_{\mu}\equiv(\hat{D}_{\mu}\hat{U})\hat{U}^{\dagger}$. Here the Higgs
field $h$ is treated as $p^{0}$ order and

$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{2}\mathrm{tr}[W_{\mu\nu}W^{\mu\nu}]-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}\;$
$\displaystyle\mathcal{L}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha_{1}gg^{\prime}B_{\mu\nu}\mathrm{tr}[TW^{\mu\nu}]+\frac{i}{2}\alpha_{2}g^{\prime}B_{\mu\nu}\mathrm{tr}[T[\hat{V}^{\mu},\hat{V}^{\nu}]]+i\alpha_{3}g\mathrm{tr}[W^{\mu\nu}[\hat{V}^{\mu},\hat{V}^{\nu}]]+\ldots$
$\displaystyle\mathcal{L}_{H}$ $\displaystyle=$
$\displaystyle(\partial_{\mu}h)\Big{\\{}\alpha_{H,1}\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[\hat{V}_{\nu}\hat{V}^{\nu}]+\alpha_{H,2}\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[\hat{V}^{\mu}\hat{V}^{\nu}]+\alpha_{H,3}\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[T[\hat{V}^{\mu},\hat{V}^{\nu}]]+\ldots\Big{\\}}\;.$

All coefficients in above Lagrangian are functions of Higgs field $h$.
Detailed expressions can be found in Ref.Z'our .

Mixings among $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ come from the gauge boson mass
term $\mathcal{L}_{M}$ and kinetic term $\mathcal{L}_{K}$. In the unitary
gauge $\hat{U}=1$, they become

$\displaystyle\mathcal{L}_{M,Z^{\prime}-Z-\gamma}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{8}(1\\!-\\!2\beta_{1})(gW^{3}_{\mu}\\!-g^{\prime}B_{\mu})^{2}+\frac{f^{2}}{2}(1\\!-\\!2\beta_{3})(g^{\prime\prime}X_{\mu}\\!+\tilde{g}^{\prime}B_{\mu})^{2}$
(8)
$\displaystyle+\frac{f^{2}}{2}\beta_{2}(g^{\prime\prime}X_{\mu}+\tilde{g}^{\prime}B_{\mu})(gW^{3,\mu}-g^{\prime}B^{\mu})\;,$
$\displaystyle\mathcal{L}_{K,Z^{\prime}-Z-\gamma}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B_{\mu\nu}-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{1}{4}(1\\!-\\!\alpha_{8}g^{2})(\partial_{\mu}W^{3}_{\nu}\\!-\partial_{\nu}W^{3}_{\mu})^{2}$
$\displaystyle+\frac{1}{2}\alpha_{1}gg^{\prime}B_{\mu\nu}(\partial_{\mu}W^{3}_{\nu}\\!-\partial_{\nu}W^{3}_{\mu})+gg^{\prime\prime}\alpha_{24}X^{\mu\nu}(\partial_{\mu}W^{3}_{\nu}-\partial_{\nu}W^{3}_{\mu})+g^{\prime}g^{\prime\prime}\alpha_{25}B_{\mu\nu}X^{\mu\nu}\;.$

Apart from the four gauge couplings
$g,g^{\prime},g^{\prime\prime},\tilde{g}^{\prime}$, seven extra dimensionless
parameters $\beta_{1},\beta_{2},\beta_{3}$ and
$\alpha_{1},\alpha_{8},\alpha_{24},\alpha_{25}$ determine the mixing terms. Of
these eleven, $\alpha_{8}$ can be absorbed into the redefinition of field
$W^{3}_{\mu}$ and coupling constant $g$ by

$\displaystyle
W^{3}_{\mu}\rightarrow\frac{W^{3}_{\mu}}{\sqrt{1-\alpha_{8}g^{2}}}\hskip
56.9055ptg\rightarrow g\sqrt{1-\alpha_{8}g^{2}}\;.$ (10)

Hence we are left with ten parameters, and on eliminating the three gauge
couplings $g,g^{\prime},g^{\prime\prime}$, leaves us seven independent
parameters
$\tilde{g}^{\prime},\beta_{1},\beta_{2},\beta_{3},\alpha_{1},\alpha_{24},\alpha_{25}$
that are related to mixings. However, the mixing masses and kinetic terms
given by (8) and (II) are so complex that to diagonalize them we must exploit
the general $3\times 3$ rotation matrix $U_{ij}$

$\displaystyle(W^{3}_{\mu},~{}B_{\mu},~{}X_{\mu})^{T}=U(Z_{\mu},~{}A_{\mu},~{}Z^{\prime}_{\mu})^{T}\;,$
(11)

which has nine matrix elements. The fact that no correction terms arise for
the kinetic terms $-\frac{1}{4}B_{\mu\nu}B_{\mu\nu}$ and
$-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}$ leads to two constraints on the matrix
elements of $U$,

$\displaystyle(U^{-1,T}U^{-1})_{22}=(U^{-1,T}U^{-1})_{33}=1\;,$ (12)

which imply that there are only seven independent matrix elements. This is
consistent with the earlier result that there are at most seven parameters
$\tilde{g}^{\prime},\beta_{1},\beta_{2},\beta_{3},\alpha_{1},\alpha_{24},\alpha_{25}$
related to mixings. In Ref.Z'our , we had obtained a set of relations between
matrix elements $U_{ij}$ and parameters
$g,g^{\prime},g^{\prime\prime},\tilde{g}^{\prime}$,
$\beta_{1},\beta_{2},\beta_{3}$,
$\alpha_{1},\alpha_{8},\alpha_{24},\alpha_{25}$ as follows

$\displaystyle
U\equiv\left(\begin{array}[]{ccc}\frac{1}{2g}c_{\alpha}&\frac{1}{2g}&-\frac{1}{2g}s_{\alpha}\\\
-\frac{1}{2g^{\prime}}c_{\alpha}&\frac{1}{2g^{\prime}}&\frac{1}{2g^{\prime}}s_{\alpha}\\\
\frac{1}{g^{\prime\prime}}(s_{\alpha}+\frac{\tilde{g}^{\prime}}{2g^{\prime}}c_{\alpha})&-\frac{\tilde{g}^{\prime}}{2g^{\prime\prime}g^{\prime}}&\frac{1}{g^{\prime\prime}}(c_{\alpha}-\frac{\tilde{g}^{\prime}}{2g^{\prime}}s_{\alpha})\end{array}\right)\left(\begin{array}[]{ccc}\frac{c_{\beta}}{A_{1}}&0&\frac{s_{\beta}}{A_{1}}\\\
ga&gb&gc\\\
-\frac{s_{\beta}}{A_{2}}&0&\frac{c_{\beta}}{A_{2}}\end{array}\right)\left(\begin{array}[]{ccc}\frac{M_{Z}}{f}&0&0\\\
0&1&0\\\ 0&0&\frac{M_{Z^{\prime}}}{f}\end{array}\right)\;,~{}~{}~{}~{}~{}$
(22)

where $c_{\alpha}\equiv\cos\alpha_{Z^{\prime}}$,
$s_{\alpha}\equiv\sin\alpha_{Z^{\prime}}$, $s_{\beta}=\sin\beta_{Z^{\prime}}$,
$c_{\beta}=\cos\beta_{Z^{\prime}}$ as well as the following definitions

$\displaystyle
A_{1}^{2}=\frac{1}{4}(1\\!-\\!2\beta_{1})c_{\alpha}^{2}+\beta_{2}s_{\alpha}c_{\alpha}+(1\\!-\\!2\beta_{3})s_{\alpha}^{2}\hskip
28.45274ptA_{2}^{2}=\frac{1}{4}(1\\!-\\!2\beta_{1})s_{\alpha}^{2}-\beta_{2}s_{\alpha}c_{\alpha}+(1-2\beta_{3})c_{\alpha}^{2}\;,~{}~{}~{}$
(23)
$\displaystyle\tan\alpha_{Z^{\prime}}=\frac{3+2\beta_{1}-8\beta_{3}-\sqrt{(3+2\beta_{1}-8\beta_{3})^{2}+16\beta_{2}^{2}}}{4\beta_{2}}\hskip
28.45274pt\tan\beta_{Z^{\prime}}=\frac{-G_{2}+\sqrt{G_{2}^{2}+4G_{0}^{2}}}{2G_{0}}~{}~{}~{}~{}~{}~{}$
(24) $\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{gA_{1}A_{2}[{g^{\prime}}^{2}{g^{\prime\prime}}^{2}-{g}^{2}{g^{\prime}}^{2}{g^{\prime\prime}}^{2}(2\alpha_{1}+\alpha_{8})+g^{2}{g^{\prime\prime}}^{2}-4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}(\alpha_{24}+\alpha_{25})+g^{2}\tilde{g}^{\prime
2}]}$
$\displaystyle\times\Big{\\{}[g^{2}{g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}-g^{\prime 2}{g^{\prime\prime}}^{2}+g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}\alpha_{8}+4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}\alpha_{25}](s_{\alpha}s_{\beta}A_{1}+c_{\alpha}c_{\beta}A_{2})$
$\displaystyle+[2g^{2}g^{\prime}\tilde{g}^{\prime}+4g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}(\alpha_{24}+\alpha_{25})](-c_{\alpha}s_{\beta}A_{1}+s_{\alpha}c_{\beta}A_{2})\Big{\\}}\;.$
$\displaystyle b^{2}$ $\displaystyle=$
$\displaystyle\frac{4{g^{\prime}}^{2}{g^{\prime\prime}}^{2}}{(g^{2}+{g^{\prime}}^{2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}-{g}^{2}{g^{\prime}}^{2}{g^{\prime\prime}}^{2}(2\alpha_{1}+\alpha_{8})+4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}(\alpha_{24}+\alpha_{25})}\;.$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{1}{gA_{1}A_{2}[{g^{\prime}}^{2}{g^{\prime\prime}}^{2}-{g}^{2}{g^{\prime}}^{2}{g^{\prime\prime}}^{2}(2\alpha_{1}+\alpha_{8})+g^{2}{g^{\prime\prime}}^{2}-4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}(\alpha_{24}+\alpha_{25})+g^{2}\tilde{g}^{\prime
2}]}$
$\displaystyle\times\Big{\\{}[g^{2}{g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}-g^{\prime 2}{g^{\prime\prime}}^{2}+g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}\alpha_{8}+4g^{2}g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}\alpha_{25}](-s_{\alpha}c_{\beta}A_{1}+c_{\alpha}s_{\beta}A_{2})$
$\displaystyle+[2g^{2}g^{\prime}\tilde{g}^{\prime}+4g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}(\alpha_{24}+\alpha_{25})](c_{\alpha}c_{\beta}A_{1}+s_{\alpha}s_{\beta}A_{2})\Big{\\}}\;.~{}~{}~{}~{}$
$\displaystyle G_{0}$ $\displaystyle=$ $\displaystyle-
A_{1}A_{2}\Big{\\{}(-g^{2}-g^{\prime
2}+{g^{\prime\prime}}^{2}+(\tilde{g}^{\prime})^{2})c_{\alpha}s_{\alpha}+g^{\prime}\tilde{g}^{\prime}(s_{\alpha}^{2}-c_{\alpha}^{2})+g^{2}[2g^{\prime
2}c_{\alpha}s_{\alpha}+g^{\prime}\tilde{g}^{\prime}(c_{\alpha}^{2}-s_{\alpha}^{2})]\alpha_{1}$
$\displaystyle+g^{2}[(g^{\prime
2}-{g^{\prime\prime}}^{2}-(\tilde{g}^{\prime})^{2})c_{\alpha}s_{\alpha}-g^{\prime}\tilde{g}^{\prime}(s_{\alpha}^{2}-c_{\alpha}^{2})]\alpha_{8}+2g^{2}{g^{\prime\prime}}^{2}(c_{\alpha}^{2}-s_{\alpha}^{2})(\alpha_{24}+g^{\prime
2}\alpha_{1}\alpha_{25})$
$\displaystyle+{g^{\prime\prime}}^{2}[-4g^{\prime}\tilde{g}^{\prime}c_{\alpha}s_{\alpha}+2g^{\prime
2}(c_{\alpha}^{2}-s_{\alpha}^{2})][g^{2}(\alpha_{8}\alpha_{25}-\alpha_{1}\alpha_{24})-\alpha_{25}]+g^{2}{g^{\prime\prime}}^{2}[8g^{\prime
2}s_{\alpha}c_{\alpha}$
$\displaystyle+4g^{\prime}\tilde{g}^{\prime}(c_{\alpha}^{2}-s_{\alpha}^{2})]\alpha_{24}\alpha_{25}+g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}s_{\alpha}c_{\alpha}(4\alpha_{25}^{2}-\alpha_{1}^{2})+4g^{2}{g^{\prime\prime}}^{2}(g^{\prime}s_{\alpha}+\tilde{g}^{\prime}c_{\alpha})(g^{\prime}c_{\alpha}-\tilde{g}^{\prime}s_{\alpha})\alpha_{24}^{2}\Big{\\}}$
$\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle
A_{1}^{2}\Big{\\{}(g^{2}+g^{\prime
2})c_{\alpha}^{2}+({g^{\prime\prime}}^{2}+(\tilde{g}^{\prime})^{2})s_{\alpha}^{2}(1-g^{2}\alpha_{8})-g^{2}g^{\prime
2}c_{\alpha}^{2}(2\alpha_{1}+\alpha_{8})+4g^{\prime}{g^{\prime\prime}}^{2}\tilde{g}^{\prime}s_{\alpha}^{2}\alpha_{25}$
(25) $\displaystyle-4g^{2}g^{\prime
2}{g^{\prime\prime}}^{2}c_{\alpha}^{2}(\alpha_{24}^{2}+\alpha_{25}^{2}+2\alpha_{24}\alpha_{25})-g^{2}{g^{\prime\prime}}^{2}s_{\alpha}^{2}[g^{\prime
2}\alpha_{1}^{2}+4(\tilde{g}^{\prime})^{2}\alpha_{24}^{2}+4g^{\prime}\tilde{g}^{\prime}(\alpha_{8}\alpha_{25}-\alpha_{1}\alpha_{24})]\Big{\\}}$
$\displaystyle-[A_{1}\rightarrow A_{2},c_{\alpha}\leftrightarrow
s_{\alpha}]+s_{\alpha}c_{\alpha}(A_{1}^{2}+A_{2}^{2})\Big{\\{}-2g^{\prime}\tilde{g}^{\prime}[1-g^{2}(\alpha_{1}+\alpha_{8})]$
$\displaystyle+4g^{2}{g^{\prime\prime}}^{2}[(\alpha_{24}-\alpha_{25})(1-{g^{\prime\prime}}^{2}\alpha_{1})+2g^{\prime}\tilde{g}^{\prime}\alpha_{24}^{2}+{g^{\prime\prime}}^{2}\alpha_{8}\alpha_{25}]\Big{\\}}\;.$

Finally the masses of $Z$ and $Z^{\prime}$ bosons are determined from

$\displaystyle\mathbf{K}_{11}=-\frac{1}{4}\hskip
28.45274pt\mathbf{K}_{33}=-\frac{1}{4}\;,$ (26)

with

$\displaystyle\mathbf{K}\equiv
U^{T}\left(\begin{array}[]{ccc}-\frac{1}{4}(1-\alpha_{8}g^{2})&\frac{1}{4}\alpha_{1}gg^{\prime}&\frac{1}{2}gg^{\prime\prime}\alpha_{24}\\\
\frac{1}{4}\alpha_{1}gg^{\prime}&-\frac{1}{4}&\frac{1}{2}g^{\prime}g^{\prime\prime}\alpha_{25}\\\
\frac{1}{2}gg^{\prime\prime}\alpha_{24}&\frac{1}{2}g^{\prime}g^{\prime\prime}\alpha_{25}&-\frac{1}{4}\\\
\end{array}\right)U\;.~{}~{}~{}~{}~{}~{}~{}~{}$ (30)

General expressions for the mixing matrix elements $U_{ij}$ are too
complicated to be written analytically. In Ref.Z'our , we listed results for
$U_{ij}$, $M_{Z}$ and $M_{Z^{\prime}}$ expanded up to order $p^{4}$ and linear
in $\tilde{g}^{\prime}$. In real new physics models appearing in the
literature, the $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings are often not so
complex. In the next section, we identify and discuss typical
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings appearing in various new physics
models.

##  III Classification of models in terms of their
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings

In this section, we organize the various new physics models that can be found
in the literature involving the $Z^{\prime}$ boson according to their
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings. Unlike the most general case
reviewed in the last section, these mixings are special
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings for which the mixing matrix elements
$U_{ij}$ and $M_{Z}$, $M_{Z^{\prime}}$ can all be work out exactly. Below we
consider five situations.

  1. 1.

Minimal $Z^{\prime}\\!-\\!Z$ mass mixing RizzoARXIV2006 ; FranziniPRD1987 ;
RizzoPRD1991 ; LangackerPRD1992 ; ChiappettaPRD1996 ; FramptonPRD1996 ;
ErlerPRL2000 ; AnokaNPB2004 ; KozlovPRD2005 ; BassoARXIV2008 ;
ChanowitzARXIV2008 ; AppelquistPRD2003 ; FerrogliaAP2007 ; CarenaPRD2004 :
This kind of model provides minimal mixing by ignoring all mixings in the
kinetic terms and $Z\\!-\\!\gamma$, $Z^{\prime}\\!-\\!\gamma$ mixings in the
mass terms. They correspond to the vanishing five parameters

$\displaystyle\tilde{g}^{\prime}=\alpha_{1}=\alpha_{8}=\alpha_{24}=\alpha_{25}=0\;.$
(31)

With the exception of gauge couplings $g,g^{\prime},g^{\prime\prime}$, the
remaining three nontrivial parameters are denoted by the $Z^{\prime}\\!-\\!Z$
mass matrix

$\displaystyle\mathcal{M}^{2}=\left(\begin{array}[]{cc}M_{Z_{0}}^{2}&M_{ZZ^{\prime}}^{2}\\\
M_{ZZ^{\prime}}^{2}&M_{Z^{\prime}_{0}}^{2}\end{array}\right)\hskip
28.45274ptZ_{0}^{\mu}\equiv\frac{gW^{3}_{\mu}\\!-g^{\prime}B_{\mu}}{\sqrt{g^{2}\\!+g^{\prime
2}}}\hskip
14.22636ptA_{0}^{\mu}\equiv\frac{g^{\prime}W^{3}_{\mu}\\!+gB_{\mu}}{\sqrt{g^{2}\\!+g^{\prime
2}}}\hskip 14.22636ptZ_{0}^{\prime\mu}\equiv X^{\mu},~{}~{}~{}~{}$ (34)

where mass parameters $M_{Z_{0}}^{2}$, $M_{Z^{\prime}_{0}}^{2}$ and
$M_{ZZ^{\prime}}^{2}$ are related to $\beta_{1},\beta_{2},\beta_{3}$ as

$\displaystyle\frac{f^{2}}{4}(1\\!-\\!2\beta_{1})(g^{2}\\!+\\!g^{\prime
2})\equiv M_{Z_{0}}^{2}\hskip
17.07182ptf^{2}(1\\!-\\!2\beta_{3})g^{\prime\prime 2}\equiv
M_{Z_{0}^{\prime}}^{2}\hskip
17.07182pt\frac{f^{2}}{2}\beta_{2}g^{\prime\prime}\sqrt{g^{2}\\!+\\!g^{\prime
2}}\equiv M_{ZZ^{\prime}}^{2}\;.~{}~{}~{}~{}~{}$ (35)

Refs.AppelquistPRD2003 ; FerrogliaAP2007  use an alternative expression which
corresponds to setting

$\displaystyle f=v_{H}\hskip 14.22636ptg^{\prime}=g_{Y}\hskip
14.22636ptg^{\prime\prime}=g_{z}\hskip 14.22636pt\beta_{1}=0\hskip
14.22636pt\beta_{2}=-\frac{1}{2}z_{H}\hskip
14.22636pt1-2\beta_{3}=\frac{1}{4}(z_{H}^{2}+\frac{v_{\phi}^{2}}{f^{2}})\;.~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Ref.CarenaPRD2004  further generalizes this which leads then to

$\displaystyle g^{\prime}\\!=g_{Y}\hskip
11.38092ptg^{\prime\prime}\\!\\!=g_{z}\hskip
11.38092pt1-2\beta_{1}=\frac{v_{H_{1}}^{2}+v_{H_{2}}^{2}}{f^{2}}~{}~{}~{}\beta_{2}\\!=-\frac{z_{H_{1}}v_{H_{1}}^{2}\\!\\!+\\!z_{H_{2}}v_{H_{2}}^{2}}{2f^{2}}\hskip
11.38092pt$ $\displaystyle
1\\!-\\!2\beta_{3}\\!=\frac{1}{4f^{2}}(z_{H_{1}}^{2}v_{H_{1}}^{2}\\!\\!+\\!z_{H_{2}}^{2}v_{H_{2}}^{2}\\!\\!+\\!v_{\phi}^{2})\;.~{}~{}~{}~{}~{}~{}~{}~{}~{}$

In this kind of model, the key $Z^{\prime}\\!-\\!Z$ mixing parameter is
$\beta_{2}$ which yields a non-vanishing off-diagonal element
$M_{ZZ^{\prime}}^{2}$ in the $Z^{\prime}\\!-\\!Z$ mass matrix. This element
further generates the seesaw splitting between the original $Z$ and
$Z^{\prime}$ masses,

$\displaystyle M_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[M_{Z_{0}}^{2}\\!+M_{Z_{0}^{\prime}}^{2}\\!-\sqrt{(M_{Z_{0}}^{2}\\!-M_{Z_{0}^{\prime}}^{2})^{2}\\!+4M_{ZZ^{\prime}}^{4}}]\approx
M_{Z_{0}}^{2}-\frac{M_{ZZ^{\prime}}^{4}}{M_{Z_{0}^{\prime}}^{2}\\!-M_{Z_{0}}^{2}}$
(36) $\displaystyle M_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[M_{Z_{0}}^{2}\\!+M_{Z_{0}^{\prime}}^{2}\\!+\sqrt{(M_{Z_{0}}^{2}\\!-M_{Z_{0}^{\prime}}^{2})^{2}\\!+4M_{ZZ^{\prime}}^{4}}]\approx
M_{Z_{0}^{\prime}}^{2}+\frac{M_{ZZ^{\prime}}^{4}}{M_{Z_{0}^{\prime}}^{2}\\!-M_{Z_{0}}^{2}}\;.~{}~{}~{}~{}$
(37)

Meanwhile the $Z^{\prime}\\!-\\!Z$ mixing can be parameterized by mixing angle
$\theta^{\prime}$

$\displaystyle\left(\begin{array}[]{c}Z_{0}^{\mu}\\\
Z_{0}^{\prime\mu}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta^{\prime}&\sin\theta^{\prime}\\\
-\sin\theta^{\prime}&\cos\theta^{\prime}\end{array}\right)\left(\begin{array}[]{c}Z^{\mu}\\\
Z^{\prime\mu}\end{array}\right)\hskip 28.45274pt\tan
2\theta^{\prime}=\frac{2M_{ZZ^{\prime}}^{2}}{M_{Z_{0}^{\prime}}^{2}\\!-M_{Z_{0}}^{2}}\;.~{}~{}~{}~{}$
(44)

leading to a rotation matrix introduced in (11) of the form

$\displaystyle U_{\mbox{\tiny Minimal $Z^{\prime}\\!\\!\\!-\\!\\!Z$ mass
mixing}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cos\theta_{W}&\sin\theta_{W}&0\\\
-\sin\theta_{W}&\cos\theta_{W}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{ccc}\cos\theta^{\prime}&0&\sin\theta^{\prime}\\\
0&1&0\\\ -\sin\theta^{\prime}&0&\cos\theta^{\prime}\end{array}\right)$ (51)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cos\theta_{W}\cos\theta^{\prime}&\sin\theta_{W}&\cos\theta_{W}\sin\theta^{\prime}\\\
-\sin\theta_{W}\cos\theta&\cos\theta_{W}&-\sin\theta_{W}\sin\theta^{\prime}\\\
-\sin\theta^{\prime}&0&\cos\theta^{\prime}\end{array}\right)\;,$ (55)

with an electroweak mixing angle $\tan\theta_{W}=g^{\prime}/g$.

  2. 2.

Minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing RizzoARXIV1998 ; LangackerPRD2008
; LangackerPRL2008 :
This kind of model provides minimal mixing by ignoring all mixings in the mass
terms and $Z\\!-\\!\gamma$, $Z^{\prime}\\!-\\!\gamma$ mixings in the kinetic
terms leading to the vanishing of seven parameters

$\displaystyle\tilde{g}^{\prime}=\beta_{1}=\beta_{2}=\beta_{3}=\alpha_{1}=\alpha_{8}=\alpha_{24}=0\;.$
(56)

Again with the exception of gauge couplings $g,g^{\prime},g^{\prime\prime}$,
the one remaining nontrivial parameter is denoted by

$\displaystyle
g^{\prime}g^{\prime\prime}\alpha_{25}\equiv-\frac{\sin\chi}{2}\;.$ (57)

following Ref.RizzoARXIV1998 , we redefine the gauge fields as

$\displaystyle B^{\mu}=B_{0}^{\mu}-\tan\chi Z^{\prime\mu}_{0}\hskip
56.9055ptX^{\mu}=\frac{Z^{\prime\mu}_{0}}{\cos\chi}\;,$ (58)

in terms of the fields $B_{0}^{\mu},Z_{0}^{\prime\mu},W^{3\mu}$, the kinetic
term appears diagonalized and the model reduces to a minimal $Z^{\prime}-Z$
mass mixing model discussed above444This detail was not pointed out in
Ref.RizzoARXIV1998 . with

$\displaystyle M_{Z_{0}}^{2}=\frac{f^{2}}{4}(g^{2}\\!+\\!g^{\prime 2})\hskip
17.07182ptM_{Z_{0}^{\prime}}^{2}=\frac{f^{2}[g^{\prime
2}\sin^{2}\chi\\!+\\!4g^{\prime\prime 2}]}{4\cos^{2}\chi}\hskip
17.07182ptM_{ZZ^{\prime}}^{2}=\frac{f^{2}}{4}g^{\prime}\sqrt{g^{2}\\!+\\!g^{\prime
2}}\tan\chi\;.~{}~{}~{}~{}~{}$ (59)

The rotation matrix introduced in (11) takes the form

$\displaystyle U_{\mbox{\tiny Minimal $Z^{\prime}\\!\\!\\!-\\!\\!Z$ kinetic
mixing}}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&-\tan\chi\\\
0&0&\frac{1}{\cos\chi}\end{array}\right)\times U_{\mbox{\tiny Minimal
$Z^{\prime}\\!\\!\\!-\\!\\!Z$ mass mixing}}$ (63)
$\displaystyle=\left(\begin{array}[]{ccc}\cos\theta^{\prime}\cos\theta_{W}&\sin\theta_{W}&\cos\theta_{W}\sin\theta^{\prime}\\\
-\sin\theta_{W}\cos\theta^{\prime}+\tan\chi\sin\theta^{\prime}&\cos\theta_{W}&-\sin\theta_{W}\sin\theta^{\prime}-\tan\chi\cos\theta^{\prime}\\\
-\sin\theta^{\prime}/\cos\chi&0&\cos\theta^{\prime}/\cos\chi\end{array}\right)\;.~{}~{}~{}~{}~{}$
(67)

  3. 3.

General $Z^{\prime}\\!-\\!Z$ mixing RizzoARXIV2006 ; Hill ; Lane ; Chiv ;
CasselARXIV2009 ; Holdom1986 ; PDG2006 ; BabuPRD1996 :
This kind of model is combinations of minimal $Z^{\prime}\\!-\\!Z$ mass mixing
model and minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing model discussed above
which correspond to

$\displaystyle\tilde{g}^{\prime}=\alpha_{1}=\alpha_{8}=\alpha_{24}=0\hskip
28.45274ptg^{\prime}g^{\prime\prime}\alpha_{25}\equiv-\frac{\sin\chi}{2}\;.$
(68)

In a similar manner as for minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing model,
we can use (58) to remove the mixing in the kinetic term and then, in terms of
the fields $B_{0}^{\mu},Z_{0}^{\prime\mu},W^{3\mu}$, the model can be changed
into a minimal $Z^{\prime}\\!-\\!Z$ mass mixing model with identifications

$\displaystyle M_{Z_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}(1-2\beta_{1})(g^{2}\\!+g^{\prime 2})$
$\displaystyle M_{Z_{0}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}[g^{\prime
2}(1-2\beta_{1})\sin^{2}\chi+4g^{\prime\prime
2}(1-2\beta_{3})+4\beta_{2}g^{\prime}g^{\prime\prime}\sin\chi]}{4\cos^{2}\chi}$
$\displaystyle M_{ZZ^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}\frac{(1-2\beta_{1})g^{\prime}\sin\chi+2\beta_{2}g^{\prime\prime}}{\cos\chi}\sqrt{g^{2}\\!+g^{\prime
2}}\;.~{}~{}~{}~{}~{}$ (69)

The resulting rotation matrix has the same form as in (67), the only change is
that now the $\theta^{\prime}$ as determined through (44) is different due to
the new expressions for
$M_{Z_{0}}^{2},M_{Z_{0}^{\prime}}^{2},M_{ZZ^{\prime}}^{2}$ given by (69). In
some dynamical models such as TC2 models, the general $Z^{\prime}\\!-\\!Z$
mixings are generated by technicolor and topcolor dynamics, as in Refs.Hill1 ;
Lane1 ; Chiv1 , while mixing parameters are given through dynamical
computations depending on the nature of the TC2 models and results in the
following expressions

$\displaystyle g^{\prime}g^{\prime\prime}\alpha_{25}=\frac{g^{\prime
2}\gamma}{2c_{Z^{\prime}}}\hskip
56.9055pt\frac{f^{2}}{2}\beta_{2}g^{\prime\prime}=\frac{g^{\prime}}{4c_{Z^{\prime}}}\times\left\\{\begin{array}[]{lll}(F_{0}^{\mathrm{TC2}})^{2}\tan\theta^{\prime}&{}{}{}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hill,Hill1}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
3(F_{0}^{\mathrm{1D}})^{2}\tan\theta^{\prime}&{}{}{}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Lane,Lane1}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
-3(F_{0}^{\mathrm{1D}})^{2}\cot\theta^{\prime}&{}{}{}&\mbox{Ref.\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Chiv,Chiv1}{\@@citephrase{(}}{\@@citephrase{)}}}}\end{array}\right.\;,$
(73)

where all symbols appearing on the right-hand side of these results are
parameters pertaining to the TC2 models.

  4. 4.

$Z^{\prime}\\!-\\!\gamma$ kinetic and $Z^{\prime}\\!-\\!Z$ mixing
HoldomPLB1991 :
B. Holdom extends the conventional $Z^{\prime}\\!-\\!Z$ mixing by further
adding in model a $Z^{\prime}\\!-\\!\gamma$ kinetic mixing term. His model
corresponds to having

$\displaystyle\tilde{g}^{\prime}=\alpha_{1}=\alpha_{8}=0\hskip
14.22636pt\frac{f^{2}}{4}(1\\!-\\!2\beta_{1})=m_{Z}^{2}\hskip
14.22636pt\frac{f^{2}}{2}\beta_{2}g^{\prime\prime}\sqrt{g^{2}\\!+g^{\prime
2}}=xm_{Z}^{2}\hskip 14.22636ptf^{2}(1\\!-\\!2\beta_{3})g^{\prime\prime
2}=m_{X}^{2}$ $\displaystyle gg^{\prime\prime}\sqrt{g^{2}\\!+g^{\prime
2}}\alpha_{24}=-\frac{1}{2}(gy+g^{\prime}w)\hskip
28.45274ptgg^{\prime\prime}\sqrt{g^{2}\\!+g^{\prime
2}}\alpha_{25}=\frac{1}{2}(g^{\prime}y-gw)\;.$ (74)

We can diagonalize the kinetic terms by redefining the $B^{\mu}$ and
$W^{3\mu}$ fields as

$\displaystyle
B^{\mu}=B_{0}^{\mu}-\frac{\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}Z_{0}^{\prime\mu}\hskip
28.45274ptW^{3\mu}=W^{3\mu}_{0}-\frac{\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}Z_{0}^{\prime\mu}~{}~{}~{}~{}~{}~{}~{}$
(75) $\displaystyle
X^{\mu}=\frac{Z_{0}^{\prime\mu}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\hskip
28.45274pt-\frac{\sin\overline{\chi}}{2}\equiv
g^{\prime}g^{\prime\prime}\alpha_{24}\hskip
28.45274pt-\frac{\sin\chi}{2}\equiv g^{\prime}g^{\prime\prime}\alpha_{25}$

and then in terms of fields $B_{0}^{\mu},Z_{0}^{\prime\mu},W^{3\mu}_{0}$, the
model becomes the minimal $Z^{\prime}\\!-\\!Z$ mass mixing model with

$\displaystyle M_{Z_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}(1-2\beta_{1})(g^{2}\\!+g^{\prime 2})$
$\displaystyle M_{Z_{0}^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}[\frac{1}{4}(1-2\beta_{1})(g^{\prime}\sin\chi-g\sin\overline{\chi})^{2}+(1-2\beta_{3})g^{\prime\prime
2}+\beta_{2}g^{\prime\prime}(g^{\prime}\sin\chi-g\sin\overline{\chi})]}{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}$
$\displaystyle M_{ZZ^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}\frac{[(1-2\beta_{1})(g^{\prime}\sin\chi-g\sin\overline{\chi})+2\beta_{2}g^{\prime\prime}]}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\sqrt{g^{2}\\!+g^{\prime
2}}\;.~{}~{}~{}~{}~{}$ (76)

for which the rotation matrix introduced in (11) takes the form

$\displaystyle U_{\mbox{\tiny$Z^{\prime}\\!\\!\\!-\\!\\!\gamma$ kinetic and
$Z^{\prime}\\!\\!\\!-\\!\\!Z$
mixing}}=\left(\begin{array}[]{ccc}1&0&-\frac{\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
0&1&-\frac{\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
0&0&\frac{1}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\end{array}\right)\times
U_{\mbox{\tiny Minimal $Z^{\prime}\\!\\!\\!-\\!\\!Z$ mass mixing}}$ (80)
$\displaystyle=\left(\begin{array}[]{ccc}\frac{g\cos\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}+\frac{\sin\theta^{\prime}\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}&~{}~{}~{}\frac{g^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}{}{}{}&\frac{g\sin\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}-\frac{\cos\theta^{\prime}\sin\overline{\chi}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
-\frac{g^{\prime}\cos\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}+\frac{\sin\theta^{\prime}\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}&\frac{g}{\sqrt{g^{2}+g^{\prime
2}}}&-\frac{g^{\prime}\sin\theta^{\prime}}{\sqrt{g^{2}+g^{\prime
2}}}-\frac{\cos\theta^{\prime}\sin\chi}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\\\
-\frac{\sin\theta^{\prime}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}&0&\frac{\cos\theta^{\prime}}{\sqrt{1-\sin^{2}\chi-\sin^{2}\overline{\chi}}}\end{array}\right)\;.~{}~{}~{}~{}~{}$
(84)

  5. 5.

Stueckelberg-type mixing KorsJHEP2005 ; FeldmanPRL2006 ; FeldmanPRD2007 :
This kind of model provides mixing through the nonzero coupling constant
$\tilde{g}^{\prime}$ and except for gauge coupling
$g,g^{\prime},g^{\prime\prime}$, a typical choice as given in
Refs.KorsJHEP2005 ; FeldmanPRL2006  is the vanishing of all other parameters

$\displaystyle\beta_{1}=\beta_{2}=\beta_{3}=\alpha_{1}=\alpha_{8}=\alpha_{24}=\alpha_{25}=0\;,$
(85)

leading to diagonal kinetic terms and mixing occurring only in the mass terms.
After rotating the standard electroweak mixing angle $\theta_{W}$, we can
redefine the gauge fields

$\displaystyle\bar{B}^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime
2}}}{(g^{2}+g^{\prime 2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}B_{0}^{\mu}+\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}{Z^{\prime}}_{0}^{\mu}$
$\displaystyle\bar{Z}^{\prime\mu}$ $\displaystyle=$
$\displaystyle\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}B_{0}^{\mu}+\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime
2}}}{(g^{2}+g^{\prime 2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}{Z^{\prime}}_{0}^{\mu}$ (86)

thereby changing the present model to a minimal $Z^{\prime}-Z$ mass mixing
model with

$\displaystyle M_{Z_{0}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{4}(g^{2}+g^{\prime 2}+\frac{4g^{\prime
2}\tilde{g}^{\prime 2}}{g^{2}+g^{\prime 2}})$ $\displaystyle
M_{Z^{\prime}_{0}}^{2}$ $\displaystyle=$ $\displaystyle
f^{2}({g^{\prime\prime}}^{2}+\frac{g^{2}\tilde{g}^{\prime 2}}{g^{2}+g^{\prime
2}})$ $\displaystyle M_{ZZ^{\prime}}$ $\displaystyle=$
$\displaystyle-\frac{f^{2}g^{\prime}\tilde{g}^{\prime}\sqrt{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}}{g^{2}+g^{\prime 2}}\;.$
(87)

The overall rotation matrix then becomes

$\displaystyle U_{\mbox{\tiny Stuekckelberg type mixing}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\cos\theta_{W}&\sin\theta_{W}&0\\\
-\sin\theta_{W}&\cos\theta_{W}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&-\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime 2}}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}&\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}\\\
0&\frac{g\tilde{g}^{\prime}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime
2}}&\frac{{g^{\prime\prime}}\sqrt{g^{2}+g^{\prime 2}}}{(g^{2}+g^{\prime
2}){g^{\prime\prime}}^{2}+g^{2}\tilde{g}^{\prime 2}}\end{array}\right)$ (98)
$\displaystyle\times\left(\begin{array}[]{ccc}\cos\theta^{\prime}&0&\sin\theta^{\prime}\\\
0&1&0\\\ -\sin\theta^{\prime}&0&\cos\theta^{\prime}\end{array}\right)$

with $\theta^{\prime}$ evaluated from the second equation of (44) and those of
(87). In Ref.FeldmanPRD2007 , the Stueckelberg-type mixing is further
generalized to include kinetic mixing by relaxing the original condition
$\alpha_{25}=0$. This kinetic mixing can be diagonalized by applying (58) and
following a similar procedure to that leading to (86) in diagonalizing the
mass terms.

##  IV the $Z^{\prime}$ boson charges to quark and leptons

The charges for the $Z^{\prime}$ boson with respect to ordinary quarks and
leptons can be expressed in terms of the gauge interaction as

$\displaystyle\mathcal{L}_{\mathrm{gauge~{}coupling}}={g^{\prime\prime}}X_{\mu}J^{\mu}_{X}\hskip
56.9055ptJ^{\mu}_{X}=\sum_{i}\bar{f}_{i}\gamma^{\mu}[y^{\prime}_{iL}P_{L}+y^{\prime}_{iR}P_{R}]f_{i}\;,$
(99)

where index $i$ distinguishes the three generations associated with the six
quarks $u,c,t,d,s,b$ and six leptons
$e,\mu,\tau,\nu_{e},\nu_{\mu},\nu_{\tau}$, and
$y_{i,L}^{\prime},y_{i,R}^{\prime}$ are the corresponding left- and right-hand
charges555Phenomenologically, we need to further express the gauge interaction
given in Eq.(99) in terms of mass eigenstate of $Z^{\prime}$, for then the
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixings discussed in the last section set
in.. The $SU(2)_{L}$ symmetry requires equating $U(1)$ charges of the two
components of the left-hand fermion doublet, i.e.
$y^{\prime}_{u,L}=y^{\prime}_{d,L}\equiv y^{\prime}_{q}$ for quark and
$y^{\prime}_{\nu,L}=y^{\prime}_{e,L}\equiv y^{\prime}_{l}$ for lepton. Thus,
we can parameterize the fermionic $U(1)^{\prime}$ charges by $y^{\prime}_{q}$,
$y^{\prime}_{u}$, $y^{\prime}_{d}$, $y^{\prime}_{l}$, $y^{\prime}_{e}$ and
$y^{\prime}_{\nu}$. In general, the assignments of $U(1)^{\prime}$ charges are
generation-dependent, but in its simplest form $U(1)^{\prime}$ charges can be
generation-independent, much like hypercharge assignments in SM. TABLE.1 lists
four sets of more common assignments for the generation-independent
$U(1)^{\prime}$ charges of fermions in new physics models involving
$Z^{\prime}$ boson CarenaPRD2004 ; PDG2008 . In the $U(1)_{B-xL}$ model (see
column 3 of TABLE.1), $Z^{\prime}$ charges are determined by the baryon number
and lepton number from $y^{\prime}_{i}=B_{i}-xL_{i}$ with a free rational
parameter $x$. Leptophobic and hadrophobic $Z^{\prime}$ models correspond to
$x=\infty$ and $x=0$, respectively. The second set of charges comes from grand
unified theories. Parameter $x$ establishes the mixing of the two extra $U(1)$
groups in the $E_{6}$ symmetry breaking patterns $E_{6}\rightarrow SU(5)\times
U(1)\times U(1)$. $Z_{\chi}$, $Z_{\psi}$ and $Z_{\eta}$ of Ref.GUTs
correspond to the special case with $x=-3$, $x=1$ and $x=-1/2$, respectively.
The third set, $U(1)_{d-xu}$ results in the vanishing of the left-hand quark
doublet charge and the ratio of right-hand up quark charges to down quark
charges is controlled by $-x$. In the last set, the free parameter $x$ is the
ratio of the charges of the left-hand quark doublet and right-hand up quark
singlet and reduces to the $U(1)_{B-L}$ model for $x=1$.

Table 1: generation-independent $U(1)^{\prime}$ charges for quarks and leptons models |  $Z^{\prime}$ EWCL | $U(1)_{B-xL}$ | $U(1)_{10+x\bar{5}}$ | $U(1)_{d-xu}$ | $U(1)_{q+xu}$
---|---|---|---|---|---
$(u_{L},d_{L})$ | $y^{\prime}_{q}$ | $1/3$ | $1/3$ | $0$ | $1/3$
$u_{R}$ | $y^{\prime}_{u}$ | $1/3$ | $-1/3$ | $-x/3$ | $x/3$
$d_{R}$ | $y^{\prime}_{d}$ | $1/3$ | $-x/3$ | $1/3$ | $(2-x)/3$
$(\nu_{L},e_{L})$ | $y^{\prime}_{l}$ | $-x$ | $x/3$ | $(x-1)/3$ | $-1$
$e_{R}$ | $y^{\prime}_{e}$ | $-x$ | $-1/3$ | $x/3$ | $-(2+x)/3$
$\nu_{R}$ | $y^{\prime}_{\nu_{R}}$ | $-1$ | $(x-2)/3$ | $-x/3$ | $(x-4)/3$

Theoretically, the charges of quarks and leptons must satisfy the anomaly
cancellation conditions to preserve the gauge symmetry. We now examine the
constraints on generation-independent $U(1)^{\prime}$ charges arising as a
consequence of these anomaly cancellation conditions. Davidson et.al.
DavidsonPRD1979  have studied anomaly cancellation for additional
$U(1)^{\prime}$ gauge group and derived the following anomaly cancellation
conditions for $U(1)_{Y}\otimes U(1)^{\prime}$ gauge groups

$\displaystyle\sum y^{\alpha}_{L}=\sum
Q^{2}(y^{\alpha}_{L}\\!-\\!y^{\alpha}_{R})=0\hskip 17.07182pt\sum
Q(y^{\alpha}_{L}y^{\beta}_{L}\\!-\\!y^{\alpha}_{R}y^{\beta}_{R})=0\hskip
17.07182pt\sum(y^{\alpha}_{L}y^{\beta}_{L}y^{\gamma}_{L}\\!-\\!y^{\alpha}_{R}y^{\beta}_{R}y^{\gamma}_{R})=0\;,~{}~{}~{}~{}$
(100)

where $\alpha,\beta,\gamma$ indexes $U(1)_{Y}$ and $U(1)^{\prime}$ charges.
Substituting the $U(1)_{Y}$ charges for ordinary quarks and leptons and
assuming the generation-independence of $U(1)^{\prime}$ charges, we find that
above equations imply

$\displaystyle\left\\{\begin{array}[]{l}y^{\prime}_{l}+3y^{\prime}_{q}=0\\\
3y^{\prime}_{l}+5y^{\prime}_{q}-3y^{\prime}_{e}-4y^{\prime}_{u}-y^{\prime}_{d}=0\\\
-{y^{\prime}_{l}}^{2}+{y^{\prime}_{q}}^{2}+{y^{\prime}_{e}}^{2}-2{y^{\prime}_{u}}^{2}+{y^{\prime}_{d}}^{2}=0\\\
3y^{\prime}_{l}+y^{\prime}_{q}-6y^{\prime}_{e}-8y^{\prime}_{u}-2y^{\prime}_{d}=0\\\
2{y^{\prime}_{l}}^{3}+6{y^{\prime}_{q}}^{3}-{y^{\prime}_{e}}^{3}-3{y^{\prime}_{u}}^{3}-3{y^{\prime}_{d}}^{3}-{y^{\prime}_{\nu_{R}}}^{3}=0\end{array}\right.\;.$
(106)

The last equation in (106) can be satisfied by assigning
$y^{\prime}_{\nu_{R}}$ a proper value or adding in our theory some other new
fermions. Solving the above equations, we obtain two sets of solutions which
satisfy the anomaly cancellation conditions

$\displaystyle\left\\{\begin{array}[]{l}y^{\prime}_{l}=-3y^{\prime}_{q}\\\
y^{\prime}_{d}=2y^{\prime}_{q}-y^{\prime}_{u}\\\
y^{\prime}_{e}=-2y^{\prime}_{q}-y^{\prime}_{u}\\\
y^{\prime}_{\nu_{R}}=-4y^{\prime}_{q}+y^{\prime}_{u}\end{array}\right.\hskip
56.9055pt{\rm or}\hskip
56.9055pt\left\\{\begin{array}[]{l}y^{\prime}_{l}=-3y^{\prime}_{q}\\\
y^{\prime}_{d}=-\frac{14}{5}y^{\prime}_{q}+\frac{1}{5}y^{\prime}_{u}\\\
y^{\prime}_{e}=-\frac{2}{5}y^{\prime}_{q}-\frac{7}{5}y^{\prime}_{u}\\\
y^{\prime}_{\nu_{R}}=\frac{\sqrt[3]{35}}{5}(4y^{\prime}_{q}-y^{\prime}_{u})\end{array}\right.\;.$
(115)

Of the six of $U(1)^{\prime}$ charges, only two of them $y^{\prime}_{q}$ and
$y^{\prime}_{u}$ are independent; the other four being linear combinations of
these two. In addition, there are two kinds of linear combinations: the first
of Eq.(115) which was given and discussed in detail in Ref.AppelquistPRD2003 ,
while the second is a new solution having not yet appeared in the literature.
We can utilize the values of $y^{\prime}_{q}$ and $y^{\prime}_{u}$ to classify
the new physics models and in the following we list some typical cases:

  1. 1.

Left Handed:
$y^{\prime}_{u}=y^{\prime}_{d}=y^{\prime}_{e}=y^{\prime}_{\nu_{R}}=0~{}\Rightarrow~{}y^{\prime}_{q}=y^{\prime}_{l}=0$

  2. 2.

Right Handed:
$y^{\prime}_{q}=y^{\prime}_{l}=0~{}\Rightarrow~{}y^{\prime}_{d}\\!=-y^{\prime}_{u}\\!=y^{\prime}_{e}\\!=-y^{\prime}_{\nu_{R}}$
or
$y^{\prime}_{d}\\!=\frac{1}{5}y^{\prime}_{u}\\!=-\frac{1}{7}y^{\prime}_{e}\\!=-\frac{1}{\sqrt[3]{35}}y^{\prime}_{\nu_{R}}$

  3. 3.

Left-Right symmetric:
$y^{\prime}_{q}=y^{\prime}_{u}=y^{\prime}_{d}~{}\Rightarrow~{}y^{\prime}_{l}=y^{\prime}_{e}=y^{\prime}_{\nu_{R}}=-3y^{\prime}_{q}$

  4. 4.

$\nu_{R}$ decouple:
$y^{\prime}_{\nu_{R}}=0~{}\Rightarrow~{}y^{\prime}_{u}=4y^{\prime}_{q},~{}y^{\prime}_{e}=2y^{\prime}_{l}=3y^{\prime}_{d}=-6y^{\prime}_{q}$

Checking the assignments given in TABLE.1 against the two solutions in (115),
we find that the $U(1)_{B-xL}$, $U(1)_{d-xu}$ and $U(1)_{q+xu}$ models are
anomaly-free when parameter $x=1$ with the right-hand neutrino charge
$y^{\prime}_{\nu_{R}}=-1$, $y^{\prime}_{\nu_{R}}=-\frac{1}{3}$ and
$y^{\prime}_{\nu_{R}}=-1$, respectively. Furthermore, the $U(1)_{10+x\bar{5}}$
model is anomaly-free when $x=-3$ with $y^{\prime}_{\nu_{R}}=-5/3$. Even
though the anomaly cancellation condition can not be satisfied with the
present quarks and leptons, we still have the possibility of canceling the
anomalies by adding some extra fermions into theory.

If we relax the generation-independence criterion on the $U(1)^{\prime}$
charges, we need to add generation indices to each of the charges in Eq.(106)
and sum over the generations on the left-hand side of Eq.(106). In this case,
there are too many free parameters and solutions. We list several possible
solutions in TABLE.2, in which the first and last columns are the two
solutions given in Ref.PDG2008 , and the remaining solutions can be seen to be
some kind of generation-dependent generalization of charge assignments given
in the third, fourth and fifth columns in TABLE.1. The typical feature of
these solutions is that for the solutions given in the first four columns of
TABLE.2, the charges for the first two generations are parameterized in a like
manner as those in the generation-independent situation by $x$ or $y$
separately, and differences appear only in the third generation of quarks and
leptons. Of special note is that for the solution to $U(1)_{q+xu+yc+zt}$, the
anomaly cancellation condition is satisfied for each generation independently.

Table 2: generation-dependent charge models | $U(1)_{B-xL_{e}\\!-yL_{\mu}\\!}$ | $U(1)_{10+x\bar{5}}~{}\mathrm{\tiny gen\\!\\!-\\!dep}$ | $U(1)_{d-xu}~{}\mathrm{\tiny gen\\!\\!-\\!dep}$ | $U(1)_{q+xu+yc+zt}$ | $2\\!+\\!1~{}\mathrm{\tiny leptocratic}$
---|---|---|---|---|---
$q_{1,L}$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $1/3$
$u_{R}$ | $1/3$ | $-1/3$ | $-x/3$ | $x/3$ | $x/3$
$d_{R}$ | $1/3$ | $-x/3$ | $1/3$ | $(2-x)/3$ | $(2-x)/3$
$q_{2,L}$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $1/3$
$c_{R}$ | $1/3$ | $-1/3$ | $-y/3$ | $y/3$ | $x/3$
$s_{R}$ | $1/3$ | $-y/3$ | $1/3$ | $(2-y)/3$ | $(2-x)/3$
$q_{3,L}$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $1/3$
$t_{R}$ | $1/3$ | $-1/3$ | $2\\!-\\!\frac{2}{3}(x\\!\\!+\\!y)\\!\pm\\!\\!\sqrt{3\\!-\\!x^{2}\\!\\!-\\!y^{2}}$ | $z/3$ | $x/3$
$b_{R}$ | $1/3$ | $3+\frac{x+y}{3}$ | $1/3$ | $(2-z)/3$ | $(2-x)/3$
$(\nu^{e}_{L},e_{L})$ | $-x$ | $x/3$ | $(x-1)/3$ | $-1$ | $-1-2y$
$e_{R}$ | $-x$ | $-1/3$ | $x/3$ | $-(2+x)/3$ | $-(2\\!+\\!x)/3-2y$
$(\nu^{\mu}_{L},\mu_{L})$ | $-y$ | $y/3$ | $(y-1)/3$ | $-1$ | $y-1$
$\mu_{R}$ | $-y$ | $-1/3$ | $y/3$ | $-(2+y)/3$ | $-(2\\!+\\!x)/3+y$
$(\nu^{\tau}_{L},\tau_{L})$ | $x+y-3$ | $3+\frac{x+y}{3}$ | $\frac{2}{3}-\frac{1}{3}(x+y)$ | $-1$ | $y-1$
$\tau_{R}$ | $x+y-3$ | $-1/3$ | $x\\!+\\!y\\!-\\!3\\!\mp\\!\frac{4}{3}\sqrt{3\\!-\\!x^{2}\\!-\\!y^{2}}$ | $-(2+z)/3$ | $-(2+x)/3+y$

##  V Summary

In this paper, we have classified various new physics models involving the
$Z^{\prime}$ boson in two different ways: one according to $Z^{\prime}$ boson
mixings with $Z$ and $\gamma$, and the other according to $Z^{\prime}$ boson
charges with respect to quarks and leptons. In regard to the former, we based
the general description for the $Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixing
derived from the EWCL on our previous workZ'our , characterizing these new
physics models into five classes: 1. Models with minimal $Z^{\prime}\\!-\\!Z$
mass mixing; 2.Models with minimal $Z^{\prime}\\!-\\!Z$ kinetic mixing;
3.Models with general $Z^{\prime}\\!-\\!Z$ mixing; 4.Models with
$Z^{\prime}\\!-\\!\gamma$ kinetic and $Z^{\prime}\\!-\\!Z$ mixing; and
5.Models with Stueckelberg-type mixing. Although the general
$Z^{\prime}\\!-\\!Z\\!-\\!\gamma$ mixing is complicated and there is no exact
analytical expression for the mixing matrix $U$ and masses
$M_{Z},M_{Z^{\prime}}$, we obtain explicit analytical expressions for each of
our five simplifying classes. We find that the most elementary mixing is the
minimal $Z^{\prime}\\!-\\!Z$ mass mixing, the other four classes of mixings
can be transformed into the minimal $Z^{\prime}\\!-\\!Z$ mass mixing through
field transformations. In regard to the latter classification, we exploit the
anomaly cancellation conditions to constrain the $U(1)^{\prime}$ charges. For
generation-independent $U(1)^{\prime}$ charges, there are six charges
$y^{\prime}_{q}$,$y^{\prime}_{u}$,$y^{\prime}_{d}$,$y^{\prime}_{l}$,$y^{\prime}_{e}$,$y^{\prime}_{\nu}$
for which anomaly cancellation requires that only two are independent
parameters while the other four can depend on these two parameters in two
different ways. While one appears already in the literature, the other is new.
For generation-dependent $U(1)^{\prime}$ charges, we have listed some possible
special solutions.

## Acknowledgments

This work was supported by National Science Foundation of China (NSFC) under
Grant No. 10875065.

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# RoadAtlas: Intelligent Platform for Automated Road Defect Detection and
Asset Management

Zhuoxiao Chen1, Yiyun Zhang1, Yadan Luo1, Zijian Wang1, Jinjiang Zhong2,
Anthony Southon2 1The University of Queensland, 2Logan City Council, Australia
zhuoxiao.chen, yiyun.zhang, y.luo<EMAIL_ADDRESS>JinjiangZhong,
<EMAIL_ADDRESS>
(2018)

###### Abstract.

With the rapid development of intelligent detection algorithms based on deep
learning, much progress has been made in automatic road defect recognition and
road marking parsing. This can effectively address the issue of an expensive
and time-consuming process for professional inspectors to review the street
manually. Towards this goal, we present RoadAtlas, a novel end-to-end
integrated system that can support 1) road defect detection, 2) road marking
parsing, 3) a web-based dashboard for presenting and inputting data by users,
and 4) a backend containing a well-structured database and developed APIs.

semantic segmentation, object detection, road defect, road marking

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††conference: Gold Coast ’21: ACM Multimedia Asia; Dec
01–03, 2021; Gold Coast, Australia††booktitle: Gold Coast ’21: ACM Multimedia
Asia, Dec 01–03, 2018, Gold Coast, Australia††price: 15.00††isbn:
978-1-4503-XXXX-X/18/06††ccs: Computing methodologies Image segmentation

##  1\. Introduction

Assessing the condition of the road network is vital for ensuring its
continued functionality. The majority of road agencies in Australia conduct
road network condition surveys through employing specialised machines such as
laser profiler meters to obtain performance and condition data of the road
network. There are also a number of road agencies that rely on experience
inspectors via walking or driving the road network to collect defects data.
Being a costly and time-consuming procedure (around $350K to $400K per
survey), it prevents the local government from conducting frequent surveys to
keep the data up to date. The cracks and destroyed road markings not only
affect the aesthetic value of the city, but also increase the risks of injury
and fatality faced by pedestrians. Hence, there is an urgent need for
automation in road asset management by adopting data science for predictive
analytics, visual recognition, and interactive visualisation.

The framework’s purpose is to forge a mutually beneficial program by forming a
novel AI-powered road defects and assets management platform, namely
RoadAtlas. Our vision is to deliver the intelligent system for jointly road
defect segmentation and road marking parsing, being able to grow and adapt to
the new data or novel tasks (e.g., detecting a new crack type). The RoadAtlas
will afford an unprecedented capacity to automatically demonstrate and report
types, severity, location and evidence of road anomalies in real-time for
planning the follow-up repairs. The RoadAtlas also enhances local government
employees’ data analytic skillsets and help road asset managers monitor the
functional and structural performance of road networks efficiently and
effectively.

##  2\. System Overview

Figure 1. The overall framework of the RoadAtlas.

The system consists of three modules: the dashboard, deep visual models and
the backend, as shown in Figure 1. The dashboard is a user interaction
interface that supports various front-end functions. The deep models are
designed for road defect and road marking recognition. The backend is built
with Python Flask to exchange data with the front-end and save data in the
MongoDB database.

Figure 2. Dashboard (top) and Road Defects View (bottom).

###  2.1. Front-end and Back-end

The Dashboard is the primary user interface designed as a web application,
illustrated in Figure 2. It features four primary functions: 1) Road Defect,
2) Road Marking, 3) Database Update and 4) Report Export. Road Defect and Road
Marking allow users to inspect and validate labelled images from the database.
The Road Defect example shown in Figure 2 displays an interactive map with map
pins representing individual defects. When the user selects a map pin, a
validation panel displays on the right, providing defect details such as the
labelled images and coordinates. Next, the user can validate the correctness
of defects and mark them as checked. The Database Update function is a gateway
for users to upload unprocessed images to the deep model for defect and
marking detection. Additional Report Export function supports saving validated
images data as CSV or JSON-formatted files. The backend aims to provide
functional APIs to exchange data with the dashboard side. Our developers set
up multiple APIs according to the need of the dashboard via the Python Flask
framework. Also, we construct the MongoDB database for managing data. We
further provide 20TB hard disk storage on our server for storing huge amounts
of images.

###  2.2. Road Defect Detection and Segmentation

When users utilise the database update function by inserting new road images,
the proposed deep models can perform recognition for road defects.

Dataset Annotation. To train models for road defect detection following
supervised learning principles, labels are required for training images.
However, the 10,000 raw images provided by the local government are
unlabelled. These images represent the foreground of the car, taken in Logan
City, Australia. We arranged five annotators to label all images with various
classes of road defects, including background, Kerb_Cracking, Road_Crocodile,
Road_Longitudinal, Road_Transverse, Road_Block and Sealed_Crack. Each of the
road defects is labelled in an irregular polygon.

Figure 3. Examples of road defect detection and segmentation results.

Crack Detection and Segmentation. We developed a pair of deep models to
effectively capture road defects consisting of a crack detector and a crack
segmenter. There are four steps for crack recognition. First, we utilise the
anonymiser (understand ai, 2019) to protect the privacy of car owners via
automatically hiding the license plate numbers. Second, we convert labelled
polygons to the minimised bounding boxes that cover the polygon. Third, we
proposed a crack detector inspired by (Jocher, 2020), fully trained with
converted labelled bounding boxes. Thus, the crack detector can accurately
produce bounding boxes for different defects. The last step is to feed
predicted bounding boxes into the crack segmenter to produce pixel-level
predictions for each road defect. We follow (Ronneberger et al., 2015) to
construct the crack segmenter, which is specially trained for segmenting road
defects. Figure 3 shows one detection result on the top, and two segmentation
pixel-level results (shown in two rows) at the bottom. In the bottom
segmentation result, the left image is the detected bounding boxes by the
crack detector. The image at the mid is the prediction feature map by the
crack segmenter. The right image maps predictions back to the original image
for visualisation.

###  2.3. Road Marking Parsing

We develop a road marking segmenter to detect different categories of road
markings at pixel level accurately. The road marking segmenter follows (Zhao
et al., 2017) and is thoroughly trained with the existing benchmark dataset
(Huang et al., 2020) containing various labelled road markings. After
prediction, we performed an extra refinement process to fit the results back
to the original image. The refinement process can be summarised into three
steps. The first step is to detect the region of interest (ROI), which is the
road in our case. Then we project both original images and prediction maps
from street view into bird eye view to obtain parallel or perpendicular road
markings. The second step is to implement low-level contour detection, which
focuses on generating the boundaries of objects. The low-level contour of road
markings is regular. The predicted pixels are irregular but cover most areas
of the road markings. Thus, we compare the prediction map with low-level
contours. If there is a high overlap between them, the low-level contours are
kept and mapped back to the original image.

##  3\. CONCLUSION

In this paper, we present a fully developed system - RoadAtlas, that
comprehensively provides functions of road defect recognition, road marking
parsing integrated into an end-to-end platform. It can achieve faster
automated recognition of road defects than manually reviewing images by well-
trained inspectors, allowing local governments to repair roads in time.
Besides, a great portion of budgets can be saved by full automation. The local
governments can effortlessly grasp and manage road conditions with RoadAtlas.

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