# Axigluon on like-sign charge asymmetry ${\cal A}^{b}_{s\ell}$, FCNCs and CP
asymmetries in $B$ decays

Chuan-Hung Chen1,2111Email<EMAIL_ADDRESS>and Gaber
<EMAIL_ADDRESS>1Department of Physics, National
Cheng-Kung University, Tainan, 701 Taiwan
2National Center for Theoretical Sciences, Hsinchu 300, Taiwan
3 Egyptian Center for Theoretical Physics, Modern University for Information
and Technology, Cairo, Egypt
4Physics Department, Faculty of Education, Thamar University, Thamar ,Yemen

###### Abstract

A non-universal axigluon in generalized chiral color models leads to flavor
changing neutral currents (FCNCs) at tree level. We analyze phenomenologically
the new contributions to $B_{q}$ (q=d, s) mixing and the related CP
asymmetries (CPAs) that are generated by axigluon exchange. We find that
although $\Delta m_{B_{q}}$ can give a strict constraint on the parameters of
$b\to q$ transition, the precise measurement of $\sin 2\beta_{J/\Psi K^{0}}$
can further exclude the parameter space of $b\to d$ transition. The axigluon-
mediated effects can enhance the like-sign dimuon charge asymmetry ${\cal
A}^{b}_{s\ell}$ by one order of magnitude larger than the standard model
prediction. Accordingly, large CPA $\sin 2\beta^{J/\Psi\phi}_{s}$ and CPA
difference $\sin 2\beta_{J\Psi K^{0}}-\sin 2\beta_{\phi K^{0}}$ are achieved.

##  I Introduction

In the standard model (SM), with three families of quarks, the unique CP
violating phase of the Cabibbo-Kobayashi-Maskawa (CKM) matrix can explain some
of the observed CP violating phenomena in $K$ and $B$ systems. However, the
failure of the KM phase in explaining the matter-antimatter asymmetry and some
recent measurements of CP violating observations in $B$ meson mixings and
decays motivates the search for new source of CP violation (CPV). Therefore,
it is an important issue to explore and to find new CP violating effects in
various systems, such as cosmos, Large Hadron Collider (LHC), Tevatron, $B$
factories etc.

Recently, several hints for the existence of new CP violating sources are
revealed in experiments. The first hint is observed in the CP asymmetries
(CPAs) of $B\to\pi K$ decays where by naive SM estimation, one expects that
$\bar{B}_{d}\to\pi^{+}K^{-}$ and $B^{-}\to\pi^{0}K^{-}$ decays have similar
CPAs. However, it is surprising that the world average difference between the
two CPAs contradicts the expectation as the experimental result is
TheHeavyFlavorAveragingGroup:2010qj

$\displaystyle\Delta
A_{CP}=A_{CP}(\pi^{+}K^{-})-A_{CP}(\pi^{0}K^{-})=-(14.8_{-1.4}^{+1.3})\%\,,$
(1)

whereas the SM prediction is $\Delta A_{CP}(SM)=0.025\pm 0.015$ Beneke:2003zv
. The large deviation from the SM prediction indicates a puzzle in the
asymmetries and it is introduced in the literature as $B\to\pi K$ puzzle
pikpuz . The second hint is observed in the time-dependent CPA of $B_{s}$
system, where CDF and DØ have shown an unexpected large CP phase in the
mixing-induced CPA for $B_{s}\to J/\Psi\phi$ and the two possible solutions
are given by TheHeavyFlavorAveragingGroup:2010qj

$\displaystyle 2\beta^{J/\Psi\phi}_{s}=2\beta_{s}+2\phi^{\rm
NP}_{s}=-0.75^{+0.32}_{-0.21}\ {\texttt{or}}\ -2.38^{+0.25}_{-0.34}$ (2)

at $90\%$ confidence level (CL). Here, $\beta_{s}\approx-0.019$ Chen:2008ug
is the SM CP violating phase and $\phi^{\rm NP}_{s}$ is the CP violating phase
of new physics. The significant deviation from the SM prediction could be
speculated by the contributions of new physics.

The third hint is observed in the like-sign charge asymmetry which is defined
as Abazov:2010hv

$\displaystyle{\cal A}^{b}_{s\ell}$ $\displaystyle=$
$\displaystyle\frac{N^{++}_{b}-N^{--}_{b}}{N^{++}_{b}+N^{--}_{b}}\,,$ (3)

where $N^{++(--)}_{b}$ denotes the number of events that $b$\- and
$\bar{b}$-hadron semileptonically decay into two positive (negative) muons.
Recently, DØ Collaboration has announced the measurement on ${\cal
A}^{b}_{s\ell}$ in the dimuon events Abazov:2010hv  with

$\displaystyle{\cal A}^{b}_{s\ell}=\left(-9.57\pm 2.51({\rm stat})\pm
1.46({\rm syst})\right)\times 10^{-3}\,.$ (4)

The SM prediction is ${\cal A}^{b}_{s\ell}=(-2.3^{+0.5}_{-0.6})\times 10^{-4}$
Abazov:2010hv ; Lenz:2006hd . If the semileptonic b-hadron decays do not
involve CP violating phase, then the charge asymmetry is directly related to
the mixing-induced CPAs in $B_{d}$\- and $B_{s}$-meson oscillations. Although
the errors of the data are still large, however the $3.2$ standard deviations
from the SM prediction can be attributed to new CP violating phases in $b\to
d$ and $b\to s$ transitions Randall:1998te ; Dighe:2010nj ; Dobrescu:2010rh ;
Choudhury:2010ya .

In order to explore the new physics and to avoid the uncontrollable QCD
uncertainties, we will concentrate our study on the mixing parameter $\Delta
m_{B_{q}}$, the charge asymmetry ${\cal A}^{b}_{s\ell}$ and the time-dependent
CPA in $B_{q}$ oscillation, where QCD effects can be controlled well by
Lattice QCD.

In the literature, many extensions of the SM such as chiral color models
chiral ; nonuni ; Sehgal:1987wi ; Doncheski:1997yj ; Giordani:2003ib ;
Choudhury:2007ux , $Z^{\prime}$ models LL_PRD45 ; BZ1 ; BZ2  etc have been
proposed. The flavor non-universal axigluon in the generalized chiral color
models AKR ; Frampton:2009rk  has been studied for solving the anomalous
forward-backward asymmetry (FBA) in the top-quark pair production at the
Tevatron D0_PRL100 ; CDF_PRL101 . Although other models such as Z’, diquarks
models Arhrib:2009hu  etc may have significant contributions to the FBA,
however, large gauge couplings and flavor changing effects should be
introduced in which chiral color model does not need. Inspired by the effects
of the axigluon on the top-quark FBA, we study the axigluon-mediated phenomena
in $B$-meson system.

A flavor universal axigluon has flavor-conserving effects only. For non-
universal axigluon which has different couplings to different quarks, flavor
changing neutral currents (FCNCs) can be generated at tree level. This is
achieved after transforming the weak eigenstates of the quarks into their
physical eigenstates. As a consequence, many phenomena will be affected by
these FCNC effects. In this paper, we analyze in detail the non-universal
axigluon contributions to the time-dependent CPAs in $B_{q}$ oscillation after
taking into account the constraint from the mixing parameter $\Delta
m_{B_{q}}$.

This paper is organized as follows. In Sec. II, we formulate the interactions
of $b\to q$ transitions which are induced by flavor non-universal axigluon
exchange. Accordingly, we derive the corresponding effective Hamiltonian for
$\Delta B=1,2$ processes. Furthermore, we discuss the contributions of the
axigluon to the charge asymmetry ${\cal A}^{b}_{s\ell}$ and the time-dependent
CPAs for $B_{d}\to J/\psi K^{0}$, $B_{d}\to\phi K^{0}$, $B_{s}\to J/\Psi\phi$
decays. The detailed numerical analysis is presented in Sec. III. We give the
conclusion in Sec. IV.

##  II Formalism

In order to study the contributions of the non-universal axigluon to the FCNC
processes, we start by writing the interactions of the massive color-octet
gauge boson with quarks as

$\displaystyle{\cal L}_{A}$ $\displaystyle=$ $\displaystyle
g_{V}\bar{q}^{\prime}\gamma_{\mu}T^{b}q^{\prime}G^{b\mu}_{A}+g_{A}\bar{q}^{\prime}\gamma_{\mu}\gamma_{5}{\bf
Z}T^{b}q^{\prime}G^{b\mu}_{A}\,,$ (5)

where we have suppressed the flavor and color indices, $g_{V,A}$ are the gauge
couplings of the new gauge group $SU(3)_{A}\times SU(3)_{B}$, $T^{b}$ are the
Gell-Mann matrices which are normalized by $Tr(T^{b}T^{c})=\delta^{ac}/2$ and
$\bf Z$ is $3\times 3$ diagonalized matrix with diag(Z)=(1, 1, $\zeta$). Here
$\zeta=\tilde{g}_{A}/g_{A}$ where $\tilde{g}_{A}$ denotes the gauge coupling
of the third-generation quark and its value depends on a specific model, e.g.
$\zeta=-1$ in Ref. Frampton:2009rk . For simplicity, we assume that the new
exotic quarks which are required for anomaly free are very heavy and their
effects are negligible. Hence, we still focus on three flavors for each up and
down type quarks. Following the scenario in Refs. AKR ; Frampton:2009rk  for
solving the large top-quark FBA, we assume that the axigluon couplings to the
third generation are different from their couplings to the first two
generations. The left- and right-handed quarks are $SU(2)$ doublet and singlet
respectively. Thus, after spontaneous symmetry breaking, the interacting and
physical eigenstates can be related by unitary matrices as
$q_{\chi}=V^{Q}_{\chi}q^{\prime}$ with $\chi$ being the chiralities $L$ and
$R$ and $Q$ being up or down type quarks. Since $\bf Z$ is not a unit matrix,
the FCNCs are arisen from the axial-vector currents and the corresponding
Lagrangian is given by

$\displaystyle{\cal L}_{FCNC}$ $\displaystyle=$ $\displaystyle
g_{A}\bar{q}\gamma_{\mu}(V^{Q}_{R}{\bf Z}V^{Q\dagger}_{R}P_{R}-V^{Q}_{L}{\bf
Z}V^{Q\dagger}_{L}P_{L})T^{b}qG^{b\mu}_{A}$ (6)

with $P_{L(R)}=(1\mp\gamma_{5})/2$. Since $V^{Q}_{\chi}$ are unknown matrices,
the FCNCs are associated with left and right-handed currents generally.
Nevertheless, if $V^{Q}_{R}=V^{Q}_{L}$, from Eq. (6) we see that the FCNCs are
only associated with axial-vector currents. In terms of the flavor indices,
the matrix $V^{q}_{\chi}{\bf Z}V^{q\dagger}_{\chi}$ can be decomposed as

$\displaystyle\left(V^{Q}_{\chi}{\bf Z}V^{Q\dagger}_{\chi}\right)_{ij}$
$\displaystyle=$ $\displaystyle\delta_{ij}+\left(V^{Q}_{\chi}({\bf
Z-1})V^{Q\dagger}_{\chi}\right)_{ij}=\delta_{ij}+(\zeta-1)(V^{Q}_{\chi})_{i3}(V^{Q*}_{\chi})_{j3}\,.$
(7)

Therefore, the Lagrangian of $b\to q$ transition can be written as

$\displaystyle{\cal L}_{b\to q}$ $\displaystyle=$ $\displaystyle
g_{A}\bar{q}\gamma_{\mu}(F^{QR}_{qb}P_{R}-F^{QL}_{qb}P_{L})T^{b}bG^{b\mu}_{A}$
(8)

with $F^{Q\chi}_{qb}=(\zeta-1)(V^{Q}_{\chi})_{i3}(V^{Q*}_{\chi})_{33}$ where
$i=(1,2,3)$ denotes the family order of the same type $Q$ quark. Based on Eq.
(8), we study the impacts of non-universal axigluon exchange on $\Delta B=2$
processes and the time-dependent CPAs in $B_{q}$ system.

By Eq. (8), the effective Hamiltonian for $\Delta B=2$ transitions which is
generated by the tree-level axigluon mediation can be written as

$\displaystyle{\cal H}^{A}_{\Delta B=2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{A}}{4m^{2}_{V}}\left[-\frac{1}{N_{C}}\left(\bar{q}\gamma_{\mu}(F^{DR}_{qb}P_{R}+F^{DL}_{qb}P_{L})b\right)^{2}\right.$
(9) $\displaystyle+$
$\displaystyle\left.\bar{q}_{\alpha}\gamma_{\mu}\left(F^{DR}_{qb}P_{R}+F^{DL}_{qb}P_{L}\right)b_{\beta}\bar{q}_{\beta}\gamma^{\mu}\left(F^{DR}_{qb}P_{R}+F^{DL}_{qb}P_{L}\right)b_{\alpha}\right]\,,$

where $N_{C}$ denotes the number of colors and we have used the identity

$\displaystyle
T^{b}_{ij}T^{b}_{k\ell}=-\frac{1}{2N_{C}}\delta_{ij}\delta_{k\ell}+\frac{1}{2}\delta_{i\ell}\delta_{jk}\,.$
(10)

In order to calculate the $B_{q}-\bar{B}_{q}$ mixing, we write the relevant
hadronic matrix elements to be

$\displaystyle\langle
B_{q}|\bar{q}\gamma_{\mu}P_{L(R)}b\bar{q}\gamma_{\mu}P_{L(R)}b|\bar{B}_{q}\rangle=\frac{1}{3}m_{B_{q}}f^{2}_{B_{q}}\hat{B}_{q}\,,$
$\displaystyle\langle
B_{q}|\bar{q}\gamma_{\mu}P_{R}b\bar{q}\gamma_{\mu}P_{L}b|\bar{B}_{q}\rangle=-\frac{5}{12}m_{B_{q}}f^{2}_{B_{q}}\hat{B}^{RL}_{1q}\,,$
$\displaystyle\langle
B_{q}|\bar{q}_{\alpha}\gamma_{\mu}P_{L}b_{\beta}\bar{q}_{\beta}\gamma^{\mu}P_{R}b_{\alpha}|\bar{B}_{q}\rangle=-\frac{7}{12}m_{B_{q}}f^{2}_{B_{q}}\hat{B}^{RL}_{2q}\,.$
(11)

To estimate the new physics effects, we employ the vacuum insertion method to
calculate the above matrix elements, i.e.
$\hat{B}_{q}\sim\hat{B}^{RL}_{1q}\sim\hat{B}^{RL}_{2q}\sim 1$ Gabbiani:1996hi
; Badin:2007bv . Additionally, in the heavy quark limit, we take $m_{b}\sim
m_{B_{q}}$. As a result, the transition matrix element for $B_{q}-\bar{B}_{q}$
oscillation mediated by axigluon exchange becomes

$\displaystyle M^{A,q}_{12}$ $\displaystyle=$ $\displaystyle\langle
B_{q}|{\cal H}^{A}_{\Delta
B=2}|\bar{B}_{q}\rangle=\frac{g^{2}_{A}}{18m^{2}_{V}}m_{B_{q}}f^{2}_{B_{q}}U^{D}_{qb}\,,$
$\displaystyle U^{D}_{qb}$ $\displaystyle=$
$\displaystyle(F^{DR}_{qb})^{2}+(F^{DL}_{qb})^{2}+4F^{DR}_{qb}F^{DL}_{qb}\,.$
(12)

For reducing the number of free parameters, we will take the approximation
$V^{Q}_{R}\approx V^{Q}_{L}=V^{D}$ in our analysis, i.e. $F^{DR}_{qb}\approx
F^{DL}_{qb}=F^{D}_{qb}$, then $U^{D}_{qb}=6(F^{D}_{qb})^{2}$. We note that the
approximation $V^{Q}_{R}\approx V^{Q}_{L}$ can be realized in hermitian Yukawa
matrices Chen:2001cv .

By combining the contributions of SM and axigluon, the transition matrix
element for $\Delta B=2$ can be formulated as

$\displaystyle M^{B_{q}}_{12}$ $\displaystyle=$ $\displaystyle|M^{\rm
SM,q}_{12}|R^{q}_{A}e^{i2(\beta_{q}+\phi^{\rm NP}_{q})}\,,$ (13)

where the new parameters are defined by

$\displaystyle R^{q}_{A}$ $\displaystyle=$
$\displaystyle\left(1+(r^{q}_{A})^{2}+2r^{q}_{A}\cos 2(\beta^{\rm
NP}_{q}-\beta_{q})\right)^{1/2}\,,$ $\displaystyle 2\beta^{\rm NP}_{q}$
$\displaystyle=$ $\displaystyle{\rm arg}(M^{A,q}_{12})\,,$ $\displaystyle
r^{q}_{A}$ $\displaystyle=$ $\displaystyle\frac{|M^{A,q}_{12}|}{|M^{\rm
SM,q}_{12}|}\,,$ $\displaystyle\tan 2\phi^{\rm NP}_{q}$ $\displaystyle=$
$\displaystyle\frac{r^{q}_{A}\sin 2(\beta^{\rm
NP}_{q}-\beta_{q})}{1+r^{q}_{A}\cos 2(\beta^{\rm NP}_{q}-\beta_{q})}\,,$ (14)

and $M^{SM,q}_{12}$ is given by BBL

$\displaystyle
M^{SM,q}_{12}=\frac{G^{2}_{F}m^{2}_{W}}{12\pi^{2}}\eta_{B}m_{B_{q}}f^{2}_{B_{q}}\hat{B}_{q}(V^{*}_{tq}V_{tb})^{2}S_{0}(x_{t})$
(15)

with $S_{0}(x_{t})=0.784x_{t}^{0.76}$, $x_{t}=(m_{t}/m_{W})^{2}$ and
$\eta_{B}\approx 0.55$ is the QCD correction to $S_{0}(x_{t})$ Hence, the mass
difference between heavy and light $B_{q}$ is $\Delta
m_{B_{q}}=2|M^{B_{q}}_{12}|=\Delta m^{\rm SM}_{B_{q}}R^{q}_{A}$. After
obtaining $M^{B_{q}}_{12}$, the time-dependent CPA through inclusive
semileptonic decays can be defined as Nakamura:2010zzi

$\displaystyle a^{q}_{s\ell}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\bar{B}_{q}(t)\to\ell^{+}X)-\Gamma(B_{q}(t)\to\ell^{-}X)}{\Gamma(\bar{B}_{q}(t)\to\ell^{+}X)+\Gamma(B_{q}(t)\to\ell^{-}X)}\,,$
(16) $\displaystyle=$ $\displaystyle\frac{1-|q/p|^{4}}{1+|q/p|^{4}}$

with

$\displaystyle\left(\frac{q}{p}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{M^{B_{q}^{*}}_{12}-i\Gamma^{B_{q}^{*}}_{12}/2}{M^{B_{q}}_{12}-i\Gamma^{B_{q}}_{12}/2}\,,$
(17)

where $\Gamma^{B_{q}}_{12}$ denotes the absorptive part of
$B_{q}\leftrightarrow\bar{B}_{q}$ transition. Due to $\Gamma^{B_{q}}_{12}\ll
M^{B_{q}}_{12}$, the wrong-sign charge asymmetry can be simplified as

$\displaystyle a^{q}_{s\ell}$ $\displaystyle=$ $\displaystyle
Im\left(\frac{\Gamma^{B_{q}}_{12}}{M^{B_{q}}_{12}}\right)\approx\frac{\Delta\Gamma^{\rm
SM}_{B_{q}}}{\Delta m_{B_{q}}}\sin(2\beta_{q}+2\phi^{\rm
NP}_{q}-\theta^{\Gamma}_{q})\,.$ (18)

Here, $\theta^{\Gamma}_{q}$ stands for the phase of $\Gamma^{B_{q}}_{12}$.
Since the absorptive part is dominated by the SM contribution, we will assume
that $\Gamma^{B_{q}}_{12}=\Gamma^{q,SM}_{12}$ in our numerical analysis. A
detailed discussions about new physics effects on $\Gamma^{B_{q}}_{12}$ can be
found in Refs. Dighe:2010nj ; Choudhury:2010ya . Since $a^{q}_{s\ell}$ is
associated with the CP phases directly, a non-zero charge asymmetry will be an
indication of CP violation. Accordingly, the like-sign charge asymmetry
defined in Eq. (3) can be written as Abazov:2010hv ; Grossman:2006ce

$\displaystyle{\cal A}^{b}_{s\ell}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(b\bar{b}\to\ell^{+}\ell^{+}X)-\Gamma(b\bar{b}\to\ell^{-}\ell^{-}X)}{\Gamma(b\bar{b}\to\ell^{+}\ell^{+}X)+\Gamma(b\bar{b}\to\ell^{-}\ell^{-}X)}\,,$
(19) $\displaystyle=$
$\displaystyle\frac{f_{d}Z_{d}a^{d}_{s\ell}+f_{s}Z_{s}a^{s}_{s\ell}}{f_{d}Z_{d}+f_{s}Z_{s}}\,,$

where $f_{q}$ is the production fraction of $B_{q}$ and

$\displaystyle Z_{q}$ $\displaystyle=$
$\displaystyle\frac{1}{1-y^{2}_{q}}-\frac{1}{1-x^{2}_{q}}\,,$ $\displaystyle
y_{q}$ $\displaystyle=$
$\displaystyle\frac{\Delta\Gamma_{B_{q}}}{2\Gamma_{B_{q}}}\,,\ \ \
x_{q}=\frac{\Delta m_{B_{q}}}{\Gamma_{B_{q}}}\,.$ (20)

Using $f_{d}=0.323(37)$, $f_{s}=0.118(15)$, $x_{d}=0.774(37)$, $y_{d}\sim 0$,
$x_{s}=26.2(5)$ and $y_{s}=0.046(27)$, the asymmetry can be rewritten as

$\displaystyle{\cal A}^{b}_{s\ell}=c_{d}a^{d}_{s\ell}+c_{s}a^{s}_{s\ell}$ (21)

with $c_{d}=0.506(43)$ and $c_{s}=0.494(43)$ Abazov:2010hv .

Another important time dependent CPA can be defined by Nakamura:2010zzi

$\displaystyle A_{f_{CP}}(t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\bar{B}_{q}(t)\to f_{CP})-\Gamma(B_{q}(t)\to
f_{CP})}{\Gamma(\bar{B}_{q}(t)\to f_{CP})+\Gamma(B_{q}(t)\to f_{CP})}\,,$
$\displaystyle=$ $\displaystyle S_{f_{CP}}\sin\Delta
m_{B_{q}}t-C_{f_{CP}}\cos\Delta m_{B_{q}}t\,,$ $\displaystyle S_{f_{CP}}$
$\displaystyle=$
$\displaystyle\frac{2Im\lambda_{f_{CP}}}{1+|\lambda_{f_{CP}}|^{2}}\,,\ \ \
C_{f_{CP}}=\frac{1-|\lambda_{f_{CP}}|^{2}}{1+|\lambda_{f_{CP}}|^{2}}$ (22)

with

$\displaystyle\lambda_{f_{CP}}$ $\displaystyle=$
$\displaystyle-\left(\frac{M^{B_{q}^{*}}_{12}}{M^{B_{q}}_{12}}\right)^{1/2}\frac{A(\bar{B}\to
f_{CP})}{A(B\to f_{CP})}=-e^{-2i(\beta_{q}+\phi^{\rm
NP}_{q})}\frac{\bar{A}_{f_{CP}}}{A_{f_{CP}}}\,,$ (23)

where $f_{CP}$ denotes the final CP eigenstate, $S_{f_{CP}}$ and $C_{f_{CP}}$
are the so-called mixing-induced and direct CPAs, $A_{f_{CP}}$ and
$\bar{A}_{f_{CP}}$ are the amplitudes of $B$ and $\bar{B}$ mesons decaying to
$f_{CP}$ and
$\bar{A}_{f_{CP}}/A_{f_{CP}}=-\eta_{f_{CP}}A_{f_{CP}}(\theta_{W}\to-\theta_{W})/A_{f_{CP}}(\theta_{W})$
with $\eta_{f_{CP}}$ and $\theta_{W}$ are the CP eigenvalue of $f_{CP}$ and
the weak CP phase respectively. Clearly, besides $\Delta B=2$ effects, the
mixing-induced CPA is also related to the $\Delta B=1$ process. In this paper,
we will concentrate on $f_{CP}=J/\Psi K_{S}$ and $\phi K_{S}$ for $q=d$ and on
$f_{CP}=J/\Psi\phi$ for $q=s$.

To calculate the decay amplitude of $B(\bar{B})\to f_{CP}$, we need to discuss
the interactions of $\Delta B=1$ processes. With the approximation
$V^{Q}_{R}\approx V^{Q}_{L}$, the effective Hamiltonian of $b\to
qq^{\prime}q^{\prime}$ can be expressed as

$\displaystyle{\cal H}_{b\to qq^{\prime}q^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{g_{A}}{m^{2}_{V}}F^{D}_{qb}\bar{q}\gamma_{\mu}\gamma_{5}T^{b}b\sum_{q^{\prime}=u,d,s,c}\bar{q}^{\prime}\gamma^{\mu}\left(g_{+}P_{R}+g_{-}P_{L}\right)T^{b}q^{\prime}$
(24)

with $g_{\pm}=g_{V}\pm g_{A}$. Using Eq. (10), we can rewrite the last
equation as

$\displaystyle{\cal H}^{A}_{b\to qq^{\prime}q^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V^{*}_{tq}V_{tb}\left[C^{\prime}_{q3}O^{q}_{3}+C^{\prime}_{q4}O^{q}_{4}+C^{R}_{q3}O^{qR}_{3}+C^{R}_{q4}O^{qR}_{4}\right.$
(25) $\displaystyle+$
$\displaystyle\left.C^{\prime}_{q5}O^{q}_{5}+C^{\prime}_{q6}O^{q}_{6}+C^{L}_{q5}O^{qL}_{5}+C^{qL}O^{qL}_{6}\right]$

in which the new Wilson coefficients are expressed by

$\displaystyle C^{\prime}_{q3}$ $\displaystyle=$
$\displaystyle\frac{1}{8N_{C}}\frac{\sqrt{2}F^{D}_{qb}}{G_{F}V^{*}_{tq}V_{tb}}\frac{g_{A}g_{-}}{m^{2}_{V}}\,,\
\ \ C^{\prime}_{q4}=-N_{C}C^{\prime}_{q3}\,,$ $\displaystyle C^{L}_{q5}$
$\displaystyle=$ $\displaystyle-C^{\prime}_{q3}\,,\ \ \
C^{L}_{q6}=-N_{C}C^{L}_{q5}\,,$ $\displaystyle C^{\prime}_{q5}$
$\displaystyle=$
$\displaystyle\frac{1}{8N_{C}}\frac{\sqrt{2}F^{D}_{qb}}{G_{F}V^{*}_{tq}V_{tb}}\frac{g_{A}g_{+}}{m^{2}_{V}}\,,\
\ \ C^{\prime}_{q6}=-N_{C}C^{\prime}_{q5}\,,$ $\displaystyle C^{R}_{q3}$
$\displaystyle=$ $\displaystyle-C^{\prime}_{q5}\,,\ \ \
C^{R}_{q4}=-N_{C}C^{R}_{q3}$ (26)

and the associated operators are

$\displaystyle O^{q}_{3}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V-A}\,,\
\ \
O^{q}_{4}=(\bar{q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A}\,,$
$\displaystyle O^{q}_{5}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V+A}\,,\
\ \
O^{q}_{6}=(\bar{q}_{\alpha}b_{\beta})_{V-A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A}\,,$
$\displaystyle O^{qR}_{3}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V+A}\,,\
\ \
O^{qR}_{4}=(\bar{q}_{\alpha}b_{\beta})_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V+A}\,,$
$\displaystyle O^{qL}_{5}$ $\displaystyle=$
$\displaystyle(\bar{q}b)_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}q^{\prime})_{V-A}\,,\
\ \
O^{qR}_{6}=(\bar{q}_{\alpha}b_{\beta})_{V+A}\sum_{q^{\prime}}(\bar{q}^{\prime}_{\beta}q^{\prime}_{\alpha})_{V-A}$
(27)

with $(\bar{f}^{\prime}f)_{V\pm
A}=\bar{f}^{\prime}\gamma_{\mu}(1\pm\gamma_{5})f$. Besides the new free
parameters that are introduced earlier, the non-leptonic $B$ decays suffer
from large uncertain QCD effects such as $\langle f_{CP}|{\cal H}_{b\to
qq^{\prime}q^{\prime}}|B\rangle$. For estimating the new physics effects, we
employ the naive factorization approach (NFA). Under the NFA, we find that the
related effective Wilson coefficients for $\bar{B}_{d}\to J/\Psi\bar{K}^{0}$
and $\bar{B}_{s}\to J/\Psi\phi$ are

$\displaystyle
C^{\prime}_{s3}+\frac{C^{\prime}_{s4}}{N_{C}}+C^{L}_{s5}+\frac{C^{R}_{s6}}{N_{C}}+C^{\prime}_{s5}+\frac{C^{\prime}_{s6}}{N_{C}}+C^{R}_{s3}+\frac{C^{R}_{s4}}{N_{C}}\,.$
(28)

With the results in Eq. (26), we clearly see that the influence of axigluon-
mediated effects on $J/\Psi(\bar{K}^{0},\phi)$ modes vanishes. In our analysis
we neglect the nonfactorizable contributions as they are subleading and
difficult to estimate. Now, only $B\to\phi K$ can display the axigluon-
mediated effects. Using NFA and the interactions of Eq. (25), the total decay
amplitude of $B\to\phi K$ is written as

$\displaystyle\bar{A}_{\phi\bar{K}^{0}}$ $\displaystyle=$
$\displaystyle\langle\phi\bar{K}^{0}|{\cal H}_{b\to
ss\bar{s}}|\bar{B}^{0}\rangle\,,$ (29) $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V^{*}_{ts}V_{tb}(a^{\rm
SM}+a^{\prime}_{s4}+a^{R}_{s4})\langle\phi|\bar{s}\gamma_{\mu}s|0\rangle\langle\bar{K}^{0}|\bar{s}\gamma^{\mu}b|\bar{B}\rangle$

where ${\cal H}_{b\to ss\bar{s}}$ is the sum of the SM and axigluon effective
Hamiltonian and $a^{\rm SM}=a_{3}+a_{4}+a_{5}$ with

$\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle
C_{3}+\frac{C_{4}}{N_{C}}\,,\ \ \ a_{4}=C_{4}+\frac{C_{3}}{N_{C}}\,,\ \ \
a_{5}=C_{5}+\frac{C_{6}}{N_{C}}\,,$ $\displaystyle
a^{\prime}_{s4}=C^{\prime}_{s4}+C^{\prime}_{s3}/N_{C}\,,\ \ \
a^{R}_{s4}=C^{R}_{s4}+C^{R}_{s3}/N_{C}\,.$

Here, $C_{3-6}$ are the effective Wilson coefficients from the gluon penguin
in the SM BBL . We note that the electroweak penguin contributions are very
small and thus we neglect them. Using $V_{ts}=-|V_{ts}|e^{-i\beta_{s}}$
Nakamura:2010zzi , we can write

$\displaystyle\frac{\bar{A}_{\phi\bar{K}^{0}}}{A_{\phi
K^{0}}}=-e^{2i\beta_{s}}\frac{a^{SM}+a^{R}_{s4}}{a^{SM}+a^{R^{*}}_{s4}}=-e^{2i(\beta_{s}+\theta^{\rm
NP}_{s})}$ (30)

with

$\displaystyle\tan\theta^{\rm NP}_{s}$ $\displaystyle=$
$\displaystyle\frac{|a^{R}_{s4}|\sin(\beta^{\rm
NP}_{s}-\beta_{s})}{a^{SM}+|a^{R}_{s4}|\cos(\beta^{\rm NP}_{s}-\beta_{s})}\,.$

By Eqs. (22) and (23), the mixing-induced CPA via $B_{d}\to\phi K^{0}$ decay
is obtained as

$\displaystyle S_{\phi K^{0}}$ $\displaystyle\equiv$ $\displaystyle\sin
2\beta_{\phi K^{0}}=\sin 2(\beta_{d}+\phi^{\rm NP}_{d}-\beta_{s}-\theta^{\rm
NP}_{s})\,,$ (31)

while the CPAs through $B_{d,s}\to J/\Psi(K_{S},\phi)$ decays are given by

$\displaystyle S_{J/\Psi K^{0}}$ $\displaystyle\equiv$ $\displaystyle\sin
2\beta_{J/\Psi K^{0}}=\sin 2(\beta_{d}+\phi^{\rm NP}_{d})\,,$ $\displaystyle
S_{J/\Psi\phi}$ $\displaystyle\equiv$ $\displaystyle\sin
2\beta^{J/\Psi\phi}_{s}=\sin 2(\beta_{s}+\phi^{\rm NP}_{s})\,.$ (32)

Although the measurement of $\sin 2\beta_{J/\Psi K^{0}}$ has approached to the
precision level, however, it might be difficult to tell if there exists new
physics by measuring $\sin 2\beta_{J/\Psi K^{0}}$ only. Nevertheless, one can
investigate a new asymmetry defined by Grossman:1996ke

$\displaystyle\Delta_{\beta_{d}}=\sin 2\beta_{J/\Psi K^{0}}-\sin 2\beta_{\phi
K^{0}}$ (33)

which is less than $5\%$ in the SM Grossman:1996ke . If a large value of
$\Delta_{\beta_{d}}$ is measured, it will be a strong hint for new physics
beyond SM.

##  III Numerical Analysis

So far, we have introduced seven new free parameters in the general chiral
color models and they are: two gauge couplings $g_{V,A}$, four parameters in
the two complex quantities $F^{D}_{qb}$ and $m_{V}$. In order to display the
dependence of $\Delta_{\beta_{d}}$ on $m_{V}$, we use the results in Ref.
Frampton:2009rk  and take $g_{V}=-0.577g_{s}$ and $g_{A}=-1.155g_{s}$ with
$\alpha_{s}=g_{s}^{2}/4\pi=0.119$. Thus, the five remaining parameters are
$|F^{D}_{qb}|$, $\beta^{\rm NP}_{q}$ for q=d, s and $m_{V}$. We list the input
values used for numerical calculations in Table 1, where the relevant CKM
matrix elements $V_{tq}=\bar{V}_{tq}\exp(-i\beta_{q})$ are obtained from the
UTfit Collaboration Bona:2009tn , the decay constant of $B_{q}$ is referred to
the result given by HPQCD Collaboration Gamiz:2009ku , the CDF and D$\O$
average value of $\Delta m_{B_{s}}$ is from Ref.
TheHeavyFlavorAveragingGroup:2010qj  and the SM Wilson coefficients of $b\to
qq^{\prime}\bar{q}^{\prime}$ are obtained from Ref. BBL . Other inputs are
obtained from particle data group (PDG) Nakamura:2010zzi .

Table 1: Numerical inputs for the parameters in the SM.  $\bar{V}_{td}$ | $\beta_{d}$ | $\bar{V}_{ts}$ | $\beta_{s}$ | $m_{B_{d}}$ | $m_{B_{s}}$
---|---|---|---|---|---
$8.51(22)\times 10^{-3}$ | $(22\pm 0.8)^{\circ}$ | $-4.07(22)\times 10^{-2}$ | $-(1.03\pm 0.06)^{\circ}$ | 5.28 GeV | 5.37 GeV
$f_{B_{d}}\sqrt{\hat{B}}_{d}$ [MeV] |  $f_{B_{s}}\sqrt{\hat{B_{s}}}$ [MeV] |  $f_{B_{d}}$ [MeV] |  $f_{B_{s}}$ [MeV] | $S^{\rm exp}_{J/\Psi K^{0}}$ | $\bar{m}_{t}(\bar{m}_{t})$
$216\pm 15$ | $266\pm 18$ | $190\pm 13$ | $231\pm 15$ | $0.655\pm 0.024$ | 163.8 GeV
$(\Delta m_{B_{d}})^{\rm exp}$ | $(\Delta m_{B_{s}})^{\rm exp}$ | $C_{3}$ | $C_{4}$ | $C_{5}$ | $C_{6}$
$0.507\pm 0.005$ ps-1 |  $17.78\pm 0.12$ ps-1 | $0.013$ | $-0.0335$ | 0.0095 | $-0.0399$

After setting up the inputs, we study the contributions of the axigluon to
FCNC processes and their associated CPAs that are defined earlier. We start by
exploring the allowed parameter space. Since the non-universal axigluon
induces FCNCs at tree level, the observed $B_{q}-\bar{B}_{q}$ mixing parameter
$\Delta m_{B_{q}}$ will give a strict constraint on the parameter space. In
Fig. 1(a)[(b)], the allowed range for $\beta^{\rm NP}_{d[s]}$ and
$|F^{D}_{d(s)b}|/m_{V}$ (in units of $10^{-6}$) is drawn by the down-left
hatched lines where we have taken the SM contributions ($\Delta m^{\rm
SM}_{B_{d}}$, $\Delta m^{\rm SM}_{B_{s}}$) to be $(0.506,\,17.76)$ ps-1.
Furthermore, since the observed $S_{J/\Psi K^{0}}$ has been a precise
measurement, it is plausible that the current data can further exclude the
values of the parameter space which are allowed by $\Delta m_{B_{d}}$. Taking
$2\sigma$ errors of $S^{\rm exp}_{J/\Psi K^{0}}$ as the experimental bound,
the allowed region for $\beta^{\rm NP}_{d}$ and $|F^{D}_{db}|/m_{V}$ sketched
by down-right hatched lines is plotted in Fig. 1(a). Clearly, $S^{\rm
exp}_{J/\Psi K^{0}}$ gives a strong constraint on the parameters that
contribute to $M^{B_{d}}_{12}$. From Fig. 1, we see that, except the two
narrow regions correspond to $|F^{D}_{db}|/m_{V}>1\times 10^{-6}$ GeV-1, the
allowed values of $|F^{D}_{db}|/m_{V}$ are limited to be
$|F^{D}_{db}|/m_{V}\leq 0.4\times 10^{-6}$ GeV-1, whereas the allowed values
of $|F^{D}_{sb}|/m_{V}$ can be one order of magnitude larger than those of
$|F^{D}_{db}|/m_{V}$. In general, the range of the CP violating phase
$\beta^{\rm NP}_{q}$ is $[-\pi,\pi]$, for illustration, we just show the
results within $[-\pi,0]$. The pattern of the constraint in $[0,\pi]$ is
similar to that in $[-\pi,0]$. In order to illustrate the influence of the
uncertainties of the SM on the free parameters, in Fig. 2 we plot the allowed
values of $|F^{D}_{sb}|/m_{V}$ and $\beta^{\rm NP}_{s}$ by including the
errors of $f_{B_{s}}\sqrt{\hat{B}_{s}}$ and $V_{ts}$. Comparing with Fig.
1(b), we see that the allowed range is extended slightly. We note that due to
the strict constraint of $S^{\rm exp}_{J/\Psi K^{0}}$, the bounds on the
parameters for $b\to d$ transition are not changed significantly, therefore,
we don’t show the corresponding diagram for $b\to d$ transition.

Figure 1: (a)[(b)] Constraints on $\beta^{\rm NP}_{d[s]}$ and
$|F^{D}_{d[s]b}|/m_{V}$ (in units of $10^{-6}$) obtained from
$B_{d[s]}-\bar{B}_{d[s]}$ mixing (down-left hatched lines) and $\sin
2\beta_{J/\Psi K^{0}}$ (down-right hatched lines).  Figure 2: Legend is the
same as Fig. 1(b), but the errors of $\Delta m_{B_{s}}$ in the SM are
included.

According to Eq. (19), if we assume no new CP violating phase in semi-leptonic
decays, we will see that the charge asymmetry ${\cal A}^{b}_{s\ell}$ depends
on two kinds of CP violating phases. One of the two phases is originated from
$B_{d}-\bar{B}_{d}$ mixing which is a $b\to d$ transition, and the other phase
is originated from $B_{s}-\bar{B}_{s}$ which is associated with $b\to s$
transition. In other words, we have to consider four parameters $\beta^{\rm
NP}_{(d,s)}$ and $|F^{D}_{(d,s)b}|/m_{V}$ simultaneously. However, if we
consider $b\to(d,s)$ transitions at the same time, we may induce a large
effect on $s\to d$ because the $\Delta K=2$ process is associated with
$F^{D}_{ds}=(\zeta-1)V^{D}_{13}V^{D*}_{23}$, i.e. $B_{d}-\bar{B}_{d}$,
$B_{s}-\bar{B}_{s}$ and $K^{0}-\bar{K}^{0}$ mixings have strong correlations.
In order to avoid inducing a large $K^{0}-\bar{K}^{0}$ mixing, we set a small
value for $V^{D}_{13}$. This is consistent with the results shown in Fig. 1(a)
where $\Delta m_{B_{d}}$ and $S_{J/\Psi K^{0}}$ strongly constrain
$|F^{D}_{db}|/m_{V}$. Hence, we assume that $a^{d}_{s\ell}$ is dominated by
the SM contribution where $a^{d}_{s\ell}(SM)=-4.8\times 10^{-4}$ Lenz:2006hd .
Consequently, the enhanced $|{\cal A}^{b}_{s\ell}|$ can be attributed to $b\to
s$ transition. With Eqs. (14), (18) and (21) and the values given in Table 1,
the contours of ${\cal A}^{b}_{s\ell}$ as a function of $\beta^{\rm NP}_{s}$
and $|F^{D}_{sb}|/m_{V}$ are shown in Fig. 3(a) where the values of the
contours are in units of $10^{-4}$. As can be seen from the figure, not only
the sign of ${\cal A}^{b}_{s\ell}$ can fit the data, but also its magnitude
can be enhanced by axigluon-mediated effects. By combining with the constraint
of $\Delta m_{B_{s}}$, the region of $\beta^{\rm NP}_{s}$ for large $|{\cal
A}^{b}_{s\ell}|$ is limited. In Fig. 3(b), we display ${\cal A}^{b}_{s\ell}$
as a function of $\beta^{\rm NP}_{s}$ where the solid, dashed and dash-dotted
line represents $|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$ GeV-1,
respectively. As shown in the figure, negative and positive values of
$\beta^{\rm NP}_{s}$ can enhance ${\cal A}^{b}_{s\ell}$. It should be noted
that, although the axigluon-mediated effect can not enhance the like-sign
charge asymmetry to be as large as the central value of DØ data, however,
$|{\cal A}^{b}_{s\ell}|$ is enhanced by one order of magnitude larger than the
SM prediction.

Figure 3: (a) Contours of ${\cal A}^{b}_{s\ell}$ as a function of $\beta^{\rm
NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ (in units of $10^{-6}$). (b) ${\cal
A}^{b}_{s\ell}$ as a function of $\beta^{\rm NP}_{s}$, where the solid, dashed
and dash-dotted line stands for $|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$,
respectively. The values on the plot (a) are ${\cal A}^{b}_{s\ell}$ in units
of $10^{-4}$.

Unlike the case of the charge asymmetry, the time-dependent CPA of $B_{s}\to
J/\Psi\phi$ decay depends only on the CP phase in $b\to s$ transition. As a
consequence, when the new CP violating effects are small in $M^{B_{d}}_{12}$,
${\cal A}^{b}_{s\ell}$ and $S_{J/\Psi\phi}$ defined in Eq. (22) can have a
strong correlation. By using Eq. (32), the contours of $S_{J/\Psi\phi}$ are
plotted as a function of $\beta^{\rm NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ in Fig.
4(a). From the figure, we find that large $S_{J/\Psi\phi}$ can be archived
when ${\cal A}^{b}_{s\ell}$ is one order of magnitude larger than the SM
prediction. Moreover, we also plot $S_{J/\Psi\phi}$ as a function of
$\beta^{\rm NP}_{s}$ in Fig. 4(b), where the solid, dashed and dash-dotted
line denotes $|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$ GeV-1, respectively.
Clearly, a large ${\cal A}^{b}_{s\ell}$ indicates a large $S_{J/\Psi\phi}$.
Although the measured values of ${\cal A}^{b}_{s\ell}$ and $S_{J/\Psi\phi}$
contain large errors, however, a few sigma deviations from the SM prediction
can be considered as a hint for new physics effect.

Figure 4:  (a) Contours of $S_{J/\Psi\phi}$ as a function of $\beta^{\rm
NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ (in units of $10^{-6}$). (b)
$S_{J/\Psi\phi}$ as a function of $\beta^{\rm NP}_{s}$, where the solid,
dashed and dash-dotted line represents $|F^{D}_{sb}|/m_{V}=(3,4,5)\times
10^{-6}$, respectively.

It is well known that the golden process to measure the angle $\beta_{d}$ in
the SM is $B_{d}\to J/\Psi K^{0}$ which is dominated by tree diagram. Although
new physics can also affect this decay mode via $b\to sc\bar{c}$ transition,
however, as discussed in Eq. (28), the axigluon contributions to $B_{d}\to
J/\Psi K^{0}$ vanish. Hence, the source of the time-dependent CPA in $B_{d}\to
J/\Psi K^{0}$ decay is only originated from the $B_{d}$ oscillation. Since
$\beta_{d}$ is also a parameter in the SM, a single measurement of $S_{J/\Psi
K^{0}}$ or $\sin 2\beta_{J/\Psi K^{0}}$ is hard to uncover the new physics. To
probe the new physics, the best way is to compare the CPA of $J/\Psi K^{0}$
with that of $\phi K_{S}$. Therefore, we do not discuss each of $S_{J/\Psi
K^{0}}$ and $S_{\phi K^{0}}$ separately. Instead, we focus on the CPA
difference $\Delta_{\beta_{d}}$ which is defined in Eq. (33) and it is only
few percent in the SM. By Eqs. (31) and (33), we see that although
$\Delta_{\beta_{d}}$ is insensitive to $F^{D}_{db}$ however it is strongly
dependent on $F^{D}_{sb}$. To see the contributions of the axigluon to
$\Delta_{\beta_{d}}$, we present the contours of $\Delta_{\beta_{d}}$ as a
function of $\beta^{\rm NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ in Fig. 5(a)[(b)],
where we have set $|F^{D}_{db}|/m_{V}=0$ and figure (a)[(b)] corresponds to
$m_{V}=0.5[1]$ TeV. Since the decay amplitude of $B\to\phi K$ depends on
$F^{D}_{sb}/m^{2}_{V}$ while $\Delta m_{B_{s}}$ is $(F^{D}_{sb}/m_{V})^{2}$,
thus a specific value for $m_{V}$ has to be given when calculating the
contours of $\Delta_{\beta_{d}}$. For further understanding the $\beta^{\rm
NP}_{s}$-dependence, we display $\Delta_{\beta_{d}}$ as a function of
$\beta^{\rm NP}_{s}$ in Fig. 6, where figure (a)[(b)] is for $m_{V}=0.5[1]$
TeV and the solid, dashed and dash-dotted line stands for
$|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$ GeV-1, respectively. It is clear
that the axigluon contributions to $\Delta_{\beta_{d}}$ are larger than that
of the SM.

Figure 5:  Contours of $\Delta_{\beta_{d}}$ as a function of $\beta^{\rm
NP}_{s}$ and $|F^{D}_{sb}|/m_{V}$ (in units of $10^{-6}$) with (a) $m_{V}=0.5$
TeV and (b) $m_{V}=1$ TeV. Figure 6:  $\Delta_{\beta_{d}}$ as a function of
$\beta^{\rm NP}_{s}$ with (a) $m_{V}=0.5$ TeV and (b) $m_{V}=1$ TeV, where the
solid, dashed and dash-dotted line represents
$|F^{D}_{sb}|/m_{V}=(3,4,5)\times 10^{-6}$, respectively.

In order to comprehend further the correlations among various physical
observables under the influence of the axigluon, we display the scatter plots
of ${\cal A}^{b}_{s\ell}$, $S_{J/\Psi\phi}$ and $\Delta_{\beta_{d}}$ with
$m_{V}=0.5(1)$ TeV versus $\Delta m_{B_{s}}$ in Fig. 7, where we have chosen
the range of $\beta^{\rm NP}_{s}$ to be $[-\pi,0]$. As an illustration, we
also show the scatter plots of ( ${\cal A}^{b}_{s\ell}$, $S_{J/\Psi\phi}$) and
(${\cal A}^{b}_{s\ell}$, $\Delta_{\beta_{d}}$) with $m_{V}=1$ TeV in Fig. 8,
in which the constraint of $\Delta m_{B_{s}}$ has been included and
$\beta^{\rm NP}_{s}$ belongs to $[-\pi,0]$. By Fig. 8(a), we see that the
correlation between ${\cal A}^{b}_{s\ell}$ and $S_{J/\Psi\phi}$ is linear,
where this behavior can be understood by the linear dependence between the
like-sign charge asymmetry and the mixing-induced CPA of $B_{s}$. Due to the
linearity, we expect that the correlation between $S_{J/\Psi\phi}$ and
$\Delta_{\beta_{d}}$ should be similar to that between ${\cal A}^{b}_{s\ell}$
and $\Delta_{\beta_{d}}$. Therefore, we just show the latter case in Fig.
8(b).

Figure 7:  Correlations between $\Delta m_{B_{s}}$ and (a) ${\cal
A}^{b}_{s\ell}$, (b) $S_{J/\Psi\phi}$, (c)[(d)] $\Delta_{\beta_{d}}$ with
$m_{V}=0.5[1]$ TeV, where the angle $\beta^{\rm NP}_{s}$ belongs to
$[-\pi,0]$. Figure 8:  (a) Correlation between ${\cal A}^{b}_{s\ell}$ and
$S_{J/\Psi\phi}$ and (b) correlation between ${\cal A}^{b}_{s\ell}$ and
$\Delta_{\beta_{d}}$ with $m_{V}=1$ TeV, where the constraint of $\Delta
m_{B_{s}}$ has been included and the angle $\beta^{\rm NP}_{s}$ belongs to
$[-\pi,0]$.

##  IV Conclusion

In general, a flavor non-universal axigluon in generalized chiral color models
can induce FCNCs at tree level. We study phenomenologically the axigluon-
mediated effects on $\Delta B=2$ FCNC processes and the associated CPAs. We
find that although $\Delta m_{B_{q}}$ strongly constrain the free parameters,
the precise measurement of $S_{J/\Psi K^{0}}$ can further exclude the
parameter space of $b\to d$ transition. Furthermore, for avoiding inducing
large $K^{0}-\bar{K}^{0}$ mixing, the parameter $V^{D}_{13}$ is chosen to be
small so that the like-sign charge asymmetry ${\cal A}^{b}_{s\ell}$ and
$\Delta_{\beta_{d}}$ are insensitive to the parameters of $b\to d$ transition.
As a result, the CP violating observables ${\cal A}^{b}_{s\ell}$,
$S_{J\Psi\phi}$ and $\Delta_{\beta_{d}}$ are strongly correlated and are only
sensitive to the parameters of $b\to s$ transition.

By the study, we find that the axigluon effects do not only preserve the
negative sign in ${\cal A}^{b}_{s\ell}$, but also enhance its magnitude by one
order of magnitude larger than the SM prediction. Subsequently, the associated
values of the parameters can also enhance the CPA $S_{J/\Psi\phi}$ and the CPA
difference $\Delta_{\beta_{q}}$ largely although they are only few percent in
the SM.

## Acknowledgement

This work is supported by the National Science Council of R.O.C. under Grant
No. NSC-97-2112-M-006-001-MY3. The author C.H.C would like to thank Prof.
Young-Chung Hsue for his help on using plot tool. G. Faisel would like thank
the National Center for Theoretical Sciences (NCTS) at Cheng Kung University
for the hospitality where this work has been done.

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