b

ts

model

ans tos. The

or of

tree-

Physics Letters B 596 (2004) 229–239

www.elsevier.com/locate/physlet

Bs–Bs mixing in Z′ models with flavor-changing neutral curren

Vernon Bargera, Cheng-Wei Chianga, Jing Jiangb, Paul Langackerc

a Department of Physics, University of Wisconsin, Madison, WI 53706, USAb HEP Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA

c Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA

Received 19 May 2004; received in revised form 22 June 2004; accepted 25 June 2004

Editor: M. Cvetic

Abstract

In models with an extraU(1)′ gauge boson family non-universal couplings to the weak eigenstates of the standardfermions generally induce flavor-changing neutral currents. This phenomenon leads to interesting results in variousB mesondecays, for which recent data indicate hints of new physics involving significant contributions fromb → s transitions. Weanalyze theBs system, emphasizing the effects of aZ′ on the mass difference andCP asymmetries. 2004 Elsevier B.V. All rights reserved.

1. Introduction

The study ofB physics and the associatedCP-violating observables has been suggested as a good meextract information on new physics at low energy scales[1–7]. SinceB–B mixing is a loop-mediated proceswithin the standard model (SM), it offers an opportunity to see the footprints of physics beyond the SMcurrently observed�Md = 0.489± 0.008 ps−1 [8] and its mixing phase sin2β = 0.736± 0.049 extracted fromtheBd → J/ψKS mode[9] agree well with constraints obtained from other experiments[10]. However, no suchinformation other than a lower bound�Ms > 14.5 ps−1 [11] is available for theBs meson yet.

Based upon SM predictions,�MBs is expected to be about 18 ps−1 and its mixing phaseφs only a coupleof degrees. In contrast to theBd system, the more than 25 times larger oscillation frequency and a factfour lower hadronization rate fromb quarks pose the primary challenges in the study ofBs oscillation andCPasymmetries. Since theBs → J/ψφ decay is dominated by a Cabibbo–Kobayashi–Maskawa (CKM) favored

E-mail addresses: barger@oriole.physics.wisc.edu(V. Barger),chengwei@physics.wisc.edu(C.-W. Chiang),jiangj@hep.anl.gov(J. Jiang),pgl@electroweak.hep.upenn.edu(P. Langacker).

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2004.06.105

230 V. Barger et al. / Physics Letters B 596 (2004) 229–239

most

t ofsicse

ermions

nglet)

ve

ls

al.theing in a

dion

darized in

sed

level process,b → ccs, that does not involve a different weak phase in the SM, its asymmetry provides thereliable information about the mixing phaseφs .

Although new physics contributions may not compete with the SM processes in most of theb → ccs decays,they can play an important role inBs–Bs mixing because of its loop nature in the SM[12]. In particular, the mixingcan be significantly modified in models in which a tree-levelb → s transition is present. Thus, measurementhe properties ofBs meson mixing is of high interest in futureB physics studies as a means to reveal new phy[13,14]. Since the currentB factories do not run at theΥ (5S) resonance to produceBs mesons, it is one of thprimary objectives of hadronic colliders to studyBs oscillation and decay in the coming years[15,16].

Flavor changing neutral currents (FCNC) coupled to an extraU(1)′ gauge boson arise when theZ′ couplings tophysical fermion eigenstates are non-diagonal. One way for this to happen is by the introduction of exotic fwith differentU(1)′ charges that mix with the SM fermions[17–21]as occurs inE6 models. In theE6 case, mixingof the right-handed ordinary and exotic quarks, allSU(2)L singlets, induces FCNC mediated by a heavyZ′ or by(small) Z–Z′ mixing, so the quark mixing can be large. Mixing between ordinary (doublet) and exotic (sileft-handed quarks induces FCNC mediated by the SMZ boson[21]. We will also allow for this possibility, but inthis case the quark mixing must be very small.

Another possibility involves family non-universal couplings. It is well-known that string models naturally giextraU(1)′ groups, at least one of which has family non-universal couplings to the SM fermions[22–25]. Generi-cally, the physical and gauge eigenstates do not coincide. Here, unlike the above-mentionedE6 case, off-diagonacouplings of fermions to theZ′ boson can be obtained without mixing with additional fermion states. In these typeof models, both left-handed and right-handed fermions can have family non-diagonal couplings with theZ′, whilecouplings to theZ are family diagonal (up to small effects fromZ–Z′ mixing).

TheZ′ contributions toBs–Bs mixing are related to those for hadronic, semileptonic, and leptonicB decays inspecific models in which the diagonalZ′ couplings toqq, �+�−, etc., are known, but are independent in gener1

We have found that in specific models,Bs–Bs mixing effects can be significant while being consistent withother constraints; these results will be presented elsewhere. In the present Letter, we will treat the mixmodel-independent way.

Recently, we have studied the implications of a sizeable off-diagonalZ′ coupling between the bottom anstrange quark in the indirectCP asymmetry ofB → φKS decay[28], which appears to show a significant deviatfrom the SM prediction[5,6,29,30]. Here we extend our analysis toBs–Bs mixing where theZ′ contributions alsoenter at the tree level. Applications to theB → πK anomaly are under investigation[31].

The Letter is organized as follows. In Section2, we review the basic formalism ofBs–Bs mixing. In Sec-tion 3, we evaluate�Ms in the SM. In Section4, we include theZ′ contributions, allowing both left-handed anright-handed couplings in the mixing, and study their effects on observables. Our main results are summSection5.

2. Bs–Bs mixing

In the conventional decomposition of the heavy and light eigenstates

(1)|Bs〉L = p∣∣B0

s

⟩ + q∣∣B0

s

⟩, |Bs〉H = p

∣∣B0s

⟩ − q∣∣B0

s

⟩,

the mixing factor

(2)

(q

p

)SM

�√

MSM∗12

MSM12

,

1 Bs–Bs mixing in leptophobicE6 models was considered in Ref.[21]. A model with non-universal right-handed couplings was discusin [26]. Mechanisms for flavor change in dynamical symmetry breaking models are described in[27].

V. Barger et al. / Physics Letters B 596 (2004) 229–239 231

-

onent,

Unlike

d

as the

L)

has a phase

(3)φSMs = 2 arg(VtbV

∗t s) = −2λ2η � −2◦, sin 2φSM

s � −0.07,

where the theoretical expectationΓ SM12 � MSM

12 is used. The approximate formula Eq.(2) receives a small correction onceΓ SM

12 is included. Model independently, this only shiftsφs is at the few percent level. With errors onλandη included, we have the SM expectation that sin 2φSM

s � −0.07± 0.01.The off-diagonal element of the decay matrix,Γ SM

12 , is evaluated by considering decay channels that are commto bothBs andBs mesons, andM12 is the off-diagonal element of the mass matrix. Due to the CKM enhancemΓ SM

12 is dominated by the charm-quark contributions over the up-quark contribution in a box diagram.the Kaon system,Γ SM

12 is much smaller thanMSM12 for B mesons because the former is related to theB meson

decays and set by the scale of its mass, whereas the latter is proportional tom2t . We can safely assume thatΓ12 is

not significantly modified by new physics becauseΓ12 receives major contributions from CKM favoredb → ccs

decays in the SM, and the SM resultΓ12 � M12 is unlikely to change.The mass difference of the two physical states is

(4)�Ms ≡ MH − ML � 2|M12| .

The width difference is

(5)�Γ ≡ ΓH − ΓL = 2 Re(M∗12Γ12)

|M12| = 2|Γ12|cosθ,

where the relative phase isθ = arg(M12/Γ12). SinceΓ12 is dominated by the contributions from CKM favoreb → ccs decays, we haveθ = arg

(−(VtbV∗t s)

2/(VcbV∗cs)

2) � π [32], and thus�Γ � −2|Γ12| is negative in the

SM. AlthoughΓ12 is unlikely to be affected by new physics, the width difference always increases as longweak phase ofM12 gets modified[33].

The observability ofBs–Bs oscillations is often indicated by the parameter

(6)xs ≡ �Ms

Γs

,

whereΓs = (4.51± 0.18) × 10−13 GeV, converted from the world average lifetimeτs = 1.461± 0.057 ps[8].The expected large value ofxs is a challenge for experimental searches. Currently, the result from all ALEPH[34],CDF [35], DELPHI [36], OPAL [37], and SLD[38] studies of�Ms with a combined 95% confidence level (Csensitivity on�Ms of 18.3 ps−1 gives[11]

(7)�Ms > 14.5 ps−1 and xs > 20.8.

It is also measured thatmBs = 5369.6± 2.4 MeV [8] and�Γs/Γs = −0.16+0.15−0.16(|�Γs|/Γs < 0.54) (with the 95%

CL upper bound given in parentheses[11]) consistent with recent next-to-leading-order (NLO) QCD estimates[39].In comparison, theBd system hasmBd = 5279.4± 0.5 MeV, �Md = (0.489± 0.008) ps−1, xd = 0.755± 0.015,andτBd = 1.542± 0.076 ps[8].

3. �Ms in the SM

The |�B| = 2 and|�S| = 2 operators relevant for our discussions are:

OLL = [sγµ(1− γ5)b

][sγ µ(1− γ5)b

], OLR

1 = [sγµ(1− γ5)b

][sγ µ(1+ γ5)b

],

(8)OLR2 = [

s(1− γ5)b][

s(1+ γ5)b], ORR = [

sγµ(1+ γ5)b][

sγ µ(1+ γ5)b].

232 V. Barger et al. / Physics Letters B 596 (2004) 229–239

r-n,

d in

a-ng

, we

rage.

siveme

ate

Because of theV –A structure, only the operatorOLL contributes toBs–Bs mixing in the SM. The other three opeators appear in theZ′ models because of the right-handed couplingsand operator mixing through renormalizatioas considered in the next section.

In the SM the contributions to

(9)MSM12 � 1

2mBs

⟨B0

s

∣∣HSMeff

∣∣B0s

⟩are dominated by the top quark loop. The result, accurate to NLO in QCD, is given by[40]

(10)MSM12 = G2

F

12π2M2

WmBs f2Bs

(VtbV∗t s)

2η2BS0(xt )[αs(mb)

]−6/23[1+ αs(mb)

4πJ5

]BLL(mb),

wherext = (mt(mt)/MW )2 and

(11)S0(x) = 4x − 11x2 + x3

4(1− x)2 − 3x3 ln x

2(1− x)3 .

Usingmt(mt ) = 170± 5 GeV, we findS0(xt ) = 2.463. The NLO short-distance QCD corrections are encodethe parametersη2B � 0.551 andJ5 � 1.627[40]. The bag parameterBLL(µ) is defined through the relation

(12)〈Bs |OLL|Bs〉 ≡ 8

3m2

Bsf 2

BsBLL(µ).

In the following numerical analysis, we will useGF = 1.16639× 10−5 GeV−2 andMW = 80.423± 0.039 GeV[8], and write the SM part of�Ms as

(13)�MSMs = 1.19

∣∣∣∣VtbV∗t s

0.04

∣∣∣∣2( fBs

23 MeV

)2(BLL(mb)

0.872

)× 10−11 GeV.

Current lattice calculations still show quite large errors on the hadronic parametersfBs = 230± 30 MeV andBLL(mb) = 0.872± 0.005[41–43]. However, the ratio

(14)ξ ≡ fBs

√BBs

fBd

√BBd

can be determined with a much smaller theoretical error, whereBBq is the renormalization-independent bag prameter for theBq meson (q = d, s). Therefore, the error on�Ms within the SM can be evaluated by compariwith �Md , i.e.,

(15)�MSMs = �MSM

d ξmBs

mBd

(1− λ2)2

λ2[(1− ρ)2 + η2] .Using the measured values of the Wolfenstein parameters[44] λ = 0.2265± 0.0024,A = 0.801± 0.025, ρ =0.189± 0.079, andη = 0.358± 0.044 [10], ξ = 1.24± 0.07 [45], and the mass parameters quoted aboveobtain the SM predictions

(16)�MSMs = (1.19± 0.24) × 10−11 GeV = 18.0± 3.7 ps−1, xSM

s = 26.3± 5.5.

As noted above, the central value ofxs is slightly larger than the current sensitivity based upon the world aveRecent LHC studies show that with one year of data,�Ms can be explored up to 30 ps−1 (ATLAS), 26 ps−1

(CMS), and 48 ps−1 (LHCb) (corresponding toxs up to 46, 42, and 75); the LHCb result is based on excluhadronic decay modes[16]. The sensitivity of both CDF and BTeV onxs can also reach up to 75 using the samodes[15], for a luminosity of 2 fb−1. The sensitivity on sin2φs is correlated with the value ofxs , and it becomesworse asxs increases. A statistical error of a few times 10−2 can be reached at CMS and LHCb for moderxs � 40 [16].

V. Barger et al. / Physics Letters B 596 (2004) 229–239 233

ee

me

ratioce

-

of

4. Z′ contributions

For simplicity, we assume that there is no mixing between the SMZ and theZ′ (small mixing effects can beasily incorporated[17]). A purely left-handed off-diagonalZ′ coupling tob ands quarks results in an effectiv|�B| = 2, |�S| = 2 Hamiltonian at theMW scale of

(17)HZ′eff = GF√

2

(g2MZ

g1MZ′BL

sb

)2

OLL(mb) ≡ GF√2

ρ2Le2iφLOLL(mb),

whereg2 is theU(1)′ gauge coupling,g1 = e/(sinθW cosθW ), MZ′ is the mass of theZ′, andBLsb is the FCNCZ′

coupling to the bottom and strange quarks. The parametersρL and the weak phaseφL in theZ′ model are definedby the second equality. Generically, we expect thatg2/g1 ∼ 1 if both U(1) groups have the same origin frosome grand unified theory, andMZ/MZ′ ∼ 0.1 for a TeV-scaleZ′. If |BL

sb| ∼ |VtbV∗t s|, then an order-of-magnitud

estimate gives usρL ∼ O(10−3), which is in the ballpark of giving significant contributions to theBs–Bs mixing.TheZ′ does not contribute toΓ12 at tree level because the intermediateZ′ cannot be on shell. After evolving fromtheMW scale tomb, the effective Hamiltonian becomes[40]

(18)HZ′eff = GF√

2

[1+ αs(mb) − αs(MW)

4πJ5

]R6/23ρ2

Le2iφLOLL(mb),

where R = αs(MW)/αs(mb). Although the above effective Hamiltonian is largely suppressed by the(g2MZ)/(g1MZ′), it contains only one power ofGF in comparison with the corresponding quadratic dependenin the SM because theZ′-mediated process occurs at tree level.

The full description of the running of the Wilson coefficient from theMW scale tomb can be found in[40]. Weonly repeat the directly relevant steps here. The renormalization group equation for the Wilson coefficients�C,

(19)d

d lnµ�C = γ T (g) �C(µ),

can be solved with the help of theU matrix

(20)�C(µ) = U(µ,MW ) �C(MW),

in which γ T (g) is the transpose of the anomalous dimension matrixγ (g). With the help ofdg/d lnµ = β(g), U

obeys the same equation as�C(µ). We expandγ (g) to the first two terms in the perturbative expansion,

(21)γ (αs) = γ (0) αs

4π+ γ (1)

(αs

4π

)2

.

To this order the evolution matrixU(µ,m) is given by

(22)U(µ,m) =(

1+ αs(µ)

4πJ

)U(0)(µ,m)

(1− αs(m)

4πJ

),

whereU(0) is the evolution matrix in leading logarithmic approximation and the matrixJ expresses the next-toleading corrections. We have

(23)U(0)(µ,m) = V

([αs(m)

αs(µ)

] �γ (0)/2β0)

D

V −1,

whereV diagonalizesγ (0)T , i.e.,γ (0)D = V −1γ (0)T V , and �γ (0) is the vector containing the diagonal elements

the diagonal matrixγ (0)D . In terms ofG = V −1γ (1)T V and a matrixH whose elements are

(24)Hij = δij γ(0)i

β1

2β20

− Gij

2β0 + γ(0) − γ

(0),

i j

234 V. Barger et al. / Physics Letters B 596 (2004) 229–239

low

GrG

e

ue

Fig. 1. Three-dimensional plot ofxs (a) and sin 2φs (b) versusρL andφL with aZ′-mediated FCNC for left-handedb ands quarks. The colorshadings in both plots have no specific physical meaning.

the matrixJ is given byJ = V HV −1.The operatorsOLL andORR do not mix with others under renormalization. Their Wilson coefficients fol

exactly the same RGEs, where the above-mentioned matrices are all simple numbers. The factor

(25)

[1+ αs(mb) − αs(MW)

4πJ5

]R6/23

in Eq. (18) reflects the RGE running. On the other hand,OLR1 andOLR

2 form a sector that is mixed under Rrunning. Although theZ′ boson only induces the operatorOLR

1 at high energy scales,OLR2 is generated afte

evolution down to low energy scales and, in particular, its Wilson coefficientCLR2 is strongly enhanced by the R

effects[46].With contributions from both the SM and theZ′ boson with only left-handed FCNC couplings included, theBs

mass difference is

(26)�Ms = �MSMs

(1+ �MZ′

s

�MSMs

)= 18.0

∣∣1+ 3.858× 105ρ2Le2iφL

∣∣ ps−1.

The corresponding result for the oscillation parameter is

(27)xs = 26.3∣∣1+ 3.858× 105ρ2

Le2iφL∣∣.

With couplings of only one chirality, the physical observables�Ms , xs , and sin2φs are periodic functions of thnew weak phaseφL with a period of 180◦.

Fig. 1(a) shows the effects of including aZ′ with left-handed coupling. We see that ifρL is small,xs is domi-nated by the SM contribution and has a value∼ 26. ForφL around 90◦ andρL between 0.001 and 0.002, theZ′contribution tends to cancel that of the SM and reducesxs to be smaller than the SM value of 26.3. In Eq.(27)andFig. 1(a), we see that theZ′ has a comparable contribution to the SM ifρL � 0.002, independent of the actual valof φL. The planned resolution of Fermilab run II and LHCb are both aboutxs � 75 [15,16]. Thus, aρL greater

V. Barger et al. / Physics Letters B 596 (2004) 229–239 235

rsion)

anasily

n

t

n of

s

Fig. 2. Contour plot ofxs and sin 2φs with a Z′-mediated FCNC for left-handedb ands quarks. The shaded region is forxs < 20.6, which isexcluded at 95% CL by experiments.The hatched region corresponds to 1σ ranges around the SM valuexs = 26.3 ± 5.5 (black curve). Thesolid curves open to the left are contours forxs = 50 (red in the web version), 70 (green in the web version) and 90 (blue in the web vefrom left to right. The curves open to the right are contours for sin 2φs = 0.5 (dotted),−0.5 (solid) and the SM value−0.07± 0.01 (dashed).

than about 0.003 will result in anxs beyond the planned sensitivity. Ifxs is measured to fall within a range, one cread from the plot what the allowed region is for the chiralZ′-model parameters. The same discussion can ebe applied to aZ′ model with only right-handed couplings.Fig. 1(b) shows sin 2φs as a function ofρL andφL. AsρL increases, sin2φs goes through more oscillations whenφL varies from 0 toπ .

In Fig. 2, we show the overlayed plot of the contours of fixedxs and those of fixed sin 2φs . The shaded regioin the center shows the experimentally excluded points in theφL–ρL plane that inducexs values smaller than 20.6.The hatched area corresponds to the parameter space points that producexs values falling within the 1σ range ofthe SM value of 26.3. Contours for higher values ofxs are also shown. The SM predicted sin2φs � −0.07± 0.01would appear as narrow bands around the sin 2φs = −0.07 curves. Note that even if thexs measurement turns outo be consistent with the SM expectation, it is still possible that the new physics contributions, such as theZ′ modelconsidered here, can alter the sin2φs value significantly. It is therefore important to have a clean determinatioboth quantities simultaneously. Oncexs and sin2φs are extracted fromBs decays, one can determineρL up to atwo-fold ambiguity andφL up to a four-fold ambiguity, except for the special case with sin2φs � 0.

�Γs can be determined with a high sensitivity by measuring the lifetime difference between decays intoCP-specific states and into flavor-specific states. Using theJ/ψφ mode, simulations determine[16] that the LHC canmeasure the ratio�Γs/Γs with a relative error� 10% for an actual value around−0.15. Tevatron simulationshow that�Γs/Γs can be measured with a statistical error of∼ 0.02. For a sufficiently largeρL, the cosθ factor inEq.(5) increases from−1 atφL = 0◦ (mod 180◦) to the maximum 1 atφL = 90◦ (mod 180◦). We are left with thephase ranges 0◦ � φL � 30◦, 60◦ � φL � 120◦, and 150◦ � φL � 180◦ (mod 180◦) where a 3σ determination of�Γs can be made.

Once the right-handedZ′ couplings are introduced, we also have to include the new|�B| = 2 operatorsOLR1 ,

OLR2 , andORR defined in Eq.(8) into the effective Hamiltonian that contributes to theBs–Bs mixing. The matrix

element ofORR is the same as that ofOLL, while those ofOLR1 andOLR

2 are

(28)〈Bs |OLR1 |Bs〉 = −4

3

(mBs

m (m ) + m (m )

)2

m2Bs

f 2Bs

BLR1 (mb),

b b s b

236 V. Barger et al. / Physics Letters B 596 (2004) 229–239

rrents,

on

e

ngt in

e

eusthe

meak

(29)〈Bs |OLR2 |Bs〉 = 2

(mBs

mb(mb) + ms(mb)

)2

m2Bs

f 2Bs

BLR2 (mb).

For theZ′ coupling to right-handed currents, we define new parametersρR and the associated weak phaseφR :

(30)ρReiφR ≡ g2MZ

g1MZ′BR

sb.

At the MW scale, we have additional contributions to the effective Hamiltonian due to the right-handed cusimilar to Eq.(17). The terms due to the left–right mixing are

(31)HZ′eff ⊃ GF√

2ρLρRei(φL+φR)

(OLR

1 + ORL1 ,OLR

2 + ORL2

)(10

).

In the RGE running, the Wilson coefficient ofOLR1 mixes with that ofOLR

2 ; the relevant anomalous dimensimatrices are[46]

(32)γ (0) =( 6

Nc12

0 −6Nc + 6Nc

),

(33)γ (1) =( 137

6 + 152N2

c− 22

3Ncf 200

3 Nc − 6Nc

− 443 f

714 + 9

Nc− 2f −203

6 N2c + 479

6 + 152N2

c+ 10

3 Ncf − 223Nc

f

),

whereNc is the number of colors andf is the number of active quarks. At the scale of theB meson masses, thvalue off is 5.

We takemb(mb) = 4.4 GeV,ms(mb) = 0.2 GeV, andΛ(5)

MS= 225 MeV. Following Eqs.(23) and (24), we find

the effective Hamiltonian terms for the operatorsOLR1,2 atmb to be

(34)HZ′eff ⊃ GF√

2ρRρLei(φL+φR)

(OLR

1 + ORL1 ,OLR

2 + ORL2

)(0.930

−0.711

).

Note that there is no contribution of the operatorOLR2 at theMW scale. It is induced through the operator mixi

in RGE running and actually has an important effect at themb scale, as one can see from its Wilson coefficienEq.(34).

In the numerical analysis, we use the central values ofBLR1 (mb) = 1.753±0.021 andBLR

2 (mb) = 1.162±0.007given in Ref.[42] with the decay constantfBs the same as before. The predicted mass difference with all thZ′contributions included is then

(35)�Ms = 18.0∣∣1+ 3.858× 105(ρ2

Le2iφL + ρ2Re2iφR

) − 2.003× 106ρLρRei(φL+φR)∣∣ ps−1.

The overall contribution toxs from the SM andZ′ is

(36)xs = 26.3∣∣1+ 3.858× 105(ρ2

Le2iφL + ρ2Re2iφR

) − 2.003× 106ρLρRei(φL+φR)∣∣.

To illustrate the interference among different contributions, we setρL = ρR = 0.001 and plotxs and sin2φs versusthe weak phasesφL andφR in Fig. 3(a) and (b), respectively.

First, we note that after the RGE running the operatorsOLR1 andOLR

2 interfere constructively. This can bseen from the relative minus sign between the Wilson coefficients in Eq.(34) and a corresponding relative minsign in the hadronic matrix elements given in Eqs.(28) and (29). Because of the constructive interference andfact that the bag parametersBLR

1 andBLR2 are twice as large asBLL, the LR and RL operators together beco

the dominant contributions. The interference of all the terms makesxs reach its maximum when one of the wephases is 180◦ and the other is 0◦ (mod 360◦). If ρL andρR are both much smaller than 10−3, the variation inxs in theφL–φR space will be indistinguishable from the predicted SM range. Compared toFig. 1(a) for Z′ with

V. Barger et al. / Physics Letters B 596 (2004) 229–239 237

2of

-handed

. Thishrough

Fig. 3.xs (a) and sin 2φs (b) as functions ofφL andφR for ρL = ρR = 0.001. The color shadings have no specific physical meaning.

only LL couplings,Fig. 3(a) shows that even for large values ofρL andρR , xs can be smaller than 20.6 due to theinterference among all the terms in Eq.(36). The currentxs � 20.6 bound excludes the regions withφL + φR � 0◦(mod 360◦). Because of the assumed equal values ofρL andρR , the parameter space points with the same sinφ

output lie along directions that are roughly parallel to theφL + φR = 360◦ line. For the more general casesdifferentρL andρR values, the crests and troughs inFig. 3(b) are no longer parallel to theφL + φR = 360◦ line.

5. Conclusions

In this Letter we discuss the effects of aZ′ gauge boson with FCNC couplings to quarks on theBs–Bs mixing.We show how the mass difference andCP asymmetry are modified by the left-handed and right-handedb–s–Z′couplings that may involve new weak phasesφL andφR. In the particular case of a left-chiral (right-chiral)Z′model, one can combine the measurements of�Ms (or xs) and sin 2φs to determine the coupling strengthρL

(ρR) and the weak phaseφL (φR) up to discrete ambiguities. Once comparable left- and right-chiral couplings areconsidered at the same time, we find the left–right interference terms dominate over the purely left- or rightterms, partly because of the renormalization running effects and partly because of the larger bag parameters.

Acknowledgements

C.-W.C. would like to thank the hospitality of the high energy theory group at University of Pennsylvaniawork was supported in part by the United States Department of Energy, High Energy Physics Division, tGrant Contract Nos. DE-FG02-95ER40896, W-31-109-ENG-38, and EY-76-02-3071.

238 V. Barger et al. / Physics Letters B 596 (2004) 229–239

nd on the

/

uly

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