B_s-anti-B_s Mixing in Z' Models with Flavor-Changing Neutral Currents

In models with an extra U(1)' gauge boson family non-universal couplings to the weak eigenstates of the standard model fermions generally induce flavor-changing neutral currents. This phenomenon leads to interesting results in various B meson decays, for which recent data indicate hints of new physics involving significant contributions from b ->s transitions. We analyze the B_s system, emphasizing the effects of a Z' on the mass difference and CP asymmetries.


I. INTRODUCTION
The study of B physics and the associated CP violating observables has been suggested as a good means to extract information on new physics at low energy scales [1,2,3,4,5,6,7].
Since B-B mixing is a loop-mediated process within the standard model (SM), it offers an opportunity to see the footprints of physics beyond the SM.The currently observed ∆M d = 0.489 ± 0.008 ps −1 [8] and its mixing phase sin 2β = 0.736 ± 0.049 extracted from the B d → J/ψK S mode [9] agree well with constraints obtained from other experiments [10].
However, no such information other than a lower bound ∆M s > 14.5 ps −1 [11] is available for the B s meson yet.
Based upon SM predictions, ∆M Bs is expected to be about 18 ps −1 and its mixing phase φ s only a couple of degrees.In contrast to the B d system, the more than 25 times larger oscillation frequency and a factor of four lower hadronization rate from b quarks pose the primary challenges in the study of B s oscillation and CP asymmetries.Since the B s → J/ψφ decay is dominated by a Cabibbo-Kobayashi-Maskawa (CKM) favored tree-level process, b → ccs, that does not involve a different weak phase in the SM, its asymmetry provides the most reliable information about the mixing phase φ s .
Although new physics contributions may not compete with the SM processes in most of the b → ccs decays, they can play an important role in B s -B s mixing because of its loop nature in the SM [12].In particular, the mixing can be significantly modified in models in which a tree-level b → s transition is present.Thus, measurement of the properties of B s meson mixing is of high interest in future B physics studies as a means to reveal new physics [13,14].Since the current B factories do not run at the Υ(5S) resonance to produce B s mesons, it is one of the primary objectives of hadronic colliders to study B s oscillation and decay in the coming years [15,16].
Flavor changing neutral currents (FCNC) coupled to an extra U(1) ′ gauge boson arise when the Z ′ couplings to physical fermion eigenstates are non-diagonal.One way for this to happen is by the introduction of exotic fermions with different U(1) ′ charges that mix with the SM fermions [17,18,19,20,21] as occurs in E 6 models.In the E 6 case, mixing of the right-handed ordinary and exotic quarks, all SU(2) L singlets, induces FCNC mediated by a heavy Z ′ or by (small) Z-Z ′ mixing, so the quark mixing can be large.Mixing between ordinary (doublet) and exotic (singlet) left-handed quarks induces FCNC mediated by the SM Z boson [21].We will also allow for this possibility, but in this case the quark mixing must be very small.Another possibility involves family non-universal couplings.It is well-known that string models naturally give extra U(1) ′ groups, at least one of which has family non-universal couplings to the SM fermions [24,25,26,27].Generically, the physical and gauge eigenstates do not coincide.Here, unlike the above-mentioned E 6 case, off-diagonal couplings of fermions to the Z ′ boson can be obtained without mixing with additional fermion states.In these types of models, both left-handed and right-handed fermions can have family non-diagonal couplings with the Z ′ , while couplings to the Z are family diagonal (up to small effects from The Z ′ contributions to B s -B s mixing are related to those for hadronic, semileptonic, and leptonic B decays in specific models in which the diagonal Z ′ couplings to q q, ℓ + ℓ − , etc. are known, but are independent in general [46].We have found that in specific models, B s -B s mixing effects can be significant while being consistent with the other constraints; these results will be presented elsewhere.In the present paper, we will treat the mixing in a model-independent way.Recently, we have studied the implications of a sizeable off-diagonal Z ′ coupling between the bottom and strange quark in the indirect CP asymmetry of B → φK S decay [28], which appears to show a significant deviation from the SM prediction [5,6,29,30].Here we extend our analysis to B s -B s mixing where the Z ′ contributions also enter at the tree level.
Applications to the B → πK anomaly are under investigation [31].
The paper is organized as follows.In Section II, we review the basic formalism of B s -B s mixing.In Section III, we evaluate ∆M s in the SM.In Section IV, we include the Z ′ contributions, allowing both left-handed and right-handed couplings in the mixing, and study their effects on observables.Our main results are summarized in Section V.
In the conventional decomposition of the heavy and light eigenstates The mass difference of the two physical states is The width difference is where the relative phase is θ = arg(M 12 /Γ 12 ).Since Γ 12 is dominated by the contributions from CKM favored b → ccs decays, we have θ = arg ((V tb V * ts )/(V cb V * cs )) ≃ π, and thus ∆Γ ≃ −2|Γ 12 | is negative in the SM.Although Γ 12 is unlikely to be affected by new physics, the width difference always increases as long as the weak phase of M 12 gets modified [32].
The observability of B s -B s oscillations is often indicated by the parameter where Γ s = (4.51± 0.18) × 10 −13 GeV, converted from the world average lifetime τ s = 1.461 ± 0.057 ps [8].The expected large value of x s is a challenge for experimental searches.

III. ∆M s IN THE SM
The |∆B| = 2 and |∆S| = 2 operators relevant for our discussions are: Because of the V − A structure, only the operator O LL contributes to B s -B s mixing in the SM.The other three operators appear in the Z ′ models because of the right-handed couplings and operator mixing through renormalization, as considered in the next section.
In the SM the contributions to are dominated by the top quark loop.The result, accurate to NLO in QCD, is given by [39] where Using m t (m t ) = 170 ± 5 GeV, we find S 0 (x t ) = 2.463.The NLO short-distance QCD corrections are encoded in the parameters η 2B ≃ 0.551 and J 5 ≃ 1.627 [39].The bag parameter B LL (µ) is defined through the relation In the following numerical analysis, we will use G F = 1.16639 × 10 −5 GeV −2 and M W = 80.423 ± 0.039 GeV [8], and write the SM part of ∆M s as Current lattice calculations still show quite large errors on the hadronic parameters f Bs = 230 ± 30 MeV and B LL (m b ) = 0.872 ± 0.005 [40,41,42].However, the ratio can be determined with a much smaller theoretical error, where BBq is the renomalizationindependent bag parameter for the B q meson (q = d, s).Therefore, the error on ∆M s within the SM can be evaluated by comparing with ∆M d , i.e., Using the measured values of the Wolfenstein parameters [43] λ = 0.2265 ± 0.0024, A = 0.801 ± 0.025, ρ = 0.189 ± 0.079, and η = 0.358 ± 0.044 [10], ξ = 1.24 ± 0.07 [44], and the mass parameters quoted above, we obtain the SM predictions As noted above, the central value of x s is slightly larger than the current sensitivity based upon the world average.Recent LHC studies show that with one year of data, ∆M s can be explored up to 30 ps −1 (ATLAS), 26 ps −1 (CMS), and 48 ps −1 (LHCb) (corresponding to x s up to 46, 42, and 75); the LHCb result is based on exclusive hadronic decay modes [16].
The sensitivity of both CDF and BTeV on x s can also reach up to 75 using the same modes [15], for a luminosity of 2 fb −1 .The sensitivity on sin 2φ s is correlated with the value of x s , and it becomes worse as x s increases.A statistical error of a few times 10 −2 can be reached at CMS and LHCb for moderate x s ≃ 40 [16].
For simplicity, we assume that there is no mixing between the SM Z and the Z ′ (small mixing effects can be easily incorporated [17]).A purely left-handed off-diagonal Z ′ coupling to b and s quarks results in an effective |∆B| = 2, |∆S| = 2 Hamiltonian at the M W scale of where g 2 is the U(1) ′ gauge coupling, g 1 = e/(sin θ W cos θ W ), M Z ′ is the mass of the Z ′ , and sb is the FCNC Z ′ coupling to the bottom and strange quarks.The parameters ρ L and the weak phase φ L in the Z ′ model are defined by the second equality.Generically, we expect that g 2 /g 1 ∼ 1 if both U(1) groups have the same origin from some grand unified theory, and , then an order-of-magnitude estimate gives us ρ L ∼ O(10 −3 ), which is in the ballpark of giving significant contributions to the B s -B s mixing.The Z ′ does not contribute to Γ 12 at tree level because the intermediate Z ′ cannot be on shell.After evolving from the M W scale to m b , the effective Hamiltonian becomes [39] where R = α s (M W )/α s (m b ).Although the above effective Hamiltonian is largely suppressed by the ratio (g 2 M Z )/(g 1 M Z ′ ), it contains only one power of G F in comparison with the corresponding quadratic dependence in the SM because the Z ′ -mediated process occurs at tree level.
The full description of the running of the Wilson coefficient from the M W scale to m b can be found in [39].We only repeat the directly relevant steps here.The renormalization group equation for the Wilson coefficients C, can be solved with the help of the U matrix in which γ T (g) is the transpose of the anomalous dimension matrix γ(g).With the help of dg/d ln µ = β(g), U obeys the same equation as C(µ).We expand γ(g) to the first two terms in the perturbative expansion, To this order the evolution matrix U(µ, m) is given by where U (0) is the evolution matrix in leading logarithmic approximation and the matrix J expresses the next-to-leading corrections.We have where V diagonalizes γ (0)T , i.e., γ T V , and γ (0) is the vector containing the diagonal elements of the diagonal matrix γ (0) D .In terms of G = V −1 γ (1)T V and a matrix H whose elements are the matrix J is given by J = V HV −1 .
The operators O LL and O RR do not mix with others under renormalization.Their Wilson coefficients follow exactly the same RGEs, where the above-mentioned matrices are all simple numbers.The factor in Eq. ( 18) reflects the RGE running.On the other hand, O LR  is strongly enhanced by the RG effects [45].
With contributions from both the SM and the Z ′ boson with only left-handed FCNC couplings included, the B s mass difference is The corresponding result for the oscillation parameter is With couplings of only one chirality, the physical observables ∆M s , x s , and sin 2φ s are periodic functions of the new weak phase φ L with a period of 180 • .
Fig. 1 (a) shows the effects of including a Z ′ with left-handed coupling.We see that if ρ L is small, x s is dominated by the SM contribution and has a value ∼ 26.For φ L around 90 • and ρ L between 0.001 and 0.002, the Z ′ contribution tends to cancel that of the SM and reduces x s to be smaller than the SM value of 26.3.In Eq. ( 27) and Fig. 1 (a), we see that the Z ′ has a comparable contribution to the SM if ρ L > ∼ 0.002, independent of the actual value of φ L .The planned resolution of Fermilab Run II and LHCb are both about x s < ∼ 75 [15,16].Thus, a ρ L greater than about 0.003 will result in an x s beyond the planned sensitivity.If The curves open to the right are contours for sin 2φ s = 0.5 (dotted), −0.5 (solid) and the SM value −0.07 ± 0.01 (dashed).that induce x s values smaller than 20.6.The hatched area corresponds to the parameter space points that produce x s values falling within the 1σ range of the SM value of 26.3.
Contours for higher values of x s are also shown.The SM predicted sin 2φ s ≃ −0.07 ± 0.01 would appear as narrow bands around the sin 2φ s = −0.07curves.Note that even if the x s measurement turns out to be consistent with the SM expectation, it is still possible that the new physics contributions, such as the Z ′ model considered here, can alter the sin 2φ s value significantly.It is therefore important to have a clean determination of both quantities simultaneously.Once x s and sin 2φ s are extracted from B s decays, one can determine ρ L up to a two-fold ambiguity and φ L up to a four-fold ambiguity, except for the special case with sin 2φ s ≃ 0. ∆Γ s can be determined with a high sensitivity by measuring the lifetime difference between decays into CP -specific states and into flavor-specific states.Using the J/ψφ mode, simulations determine [16] that the LHC can measure the ratio ∆Γ s /Γ s with a relative error < ∼ 10% for an actual value around −0.15.Tevatron simulations show that ∆Γ s /Γ s can be measured with a statistical error of ∼ 0.02.For a sufficiently large ρ L , the cos θ factor in Eq. ( 5) increases from −1 at φ L = 0 • (mod 180 • ) to the maximum 1 at φ L = 90 • (mod 180 • ).
We are left with the phase ranges 0 For the Z ′ coupling to right-handed currents, we define new parameters ρ R and the associated weak phase φ R : At the M W scale, we have additional contributions to the effective Hamiltonian due to the right-handed currents, similar to Eq. (17).The terms due to the left-right mixing are In the RGE running, the Wilson coefficient of O LR 1 mixes with that of O LR 2 ; the relevant anomalous dimension matrices are [45] where N c is the number of colors and f is the number of active quarks.At the scale of the B meson masses, the value of f is 5.
We take m b (m b ) = 4.4 GeV, m s (m b ) = 0.2 GeV, and Λ M S = 225 MeV.Following Eqs.(23)(24), we find the effective Hamiltonian terms for the operators O LR 1,2 at m b to be Note that there is no contribution of the operator O LR In the numerical analysis, we use the central values of B LR 1 (m b ) = 1.753 ± 0.021 and B LR 2 (m b ) = 1.162 ± 0.007 given in Ref. [41] with the decay constant f Bs the same as before.The predicted mass difference with all the Z ′ contributions included is then The overall contribution to x s from the SM and Z ′ is To illustrate the interference among different contributions, we set ρ L = ρ R = 0.001 and plot x s and sin 2φ s versus the weak phases φ L and φ R in Fig. 3 (a) and (b), respectively.
First, we note that after the RGE running the operators O LR contributions from CKM favored b → ccs decays in the SM, and the SM result Γ 12 ≪ M 12 is unlikely to change.

1 and O LR 2 form a sector
that is mixed under RG running.Although the Z ′ boson only induces the operator O LR 1 at high energy scales, O LR 2 is generated after evolution down to low energy scales and, in particular, its Wilson coefficient C LR

FIG. 1 :
FIG. 1: Three-dimensional plot of x s (a) and sin 2φ s (b) versus ρ L and φ L with a Z ′ -mediated FCNC for left-handed b and s quarks.The color shadings in both plots have no specific physical meaning.

xFIG. 2 :
FIG. 2: Contour plot of x s and sin 2φ s with a Z ′ -mediated FCNC for left-handed b and s quarks.The shaded region is for x s < 20.6, which is excluded at 95% CL by experiments.The hatched region corresponds to 1σ ranges around the SM value x s = 26.3 ± 5.5 (black curve).The solid curves open to the left are contours for x s = 50 (red), 70 (green) and 90 (blue) from left to right.

2
at the M W scale.It is induced through the operator mixing in RGE running and actually has an important effect at the m b scale, as one can see from its Wilson coefficient in Eq.(34).

3 (
can be seen from the relative minus sign between the Wilson coefficients in Eq.(34) and a corresponding relative minus sign in the hadronic matrix elements given in Eqs.(28) and(29).Because of the constructive interference and the fact that the bag parameters B LR 1 and B LR 2 are twice as large as B LL , the LR and RL operators together become the dominant contributions.The interference of all the terms makes x s reach its maximum when one of the weak phases is 180 • and the other is 0 • (mod 360 • ).If ρ L and ρ R are both much smaller than 10 −3 , the variation in x s in the φ L -φ R space will be indistinguishable from the predicted SM range.Compared to Fig. 1 (a) for Z ′ with only LL couplings, Fig. a) shows that even for large values of ρ L and ρ R , x s can be smaller than 20.6 due to the interference among all the terms in Eq. (36).The current x s ≥ 20.6 bound excludes the regions with φ L + φ R ≃ 0 • (mod 360 • ).Because of the assumed equal values of ρ L and ρ R , the parameter space points with the same sin 2φ output lie along directions that are roughly parallel to the φ L + φ R = 360 • line.For the more general cases of different ρ L and ρ R values, the crests and troughs in Fig. 3 (b) are no longer parallel to the φ L + φ R = 360 • line.

FIG. 3 :
FIG.3:x s (a) and sin 2φ s (b) as functions of φ L and φ R for ρ L = ρ R = 0.001.The color shadings have no specific physical meaning.
and 150 • < ∼ φ L < ∼ 180 • (mod 180 • ) where a 3σ determination of ∆Γ s can be made.Once the right-handed Z ′ couplings are introduced, we also have to include the new|∆B| = 2 operators O LR 1 , O LR 2 ,and O RR defined in Eq. (8) into the effective Hamiltonian that contributes to the B s -B s mixing.The matrix element of O RR is the same as that of O LL , while those of O LR