Flavor and CP violating Z exchange and the rate asymmetry in B -->phi K_S

Recent measurements of time dependent CP asymmetry in B -->phi K_S, if confirmed, would indicate a new source of CP violation. We examine flavor violating tree-level Z currents in models with extra down-type quark singlets that arise naturally in string compactified gauge groups like E_6. We evaluate the new operators at the scale mu = m_b in NLO, and using QCD improved factorization to describe B -->phi K_S, find the allowed range of parameters for rho and psi, the magnitude and phase of the flavor violating parameter U_{bs}. This allowed range does satisfy the constraint from flavor changing process b -->s l+ l- . However, further improvement in measurement of these rates could severely constrain the model.


Introduction
The ongoing B physics experiments by BaBar and Belle collaborations [1,2] provide a unique opportunity to study the flavor structure of the standard model quark sector and also the origin of CP violation. In addition to this, any new physics effects in B physics can also be tested in these experiments. Recent time dependent asymmetries measured in the decay B → φK S both by BaBar and Belle collaborations [1][2][3][4] show significant deviation from the standard model and this has generated much theoretical speculation regarding physics beyond the standard model [5]. In the standard model, the process B → φK S is purely penguin dominated and the leading contribution has no weak phase. The coefficient of sin(∆m B t) in the asymmetry therefore should measure sin 2β, the same quantity that is involved in B → ψK S in the standard model. The most recent measured average values of asymmetries are [4,6] S ψK S = 0.734 ± 0.055 and S φK S = −0.15 ± 0.33. The value for S ψK S agrees with the standard model expectation. The deviation in the φK S is intriguing because a penguin process being a loop induced process is particularly sensitive to new physics which can manifest itself through exchange of heavy particles. In this article we will consider an extension of the standard model, with extra down type singlet quarks. These extra down type singlet quarks appear naturally in each 27-plet fermion generation of E 6 Grand Unification Theories (GUTs) [7][8][9][10]. The mixing of these singlet quarks with with the three SM down type quarks, provides a framework to study the deviations from the unitarity constraints of 3 × 3 CKM matrix. This model has been previously studied in connection with R b and F-B asymmetry at the Z pole as it provides a framework for violation of the unitarity of the CKM matrix [10][11][12]. This mixing also induces tree-level flavor changing neutral currents (FCNC). These tree-level FCNC couplings can have a significant effects on different CP conserving as well as CP violating B processes [11, 13 -24].
In this article we study the FCNC effect arising from the Z − b −s coupling U bs to the B → φK S process. This new FCNC coupling U bs can have a phase, which can generate the additional source of CP violation in the B → φK S process, and thus affect measured values of S φK S and C φK S . We parameterize this coupling by U bs = ρe iψ . We then study B → φK S taking into account the new interactions in the QCD improved factorization scheme (BBNS approach ) [25]. This method incorporates elements of naive factorization approach (as its leading term ) and perturbative QCD corrections (as sub-leading contributions) and allows one to compute systematic radiative corrections to the naive factorization for the hadronic B decays. Recently, several studies of B → P V , and specifically B → φK S have been performed within the frame work of QCD improved factorization scheme [26][27][28][29][30]. In our analysis of B → φK S , we follow [30] which is based on the original paper [25]. In our analysis, we only consider the contribution of the leading twist meson wave functions, and also neglect the weak annihilation contribution which is expected to be small. Inclusion of these would introduce more model dependence in the calculation through the parameterization of an integral, which is otherwise infrared divergent.

B → φK S in the QCDF Approach
In the standard model, the effective Hamiltonian for charmless B → φK S decay is given by where the Wilson coefficients C i (µ) are obtained from the weak scale down to scale µ by running the renormalization group equations. The definitions of the operators can be found in Ref. [25]. The Wilson coefficients C i can be computed using different schemes [35]. In this paper we will use the NDR scheme. The NLO values of C i (i = 1 − 10) and LO values of C 7γ , C 8g respectively at µ = m b /2 and m b used by us based on Ref. [25] are shown in Table 1.
In the QCD improved factorization scheme, the B → φK S decay amplitude due to a particular operator can be represented in following form : where < φK | O | B > f act denotes the naive factorization result.The second and third term in the bracket represent higher order α s and Λ QCD /m b correction to the hadronic transition amplitude. Following the scheme and notations presented in Ref. [30], we write down the total B → φK S amplitude, which is the sum of the standard model as well as Z exchange tree-level contribution from extra down-type quark singlets (EDQS) model in the heavy quark limit where p is summed over u and c. The coefficients a p i are given by is the sum of the standard model and the EDQS model Wilson coefficients. The quantities V φ , H φ , P p φ and Q p φ are hadronic parameters that contain all nonperturbative dynamics, are given in Ref. [25,36].
For the sake of completeness, we give the branching ratio for B → φK S decay channel in the rest frame of the B meson.
where, τ B represents the B meson lifetime and the kinematical factor | P cm | is written as

FCNC Z couplings in EDQS model
Models with extra down-type quarks (EDQS) have a long history. The earliest consideration of such models was in the context of the grand unification group E 6 which arises from string compactification. The quarks and leptons of each generation belong to the 27 representation [7][8][9][10]. Each generation has one extra quark singlet of the down type, and also one extra lepton of the electron type. The group also has extra Z bosons, which we will assume to be too heavy to have any effect on B → φK S process. The down type mass matrix is then a 6 × 6 by-unitary matrix, and in general when we rotate the quarks to their mass basis, off-diagonal couplings arise. In EDQS model, the Z mediated FCNC interactions are given by In general for n copies of extra down-type quark singlet model, U αβ is : where, N d = 3 + n represents the number of down type quark states, and U is the neutral current mixing matrix for the down quark sector. The non vanishing components of U αβ will lead to FCNC process at tree level, generating new physics contribution to the measured CP asymmetries. The new tree level FCNC Z mediated contribution to the b → sqq process is shown in Figure 1. The new operators arising from this tree-level FCNC process have been shown to lead to the following effective Hamiltonian for b → sqq process in this model [22]: where, the new Wilson coefficientsC 3 ,C 7 andC 9 at the scale M Z are given by: where, κ = U bs  Table 2.

B physics constraints on U bs
In this section, we review the constraints on the flavor violating parameter U bs from different flavor changing B processes. These processes can be classified into two classes, CP conserving  Table 2: Wilson coefficients of EDQS model in NDR scheme, without the overall multiplicative factor κ. and CP violating. Among the CP conserving processes, B(B → X s ℓ + ℓ − ), and ∆M Bs can put constraints on U bs [13,15,16]. Using recent Belle [38] measurement of B(B → X s ℓ + ℓ − ) = (6.1 ± 1.4 +1.4 −1.1 ) × 10 −6 the authors in Ref. [21] had shown that | U bs |≤ 1 × 10 −3 . However, this bound has recently been updated in Ref. [22] to which also updates their previous bounds in Ref. [17,39] The bound in Eq. (24) is based on inclusive B → X s e + e − decays at NNLO [40]. We shall adopt this bound in our analysis.

This bound is valid in both general n extra down-type singlet quark model and in the model
with a single extra down-type quark singlet [16]. Similarly, the b → sγ branching ratio provides comparable limits [14,16].
It has been shown in Ref. [20,41]  Feynman diagrams are given in Ref. [41]. Following the paper [20], with slight change in the notations, we write down the expression for the B q −B q mixing: where, (q = d, s) and where, the definitions of different parameters used above can be found in Ref. [20].
We have found that to satisfy the measured ∆M B d 4 within one sigma, where we consider both the theory error of 20% arising from the value of f 2 BqB Bq and the experimental error taken in quadrature, the FCNC coupling | U bd | should be less than ∼ (2 − 3) × 10 −4 . This is a very stringent limit.
The ∆M Bs has not been measured yet, and so only lower limit on the mass difference is available. We have found that the new contribution to ∆M Bs from EDQS model is less than 3% when compared to the standard model. It can be shown that for similar values of U bd and U bs , the FCNC effects on ∆M Bs will be suppressed by a factor ∼ λ 2 when compared with the effects on ∆M B d . This implies that in EDQS model, FCNC effects are hard to detect in B s −B s mixing [41].

B → φK S analysis
In the last section we have discussed the allowed range of the FCNC parameter U bs from different B processes. In this section we will study the effect of this FCNC parameter (U bs ) in the B → φK S process. For this we express U bs in the following form: U bs = ρe iψ . We will then vary ρ and ψ 5 in range such that Eq. (24) is satisfied. We then study the allowed region of parameters in the ρ − ψ plane from the three measured quantities . In our analysis we have considered 20% error on this parameter. Similarly, we vary C φK S and S φK S by 1σ and 2σ from their central value to get the allowed region in the ρ − ψ plane.
In Figure 2 (a) we show such allowed region in ρ − ψ plane for the scale µ = m b /2.
The whole area left of the dotted contour is allowed by saturating Eq. (24). The area outside the thick contour labeled by BR is 2σ allowed region from the branching ratio measurement. The parameter space enclosed by the thin contour marked by S φK is allowed by 2σ from data on the S φK S . This whole parameter space is allowed by 1σ from C φK S measurement, The regions (marked by Z) is the only allowed parameter space in ρ − ψ plane with 6.5 × 10 −4 ≤ ρ ≤ 10 × 10 −4 and −1.7 ≤ ψ ≤ −0.85 which satisfy the experimentally measured C φK S , S φK S and B(B → φK S ) within the errors described above. We note that only negative values of ψ give acceptable range of S φK . The Figure 2 (b) correspond to the scale µ = m b . In this case though we have larger allowed area from the S φK measurement, but the 2σ branching ratio contour pushes the allowed range towards higher values of ρ and 5 ψ in units of radian somewhat lower range of the phase ψ. This particular behavior of the branching ratio contour can be understood from that fact that for µ = m b , the SM branching ratio is 3.8 × 10 −6 , which is much smaller than the lower end of the 2σ band of the experimental number. Hence, one needs larger values of ρ to push the total branching ratio within the 2σ limit. For this reason the allowed region shrinks to a point in this case.

Conclusions
In We have shown that in the model with an arbitrary number of down type singlet quarks, the value of S φK S and C φK S can be well explained by the values of ρ and ψ in the region marked by Z in Figure 2 (a). Improvements in measurements of B → X s ℓ + ℓ − can tighten the constraints in Eq. (24) and either rule in or rule out this model.

Acknowledgments
This work was supported in part by US DOE contract numbers DE-FG03-96ER40969.
We would like to thank A. K. Giri for helpful correspondence. We would particularly like to thank G. Hiller for clarifying the operator structure of the Z mediated FCNC interactions.