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A Consistent Resolution of Possible Anomalies in B^0 -->phi K_S and B^+ -->eta' K^+ Decays

In the framework of R-parity violating (\rpv) supersymmetry, we try to find a consistent explanation for both recently measured CP asymmetry in B^0 -->phi K_S decay and the large branching ratio of B^{+/-} -->eta' K^{+/-} decay, which are inconsistent with the Standard Model (SM) prediction. We also investigate other charmless hadronic B -->PP and B -->VP decay modes whose experimental data favor the SM: for instance, recently measured CP asymmetries in B^0 -->eta^{prime} K_S and B^0 -->J / Psi K_S. We find that all the observed data can be accommodated for certain values of \rpv couplings.

The main mission of B factories, such as KEK−B and SLAC−B, is to test the Standard Model (SM) and further to find possible new physics effects in B meson systems. For this goal, a variety of useful observables can be measured to be compared with theoretical expectations. One of the important observables is CP asymmetries in various B meson decays, and another is branching ratios (BRs) for rare B decay processes.
CP violation in B system has been confirmed in measurements of time-dependent CP asymmetries in B → J/ΨK S decay. The world average of the asymmetry in B → J/ΨK S [1] is given by which is consistent with the SM expectation. However, recent measurements of sin(2β) in B 0 → φK S decay disagree with the above value. The measurements of time-dependent CP asymmetries in B 0 → φK S have been recently reported by BaBar [2] and Belle [3], respectively: sin(2β) BaBar φK S = −0.19 +0.52 −0.50 ± 0.09 , sin(2β) Belle φK S = −0.73 ± 0.64 ± 0.18 .
Both experimental results show (a tendency of) negative values of sin(2β).
Because within the SM the difference between the asymmetries in B → J/ΨK S and B 0 → φK S is expected to be [4] | sin(2β with λ ≈ 0.2, these results indicate 2.7 σ deviation from the SM prediction and may reveal new physics effects. The decay process B 0 → φK S is a pure penguin process b → sss, and in the SM the time-dependent CP asymmetry of this mode is expected to measure the same sin(2β) as in  [4,5], one might expect that other modes having the same internal quark level process b → sss (e.g., B 0 → η ′ K S ) would reveal the similar effects. Interestingly, the recent measurements of CP asymmetries in B 0 → η ′ K S by Belle [3] agree well with the results of sin(2β) J/ΨK S given in Eq. (1) (see Table I).
Therefore, any successful explanation invoking new physics should accommodate the data disfavoring the SM, such as sin(2β) φK S , and simultaneously all the data favoring the SM, such as sin(2β) η ′ K S and sin(2β) J/ΨK S . Motivated by the recent measurements of sin(2β) φK S , a few first analyses [1,[6][7][8][9] have been done. But, all of them explain only the deviation of sin(2β) φK S from the SM prediction, or in addition, make (at most) qualitative statements There also exist plenty of experimental data observed for charmless hadronic B → P P (P denotes a pseudoscalar meson) and B → V P (V denotes a vector meson) decay modes, which are well understood within the SM. However, among the B → P P decay modes, the BR of the decay mode B ± → η ′ K ± is found to be still larger than that expected within the SM 1 [10][11][12][13]: the experimental world average is B(B ± → η ′ K ± ) = (75 ± 7) × 10 −6 . The SM contribution is about 3σ smaller than the experimental world average [13]. Among the B → V P decay modes, the experimentally observed BR for the decay B 0 → ηK * 0 has been also found to be 2σ larger than the SM expectation [13].
In this letter we try to find a consistent explanation for all the observed data in charmless hadronic B → P P and B → V P decays in the framework of R-parity violating (R p / ) supersymmetry (SUSY). In R p / SUSY, the effects of R p / couplings on B decays can appear at tree level and can be in some cases comparable to the SM contribution. Our main focus is on the recent measurement of CP asymmetry in B 0 → φK S and the large BR for B ± → η ′ K ± : both data appear to be inconsistent with the SM prediction. In order to achieve this goal, we investigate all the observed B → P P and B → V P decay modes and explicitly show that indeed all the observed data can be accommodated in a consistent way for certain values of The R p / part of the superpotential of the minimal supersymmetric standard model (MSSM) can contain terms of the form where E i , U i and D i are respectively the i-th type of lepton, up-quark and down-quark singlet superfields, L i and Q i are the SU(2) L doublet lepton and quark superfields, and H 2 is the Higgs doublet with the appropriate hypercharge. From the symmetry reason, we need The bilinear terms can be rotated away with redefinition of lepton and Higgs superfields, but the effect reappears as λs, λ ′ s and lepton-number violating soft terms [15]. The first three terms of Eq. (5) violate the lepton number, whereas the fourth term violates the baryon number. We do not want all these terms to be present simultaneously due to catastrophic rates for proton decay. In order to prevent proton decay, one set needs to be forbidden.
For our purpose, we will assume only λ ′ −type couplings to be present. Then, the effective Hamiltonian for charmless hadronic B decay can be written as [16] H with where α and β are color indices and γ µ R,L ≡ γ µ (1 ± γ 5 ). The leading order QCD correction to this operator is given by a scaling factor f ≃ 2 for mν = 200 GeV.
The available data on low energy processes can be used to impose rather strict constraints on many of these couplings [17][18][19]. Most  GeV.
In our calculation, we use the effective Hamiltonian and the effective Wilson coefficients (WCs) for the processes b → sqq ′ and b → dqq ′ given in Ref. [22]. The decay amplitude of where the SM part of the amplitude is and the R p / part of the amplitude involves only d L 222 and d R 222 : with A φ = K|sγ µ (1 − γ 5 )b|B φ|sγ µ s|0 . This particular structure of A φ is obtained from the operatorsd nα γ µ L(R) d jβdkβ γ µR(L) b α given in Eq. (6), which are derived from the operators s L(R) s R(L)sR(L) b L(R) by Fierz transformation [23]. The CP asymmetry parameter sin(2β) XY for B → XY can be written as where Here β denotes the CP angle in the SM andβ denotes the effective CP angle given by 2β = 2β + θ with a new weak phase θ arising from new physics.
wherem  by A u η , A s η ′ by A s η and m η ′ by m η . Replacing a pseudoscalar meson by a vector meson, we also get similar expressions for the amplitudes of B ±(0) → η ′ K * ±(0) modes.
In Eqs. (10) and (12) which are relevant to the process b → sss. We divide our results into the following two cases.
Case 1: We use the following input parameters: the CP angle γ = 110 0 , the s quark mass m s (at m b scale) = 85 MeV, and the decay constants and the hadronic form factors used in Ref. [13]. We take β = 26 0 as our input, which corresponds to sin(2β) J/ΨK = 0.78 . We find a constructive contribution to the SM part of the amplitude for B + → η ′ K + which helps to satisfy the experimental data.
The SM BR of B + → η ′ K + is far below the experimental lower bound. Our result in this case is denoted by the dotted line, which is clearly consistent with the experimental data: B(B + → η ′ K + ) = 69.3 × 10 −6 for ξ = 0.45. In Table II, we estimate the BRs and CP (rate) asymmetries A CP for B → η (′) K ( * ) and B + → φK + , where A CP is defined by with b and f denoting b quark and a generic final state, respectively. The estimated BRs are well within the experimental limits. The BR of B 0 → φK 0 has also been measured, B(B 0 → φK 0 ) = (8.1 +3.1 −2.5 ± 0.8) × 10 −6 [2]. The theoretical value for the BR of this mode is same as that of B + → φK + and is allowed by the experimental data. Further, We note that A CP 's for B + → φK + and B 0 → ηK * 0 are large and have positive signs. This can be taken as a prediction of this model. It is hopeful that the CDF at Fermilab could measure the CP asymmetry of B + → φK + mode using their two track trigger in the ongoing RUN II experiment [24].
So far we have assumed that the magnitudes and phases are same for d L 222 and d R 222 .
We can assume them to be different and still obtain good fits. For example, if we set   In this case, we find that a consistent explanation of all the observed data for B → P P and B → V P is possible for |λ ′ 322 | = |λ ′ 332 | = |λ ′ 323 | = 0.069, tan θ ′ = 2.8, and m susy = 200 GeV. In this scenario, a large value of sin(2β) with the negative sign is possible for B 0 → φK S : sin(2β) φK S = −0.82 for ξ = 0.25, as shown in Table I The BRs of other B → P P, V P modes satisfy the experimental data for ξ = 0.25 [13].
In conclusion, we have shown that in R p violating SUSY it is possible to understand consistently both recently measured CP asymmetry sin(2β) in B 0 → φK S decay and the large branching ratio of B ± → η ′ K ± decay, which appear to be inconsistent with the SM prediction. Experimental results imply that CP asymmetry for B → φK s is different from the SM value, but for B → η ′ K s it is same as in the SM. We have found that this tendency can be consistently understood in the framework of this model. We have searched for possible parameter space and found that all the observed data for B → P P and B → V P decays can be accommodated for certain values of R p / couplings. Experimentally, these couplings can be observed through the direct production of SUSY particles at Tevatron (e.g., χ ± 1 χ 0 2 production and their subsequent decays to final states with multiple b quarks) and other rare decays like B s → µ ± µ ∓ [25].