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R-parity-violating SUSY and CP violation in B -->phi K_s

Recent measurements of CP asymmetry in B -->phi K_S appear to be inconsistent with Standard Model expectations. We explore the effect of R-parity-violating SUSY to understand the data.


Introduction
In the standard model (SM), CP violation is due to the presence of phases in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. The SM predicts large CP-violation in B decays [1] and the B-factories BaBar and Belle will test the SM explanation of CP violation. One of the CP phases of the CKM unitarity triangle has been already measured: sin 2β = 0.78 ± 0.08 [2], which is consistent with the SM.
The goal of the B-factories is not only to to test the Standard Model (SM) picture of CP violation but also to discover evidence of new physics. Decays that get significant contributions from penguins are most likely to be affected by new physics [3]. In particular the decay B → φK S is very interesting because it is pure penguin and is dominated by a single amplitude in the SM. Hence this decay can be used to measure sin(2β) and if this measurement is found to disagree with sin(2β) from other measurements, like B → J/ψK S , then it will be a clear sign of new physics [4].
This result appears to be inconsistent with SM prediction as sin(2β) from B → J/ψK s should agree with sin(2β) from B → φK s up to 0(λ 2 ) with λ ∼ 0.2. However, the measurements presented above seem to indicate a 2.8 σ deviation from SM expectation and have led to speculations about evidence of new physics [7]. In this paper we study the effect of R-parity violating SUSY for the decay B → φK S and show that the present measurements of sin(2β) can be easily accommodated in the presence of R-parity violating SUSY.

R-parity breaking SUSY and B → φK S
In supersymmetric models, R-parity invariance is often imposed on the Lagrangian in order to maintain the separate conservation of baryon number and lepton number. The R-parity of a field with spin S, baryon number B and lepton number L is defined to be R is +1 for all the SM particles and −1 for all the supersymmetric particles.
The presence of R-parity conservation implies that super particles must be produced in pairs in collider experiments and the lightest super particle (LSP) must be absolutely stable. The LSP therefore provides a good candidate for cold dark matter. There is, however, no compelling theoretical motivation, such as gauge invariance, to impose R-parity conservation.
The most general superpotential of the MSSM, consistent with SU(3) × SU(2) × U(1) gauge symmetry and supersymmetry, can be written as where W R is the R-parity conserving piece, and W R breaks R-parity. They are given by Here The λ and λ ′ couplings in [Eq. (7)], violate lepton number conservation, while the λ ′′ couplings violate baryon number conservation. There are 27 λ ′ -type couplings and 9 each of the λ and λ ′′ couplings as λ [ij]k is antisymmetric in the first two indices and λ ′′ i[jk] is antisymmetric in the last two indices. The non-observation of proton decay imposes very stringent conditions on the simultaneous presence of both the baryon-number and leptonnumber violating terms in the Lagrangian [8]. It is therefore customary to assume the existence of either L-violating couplings or B-violating couplings, but not both. The terms proportional to λ are not relevant to our present discussion and will not be considered further.
The antisymmetry of the B-violating couplings, λ ′′ i[jk] in the last two indices implies that there are no operators that can generate the b → sss transition, and hence cannot contribute to B → φK S at least at the tree level.
We now turn to the L-violating couplings. In terms of four-component Dirac spinors, these are given by [9] For the b → sss transition, the relevant Lagrangian is where γ RL = (1±γ 5 )

2
. There are a variety of sources which bound the above couplings [10] but the present bounds are fairly weak and the contribution from the L-violating couplings can significantly affect B → φK S measurements.
In the SM, the amplitude for B → φK S , can be written within factorization as 2 where f φ is the φ decay constant, F BK is the B → K semileptonic form factor. The a t,c i are the usual combination of Wilson's coefficient in the effective Hamiltonian. For a i as well as the quark masses we use the values used in Ref [12]. The R-parity contribution can be written as, We write where φ is the weak phase in the R-parity violating couplings and M is some mass scale with M ∼ m ν i . We require |X| to be such that |A SM, φK S | and |A R φK S | are of the same size, which then fixes |X| ∼ 1.5 × 10 −3 for M = 100 GeV which is smaller than the existing constraints on |X| from Ref [10].
We can now calculate sin(2β) ef f from where A = A SM, φK S + A R φK S andĀ is the amplitude for the CP-conjugate process. Note that from Eq. 10 and Eq. 11 sin(2β) ef f is independent of Z and hence free from uncertainties in the form factor and decay constants. It is also possible that non factorizable effects may be less important in sin(2β) ef f as we are taking ratios of amplitudes. In Fig. 1 we plot sin(2β) ef f versus the phase φ and it is clear from the figure that the present measurements in Eq. 1 and Eq. 2 can be easily explained. We now turn to the calculation of branching ratios and the direct CP asymmetry. The measured branching ratio for (B → ΦK 0 ) is (8.1 +3.1 −2.5 ± 0.8) × 10 −6 [13] while Belle measures a value for the direct CP asymmetry, i.e. the cosine term C = −0.56 ± 0.41 ± 0.12 [6] which is consistent with zero. The calculation of the branching ratio as well as the direct CP asymmetry is difficult and suffers from hadronic uncertainties even within factorization. The branching ratio, within naive factorization, depends on the form factors and the φ decay constants. The uncertainties in these quantities can easily change the predicted branching ratio by a factor of 2 or so. The direct CP asymmetry could potentially be large as there are two interfering amplitude of the same size. However the direct CP asymmetry crucially depends on the strong phase which can be perturbatively generated through tree level rescattering in factorization. The size of this strong phase, in this case, depends on the charm quark mass as well and the gluon momentum in the penguin diagram. Using the values of the formfactor F BK = 0.38 and the φ decay constant f φ = 0.237 [11] and taking a typical value for the phase φ = 1.5 radians(86 degrees) we obtain the branching ratio for (B → ΦK 0 ) = 9.5 × 10 −6 , sin 2β = −0.57 and the direct CP asymmetry ∼ 35 %. We have used m c = 1.4 GeV, m b = 5 GeV and the gluon momentum q 2 = m 2 b /2 to obtain these numbers. We would like to stress here that even in the absence of the strong phase one can still easily accommodate the data for sin 2β in B → φK S . If in fact the strong phases are small one could look for T-violating correlation in B → φK * or in the corresponding Λ b decays [14].
We also point out that the R-parity violating operator for b → sss is not the related to the b → sū(d)u(d), unlike some models of new physics. Hence R-parity violating effects in B → φK S can be very different than in B → Kπ for example which is a b → sūu transition. However the new R-parity violating SUSY generated b → sss operators will affect decays like B → K(K * )η(η ′ ), Λ b → Λη(η ′ ), Λ b → Λφ etc [15].
In conclusion, recent measurements of CP asymmetry in B → φK S appear to be inconsistent with Standard Model expectations. We show that the effect of R-parity-violating SUSY can easily accommodate the data.