Complex flavour couplings in supersymmetry and unexpected CP-violation in the decay B ->phi K

Complex flavour couplings (off-diagonal mass terms) in the squark sector of supersymmetric theories may drastically alter both the rate and the CP-violating asymmetry of certain B-meson decays. We consider the effects of couplings that induce b ->s transitions and lead to final state with strangeness one. We investigate the bounds that must be satisfied by the new terms and explore the possible implications on direct and mixing induced asymmetries in the charged and neutral B ->J/Psi K and B ->phi K decays.


Introduction
Certain decays of B-mesons are expected to shed light on the mechanism of CP violation and, more generally, on new physics. Indeed, the first surprising results on the CP -asymmetry a ψK in the decay B → J/ΨK raised the hope for a first new physics signal. Meanwhile, the value of the measured asymmetry has changed considerably and the present world average [1] a W A ψK = 0.79 ± 0.12 (1) almost coincides with the standard model expectation [2] a SM ψK ≃ 0.70 ± 0.10 .
Although this came as a disappointment, it reinforced the view that one should be prepared, both on the theoretical and the experimental side, for unexpected observations. In fact, the small error of the above world average and the improvements that are expected during the next years will allow to detect even small deviations from the standard model predictions.
There are of course many new physics scenarios. On the one hand, it is possible to describe their signatures in very general terms (for recent expositions, see e.g. Refs. [3][4][5]). On the other hand, one can propose a specific new physics model and investigate its consequences. The latter approach was the topic of a huge body of work. For what concerns supersymmetric models, emphasis was given recently to the so-called minimal flavour violating models (MFV) [6] and their possible variants [7].
In virtually all new physics models, new non-standard fields are introduced and, therefore, new complex (i.e. with CP -violating phases) couplings appear. This is well known in supersymmetry to which we turn for definiteness. In supersymmetric models there are several classes of phases. Those in the µ and flavour diagonal A terms do not contribute to flavour changing processes but are strongly bound by the electric dipole moment of the neutron and other particles [8]. The phases of the Yukawa couplings are the same as in the standard model. If these are the only phases and flavour changing couplings of a model, the latter belongs to the class of MFV supersymmetric models that exhibit many analogies to the standard model [6]. Therefore, the most interesting phases are those in the squark and slepton mass matrices. Their flavour diagonal elements are either real by definition (in the LL and RR sectors) or small (in the LR and RL ones) as pointed out before (restrictions from electric dipole moments). Therefore the flavour changing elements (i.e. (m 2 ) u,d 23 and (m 2 ) u,d 13 , in the sectors LL, RL, etc.) are the most interesting ones and we will focus on their effects. They contribute mainly through loop diagrams with internal gluinos and charginos. Since we are interested in contributions to Wilson coefficients that are already quite large in the SM ‡ (i.e. the coefficients of the QCD penguin and chromo-magnetic dipole moment operators), we will only discuss gluino loops; in fact, they tend to dominate over the corresponding chargino ones. ‡ Contributions to the small electroweak penguin coefficients are usually negligible because they do not appreciably change the standard model predictions.
Clearly this is not the first analysis of such SUSY diagrams. Their impact on flavour physics is known since a long time [9], and there are numerous recent investigations. To clarify the new aspects provided in this paper, we review briefly some of the most recent efforts.
In Ref. [10] a first comprehensive analysis of the gluino exchanges was given. The systematics of the perturbative expansion was investigated in Ref. [11], which also included an analysis of the mass insertion approximation: the latter was found to be sufficient in many cases and we will use it again in this study. In Ref. [12] the bounds from the rare decay b → sγ were derived; however, the various couplings were assumed to be real. In Ref. [13], the influence on the electromagnetic penguins (isospin violating terms) and the consequences on B → Kπ decays were studied; particular focus was given to the determination of the CKM angle γ. In Ref. [3] effects of new terms on direct CP -violating asymmetries in charged decays were given, but the possible values of the new coefficients were not investigated. In Ref. [5] a completely general parameterization of new physics effects in the decays B → φK and B → J/ΨK was given; however, no particular model was explicitely studied. In Ref. [14], the case of left-right symmetric models were investigated and it was found that, in these models, sizeable deviations from the relation a ψK = a φK are possible.
In this paper, we consider the complex (2, 3) entries in the down squark mass matrix and work out their consequences for the time dependent CP asymmetries in the decays B → φK and B → J/ΨK. In the standard model, these asymmetries are equal and are a measure of the phase of the B d -B d mixing amplitude (which in the SM is 2β). The complex couplings that we consider leave the B d -B d mixing phase unchanged but have a significant impact on the amplitudes of the decays. The B → J/ΨK amplitude is dominated by SM tree level contributions and, consequently, the CP -violating asymmetry is hardly modified; on the other hand, there is room for such an effect in the transition B → φK. This decay is easily accessible at the B-factory experiments [15] and an investigation of the related CP asymmetries is clearly worthwhile. In the analysis we will concentrate on this decay. This note is organized as follows. In Sec. 2 we review the effective Hamiltonian for b → sss transitions in generic extensions of the standard model and give the explicit expressions of the various CP asymmetries. We also discuss the observables B(B → X s γ) and ∆M Bs which are related to the b → s transition and provide interesting additional pieces of information. In Sec. 3 we present the SUSY model that we consider and the explicit contributions to the Wilson coefficients. The numerical analysis is presented in Sec. 4. The impact of the complex SUSY parameters on the branching ratio and CP asymmetry of B → X s γ is explored and we analyze the correlation between the mixing-induced (and direct) CP asymmetries in B → φK and the B s -B s mass difference. A brief summary of our results and some comments are found in Sec. 5.

Effective Hamiltonian
The effective Hamiltonian for b → s transitions can be written as where (e u,c = 2/3 and e d,s,b = −1/3) In Eq. (3), we did not write the operators O u 1,2 obtained by the replacements c → u in O 1,2 , and the semileptonic current-current operators which induce the transitions b → sℓ + ℓ − . The effective Hamiltonian for b → d transitions can be obtained via the substitution s → d. Also, operators with different helicity structures are not explicitely written and can be obtained from the above operator basis via the replacement L ↔ R [11]. We will comment on their impact on the numerical analysis in section 4.
The SM values of the Wilson coefficients C 1−10 (m b ) and C 7γ,8g (m b ) are given in table 1; the opposite chirality operators do not get standard model contributions if the light masses are neglected. Table 1: SM Wilson coefficients at the scale m b .
We will use the following approximate form § for the B → (J/Ψ, φ)K amplitudes [3]: 76 is the ratio of the chromo-magnetic to penguin form factors for the B → (J/Ψ, φ)K decay, q 2 ≃ m 2 b /2, and r 8 ≃ 1/12 is the ratio of colour octet and singlet matrix elements. P is an O(α s ) contribution to the matrix elements of the QCD penguin operators, it is insensitive to new short distance physics, and it is originated by loop diagrams with an internal charm quark. Since the typical q 2 is above the charm production threshold, P carries a strong phase and its numerical value is −0.0132 − i 0.0145 [16]. Finally, the coefficients C ψ,φ are where all the C i are to be evaluated at the scale m b .
New physics contribute to the above Wilson coefficients; the resulting new phases will show up in the direct and mixing induced CP asymmetries. Before writing an explicit model for such contributions, we briefly discuss the renormalization group (RG) running of The solution of the NLO RG equations reads: and the other coefficients are fixed to their SM value. It follows that the impact of the QCD penguin matching conditions (the values of the Wilson § This expression is based on naive factorization and is therefore not exact; but it correctly describes the way new physics enters. coefficients at the scale M W ) on C ψ (m b ) and C φ (m b ) is respectively of order 1% and 10%. Thus, as it is well known, their effects on C ψ (m b ) is totally negligible: the coefficients of the current-current operators totally dominate the amplitude even if their contribution is colour suppressed. Thus, the SM prediction of an extremely small direct CP asymmetry in the J/Ψ mode is unaltered by new physics contributions. On the other hand, new physics whose contributions are large enough to dominate the phases of C 3−6 (M W ), may modify the overall phase of C φ (m b ) at the 10% level. Taking into account that the CP asymmetry depends on twice the phase of C φ (m b ), we see that in principle large (O(0.2)) deviations from the SM relation a ψK = a φK are possible. Moreover, the presence of a term proportional to C 8g in Eq. (17) can have a strong impact if there are large new complex contributions to the chromo-magnetic dipole operator. In the standard model, this term is negligible.
Let us introduce the time-dependent CP asymmetry in the decays B 0 d /B 0 d → φK, given by is the state at time t which started as a pure B 0 d (B 0 d ) at t = 0. A dir CP and A mix CP are the direct and mixing-induced CP-asymmetries, respectively. Their explicit expressions are with Here, β is the inner angle of the unitarity triangle of the standard model, θ d is a possible new physics contribution to the phase of the B d −B d oscillations, and A (Ā) are the amplitudes of the decay in question (and its CP conjugate).
We turn next to the branching ratio and CP asymmetry in the decay B → X s γ and to the B s -B s mass difference. These observables are strongly affected by new physics in the FCNC b → s transition and therefore must be properly accounted for.

B → X s γ
The inclusive transition B → X s γ plays a major role in limiting possible new physics contributions to B decays both through constraints on the branching ratio and on direct CP asymmetry. The experimental information on the latter quantities is [17] B(B → X s γ) = (3.22 ± 0.40) × 10 −4 (26) A CP (B → X s γ) = (−3.5 ± 7.7) % .
In the numerical analysis we will use the NLO computation of B(B → X s γ) which, as a function of the . The parameter δ that determines the cut on the photon energy spectrum is set to 0.90 according to Ref. [18].
Similarly, the CP can be written as [19]:

∆M B s
The effective Hamiltonian for the ∆S = 2 transitions can be written as together with the operators O V RR and O SRR 1,2 , obtained from the corresponding LL ones via the substitution L ↔ R.
In order to minimize the impact of hadronic uncertainties we will consider, as usual, the ratio X sd ≡ ∆M Bs /∆M B d . Its explicit expression is [20]: where with κ s = M 2 Bs /(m b + m s ) 2 and ξ is a ratio of hadronic matrix elements, numerically equal to 1.16 ± 0.05 [2]. The coefficient C d is obtained by replacing s with d in Eq. (38). It is important to stress that the contributions from new physics are generally different for the B s and B d systems. The ratio C s /C d can therefore be sizeably different from unity and has to be taken into account in model independent analyses of the unitary triangle when using Eq. (37).
In the numerical analysis we will require X sd to lay in the interval (30 ÷ 60) in order to satisfy the lower bound ∆M Bs > 14.9 ps −1 and have an observable B s -B s mass difference at the same time.

An example: SUSY gluino contributions
In order to present a definite new physics model which contributes sizeably to the CP asymmetries in B → φK, we turn to a variant of the MSSM with complex off-diagonal squark mass terms. In particular we will focus on the following entries: The other parameters of the model are the common mass of the squarksm, and the gluino mass mg. Following a common practice, we consider the normalized insertions δ d 23 given by the ratios of the various (m 2 ) d 23 tom 2 . It is well known, and in the next section we will give a detailed quantitative analysis of the question, that B(B → X s γ) puts severe bounds of order O(10 −2 ) on the LR and RL insertions while the impact on the LL and RR ones is rather mild. This strong result follows from the mg/m b chiral enhancement of the (δ d 23 ) LR,RL contributions to the Wilson coefficient C 7γ,8g . Note that this chiral factor is absent in the in the WC's that govern the B s -B s mass difference and the B → φK amplitude. For the latter observables, in fact, all the mass insertions enter with similar weight: therefore, the only insertions that can play a role are the LL and RR ones. Since the analyses for both insertions give the same results, we will consider explicitely only (δ d 23 ) LL . The expressions of the SUSY contributions to the various coefficients are [21] where xgq ≡ m 2 g /m 2 q (we assume a common squark mass). The loop functions can be found in Refs. [20,21]. Note that we have included the phase of V tb V * ts (which is basically a minus sign) in the definition of the mass insertion.

Numerical analysis
We begin with the constraints imposed by the B → X s γ branching ratio and CP asymmetry. We assume the absence of so-called accidental cancellations between different large contributions; therefore, we present the analysis assuming the presence of only one insertion per time besides the SM. If all of the insertions are substantial at the same time, the allowed ranges of some combination of them can be sizeably enlarged (see for instance Ref. [12] where the case of multiple real mass insertions is considered). In Fig. 1 we take xgq = 1,m = 250, 500 GeV and require |δ| < 1; we also show the impact of using the 68% C.L. and 95% C.L. constraints on the B → X s γ branching ratio. The corresponding bounds induced by the CP asymmetry are much weaker and do not impact, at the moment, the allowed areas: future experimental improvements can substantially modify this picture. We plot the resulting regions in the [Reδ, Imδ] plane. Note that the contours scale withm 2 . As mentioned before, the bounds on the LL insertion are not very strong. Even for very light squarks (i.e.m = 250 GeV) it is possible to largely evade the constraint if the imaginary part of the insertion is O(1). This reflects the absence of interference with the (real) SM contribution. On the other hand, the corresponding bounds on the LR insertion are, as expected, of order O(10 −2 ); moreover, as it follows from the figure, if |δ LR | is larger than O(10 −3 ), a substantial correlation between real and imaginary parts is required. For the δ d RR and δ d RL insertions the results are similar. Next we analyze the possible deviations of the CP asymmetries in B → φK from the SM expectation and the relation to the B s -B s mass difference which is also altered by the insertions. In Fig. 2 we plot the correlation between the B s -B s mass difference and δA mix ≡ A mix − A mix,SM . We scan over the input SUSY parameters in the ranges for different values of xgq, the gluino-squark mass ratio. We require each point to satisfy the B → X s γ 95% C.L. constraint and to give a B s -B s mass difference in the range 30 ≤ X sd ≤ 60. The various regions correspond to the limiting case: for a given ratio xgq, no point lays outside them. For xgq ≥ 1 we find that the deviations from the SM expectation remain below 0.05. For smaller values of xgq much larger and thus observable contributions are possible. This strong dependence on the ratio between the gluino and squark masses is due to the particular dependence of the loop functions on xgq. Moreover, the presence of definite bands in the [X sd , δA mix ] plane is due to an interplay between the B → X s γ constraint and the requirement of fixed ∆M Bs .
As previously stated, δA mix measures the deviation of the mixing-induced CP asymmetry from the standard model prediction. Since all new physics in the B d -B d mixing amplitude will affect the CP asymmetries in the decays B → J/ΨK and B → φK in the same way, δA mix is equal to sin 2(β + θ d + θ A ) − sin 2(β + θ d ) where θ d can be generated by other SUSY couplings that we do not consider here and θ A is the phase of the B → φK decay amplitude. In other words, δA mix is the difference between the mixing-induced CP asymmetries in the decays B → J/ΨK and B → φK (see also the discussion in Sec. 2).
The new weak phases in the B → φK amplitude also leads to a non vanishing direct CP asymmetry, A dir , which can be measured for instance in decays of charged B-mesons. This asymmetry depends crucially on the presence of a strong rescattering phase, provided by the term P of Eq. (17). Nevertheless, we find that the new physics contributions to the two asymmetries are strongly correlated. This can be understood as follows. Let us parametrize the B → φK decay amplitude as A 1 e iφ 1 e iδ 1 + A 2 e iφ 2 e iδ 2 where φ i and δ i are weak and strong phases respectively; note that this parametrization is arbitrary but that all physical observables do not depend on its choice. In the SUSY model that we consider, the weak phases φ i are entirely due to the imaginary part of the mass insertions. Using the above parametrization, we find that the ratio A dir /δA mix has an extremely tiny dependence on the phases φ i ; therefore, the correspondence between A dir and δA mix is almost one-to-one. In Fig. 3, we explicitely show this correlation.

Conclusions
We considered the effects of sizeable flavour changing entries in the squark matrices on b → s transitions. In particular we allowed for complex values of the relevant off-diagonal elements.
We first investigated the bounds that these entries must obey in order to satisfy the b → sγ data. Fig. 1 shows that the inclusion of complex values for the mass insertion parameters strongly enlarges their allowed regions. In fact, the absolute values of the insertions can be much larger than in the existing literature where only real couplings were considered. Moreover, there are interesting correlations between real and imaginary parts which may give important hints on the structure of the underlying theory.
We then considered the influence of the new terms on the CP -violating asymmetry in the decays B → φK and on the B s -B s mixing (the B d -B d mixing is not affected by the terms we are interested in). In Fig. 2, we plot the deviation δA mix of the mixing-induced CP asymmetry in B → φK from the SM expectation versus the ratio X sd . We see that δA mix can reach the 20% level in some corners of the parameter space; on the other hand, deviations of order O(10%) are easily possible with moderately light squark and gluino masses. Note that such large contributions are possible only for configurations in which mg mq. In a similar fashion, Fig. 3 shows that the direct CP asymmetry A dir can receive contributions of the same order of magnitude. We stress that in this framework effects on B → J/ΨK decays are expected to be tiny.
The B s -B s ∆M Bs is very sensitive to the mass insertions we consider, while ∆M B d remains unaffected. This implies that the determination of V td from ratio ∆M Bs /∆M B d may be misleading. Moreover, as it follows from Fig. 2 and 3, an experimental determination of ∆M Bs in excess with respect to the SM prediction together with sizeable δA mix and A dir would be strong signatures in favour of this kind of models. During the next year, the Bfactories BABAR and BELLE will gather enough luminosity to study the CP asymmetries in B → φK decays and will test this class of SUSY models soon.