Dipole coefficients in B ->X_s gamma in supersymmetry with large \tan\beta and explicit CP violation

We perform a detailed study of the electric and chromoelectric dipole coefficients in B ->X_s \gamma decay in a supersymmetric scheme with explicit CP violation. In our analysis, we adopt the minimal flavor violation scheme by taking into account the \tan\beta-enhanced large contributions beyond the leading order. We show that the coefficients can deviate from the SM prediction significantly in both real and imaginary directions. Experimental bounds still allow for large deviations from the SM predictions for both dipole coefficients such that the CP asymmetry is as large as \pm 8%. There are further implications of these coefficients for the charmless hadronic and semileptonic B decays. As a direct application of our analysis, we have discussed \Lambda_b ->\Lambda \gamma decay.

The Operator Product Expansion (OPE) combined with the heavy quark effective theory (HQET) forms the basic tool in analyzing the decays as well as productions of hadrons (See the review [1]). Basically, the effective Hamiltonian describing the scattering processes can be expanded in a series of local operators (whose hadronic matrix elements constitute the longdistance effects) with Wilsonian coefficients (which are generated by the short-distance effects).
Among all the hadronic scattering processes the rare ones are particularly important, as the contributions of the standard electroweak theory (SM) and those of the 'new physics (NP)' arise at the same loop level thus suffering no relative loop suppressions. Furthermore, decays of the b-flavored hadrons (B, B ⋆ , Λ b , · · ·), compared to strange and charmed ones, are especially important as for such systems the HQET is fully applicable, and via the OPE, one can both test the SM and search for possible NP effects, by confronting the associated Wilson coefficients with the experiment.
In this work we will analyze the electric and chromoelectric dipole coefficients (denoted hereafter by C 7 and C 8 , respectively) describing the short-distance physics effects in rare B decays (e.g. B → X s γ, B → K ⋆ (892)γ and B → X s ℓ + ℓ − , · · ·) [2], by taking into account existing experimental results. The electric dipole coefficient C 7 , rescaled to µ = m b level, is directly constrained by the experimental result on B → X s γ [3]. However, the situation for the coefficient C 8 is obscured by the fact that the gluonic decay b → sg is not directly accesible in experiments.
However, this very coefficient plays an important role in the charmless hadronic B decays. For example, the theoretical predictions for the inclusive semileptonic decay rate Γ(b → ceν) and the charm multiplicity in B meson decays are significantly higher than the experimental results (See [4] and references therein). The most plausable solution to this discrepancy stems from possible enhancement of the chromoelectric coefficient C 8 by NP effects. Moreover, it is the relative phase between C 8 and C 7 which determines the CP asymmetry in B → X s γ, which will be measured in near-future B factories [5]. Consequently, it is essential to determine the size and phase of the gluonic coefficient in regions of the parameter space of NP where the theoretical predictions for B → X s γ agrees with the experiment.
In what follows, we will take low-energy minimal supersymmetry (SUSY) with explicit CP violation as the NP candidate. We will adopt the minimal flavor violation scenario (MFV), take into account the tan β-enhanced large contributions beyond the leading order (LO). The inclusive mode B → X s γ has already been analyzed within such a scheme by [6] (in CPconserving SUSY), and has been furthered by [7] to the CP-violating SUSY concluding a sizable CP asymmetry, which can compete with the experiment in near future. However, in both [6] and [7] the size and phase of C 8 , its correlation with C 7 , and its interdependence with the branching ratio, as well as its effects on the CP asymmetry have not been reported in detail. In particular, given the post-LEP bound on tan β > ∼ 3.5, it is necessary to have a detailed knowledge of C 8 in this portion of the SUSY parameter space. The main goal of this work is to determine the size and phase of the chromoelectric coefficient C 8 within the CP-violating SUSY at beyondthe-leading-order (BLO) precision in regions of the parameter space allowed by the existing The inclusive decay B → X s γ is well approximated (within at most 10%) by the partonic decay b → sγ which is described by the effective Hamiltonian where V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and the operator basis O i=1,···,8 is defined in [8,2]. This OPE for the Hamiltonian separates the long-distance (the matrix elements of the local operators O i ) and short-distance (associated Wilson coefficients C i ) at any scale µ ∈ (m b , M W ). Moreover, HQET approximates the inclusive rate by the partonic one (all terms being of the order O(Λ QCD /m b ) and higher ones are negligible) [9].
Evolution of the Wilson coefficients from µ = M W down to µ = m b level is governed by the standard QCD RGEs: where η = α s (M W )/α s (m b ), and the numerical coefficients h i , r i and g i are given in [8].
The initial values for the QCD RGEs, C 2,7,8 (M W ), depend on details of the short-distance theory at µ ∼ M W . In standard electroweak theory for instance, one finds C 2 (m b ) = 1.023, 148 at BLO precision [2]. Consequently, the NP effects can appear in various ways: (i) There can be observable deviations from these numbers with or without sign change, or (ii) the coefficients can take complex values. Each type of departure from the SM prediction implies certain aspects of the weak-scale NP effects. For instance, for (ii), it is obvious that the NP brings new sources of CP violation, and necessarily, the CP asymmetry of the decay deviates from the SM prediction ( < ∼ 1%) [5,10]. In what follows, we will take SUSY with explicit CP violation as the NP candidate, and concentrate on the results of [7] where the Wilson coefficients were computed at NLO precision for those threshold effects enhanced at large tan β. Within this framework C 2 (M W ) = 1 as in the SM, but the two dipole coefficients C 7,8 (M W ) are significantly modified compared to the LO results [11]. With the MFV scheme, only chargino-top squark, charged Higgs-top quark and W -boson-top quark loops give significant contributions In CP-violating SUSY, at LO precison C W,H 7,8 (M W ) are always real; they do not contribute to CP-violating observables, such as the CP-asymmetry, in the decay. However, the chargino contribution is complex due to the µ parameter (having finite phase φ µ ) and the stop trilinear With BLO precision, however, there are finite threshold corrections to each piece in (3) such Moreover, larger the tan β larger their imaginary parts, so that even naively one expects CP-violating effects to be enhanced at large tan β. Indeed, as reported in [7], the BLO CP asymmetry is significantly larger than the LO one at sufficiently large tan β. Although the asymmetry remains < ∼ 8% in both cases [12], there occurs an enhanced sensitivity to tan β for the BLO case. In the following we will perform a numerical study of the electric and chromoelectric dipole coefficients, and discuss their phenomenological implications. In the numerical analysis, we take: (i) the light stopt 2 and the charged Higgs H ± are degenerate and weigh close to the weak scale, GeV; (ii) the sfermions of first two generations are heavy enough, so that one can neglect their contribution to B → X s γ (this is a viable way of suppressing the one-loop EDMs [7]); (iii) the SU(2) gaugino, the right-handed sbottom and the heavy stop are heavy, TeV, and form the SUSY breaking scale; (iv) the stop and sbottom trilinear couplings have the same phase, θ A b = θ At , and the latter is degenerate with the µ parameter |µ| = |A b | = 150 GeV (sbottom parameters are needed for 2-loop EDM calculations [13]); and finally (v) the light stop is dominantly right-handed to agree with the electroweak precision data, hence the stop mixing is to be sufficiently small (θ t = π/20). Moreover, we vary tan β from 10 to 50 and the phase φ A,µ from 0 to π in forming the scatter plots. The parameter space mentioned here has been determined after trying several combinations, and it is one those points yielding large CP-violation in the system. One particularly notices that the lightest chargino is a Higgsino so that CP-violation via chargino-stop contribution is enhanced.
In minimal supergravity, for instance, the lightest chargino is SU(2) gaugino and thus CPviolation is very much suppressed. Fig. 1 is the tan β dependence of Re[C 7 (m b )] and Im[C 7 (m b )] for 10 ≤ tan β ≤ 50  However, there are regions of the parameter space where C 7 is pure imaginary, is pure real or vanishes exactly. Similar to observations made for Fig. 1, one can discuss the Wilson coefficient C 8 using Fig. 2 where its real (upper window) and imaginary (lower window) parts are separately plotted against tan β when φ µ,A vary from 0 to π. One notices that, unlike C 7 , for low tan β, Re[C 8 (m b )] deviates from its SM value. This stems from the fact that Those points where both C 7 and C 8 vanish are particularly interesting, as in this case one has to saturate the experimental bounds on B → X s γ via the chirality-flipped Wilson coefficients, implying large contributions due to the gluino exchange [14]. As we are working in the MFV scheme, such effects are obviously beyond the scope of our discussion. Fig. 3   A closer comparative look at the figures suggests that the CP asymmetry, which will be measured in near-future B factories with increasing precision, is maximal when |C 8 (m b )| ∼ |C 7 (m b )|. This results confirms earlier predictions [16] where it was already shown that A CP (B →

Depicted in
Even after including the NLL corrections to the semileptonic inclusive B decay B → Xeν, the theoretical predictons for the branching ratio and the charm multiplicity turn out to be larger than the experimental result [17,4]. Therefore, it is conceivable that possible enhancements in C 8 can account for the existing discrepancy between the theory and the experiment. As the numerical analyses above show, it is possible to significantly shift this coefficient in both real There are other processes where the radiatively corrected Wilson coefficients play an im-portant role. As an application, for instance, we analyze the bottom baryon radiative decay, Λ b → Λγ, which is again dominated by the mechanism b → sγ. We will particularly concentrate on the dependence of the braching ratio on the Wilson coefficients at the weak scale.
The decay rate of Λ b → Λγ has been computed in [19], and its expression is given by: with The form factors in a and b coefficients, depend on the momentum transfer as follows [19,20]: where q 2 m = (m Λ b − m Λ ) 2 and, q 2 = p Λ b − p Λ , with n=1 and n=2 representing the monopole and dipole contributions. Here, m V and m A are the pole masses of the vector and axial vector mesons, respectively.
In the numerical analysis, following [19,20], we let can be pure real, pure imaginary as well as it just vanishes.
In the recent work [19], it has been shown that B(Λ b → Λγ) has a magnitude of 1.9 × 10 −5 , when only the monopole q 2 dependence of the baryon form factors is considered (n=1  decay has also been considered in [21] with the predicted branching ratios in the range of We would like to note that the analysis of [19] has been carried out in the context of the S.M, and C 7 (m b ) has a fixed value (C 7 (m b ) = −0.312). However, in our work, C 7 and C 8 are complex, and as we have shown in Figs. 1 and 2      behave similarly. However, it decreases by an order of magnitude in the dipole case, and the maximal values of A CP (B → X s γ) are obtained when B(Λ b → Λγ) ∼ 1.5 × 10 −5 (1.9 × 10 −6 ) for n=1 (2). In the recent work of [19], it has been shown that B(Λ b → Λγ) = 2.3 × 10 −6 for n=2.
As we can see from the upper and lower windows, the asymmetry nearly takes the largest value, when B(Λ b → Λγ) is in the range of the SM prediction [19,22,23]. When B(Λ b → Λγ) takes larger values than the SM prediction, the asymmetry gradually drops to the corresponding SM prediction ∼ 1%.
In conclusion, we have computed the dipole coefficients C 7,8 in SUSY with explicit CP viola-tion with special emphasis on large values of tan β. We have shown that the present experimental bounds on B → X s γ allows for large deviations in the Wilson coefficients (with respect to the SM prediction) in both real and imaginary directions. The CP asymmetry in the decay is enhanced by an order of magnitude, thanks to especially the SUSY threshold corrections. The allowed deviations from the SM values can account for (being a plausable hypothesis) the discrepancy between the experiment and theory for the semileptonic B decays. As an illustration, we have discussed Λ b → Λγ decay.
M. B would like to thank the Scientific and Technical Research Council of Turkey (TÜBİTAK) for partial support under the project, No:TBAG2002(100T108).