Two-loop gluino corrections to the inclusive $B\to X_{_s}\gamma$ decay in CP violating MSSM with large $\tan\beta$

We investigate two-loop gluino corrections to the effective Lagrangian for $b\to s+\gamma(g)$ in the minimal supersymmetric extension of the standard model (MSSM) at large $\tan\beta$, including the contributions in which quark flavor change is mediated by charginos. Using the translation invariant of loop momenta and the Ward-Takahashi identities (WTIs) that are required by the $SU(3)_c\times U(1)_{_{em}}$ gauge invariance, we simplify our expressions to concise forms. As an example, we discuss two-loop gluino corrections to the CP asymmetry of inclusive $B\to X_{_s}\gamma$ decay in CP violating MSSM.


I. INTRODUCTION
The measurements of the branching ratios at CLEO, ALEPH and BELLE [1] give the combined result BR(B → X s γ) = (3.11 ± 0.42 ± 0.21) × 10 −4 , which agrees with the next-to-leading order (NLO) standard model (SM) prediction [2] BR(B → X s γ) = (3.29 ± 0.33) × 10 −4 . Beside the Cabibbo-Kobayashi-Maskawa (CKM) mechanism, the soft breaking terms provide a new source of CP and flavor violation in the MSSM. Those CP violating phases can affect the important observables in the mixing of Higgs bosons [3], the lepton and neutron's electric dipole moments (EDMs) [4,5], lepton polarization asymmetries in the semi-leptonic decays [6], the production of P -wave charmonium and bottomonium [7], and CP violation in rare B-decays and in B 0B0 mixing [8]. At present, the strictest constraints on those CP violation phases originate from the lepton and neutron's EDMs. Nevertheless, if we invoke a cancellation mechanism among different supersymmetric contributions [4], or choose the sfermions of the first generation heavy enough [5], the loop inducing lepton and neutron's EDMs bound the argument of the µ parameter to be ≤ π/(5 tan β), leaving no constraints on the other explicitly CP violating phases.
The supersymmetry models at large tan β are implied by grand unified theories, where the unification of up-and down-type quark Yukawa couplings is made [9]. From the technical viewpoint, the dominant contributions to the relevant effective Lagrangian are the terms proportional to (tan β) n (n = 1, 2, · · ·) in a large tan β scenario. This will simplify our two-loop analysis drastically since we just keep those terms enhanced by tan β.
Assuming no additional sources of flavor violation other than the CKM matrix elements, the authors of [10] present an exact analysis of the two-loop gluino corrections to the rare decay b → s + γ(g) in which quark flavor change is mediated by the charged Higgs in CP conserving MSSM at large tan β. They also compare their exact result with that originating from the heavy mass expansion (HME) approximation [11]. Although the HME result approximates the exact two-loop analysis adequately when the supersymmetry energy scale is high enough, their analysis implies that the difference between the HME approximation and exact calculation is obvious in some parameter space of the MSSM. However, they do not consider the case in which quark flavor change is mediated by the charginos (the super partners of the charged Higgs and W bosons). In fact, we cannot provide any strong reason to ignore the contribution from the diagrams in which quark flavor change is induced by the charginos, even within large tan β scenarios. In this work, we present a complete analysis on the two-loop gluino corrections to the rare transitions b → s + γ(g) by including the contributions of those diagrams where quark flavor change is mediated by charginos in the framework of CP violating MSSM at large tan β. Furthermore, we also simplify our expressions to concise forms through loop momentum invariant and the WTIs that are required by the SU(3) c × U(1) em gauge invariance.
The paper is organized as follows. In Sec. II, we give all the diagrams needed to evaluate the O(α s tan β) contributions to the Wilson coefficients C 7 and C 8 entering the branching ratio BR(B → X s γ). The corresponding Wilson coefficients at the matching EW scale µ EW are also presented there. We apply the effective Lagrangian to the rare decay B → X s γ in Sec. III. By the numerical method, we show the two-loop corrections on the CP asymmetry for the process. Our conclusion is given in Sec. IV, and some long formulae are collected in appendices.

II. THE WILSON COEFFICIENTS FROM THE TWO-LOOP DIAGRAMS
In this section, we derive the relevant Wilson coefficients for the partonic decay b → sγ including two-loop gluino corrections. In a conventional form, the effective Hamilton is written as where V is the CKM matrix. The definitions of the magnetic and chromo-magnetic dipole operators are where F µν and G a µν are the field strengths of the photon and gluon respectively, and T a (a = 1, · · · , 8) are SU(3) c generators. In addition, e and g s represent the EW and strong couplings respectively. The other operators O i (i = 1, · · · , 6) are defined in [12].
In the framework of CP violation MSSM, the one-loop analysis on the CP asymmetry of inclusive B → X s γ decay has been presented elsewhere [9]. The two loop gluino diagrams, (4) at the two loop order is more challenging than that at the one loop order. Before we give those Wilson coefficient expressions explicitly, we state firstly the concrete steps required to obtain the coefficients from those two loop diagrams.
• After writing the amplitudes of those two loop triangle diagrams, we expand them in powers of external momenta to the second order.
• The even rank tensors in the loop momenta q 1 , q 2 can be replaced as follows The two-loop vacuum diagram with momenta and masses as in Eq. 5.
where D is the time-space dimension, T µνρσ = g µν g ρσ + g µρ g νσ + g µσ g ρν and D 0 = ((q 2 − . The odd rank tensors in the loop momenta can be dropped since the integrations are symmetric under the transformation q 1,2 → −q 1,2 . Here, we only retain the simplest two-loop propagator composition 1/D 0 which corresponds to the two-loop vacuum diagram (FIG. 2). Any complicated composition of two-loop propagators can be expressed as the linear combination of the simplest one 1/D 0 by use of the obvious decomposition formula with Q = q 1 , q 2 , or q 2 −q 1 . As an example, we apply the above two steps to the triangle where p, k represent the incoming momenta of the external quark b and photon t) are the mixing matrices of scalar quarks, and A q are the corresponding trilinear soft breaking parameters. Furthermore, θ 3,µ denote the CP phases of the SU(3) c gaugino mass and of the µ parameter respectively.
Since the quark mass m b from the Yukawa coupling of bottom quark is same order as the external momenta p, k in magnitude, we just expand the propagators in powers of the external momenta to the first order. In the soft breaking potential, the CP phase θ 3 is contained in the gluino mass terms where λ G denotes the gluino in a two-component Majorana spinor. With the redefini-tion of the gluino field the mass terms are transformed into with the four-component Majorana spinor Correspondingly, the CP phase θ 3 is transfered from the mass terms to the quarksquark-gluino vertex which is given by [13] − Here, α, β = 1, 2, 3 are quark and squark color indices, and ω ± = 1±γ 5 2 . This is the reason why there is a |m 3 | rather than m 3 in the gluino propagator. Using Eq. 5 and Eq. 6, the amplitude of FIG. 3 is finally formulated as In a similar way, we can obtain the other triangle diagram amplitudes.
• Using loop momentum translation invariant, we formulate the sum of those amplitudes in gauge invariance form explicitly, then extract the corresponding Wilson coefficients which are expressed by the two-loop vacuum integrals [14]. In fact, there are many identities among those two-loop integrations. In order to obtain those necessary identities which are used to simplify the sum of those triangle amplitudes, we start from the zero integration such as Under the loop momentum translation q 2 → q 2 −a where a is same order as the external momentum p in magnitude, we expand the integration in powers of the momenta p, a to the second order The above identical equation implies Similarly, we can get the following identities from the invariant of Eq. 14 under the Using the concrete expressions of two-loop vacuum integrals in Ref. [14], we can also verify those equations in Eq. 16 and Eq. 17 directly after some tedious calculations.
Replacing the numerator of Eq. 14 with other odd rank tensors in the loop momenta q 1 , q 2 , the more additional identities among two-loop integrations are gotten. In general, those identities are linearly dependent. After some simplification, we obtain those linearly independent equations in appendix A. Certainly, those linearly independent equations can also be derived from those two-loop integrations in which the numerators are even rank tensors of the loop momenta q 1 , q 2 . However, the process to derive the linearly independent equations with the numerators in even powers of the loop momenta is more complicated than that with the numerators in odd powers of the loop momenta.
After the above procedure, we finally obtain the relevant coefficients from the charged Higgs contribution up to O(α s tan β) where α s = g 2 s /4π, and the expressions of the form factors N γ, g H (1,2) can be found in appendix B. Note, Ref. [10] has also obtained the Wilson coefficients from the same diagrams. We formulate our expressions in the more concise forms using the identities from appendix A.
For the chargino contribution that is ignored by Ref. [10], we can similarly have , ). Z − , Z + are the leftand right-handed mixing matrices of charginos, and the form factors N γ, g 2,3,4) are collected in appendix B. After we simplify the sum of thesbγ (g) triangle diagram amplitudes using the identities in appendix A, we find that the effectivesbγ (g) vertices should also include the two-point operator beside the magnetic (chromo-magnetic) dipole operators O 7 (O 8 ). Here, the covariant derivative acting on the quark fields is with G µ = G a µ T a . Certainly, the Wilson coefficient of this operator does not give any contribution to the rare process b → sγ after we evolve the corresponding coefficients from the matching EW scale to the hadronic scale. Nevertheless, when we extract the Wilson coefficients of O 7 (O 8 ), it makes sense to keep this operator for the following reason. Beside the effective vertex with two quarks the operator O se can also induce the effective vertices with two quarks and one photon or in the momentum space. Here, p, k are the incoming momenta of the external quark b and gauge boson (γ or g) respectively. For the effectivesbγ (g) vertices A γ (g) µ (p, k), the corresponding WTIs required by the SU(3) c × U(1) em gauge invariance are written as where iΣ(p) represents the sum of amplitudes for the self energy diagrams (FIG. 1). Expanding the self energy amplitudes in powers of external momentum to the third order, we where the function B 0 only depends on the heavy freedoms which are integrated out. In the effective theory, there are two deductions from the WTIs: • the effectivesbγ (g) vertices can be formulated as after we expand A γ (g) µ (p, k) in powers of the external momenta to the second order, where B γ,g 1,2 are the functions of heavy freedoms only; • additionally, B 0 = B γ 1 = B g 1 .
The above deductions can be taken as the criterion to test our calculations. In Eq. 26, the functions B γ,g 2 are proportional to the Wilson coefficients of the magnetic and chromomagnetic dipole operators respectively.
As an application, we will investigate the CP asymmetry of the rare decay B → X s γ within the framework of MSSM. In the SM, the CP asymmetry of the B → X s γ process is calculated to be rather small: A CP ∼ 0.5% [15]. For experimental data, the recent measurement [16] of the CP asymmetry implies the 95% range of −0.30 ≤ A CP (B → X s γ) ≤ 0.14 .
In other words, studies of the direct CP asymmetry in B → X s γ may uncover new sources of the CP violation which lie outside the SM. Up to the NLO, the complete theoretical prediction has been presented in Ref. [17]. In order to eliminate the strong dependence on the b-quark mass, the branching ratios is usually normalized by the decay rate of the B where f (z) = 1 − 8z + 8z 3 − z 4 − 12z 2 ln z is the phase-space factor with z = (m c /m b ) 2 , and α = e 2 /(4π) is the electroweak fine-structure constant. The CP asymmetry in the rare decay B → X s γ is correspondingly formulated as [15] A with g(z) = 5 + ln z + ln 2 z − π 2 /3 + ln 2 z − π 2 /3 z + 28/9 − 4/3 ln z z 2 + O(z 3 ).  [18]. Without loss of generality, we always assume the supersymmetric parameters µ = A t e −iπ/2 = 100 GeV, m 3 e −iπ/4 = 300 GeV, A s e −iπ/2 = m t R = 200 GeV, m t L = m s L = 5 TeV here. In order to suppress the one-loop EDMs, we choose the µ parameter CP phase θ µ = 0. As for the CP phase which is contained in the SU(2) gaugino mass parameter m 2 , it is set as θ 2 = arg(m 2 ) = π/4. Taking tan β = 30, |m 2 | = 300 GeV, m s R = 500 GeV, we plot the CP asymmetry and branching ratio of the inclusive B → X s γ decay versus the charged Higgs mass in FIG.   4. Considering the experimental constraint on the branching ratio BR(B → X s γ) at 1σ tolerance, the CP asymmetry including the two-loop corrections can be larger than 3%, and the one-loop result is smaller than 1% with our chosen parameters. The choice of parameter H + ) (Q denotes loop momenta q 1 , q 2 , or or q 2 − q 1 ), and depend on the parameter |m 2 | through the chargino propagator (/ Q − m χ )/(Q 2 − m 2 χ ) (m χ denotes the chargino mass) before the loop momentum integration.
In our analysis, we do not compare the exact two-loop analysis with the HME result since the discussion has already been presented in Ref. [10].

IV. CONCLUSIONS
In this work, we present the complete two-loop gluino corrections to inclusive B → X s γ decay in explicit CP violating MSSM within large tan β scenarios. Beside the diagrams where quark flavor change is mediated by the charged Higgs, we also include those diagrams in which quark flavor change is mediated by the charginos. Using loop momentum translation invariant, we formulate our expressions fulfilling the SU(3) c × U(1) em WTIs. From the numerical analysis, we show that the two-loop corrections to the branching ratio are comparable with the one-loop predictions at large tan β. Correspondingly, the CP asymmetry can also reach about 5%, which is much larger than that predicted by the SM.

APPENDIX A: IDENTITIES AMONG THE TWO-LOOP SCALAR INTEGRALS
Here, we report the identities that are used in the process of obtaining Eq. 18 and Eq.19, they can be derived from the loop momentum translation invariant of the amplitudes.
They are , and D is the time-space dimension. In addition the two-loop vacuum integral has been discussed in Ref. [14].