Physics Letters B 635 (2006) 335–342 www.elsevier.com/locate/physletb QCD corrections to radiative B decays in the MSSM with minimal flavor violation G. Degrassi a, P. Gambino b,P.Slavichc,∗ a Dipartimento di Fisica, Università di Roma Tre and INFN, Sezione di Roma III, Via della Vasca Navale 84, I-00146 Rome, Italy b INFN, Sezione di Torino and Dipartimento di Fisica Teorica, Università di Torino, Via P. Giuria 1, I-10125 Torino, Italy c LAPTH, 9, Chemin de Bellevue, F-74941 Annecy-le-Vieux, France Received 24 January 2006; accepted 27 February 2006 Available online 13 March 2006 Editor: G.F. Giudice Abstract We compute the complete supersymmetric QCD corrections to the Wilson coefficients of the magnetic and chromomagnetic operators, relevant in the calculation of b → sγ decays, in the MSSM with minimal flavor violation. We investigate the numerical impact of the new results for different choices of the MSSM parameters and of the scale where the quark and squark mass matrices are assumed to be aligned. We find that the corrections can be important when the superpartners are relatively light, and that they depend sizeably on the scale of alignment. Finally, we discuss how our calculation can be employed when the scale of alignment is far from the weak scale. 2006 Elsevier B.V. All rights reserved. 1. Introduction The theoretical accuracy of the predictions in the context of new physics models may have important consequences on B More than a decade after their first direct observation, ra- model building. This is particularly true for radiative decays, diative B decays have become a key element in the program where higher order corrections can be enhanced by large fac- of precision tests of the Standard Model (SM) and its exten- tors: in this case the current status of theoretical calculations is not always satisfactory. While the NLO corrections have been sions. The inclusive decay B → Xsγ is particularly well suited to this precision program thanks to its low sensitivity to non- extensively studied in the context of two Higgs doublet mod- perturbative effects. The present experimental world average els [2,6,7], in the Minimal Supersymmetric Standard Model (MSSM) the complete leading order (LO) result is known [8,9] [1] for the branching ratio of B → Xsγ has a total error of less than 10% and agrees well with the SM prediction, that is but the NLO analysis is still incomplete to date. The main rea- subject to a similar uncertainty [2]. In view of the final accu- son is that new sources of flavor violation generally arise in the racy expected at the B factories, about 5%, the SM calculation MSSM, making a general analysis quite complicated even at the b → s needs to be improved. It presently includes next-to-leading or- leading order [9]. Experimental constraints on generic der (NLO) perturbative QCD corrections as well as the leading flavor violation have been recently studied in [10]: radiative de- non-perturbative and electroweak effects (see [3,4] for a concise cays play a central role in these analyses, and the constraints discussion and a complete list of references). The calculation of are strong only for some of the flavor violating parameters. next-to-next-to-leading order (NNLO) QCD effects is currently A simplifying assumption frequently employed in super- minimal flavor violation under way [5] and is expected to bring the theoretical accuracy symmetric analyses is that of (MFV), to the required level. according to which the only source of flavor (and possibly of CP) violation in the MSSM is the CKM matrix [11,12].It can be implemented by assuming that the squark and quark * Corresponding author. mass matrices can be simultaneously diagonalized and, as E-mail address: [email protected] (P. Slavich). a consequence, it implies the absence of tree-level flavor- 0370-2693/$ – see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.02.067 336 G. Degrassi et al. / Physics Letters B 635 (2006) 335–342 changing gluino (FCG) interactions. The MFV hypothesis cer- the context of realistic models of SUSY breaking and contains tainly represents a useful and predictive approximation scheme our conclusions. and seems to be favored by the present absence of deviations from the SM. However, because the weak interactions affect the 2. Gluino contribution to the Wilson coefficients squark and quark mass matrices in a different way [13], their simultaneous diagonalization is not preserved by higher order As we focus here on short-distance contributions with corrections and can be consistently imposed only at a certain MFV, we can restrict our discussion to the form of the scale µMFV, complicating the study of higher order contribu- Wilson coefficients of the B = 1 magnetic and chromo- 1 2 µν tions in this framework. The NLO study of radiative decays in magnetic operators Q7 = (e/16π )mbs¯Lσ bRFµν and Q8 = 2 ¯ µν a a the MFV scenario has been pioneered in [14] (see also [15]), (gs/16π )mbsLσ T bRGµν evaluated at the matching scale where the gluonic corrections to chargino contributions have µW in the effective Hamiltonian been computed, while those involving a gluino were computed H =−4√GF ∗ in the heavy gluino limit, in which case FCG effects can be con- VtsVtb Ci(µW )Qi(µW ), (1) 2 sistently neglected. i An alternative possibility is to include only the potentially where GF is the Fermi constant and Vts, Vtb are elements of large contributions beyond the leading order: they originate the CKM matrix. We can organize the Wilson coefficients of from terms enhanced by tan β factors, when the ratio between the operators Q7,8 in the following way: the two Higgs vacuum expectation values is large, or by log- ± = (0)SM + (0)H + (0)SUSY arithms of MSUSY/MW , when the supersymmetric particles C7,8(µW ) C7,8 (µW ) C7,8 (µW ) C7,8 (µW ) are considerably heavier than the W boson. Compact formulae α (µ ) ± + s W C(1)SM(µ ) + C(1)H (µ ) that include both kinds of higher-order effects within MFV are 4π 7,8 W 7,8 W given in Ref. [16]. Indeed, tan β-enhanced terms at the next- + C(1)SUSY(µ ) , (2) to-leading order do not only appear from the Hall–Rattazzi– 7,8 W Sarid effect (the modified relation between the bottom mass where the various LO contributions are classified according and Yukawa coupling) [18], but also from an analogous ef- to whether the corresponding one-loop diagrams contain only fect in the top-quark Yukawa coupling [11,16,17] and in effec- (0)SM SM fields (C7,8 ), a physical charged Higgs boson and an tive flavor-changing s¯ b neutral heavy Higgs vertices [11].In ± L R up-type quark (C(0)H ), or a chargino and an up-type squark the effective theory approach first employed in [19] the dom- 7,8 (0)SUSY (0)SM (0)H± inant terms enhanced by tan β can be taken into account at (C7,8 ). The expressions for C7,8 and C7,8 can be all orders. A generalization beyond MFV has been proposed (0)SUSY found, e.g., in Ref. [6], while those for C7,8 can be found, in [20]. e.g., in Eq. (4) of Ref. [14]. Neutralino and gluino exchange In the limit of heavy superpartners, in particular, the Higgs diagrams will be neglected under our MFV assumption. The sector of the MSSM is modified by non-decoupling effects relation between the LO and NLO Wilson coefficients at µW and can differ substantially from the type-II two Higgs doublet and the branching ratio for B → Xsγ is well known (see for model. The charged Higgs contribution therefore receives two- example Refs. [2,6]). loop contributions enhanced by tan β that have been computed (1)SM (1)H ± The NLO coefficients C7,8 and C7,8 contain the glu- in [11,16,17] in the limit of heavy gluino. Interestingly, the ex- onic two-loop corrections to the SM and charged Higgs loops, plicit calculation of the relevant two-loop diagrams presented in respectively, and can be found for instance in Ref. [6].AtNLO [21] has demonstrated the validity of this approximation even the supersymmetric contribution C(1)SUSY can be further de- when the charged Higgs is not much lighter than the gluino. 7,8 composed, However, there is a priori no reason why the results derived in ˜ ± the heavy gluino limit should be a good approximation of the C(1)SUSY = C(1)g + C(1)χ , (3) true result for generic values of the relevant mass parameters or 7,8 7,8 7,8 ˜ in the case of other two-loop contributions. (1)g where C7,8 contains two-loop diagrams with a gluino together In this Letter we present the results of the full NLO calcu- ± with a Higgs or W boson, while C(1)χ corresponds to two- lation of the supersymmetric QCD corrections to the Wilson 7,8 loop diagrams with a chargino together with a gluon or a gluino coefficients of the two operators that are relevant in the MFV or a quartic squark coupling. It should be recalled that, un- scenario, extending and completing the work of Ref. [14].In (1)SM (1)H ± particular, we compute all two-loop diagrams that contain a like C7,8 and C7,8 , the two-loop gluonic corrections to gluino, under the assumption that the gluino couplings to quarks the chargino loops are not UV finite: as shown in [14],inor- der to obtain a finite result one has to combine them with the and squarks are flavor conserving at the scale µMFV.