Heavy Flavours 8, Southampton, UK, 1999 PROCEEDINGS Theory of Rare B Meson Decays C. Greub∗ Institut f¨ur theoretische Physik, Universit¨at Bern, CH-3012 Bern E-mail: [email protected] Abstract: I review the NLO QCD calculations of the branching ratio for B → Xsγ in the SM. −4 Including the leading electromagnetic corrections, one obtaines BR(B → Xsγ)=(3.32 0.30)× 10 . Confronting theory with the newest data, an updated value for |Vts| is obtained: |Vts| =0.037 0.007. Theoretical progress on the photon energy spectrum is also discussed. The inclusive FCNC semilep- tonic decays in the SM are briefly summarized. Furthermore, B → Xsγ is considered in 2HDMs and in different SUSY scenarios. QCD corrections are shown to be crucial. 1. Introduction used for the determination of various CKM ma- trix elements, occurring in the SM Lagrangian. In the Standard model (SM), rare B meson de- To extract these parameters from data, it is cru- cays are induced by one-loop diagrams, where cial that the corresponding decay rates are reli- W bosons and up-type quarks are exchanged. In ably calculable. An important class of such de- many extensions of the SM, there are additional cays are the inclusive rare B decays, like B → + − contributions, where the SM particles in the loop Xsγ, B → Xs` ` , B → Xsνν¯, B → Xdγ, + − are replaced by nonstandard ones, like charged B → Xd` ` , B → Xdνν¯, which are sensitive Higgs bosons, gluinos, charginos etc. Being in- to the CKM matrix elements |Vts| and |Vtd|,re- duced also at the one loop-level, the new physics spectively. In contrast to the corresponding ex- contributions are not necessarily suppressed rel- clusive channels, these inclusive decay modes are ative to the SM one. The resulting sensitivity theoretically cleaner, in the sense that no specific for nonstandard effects implies the possibility for model is needed to describe the final hadronic an indirect observation of new physics, or allows state. Nonperturbative effects in the inclusive to put limits on the masses and coupling param- modes are well under control due to heavy quark eters of the new particles. A general overview, effective theory. For example, the decay width where many B physics observables are investi- Γ(B → Xsγ) is well approximated by the par- gated w.r.t. new physics, was given at this con- tonic decay rate Γ(b → Xsγ) which can be an- ference by A. Masiero [1]. Concerning the new alyzed in renormalization group improved per- physics aspects of rare B decays, I concentrate in turbation theory. The class of non-perturbative 2 this article (see sections 3 and 4) on recent cal- effects which scale like 1/mb is expected to be culations of the branching ratio for the process well below 10% [7]. This numerical statement B → Xsγ in a general class of two-Higgs-doublet also holds for the non-perturbative contributions 2 models (2HMDs) [2, 3] and in supersymmetric which scale like 1/mc [8, 9]. scenarios [4, 5, 6], stessing in particular the im- The framework and the NLO theoretical re- portance of leading- (LO) and next-to-leading sults for the branching ratio of the decay B → (NLO) QCD corrections. Xsγ are discussed in section 2.1; the photon en- However, also in the absence of new physics, ergy spectrum and the partially integrated branch- rare B decays are very important; they can be ing ratio for this process are reviewed in section ∗Work partially supported by Schweizerischer Nation- 2.2; an updated value for the CKM matrix ele- alfonds. ment |Vts|, extracted from the most recent mea- Heavy Flavours 8, Southampton, UK, 1999 C. Greub surements of BR(B → Xsγ) and the correspond- where Oi(µ) are local operators consisting of light ing calculations, where also the leading electro- fields, Ci(µ) are the corresponding Wilson coeffi- magnetic corrections are included, is given in sec- cients, which contain the complete top- and W - ∗ tion 2.3; the other inclusive rare decays men- mass dependence, and λt = VtbVts with Vij be- tioned above, are briefly discussed in section 2.4. ing the CKM matrix elements. The CKM depen- The exclusive analogues, B → K∗γ, B → dence globally factorizes, because we work in the (∗) + − + − K ` ` , B → ργ, B → ρ` ` etc., require the approximation λu =0. calculation of form factors. As the QCD sum rule Retaining only operators up to dimension 6 calculations and the lattice results for these form and using the equations of motion, one arrives at factors were summarized by V. Braun [10], I do the following basis not discuss these decays in the following. µ O1 =(¯c γ b )(¯s γ c ) , Finally, there is the class of non-leptonic two- Lβ Lα Lα µ Lβ µ body decays, like B → ππ, B → Kπ;thetheo- O2 =(¯cLαγ bLα)(¯sLβγµcLβ) , e retical status of these processes was discussed by µν O7 = 2 s¯α σ (mb(µ)R) bα Fµν , L. Silvestrini [11], and new CLEO results were 16π A gs µν λαβ A presented by D. Jaffe [12]. O8 = s¯ σ (m (µ)R) b G (2.2). 16π2 α b 2 β µν As the Wilson coefficients of the QCD penguin 2. Inclusive rare B meson decays in operators O3, ..., O6 are small, we do not list them the SM here. It is by now well known that a consistent 2.1 BR(B → X γ) at NLO precision s calculation for b → sγ at LO (or NLO) precision Short distance QCD effects enhance the partonic requires three steps: decay rate Γ(b → sγ) by more than a factor of 1) a matching calculation of the full standard two. Analytically, these QCD corrections contain model theory with the effective theory at large logarithms of the form αn(m )logm(m/M ), s b b the scale µ = µ to order α0 (or α1)for where M = m or M = m and m ≤ n (with W s s t W the Wilson coefficients, where µ denotes n =0,1,2, ...). In order to get a reasonable W a scale of order M or m ; prediction for the decay rate, it is evident that W t one has to resum at least the leading-log (LO) 2) a renormalization group evolution of the series (m = n). As the error of the LO re- Wilson coefficients from the matching scale sult [13] was dominated by a large renormaliza- µW down to the low scale µb = O(mb), us- tion scale dependence at the 25% level, it be- ing the anomalous-dimension matrix to or- 1 2 came clear that even the NLO terms of the form der αs (or αs); n n αs(mb)(α (mb)ln (mb/M )) have to be taken s 3) a calculation of the matrix elements of the into account systematically. operators at the scale µ = µ to order α0 To achieve the necessary resummations, one b s (or α1). usually contructs in a first step an effective low- s energy theory and then resums the large loga- At NLO precision, all three steps are rather rithms by renomalization group techniques. The involved: The most difficult part in Step 1 is the low energy theory is obtained by integrating out order αs matching of the dipole operators O7 and the heavy particles which in the SM are the top O8. Two-loop diagrams, both in the full- and quark and the W -boson. The resulting effective in the effective theory have to be worked out. Hamiltonian relevant for b → sγ in the SM and This matching calculation was first performed many of its extensions reads by Adel and Yao [14] and then confirmed by Greub and Hurth [15], using a different method. X8 W → −4√GF Later, two further recalculations of this result Heff (b sγ)= λt Ci(µ) Oi(µ) , 2 2 i=1 were presented [3, 16]. The order αs anoma- (2.1) lous matrix (Step 2) has been worked out by 2 Heavy Flavours 8, Southampton, UK, 1999 C. Greub Chetyrkin, Misiak and M¨unz [17]. The extrac- corrections presented in [22]; we then obtained tion of certain elements in this matrix involves → 0.00 0.00 0.26 × −4 the calculation of pole parts of three loop dia- BR(B Xsγ)= 3.32 0.11 0.08 0.25 10 . grams. Step 3 consists of Bremsstrahlung con- (2.4) tributions and virtual corrections. The Brems- The bulk of the change with respect to the value in equation (2.3) is due to the different value of strahlung corrections were worked out some time −1 ago by Ali and Greub [18] and have been con- αem used. firmed and extended by Pott [19]. An analysis A remark concerning the error due to the of the virtual two loop corrections was presented variation of the low scale µb in the results (2.3) by Greub, Hurth and Wyler [20]. and (2.4) is in order here: As it will be dis- cussed in more detail in section 3, it was real- Combining the NLO calculations of these 3 ized in [2] that in multi Higgs doublet models steps, leads to the following NLO QCD predic- the QCD corrections in certain regions of the pa- tion for the branching ratio [2]: rameter space are much larger than in the SM. As a consequence, the dependence on the scale 0.01 0.00 0.29 −4 BR(B → Xsγ)= 3.57 0 12 0 08 0 27 ×10 . µb of BR(B → Xsγ) is also larger. Later, Ka- (2.3) gan and Neubert pointed out very explicitly in The central value is obtained for µb =4.8GeV, their analysis [22] that the scale dependences in µW =mW and the central values of the input pa- individual contributions to the branching ratio rameters listed in [2].