Radiative and Electroweak Penguin Decays of B Mesons
aa r X i v : . [ h e p - ph ] J a n Radiative and Electroweak Penguin Decays of B Mesons
Radiative and Electroweak Penguin Decaysof B Mesons
Tobias Hurth
Inst. for Physics, Johannes Gutenberg University, D-55099 Mainz, Germany;email: [email protected] and Mikihiko Nakao
KEK, High Energy Accelerator Research Organization, Tsukuba, 305-0801,Japan and the Graduate University for Advanced Studies (Sokendai), Tsukuba,305-0801, Japan; email: [email protected]
Abstract
The huge datasets collected at the two B factories, Belle and BaBar, have madeit possible to explore the radiative penguin process b → sγ , the electroweak penguin process b → sℓ + ℓ − and the suppressed radiative process b → dγ in detail, all in exclusive channels andinclusive measurements. Theoretical tools have also advanced to meet or surpass the experi-mental precision, especially in inclusive calculations and the various ratios of exclusive channels.In this article, we review the theoretical and experimental progress over the past decade in theradiative and electroweak penguin decays of B mesons. CONTENTS
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electroweak Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbative Corrections to Inclusive Decays . . . . . . . . . . . . . . . . . . . . . . . . Hadronic Power Corrections to Inclusive Modes . . . . . . . . . . . . . . . . . . . . . . Nonperturbative Corrections due to Kinematical Cuts . . . . . . . . . . . . . . . . . . . Charmonium Resonance Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . Soft Collinear Effective Theory for Exclusive Decays . . . . . . . . . . . . . . . . . . . EXPERIMENTAL TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . Exclusive B Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inclusive Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PRESENT THEORETICAL PREDICTIONS . . . . . . . . . . . . . . . . . . . . . Inclusive Penguin Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exclusive Penguin Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PRESENT EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . Inclusive B → X s γ Branching Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . Exclusive Measurements of b → sγ Processes . . . . . . . . . . . . . . . . . . . . . . . CP and Isospin Asymmetries in b → sγ Processes . . . . . . . . . . . . . . . . . . . . easurements of b → dγ Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exclusive B → K ( ∗ ) ℓ + ℓ − Branching Fraction . . . . . . . . . . . . . . . . . . . . . . . B → K ( ∗ ) ℓ + ℓ − Asymmetries and Angular Distributions . . . . . . . . . . . . . . . . . Inclusive B → X s ℓ + ℓ − Branching Fraction . . . . . . . . . . . . . . . . . . . . . . . . OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The B meson system, which is a bound state that consists of a b quark and alight antiquark, provides an ideal laboratory for precise study of the StandardModel (SM) of particle physics, and thus facilitates the search for new physics(NP). Because the b quark mass is much larger than the typical scale of thestrong interaction, the otherwise troublesome long-distance strong interactionsare generally less important and are under better control than in other lightermeson systems. Radiative penguin decays of the B meson with the emission ofa photon ( γ ) and electroweak penguin decays with the emission of a lepton pair( ℓ + ℓ − , ℓ = e, µ ) are of particular interest in this respect. These processes revealthe flavor changing neutral current (FCNC), that is the transition of a b quarkwith an electric charge of − / s or a d quark of the same charge. Inthe SM, the FCNC occurs only via virtual loop diagrams (Fig. 1). AdditionalNP contributions to these decay rates are not necessarily suppressed with respectto the SM contribution. Examples of such NP scenarios include those in whichthe SM particles in the loop diagram are replaced by hypothetical new particlesat a high mass scale; so far, they have not been directly accessible in colliderexperiments. Radiative and electroweak penguin decays are highly sensitive toNP because they are theoretically well-understood and have been extensivelymeasured at the B factories. The search for such NP effects complements thesearch for new particles produced at collider experiments.The first generation of the B factories at KEK (the Belle experiment at theKEKB e + e − collider) (3) and at SLAC (the BaBar experiment at the PEP-II e + e − collider) (4) have collected huge samples of B meson decays and havethereby established the SM picture of CP violation and other flavor-changing pro-cesses in the quark sector. These processes are governed by a single 3 × B factories, the Tevatron B physicsprograms (namely the CDF (7) and D0 (8) experiments), and earlier kaon de-cay experiments. In other words, none of the current measurements of B mesondecays have observed any unambiguous sign of NP (9, 10). Although this exper-imental result is an impressive success of the CKM theory within the SM, thereis still room for sizable new effects from new flavor structures, given that FCNCprocesses have been tested up to only the 10% level.The nonexistence of large NP effects in the FCNC processes hints at the famousflavor problem, namely why FCNCs are suppressed. This problem must be solvedin any viable NP model. Either the mass scale of the new degrees of freedom isvery high or the new flavor-violating couplings are small for reasons that remainto be found. For example, assuming generic new flavor-violating couplings, the The name penguin decays was first introduced in Ref. (1) as the result of a bet. A moredetailed account of the name can be found in Ref. (2). adiative and Electroweak Penguin Decays of B Mesons K - K mixing implies a very high NP scale of order 10 –10 TeVdepending on whether the new contributions enter at loop-level or at tree-level.In contrast, theoretical considerations on the Higgs sector, which is responsiblefor the mass generation of the fundamental particles in the SM, call for NP atorder 1 TeV. As a consequence, any NP below the 1-TeV scale must have anongeneric flavor structure. The present measurements of B decays, especially ofFCNC processes, already significantly restrict the parameter space of NP models.For further considerations on NP, the reader is referred to another article in thisvolume (11) and to Ref. (12).Quark-level FCNC processes such as b → sγ , to which NP may contribute,cannot be directly measured because the strong interaction forms hadrons fromthe underlying quarks. Instead, the experimentally measured and theoreticallycalculated process is a B meson decay into a photon plus an inclusive hadronicfinal state X s , which includes all the hadron combinations that carry the strangequantum number s = +1 of the s quark. Exclusive final states with one or afew specific hadrons in the final state (e.g., B → K ∗ γ ) have less predictive powertheoretically; however, because the measurements are easier and better defined,there are other useful observables beyond branching fractions, in particular CP ,forward-backward, isospin, and polarization asymmetries. In the future, a largeoverconstrained set of measurements of these observables will allow us to detectspecific patterns and to distinguish between various NP scenarios.This review covers progress in radiative and electroweak decays in the pastdecade, during which a huge number of B factory results were accumulated andsignificant progress in various theoretical aspects was achieved. The pioneeringwork that led to the first observation of the b → sγ process by CLEO (13, 14)was discussed in an earlier volume of this journal (15).Our review is organized as follows. In Section 2, we describe theoretical toolsfor radiative and electroweak penguin decays, and in Section 3 we describe exper-imental techniques. We give theoretical predictions in Section 4 and summarizethe measurements of radiative and electroweak penguin decays in Section 5. Fi-nally, we briefly discuss future prospects in Section 6. Inclusive B decays are theoretically clean because they are dominated by par-tonic (perturbatively calculable) contributions. Nonperturbative corrections arein general rather small (16, 17, 18). This result can be derived with the help ofthe heavy mass expansion (HME) of the inclusive decay rates in inverse powersof the b quark mass. Up-to-date predictions of exclusive B decays are based onthe quantum chromodynamics (QCD)-improved factorization (QCDF) and softcollinear effective theory (SCET) methods. In general, exclusive modes havelarger nonperturbative QCD corrections than do inclusive modes. In this review, we use the following notations and conventions: We denote the inclusivedecay as B → X s γ when charge conjugation is implied, or as B → X s γ and B → X s γ to reflectthe quark charges of the underlying processes b → sγ and b → sγ , respectively, when CP andangular asymmetries are concerned. Here, B denotes either an isospin- and CP -averaged stateof B , B , B + and B − mesons, or an isospin averaged state of B and B + (in the latter case, B denotes B and B − ). Expressions are constructed similarly for inclusive X d γ and X s ℓ + ℓ − final states, and isospin-averaged exclusive final states. In the literature, the notation B → X s γ is also commonly used for the case that includes charge conjugation. Tobias Hurth and Mikihiko Nakao
Rare B decays are governed by an interplay between the weak and strong inter-actions. The QCD corrections that arise from hard gluon exchange bring in largelogarithms of the form α ns ( m b ) log m ( m b /M ), where M = m t or M = m W and m ≤ n (with n = 0 , , , ... ). These large logarithms are a natural feature in anyprocess in which two different mass scales are present. To obtain a reasonableresult, one must resum at least the leading-log (LL) series, n = m , with thehelp of renormalization-group techniques. Working to next-to-leading-log (NLL)or next-to-next-to-leading-log (NNLL) precision means that one resums all theterms with n = m + 1 or n = m + 2. A suitable framework in which to achievethe necessary resummations of the large logarithms is an effective low-energytheory with five quarks; this framework is obtained by integrating out the heavyparticles, which in the SM are the electroweak bosons and the top quark.This effective field theory approach serves as a theoretical framework for bothinclusive and exclusive modes. The standard method of the operator productexpansion (OPE) (19,20) allows for a separation of the B meson decay amplitudeinto two distinct parts: the long-distance contributions contained in the operatormatrix elements and the short-distance physics described by the so-called Wilsoncoefficients.The electroweak effective Hamiltonian (21, 22, 23) can be written as H eff = 4 G F √ X C i ( µ, M ) O i ( µ ) , (1)where O i ( µ ) are the relevant operators and C i ( µ, M ) are the corresponding Wil-son coefficients. As the heavy fields are integrated out, the complete top and W mass dependence is contained in the Wilson coefficients. Clearly, only withinthe observable H eff does the scale dependence cancel out. G F denotes the Fermicoupling constant.From the µ independence of the effective Hamiltonian, one can derive a renor-malization group equation (RGE) for the Wilson coefficients C i ( µ ): µ ddµ C i ( µ ) = γ ji C j ( µ ) , (2)where the matrix γ is the anomalous dimension matrix of the operators O i , whichdescribes the anomalous scaling of the operators with respect to the scaling atthe classical level. At leading order, the solution is given by˜ C i ( µ ) = (cid:20) α s ( µ W ) α s ( µ ) (cid:21) ˜ γ ii β ˜ C i ( µ W ) =
11 + β α s ( µ )4 π ln µ W µ ˜ γ ii β ˜ C i ( µ W ) , (3)where µ d/dµ α s = − β α s / (4 π ), and β and ˜ γ ii correspond to the leading anoma-lous dimensions of the coupling constant and of the operators, respectively. Thetilde indicates that the diagonalized anomalous dimension matrix is used.Although the Wilson coefficients C i ( µ ) enter both inclusive and exclusive pro-cesses and can be calculated with perturbative methods, the calculational ap-proaches to the matrix elements of the operators differ between the two cases.Within inclusive modes, one can use the quark-hadron duality to derive a well-defined HME of the decay rates in powers of Λ /m b (24, 25, 26, 27, 28, 29). In adiative and Electroweak Penguin Decays of B Mesons B → X s γ is well approximated by the partonicdecay rate, which can be calculated in renormalization-group-improved pertur-bation theory (30, 31):Γ( B → X s γ ) = Γ( b → X parton s γ ) + ∆ nonpert . . (4)In exclusive processes, however, one cannot rely on quark-hadron duality, so onemust estimate the matrix elements between meson states. A promising approachis the QCDF-method, which has been systematized for nonleptonic decays inthe heavy quark limit (32, 33, 34). In addition, a more general quantum fieldtheoretical framework for QCDF, known as SCET, has been proposed (35, 36, 37,38, 39, 40). This method allows for a perturbative calculation of QCD correctionsto na¨ıve factorization and is the basis for the up-to-date predictions for exclusiverare B decays. However, within this approach, a general quantitative method toestimate the important Λ /m b corrections to the heavy quark limit is missing. Within inclusive B decay modes, short-distance QCD effects are very important.For example, in the B → X s γ decay these effects lead to a rate enhancement bya factor of greater than two. Such effects are induced by hard-gluon exchangesbetween the quark lines of the one-loop electroweak diagrams. The correspondinglarge logarithms have to be summed as discussed above.The effective electroweak Hamiltonian that is relevant to b → s/d γ and b → s/d ℓ + ℓ − transitions in the SM reads H eff = − G F √ " λ tq X i =1 C i O i + λ uq X i =1 C i ( O i − O ui ) , (5)where the explicit CKM factors are λ tq = V tb V ∗ tq and λ uq = V ub V ∗ uq . The unitarityrelations λ cq = − λ tq − λ uq have already been used. The dimension-six operatorsare O = ( s L γ µ T a c L )( c L γ µ T a b L ) , O = ( s L γ µ c L )( c L γ µ b L ) , O u = ( s L γ µ T a u L )( u L γ µ T a b L ) , O u = ( s L γ µ u L )( u L γ µ b L ) , O = ( s L γ µ b L ) P q ( qγ µ q ) , O = ( s L γ µ T a b L ) P q ( qγ µ T a q ) , O = ( s L Γ b L ) P q ( q Γ ′ q ) , O = ( s L Γ T a b L ) P q ( q Γ ′ T a q ) , O = e π m b ( s L σ µν b R ) F µν , O = g s π m b ( s L σ µν T a b R ) G aµν , O = e π ( s L γ µ b L ) P ℓ ( ℓγ µ ℓ ) , O = e π ( s L γ µ b L ) P ℓ ( ℓγ µ γ ℓ ) , (6)where Γ = γ µ γ ν γ λ and Γ ′ = γ µ γ ν γ λ . The subscripts L and R refer to left- andright-handed components, respectively, of the fermion fields. In b → s transitionsthe contributions proportional to λ us are rather small, whereas in b → d decays,where λ ud is of the same order as λ td ; these contributions play an important rolein CP and isospin asymmetries. The semileptonic operators O and O occuronly in the semileptonic b → s/d ℓ + ℓ − modes.Among the four-quark operators, only the effective couplings for i = 1 , µ = m b [ C , ( m b ) ∼ C ( m b ) ∼ − . , C ( m b ) ∼ − .
15] and the semileptonic operators [ C ( m b ) ∼ , C ( m b ) ∼ −
4] also play a significant role.
Tobias Hurth and Mikihiko Nakao
There are three principal calculational steps that lead to the LL (NNLL) resultwithin the effective field theory approach:1. The full SM theory must be matched with the effective theory at the scale µ = µ W , where µ W denotes a scale of order m W or m t . The Wilson coeffi-cients C i ( µ W ) pick up only small QCD corrections, which can be calculatedwithin fixed-order perturbation theory. In the LL (NNLL) program, thematching has to be worked out at the O ( α s ) [ O ( α s )] level.2. The evolution of these Wilson coefficients from µ = µ W down to µ = µ b must then be performed with the help of the renormalization group, where µ b is of the order of m b . As the matrix elements of the operators evaluatedat the low scale µ b are free of large logarithms, the latter are contained inresummed form in the Wilson coefficients. For the LL (NNLL) calculation,this RGE step has to be performed using the anomalous-dimension matrixup to order α s ( α s ).3. To LL (NNLL) precision, the corrections to the matrix elements of theoperators h sγ |O i ( µ ) | b i at the scale µ = µ b must be calculated to order α s ( α s ) precision. The calculation also includes bremsstrahlung corrections. B → X s γ The error of the LL prediction of the B → X s γ branching frac-tion (41, 42, 43, 44) is dominated by a large renormalization scale dependence atthe ±
25% level, which indicates the importance of the NLL series. By conven-tion, the dependence on the renormalization scale µ b is obtained by the variation m b / < µ b < m b . The three calculational steps of the NLL enterprise—Step1 (45, 46), Step 2 (47, 48), and Step 3 (49, 50, 51, 52, 53)—have been performed bymany different groups and have been independently checked. The resulting NLLprediction had a small dependence on the scale µ b as well as on the matchingscale µ below 5%. But as first observed in Ref. (54), there was a large charmmass-scheme dependence because the charm loop vanishes at the LL level and thesignificant charm dependence begins only at the NLL level. By varying m c /m b in the conservative range 0 . ≤ m c /m b ≤ .
31, which covers both the polemass value (with its numerical error) and the running mass value m c ( µ c ) with µ c ∈ [ m c , m b ], one finds an uncertainty of almost 10% (55,56). This uncertainty isthe dominant error in the NLL prediction. The renormalization scheme for m c isan NNLL issue, and a complete NNLL calculation reduces this large uncertaintyby at least a factor of two (57). This finding motivated the NNLL calculation ofthe B → X s γ branching fraction.Following a global effort, such an NNLL calculation was recently performedand led to the first NNLL prediction of the B → X s γ branching fraction (58).This result is based on various highly-nontrivial perturbative calculations (59,60, 61, 62, 63, 65, 64, 66, 67, 68, 69): Within Step 1 the matching of the effectivecouplings C i at the high-energy scale µ ∼ M W requires a three-loop calculationfor the cases i = 7 , i = 1 , ...,
6) and theself-mixing of the dipole operators ( i = 7 ,
8) have been calculated by a three-loop calculation of anomalous dimensions (61, 62), and the mixing of the four-quark operators into the dipole operators by a four-loop calculation (63). Thesetwo steps have established the effective couplings at the low scale µ b ∼ m b toNNLL precision. Thus, large logarithms of the form α n + ps ( m b ) log n ( m b /m W ),( p = 0 , , adiative and Electroweak Penguin Decays of B Mesons O including the bremsstrahlung contributions hasbeen calculated in Refs. (65, 64, 66, 67). The other important piece is the three-loop matrix elements of the four-quark operators, which has first been calculatedwithin the so-called large- β approximation (68). A calculation that goes beyondthis approximation by employing an interpolation in the charm quark mass m c from m c > m b to the physical m c value has been presented in Ref. (69). In thisinterpolation the α s β result (68) is assumed to be a good approximation forthe complete α s result for vanishing charm mass. It is this part of the NNLLcalculation which is still open for improvement. Indeed a complete calculationof the three-loop matrix elements of the four-quark operators O , for vanishingcharm mass is work in progress (70) and will cross-check this assumption and thecorresponding error estimate due to the interpolation.Some perturbative NNLL corrections have not yet been included in the presentNNLL estimate, but they are expected to be smaller than the current perturbativeuncertainty of 3%: the virtual and bremsstrahlung contributions to the ( O , O )and ( O , O ) interferences at order α s , the NNLL bremsstrahlung contributionsin the large- β -approximation beyond the ( O , O ) interference term (which arealready available (71)), the four-loop mixing of the four-quark operators into theoperator O (63), and the exact mass dependence of various matrix elementsbeyond the large β approximation (72, 70, 73).In the present NNLL prediction (58), the reduction of the renormalization-scaledependence at the NNLL is shown in Fig. 2. The most important effect occursfor the charm mass MS renormalization scale µ c , which has been the main sourceof uncertainty at the NLL. The current uncertainty of ±
3% due to higher-order[ O ( α s )] effects can be estimated via the NNLL curves in Fig. 2. The reductionfactor of the perturbative error is greater than a factor of three. The centralvalue of the NNLL prediction is based on the choices µ b = 2 . µ c = 1 . − .
6% (74, 75, 76, 77).They are included in the present NNLL prediction. B → X s ℓ + ℓ − Compared with the B → X s γ decay, the inclusive B → X s ℓ + ℓ − decay presents a complementary and more complex test of the SM, given thatdifferent perturbative electroweak contributions add to the decay rate. This in-clusive mode is also dominated by perturbative contributions, if one eliminates cc resonances with the help of kinematic cuts. In the so-called perturbative q -windows below and above the resonances, namely in the low-dilepton-mass region1 GeV < q = m ℓℓ < as well as in the high-dilepton-mass region where q > . , theoretical predictions for the invariant mass spectrum are dom-inated by the perturbative contributions. A theoretical precision of order 10% ispossible.Compared with the decay B → X s γ , the effective Hamiltonian (Eq. 5) con-tains two additional operators of O ( α em ), the semileptonic operators O and O . Moreover, the first large logarithm of the form log( m b /m W ) already ariseswithout gluons, because the operator O mixes into O at one loop. It is thenconvenient to redefine the dipole and semileptonic operators via e O i = 16 π /g s O i , Tobias Hurth and Mikihiko Nakao e C i = g s / (4 π ) C i for i = 7 , ...,
10. With this redefinition, one can follow the threecalculational steps discussed above. In particular, after the reshufflings the one-loop mixing of the operator O with e O appears formally at order α s . To LLprecision, there is only e O with a non-vanishing tree-level matrix element and anon-vanishing coefficient.It is well-known that this na¨ıve α s expansion is problematic, since the formally-leading O (1 /α s ) term in C is accidentally small and much closer in size to an O (1) term. Thus, also specific higher order terms in the general expansion arenumerically important. The complete NLL contributions have been presented (44, 78). For the NNLLcalculation, many components were taken over from the NLL calculation of the B → X s γ mode. The additional components for the NNLL QCD precision havebeen calculated (60, 48, 61, 79, 80, 85, 83, 84, 82, 81, 86, 88, 87): Some new pieces forthe matching to NNLL precision (Step 1) have been calculated in Ref. (60). ToNNLL precision the large matching scale uncertainty of 16% at the NLL level iseliminated. In Step 2, the mixing of the four-quark operators into the semilep-tonic operator O has been calculated (48, 61). In Step 3, the four-quark matrixelements including the corresponding bremsstrahlung contributions have beencalculated for the low- q region in Refs. (79, 80, 81), bremsstrahlung contributionfor the forward-backward asymmetry in B → X s ℓ + ℓ − in Refs. (85, 83, 84), andthe four-quark matrix elements in the high- q region in Refs. (82, 81, 86). Thetwo-loop matrix element of the operator O has been estimated using the cor-responding result in the decay mode B → X u ℓν and also Pade approximationmethods (88); this estimate has been further improved in Ref. (87).More recently electromagnetic corrections were calculated: NLL quantum elec-trodynamics (QED) two-loop corrections to the Wilson coefficients are of O (2%) (88).Also, in the QED one-loop corrections to matrix elements, large collinear loga-rithms of the form log( m b /m ℓ ) survive integration if only a restricted part of thedilepton mass spectrum is considered. These collinear logarithms add anothercontribution of order +2% in the low- q region for B → X s µ + µ − (89). For thehigh- q region, one finds −
8% (90). B → X d γ and B → X d ℓ + ℓ − The perturbative QCD corrections in theinclusive decays B → X d γ (91, 55, 56) and B → X d ℓ + ℓ − (92, 93) can be treatedcompletely analogously to those in the corresponding b → s modes. The effectiveHamiltonian is the same in these processes, up to the obvious replacement of the s quark field by the d quark field. However, because λ u = V ub V ∗ ud for b → dγ isnot small with respect to λ t = V tb V ∗ td and λ c = V cb V ∗ cd , one must also account forthe operators proportional to λ u , namely O u , in Eq. 5. The matching conditions C i ( m W ) and the solutions of the RGEs, which yield C i ( µ b ), coincide with thoseneeded for the corresponding b → s processes (91, 92). The B → X s ℓ + ℓ − decay amplitude has the following structure ( κ = α em /α s ): A = κ (cid:2) A LL + α s A NLL + α s A NNLL + O ( α s ) (cid:3) with A LL ∼ α s A NLL (7)A strict NNLL calculation of the squared amplitude A should only include terms up to order κ α s . However, in the numerical calculation, one also includes the term A NLL A NNLL of order κ α s which are numerically important. These terms beyond the formal NNLL level are pro-portional to | C | and | C | and are scheme-independent. One can even argue that one picksup the dominant NNNLL QCD corrections because the missing NNNLL piece in the squaredamplitude, namely A LL A NNNLL , can safely be neglected (87). adiative and Electroweak Penguin Decays of B Mesons The inclusive modes B → X s γ and B → X s ℓ + ℓ − are dominated by the partoniccontributions. Indeed, if only the leading operator in the effective Hamiltonian( O for B → X s γ , O for B → X s ℓ + ℓ − ) is considered, the HME makes itpossible to calculate the inclusive decay rates of a hadron containing a heavyquark, especially a b quark (24, 25, 26, 27, 28, 29). The optical theorem relatesthe inclusive decay rate of a hadron H b to the imaginary part of the forwardscattering amplitude Γ( H b → X ) = 12 m H b ℑ h H b | T | H b i , (8)where the transition operator T is given by T = i R d x T [ H eff ( x ) H eff (0)]. Theinsertion of a complete set of states, | X ih X | , leads to the standard formula forthe decay rate:Γ( H b → X ) = 12 m H b X X (2 π ) δ ( p i − p f ) | h X | H eff | H b i | . (9)It is then possible to construct an OPE of the operator T , which is expressed asa series of local operators that are suppressed by powers of the b quark mass andwritten in terms of the b quark field: T [ H eff H eff ] OP E = 1 m b ( O + 1 m b O + 1 m b O + ... ) . (10)This construction is based on the parton–hadron duality. The sum is performedover all exclusive final states and that the energy release in the decay is large withrespect to the QCD scale, Λ ≪ m b . With the help of the heavy quark effectivetheory (HQET) (94,95), namely the new heavy quark spin-flavor symmetries thatarise in the heavy quark limit m b → ∞ , the hadronic matrix elements within theOPE, h H b | O i | H b i , can be further simplified. In this well-defined expansion, thefree quark model is the first term in the constructed expansion in powers of 1 /m b and, therefore, is the dominant contribution. In the applications to inclusive rare B decays, one finds no correction of order Λ /m b to the free quark model approx-imation. The corrections to the partonic decay rate begin with 1 /m b only, whichimplies the rather small numerical impact of the nonperturbative corrections onthe decay rate of inclusive modes. However, there are more subtleties to considerif other than the leading operators are taken into account (see below). B → X s γ These techniques can be used directly in the decay B → X s γ tosingle out nonperturbative corrections to the branching fraction: If one neglectsperturbative QCD corrections and assumes that the decay B → X s γ is due tothe leading electromagnetic dipole operator O alone, then the photon wouldalways be emitted directly from the hard process of the b quark decay. Onehas to consider the time-ordered product T O +7 ( x ) O (0). Using the OPE for T O +7 ( x ) O (0) and HQET methods, as discussed above, the decay width Γ( B → X s γ ) reads (30, 31) (modulo higher terms in the 1 /m b expansion):Γ ( O , O ) B → X s γ = α em G F m b π | V tb V ts | C ( m b ) δ NPrad m b ! , δ NPrad = 12 λ − λ , (11)0 Tobias Hurth and Mikihiko Nakao where λ and λ are the HQET parameters for the kinetic energy and the chro-momagnetic energy, respectively. If the B → X s γ decay width is normalized bythe (charmless) semileptonic one, the nonperturbative corrections of order 1 /m b cancel out within the ratio B ( B → X s γ ) / B ( B → X u ℓν ).However, as noted in Ref. (96), there is no OPE for the inclusive decay B → X s γ if one considers operators beyond the leading electromagnetic dipole op-erator O . Voloshin (97) has identified a contribution to the total decay ratein the interference of the b → sγ amplitude due to the electromagnetic dipoleoperator O and the charming penguin amplitude due to the current-current op-erator O . This resolved photon contribution contains subprocesses in which thephoton couples to light partons instead of connecting directly to the effectiveweak-interaction vertex (98). If one treats the charm quark as heavy, then it ispossible to expand the contribution in local operators. The first term in this ex-pansion may be the dominating one (96,99,100). This nonperturbative correctionis suppressed by λ /m c and is estimated to be of order 3% compared with theleading-order (perturbative) contribution to the decay rate Γ b → sγ which arisesfrom the electromagnetic operator O :∆Γ ( O , O ) B → X s γ Γ LL b → sγ = − C C λ m c ≃ +0 . . (12)However, if the charm mass is assumed to scale as m c ∼ Λ m b , then the charmpenguin contribution must be described by the matrix element of a nonlocaloperator (96, 99, 100, 101).Recently, another example of such nonlocal matrix elements within the power-suppressed contributions to the decay B → X s γ was identified (101)—specifically,in the interference of the b → sγ transition amplitude mediated by the electro-magnetic dipole operator O , where the b → sg amplitude is mediated by thechromo-magnetic dipole operator O , followed by the fragmentation of the gluoninto an energetic photon and a soft quark-antiquark pair. A na¨ıve dimensionalestimate of these power corrections leads to∆Γ ( O , O ) B → X s γ Γ LL b → sγ ∼ πα s C C Λ m b , (13)whereas an estimate using the vacuum insertion method for the nonlocal matrixelements indicates an effect of − B → X s γ photon spectrum canbe parameterized systematically in terms of subleading shape functions. For theinterference of the O – O pair, these nonlocal operators reduce to local operators,if one considers the total decay rate (102,103,104), whereas other resolved photoncontributions to the total decay rate—such as the previously analyzed O – O interference term (101)—cannot be described by a local OPE. A recent systematicanalysis (98) of all resolved photon contributions related to other operators inthe weak Hamiltonian establishes this breakdown of the local OPE within thehadronic power corrections as a generic result. Clearly, estimating such nonlocalmatrix elements is very difficult, and an irreducible theoretical uncertainty of ± (4 − CP averaged decay rate, defined with a photon-energycut of E γ = 1 . B → X s γ mode have reached the nonperturbative adiative and Electroweak Penguin Decays of B Mesons CP asymmetries hasnot yet been estimated. B → X s ℓ + ℓ − Hadronic power corrections in the decay B → X s ℓ + ℓ − thatscale with 1 /m b , 1 /m b (30, 31, 105, 106, 107, 108), and 1 /m c (100) have also beenconsidered. They can be calculated quite analogously to those in the decay B → X s γ . However, a systematic analysis of hadronic power corrections includingall relevant operators has yet to be performed. Thus, an additional uncertaintyof ±
5% should be added to all theoretical predictions for this mode on the basisof a simple dimensional estimate.In the high- q region, one encounters the breakdown of the HME at the endpoint of the dilepton mass spectrum: Whereas the partonic contribution vanishes,the 1 /m b and 1 /m b corrections tend towards a nonzero value. In contrast tothe end-point region of the photon-energy spectrum in the B → X s γ decay, nopartial all-order resummation into a shape function is possible. However, for anintegrated high- q spectrum an effective expansion is found in inverse powers of m eff b = m b × (1 − √ s min ) rather than m b (109, 110). The expansion converges lessrapidly, depending on the lower dilepton-mass cut s min = q /m b (81).The large theoretical uncertainties could be significantly reduced by normaliz-ing the B → X s ℓ + ℓ − decay rate to the semileptonic B → X u ℓν decay rate withthe same q cut (108): R ( s ) = Z s dˆ s dΓ( B → X s ℓ + ℓ − )dˆ s / Z s dˆ s dΓ( B → X u ℓν )dˆ s . (14)For example, the uncertainty due to the dominating 1 /m b term would be reducedfrom 19% to 9% (89). B → X d γ The nonperturbative contributions in the decay B → X d γ canbe treated analogously to those in the decay B → X s γ . The power correctionsthat scale as 1 /m b (in addition to the CKM factors) are the same for the twomodes. Also, the systematic analysis of resolved contributions given in Ref. (98)can be applied to this case. However, the long-distance contributions from theintermediate u quark in the penguin loops are critical. They are suppressed inthe B → X s γ mode by the CKM matrix elements. In B → X d γ , there is noCKM suppression, and one must account for the nonperturbative contributionsthat arise from the operator O u . The contribution due to the O u – O interferencescales with Λ /m b (106). However, this interference contribution vanishes in thetotal CP -averaged rate of B → X s γ at order Λ /m b (98). This result applies tothe total rate of B → X d γ as well. Other interference terms, namely the doubleresolved contributions O u – O and O u – O u , arise first at order 1 /m b , as they canalso be deduced from the results presented in Ref. (98). Thus, there is no powercorrection due to the operator O u in the total rate of B → X d γ at order Λ /m b ,which implies that the CP -averaged decay rate of B → X d γ is as theoreticallyclean as the decay rate of B → X s γ . B → X d ℓ + ℓ − In the case of B → X d ℓ + ℓ − long-distance contributions dueto u quark loops can be avoided in the low- q window 1 GeV < q < .The ρ and ω resonances are below, and the cc ( J/ψ , ψ ′ ) resonances are abovethis window (92). The effect of their respective tails can be taken into account2 Tobias Hurth and Mikihiko Nakao within the Kr¨uger-Sehgal (KS) approach (see Section 2.5) (111,112). In this low- q region, one can then treat the nonperturbative power corrections analogouslyto those in the decay B → X s ℓ + ℓ − , and one can expect a similar theoreticalaccuracy in this q window. There are additional subtleties in inclusive modes. Kinematical cuts induce ad-ditional sensitivities to nonperturbative physics. B → X s γ In the measurement of the inclusive mode B → X s γ one needs cutsin the photon-energy spectrum to suppress the background from other B decays(Fig. 3).These shape-function effects were taken into account in the experimental anal-ysis, and the corresponding theoretical uncertainties due to this model depen-dence are reflected in the extrapolation error of the experimental results (seeSection 5.1). The extrapolation is done from the experimental energy cut valuesdown to 1 . O , a cut around1 . m b − E γ (117).A multiscale OPE with three short-distance scales m b , √ m b ∆, and ∆ has beenproposed to connect the shape function and the local OPE region. Recently,such additional perturbative cutoff-related effects have been calculated to NNLLprecision by the use of SCET methods (118, 119, 120). Such perturbative ef-fects due to the additional scale are negligible for 1 . . . E = 1 . O ( α s )terms (121). Further work is needed to clarify this issue.There is an alternative approach to the cut effects in the photon-energy spec-trum that is based on dressed gluon exponentiation and on the incorporation ofSudakov and renormalon resummations (123, 122). The greater predictive powerof this approach is related in part to the assumption that nonperturbative powercorrections associated with the shape function follow the pattern of ambiguitiespresent in the perturbative calculation (124). In the future, these additional per-turbative cut effects could be analyzed and combined together with those alreadyincluded in the experimental average. B → X s ℓ + ℓ − In the inclusive decay B → X s ℓ + ℓ − , the hadronic and dileptoninvariant masses are independent kinematical quantities. A hadronic invariant-mass cut is imposed in the experiments (see Section 5.7). The high-dilepton-mass region is not affected by this cut, but in the low-dilepton mass region thekinematics with a jet-like X s and m X ≤ m b Λ implies the relevance of the shapefunction. A recent SCET analysis shows that by using the universality of theshape function, a 10 −
30% reduction in the dilepton-mass spectrum can be adiative and Electroweak Penguin Decays of B Mesons −
10% to +10% withequally large uncertainties. In the future it may be possible to decrease suchuncertainties significantly by constraining both the leading and subleading shapefunctions using the combined B → X s γ , B → X u ℓν and B → X s ℓ + ℓ − data (127). One must also consider the on-shell cc resonances, which have to be taken out.Whereas in the decay B → X s γ the intermediate ψ background, namely B → ψX s followed by ψ → X ′ γ , is suppressed for the high-energy cut E γ and can besubtracted from the B → X s γ decay rate, the cc resonances show up as largepeaks in the dilepton-invariant mass spectrum in the decay B → X s ℓ + ℓ − .As discussed in Section 2.2, these resonances can be removed by making appro-priate kinematic cuts in the invariant mass spectrum. However, nonperturbativecontributions away from the resonances within the perturbative windows are alsoimportant. In the KS approach (111, 112) one absorbs factorizable long-distancecharm rescattering effects (in which the B → X s cc transition can be factorizedinto the product of sb and cc color-singlet currents) into the matrix element ofthe leading semileptonic operator O . Following the inclusion of nonperturbativecorrections scaling with 1 /m c , the KS approach avoids double-counting. For theintegrated branching fractions one finds an increase of (1 − q re-gion due to the KS effect, whereas in the high- q region the increase is well belowthe uncertainty due to the 1 /m b corrections. As shown in Fig. 3, the integratedbranching fraction is dominated by this resonance background which exceeds thenonresonant charm-loop contribution by two orders of magnitude. This featureshould not be misinterpreted as a striking failure of global parton-hadron dual-ity (113), which postulates that the sum over the hadronic final states, includingresonances, should be well approximated by a quark-level calculation (128). Cru-cially, the charm-resonance contributions to the decay B → X s ℓ + ℓ − are expressedin terms of a phase-space integral over the absolute square of a correlator. Forsuch a quantity global quark-hadron duality is not expected to hold. Neverthe-less, local quark-hadron duality (which, of course, also implies global duality)may be reestablished by resumming Coulomb-like interactions (113). The Wilson coefficients of the weak effective Hamiltonian are process-independentand therefore can be used directly in the description of exclusive modes. It iscomputing of the hadronic matrix elements between meson states that is diffi-cult in the case of exclusive modes and that limits the theoretical precision. Thena¨ıve approach consists of writing the amplitude A ≃ C i ( µ b ) hO i ( µ b ) i and param-eterizing hO i ( µ b ) i in terms of form factors. A substantial improvement can beobtained by using the QCDF method (32, 33, 34) and its field-theoretical formu-lation, the SCET method (35, 36, 37, 38, 39, 40). These methods form the basisof the up-to-date predictions of exclusive B decays. Within this framework onecan show that, even if the form factors were known with infinite precision, the4 Tobias Hurth and Mikihiko Nakao description of exclusive decays would be incomplete due to the existence of so-called nonfactorizable strong interaction effects that do not correspond to formfactors.The QCDF and SCET methods were first systematized for nonleptonic decaysin the heavy quark limit. In contrast to the HQET, SCET does not correspondto a local operator expansion. Whereas HQET is applicable to B decays if theenergy transfer to light hadrons is small, for example to B → D transitions atsmall recoil to the D meson, HQET is not applicable if some of the outgoing,light particles have momenta of order m b . If so, one faces a multi-scale problemthat can be tackled within SCET. In this case, there are three relevant scales: (a)Λ = few × Λ QCD , the soft scale set by the typical energies and momenta of thelight degrees of freedom in the hadronic bound states; (b) m b , the hard scale setby both the heavy b quark mass and the energy of the final-state hadron in the B meson rest frame; and (c) the hard-collinear scale µ hc = √ m b Λ, which appearsthrough interactions between the soft and energetic modes in the initial andfinal states. The dynamics of hard and hard-collinear modes can be describedperturbatively in the heavy quark limit m b → ∞ . Thus, SCET describes B decays to light hadrons with energies much larger than their masses, assumingthat their constituents have momenta collinear to the hadron momentum. B → K ∗ γ and B → ργ The application of the QCDF formalism to radiativeand semileptonic decays was first proposed in Ref. (129). For B → K ∗ γ , or moregenerally for B → V γ , where V is a light vector meson, the QCDF formula forthe hadronic matrix element of each operator of the effective Hamiltonian in theheavy quark limit and to all orders in α s reads h V γ | O i | B i = T Ii F B → V ⊥ + Z ∞ dωω φ B ( ω ) Z du φ V ⊥ ( u ) T IIi ( ω, u ) . (15)This formula separates the process-independent nonperturbative quantities F B → V ⊥ ,a form factor evaluated at maximum recoil ( q = 0), and φ B and φ V ⊥ , thelight-cone distribution amplitudes (LCDAs) for the heavy and light mesons, re-spectively, from the perturbatively calculable quantities T I and T II . The lattercorrespond to vertex and spectator corrections, respectively, and have been cal-culated to O ( α s ) (130, 131, 132, 133). More recently, some α s terms were alsopresented (134).Light-cone wave functions of pseudo-scalar and vector mesons that enter thefactorization formula have been studied in detail through the use of light-coneQCD sum rules (135, 136, 137, 138). However, not much is known about the B meson LCDA, whose first negative moment enters the factorized amplitude at O ( α s ). Because this moment also enters the factorized expression for the B → γ form factor, it might be possible to extract its value from measurements of decayssuch as B → γeν , if the power corrections are under control.The QCDF formula also includes an important simplification in the form factordescription. The B → V form factors at large recoil have been analyzed inSCET (139, 140, 141) and are independent of the Dirac structure of the currentin the heavy quark limit; as a consequence, all B → V ⊥ form factors reduce toa single form factor up to factorizable corrections in the heavy quark and largeenergy limits (142, 129).Field-theoretical methods such as SCET make it possible to reach a deeperunderstanding of the QCDF approach. The various momentum regions are repre- adiative and Electroweak Penguin Decays of B Mesons T I and T II can be shown to be Wilson coefficients of effective field operators.Using SCET one can prove the factorization formula to all orders in α s and toleading order in Λ /m b (143). QCD is matched on SCET in a two-step procedurethat separates the hard scale µ ∼ m b and then the hard-collinear scale µ ∼ √ Λ m b from the hadronic scale Λ. The vertex correction term T I involves the hard scales,whereas the spectator scattering term T II involves both the hard and the hard-collinear scales. This is why large logarithms have to be resummed (143), whichcan be done most efficiently in SCET.In principle, the field-theoretical framework of SCET allows one to go beyondthe leading-order result in Λ /m b (144). However, a breakdown of factorization isexpected at that order (34). For example, in the analysis of B → K ∗ γ decays atsubleading order, an infrared divergence is encountered in the matrix element of O (145). In general, power corrections involve convolutions, which turn out tobe divergent. Currently, no solution to this well-analyzed problem of end-pointdivergences within power corrections is available (146, 140, 147). Thus, withinthe QCDF/SCET approach, a general, quantitative method to estimate the im-portant Λ /m b corrections to the heavy quark limit is missing, which significantlylimits the precision in phenomenological applications.Nevertheless, some very specific power corrections are still computable and areoften numerically important. Indeed, this is the case for the annihilation and weakexchange amplitudes in B → ργ . The annihilation contributions also representthe leading contribution to isospin asymmetries (145). All these corrections areincluded in recent analyses of radiative exclusive decays (148,149,150). Moreover,the method of light-cone QCD sum rules can help provide estimates of suchunknown subleading terms. For example, power corrections for the indirect CP asymmetries in B → V γ decays have been analyzed in this manner (151). B → K ( ∗ ) ℓ + ℓ − There is also a factorization formula for the exclusive semilep-tonic B decays, such as B → K ∗ ℓ + ℓ − , that are analogous to the one for the radia-tive decay B → K ∗ γ (130, 150). The simplification due to form factor relationsis even more drastic. The hadronic form factors can be expanded in the smallratios Λ /m b and Λ /E , where E is the energy of the light meson. If we neglectcorrections of order 1 /m b and α s , the seven a priori independent B → K ∗ formfactors reduce to two universal form factors ξ ⊥ and ξ k (142, 129). This reductionmakes it possible to design interesting ratios of observables in which any soft formfactor dependence cancels out for all dilepton masses q at leading order in α s and Λ /m b (152).The theoretical simplifications of the QCDF/SCET approach are restricted tothe kinematic region in which the energy of the K ∗ is of the order of the heavyquark mass; that is, q ≪ m B . Moreover, the influences of very light resonancesbelow 1 GeV question the QCDF results in this region. In addition, the longitu-dinal amplitude in the QCDF/SCET approach generates a logarithmic divergencein the limit q →
0, which indicates problems in the theoretical description below1 GeV (130). Thus, the factorization formula applies well in the dilepton massrange 1 GeV < q < .Clearly, the QCDF and SCET methods are also applicable to the phenomeno-logically important semileptonic decays such as B → Kℓ + ℓ − (130, 153), B → ρℓ + ℓ − (150), and B s → φℓ + ℓ − . The decay mode into a pseudoscalar is analogous6 Tobias Hurth and Mikihiko Nakao to the decay mode into a longitudinal vector meson.
The Υ(4 S ) resonance produced by the e + e − collision at the B factories providesa clean sample of B B and B + B − meson pairs as well as strong kinematicalconstraints that are otherwise unavailable, particularly at hadron colliders. Themain background is from continuum light quark pair production ( e + e − → qq , q = u, d, s, c ), which has a cross section only three times larger than that of BB production. Radiative and electroweak penguin B decays are efficiently measuredat the B factories thanks to their clear signatures: a high-energy photon and alepton pair, respectively. B Reconstruction A B meson decaying into an exclusive final state is reconstructed by measuring alllong-lived decay products ( π ± , K ± , e ± , µ ± and γ ), selecting intermediate statesof certain invariant masses, and calculating two standard variables: the beam-energy constrained mass M bc = q s/ − | p ∗ B | (also referred to as the beam-energysubstituted mass, M ES ) and the energy difference ∆ E = E ∗ B − √ s/
2. Here, √ s/ p ∗ B and E ∗ B are the momentum and energy, respectively,of the reconstructed B meson candidate in the Υ(4 S ) rest frame. M bc has a peak at the B meson mass and ∆ E has a peak at zero (Fig. 4).The resolution of M bc is significantly better than that of ∆ E , as the former isdominated by the spread of the beam energy, whereas the latter is dominatedby the detector resolution. The ∆ E variable is sensitive to misreconstructedbackground events, whereas M bc has little separation power for them. When akaon is misidentified as a pion, ∆ E shifts by approximately 50 MeV, and when alow momentum pion is added or missed, ∆ E shifts by more than the pion mass.An exclusive b → dγ final state is thus separated from a similar b → sγ statewith ∆ E , but this separation is marginal due to the photon energy resolution ofthe electromagnetic calorimeter. Therefore, pion to kaon separation is crucial forthe measurement of the suppressed b → dγ processes.Background events due to random combinations of particles are also reducedby correctly identifying particle species. For the b → sℓ + ℓ − processes, electronsand muons are almost completely separated from the more abundant hadrons.In addition, various techniques based on the event topology can be applied tosuppress the background from continuum qq events. A fully inclusive measurement of B → X s γ , in which the system recoiling againstthe emitted photon is not reconstructed, has been performed at the B factoriesthanks to the clean environment. The dominant background photon sources are(a) the copiously produced π → γγ decays, (b) η → γγ to a lesser extent, and(c) other secondary and initial-state radiation photons in continuum qq events.These contributions can be safely subtracted because they are measured in eventstaken 60 or 40 MeV below the Υ(4 S ) resonance (i.e., off resonance). Here, smallcorrections due to the center-of-mass energy difference are applied to the pro-duction cross section and the reconstruction efficiency. To avoid sacrificing the adiative and Electroweak Penguin Decays of B Mesons B decay sample for other studies, the size of the off-resonance data sample isonly ∼
10% of the on-resonance sample from both Belle and BaBar and is thedominant source of statistical and systematic errors (CLEO collected one-thirdof the sample as off-resonance). The second severe background source arises fromsimilar secondary photons from B decays. These contributions are subtractedfrom the expected photon spectrum on the basis of measured π and η spectrafrom B decays, various control samples, or Monte Carlo simulation.An alternative technique is to measure as many exclusive modes as possible andthen calculate their sum (i.e., the sum-of-exclusive method). Exclusive branchingfractions measured to date do not saturate the inclusive process, but one can stillinfer the total branching fraction by estimating the fraction of unmeasured modesof typically ∼
45% (or ∼
30% if K L modes are accounted for by corresponding K S modes) using simulated hadronization processes. In the simulation, a light quarkpair is generated according to the SM mass spectrum and final-state hadrons areproduced by the PYTHIA program (155). This method also provides direct in-formation about the B meson. For example, the B meson momentum defines the B meson rest frame, and charge and flavor information allows CP - and isospin-asymmetry measurements. So far, the sum-of-exclusive method is the only wayto perform inclusive measurements of B → X s ℓ + ℓ − and B → X d γ decays.Another potentially definitive method is the so-called B -reco technique, inwhich the other B meson is fully reconstructed, thereby allowing the target B decay to be measured in a very clean environment. The efficiency is as low as afraction of a percent, and will be more important in future experiments. Theoretical predictions have significantly improved in the past decade along withthe development of the theoretical tools. There have also been improvements inthe relevant experimental input quantities, as discussed below.
The inclusive radiative and electroweak penguin modes offer theoretically cleanobservables because nonperturbative corrections are small and well under control.This assessment also applies to the branching fraction of B → X d γ mode asdiscussed in Section 2.3. Inclusive B → X s γ The stringent bounds obtained from B → X s γ onvarious nonstandard scenarios are a clear example of the importance of cleanFCNC observables for discriminating NP models.The branching fraction for B → X q γ ( q = s, d ) can be parameterized as B ( B → X q γ ) E γ >E = B ( B → X c eν ) exp α em πC (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V ∗ tq V tb V cb (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h P ( E ) + N ( E ) i , (16)where α em = α onshellem (74), C = | V ub | / | V cb | × Γ[ B → X c eν ] / Γ[ B → X u eν ] and P ( E ) and N ( E ) denote the perturbative and nonperturbative contributions,respectively. The latter are normalized to the charmless semileptonic rate toseparate the charm dependence.8 Tobias Hurth and Mikihiko Nakao
The first NNLL prediction, which is based on the perturbative calculationsdiscussed in Section 2.2 and on the analyses of nonperturbative corrections pre-sented in Sections 2.3 and 2.4, for a photon-energy cut E γ > . B ( B → X s γ ) NNLL = (3 . ± . × − . (17)The overall uncertainty consists of nonperturbative (5%), parametric (3%), per-turbative (scale) (3%) and m c -interpolation ambiguity (3%), which are addedin quadrature. An additional scheme dependence in the determination of theprefactor C has been found (156); it is within the perturbative uncertainty of3% (121).Thus, the SM prediction and the experimental average (see Section 5.1) areconsistent at the 1 . σ level. This finding implies very stringent constraints onNP models, such as (a) the bound on the charged Higgs mass in the two-Higgsdoublet model (157, 158) ( M H + > /R > T -parity have also been presented (170). Finally, model-independent analysesin the effective field theory approach without (171) and with the assumption ofminimal flavor violation (172,173) also show the strong constraining power of the B → X s γ branching fraction. Inclusive B → X d γ The theoretical predictions for the branching fraction B ( B → X d γ ) for photon energies E γ > . B ( B → X d γ ) = (cid:16) . +0 . − . (cid:12)(cid:12)(cid:12) mcmb ± . CKM ± . param . ± . scale (cid:17) × − , (18)and B ( B → X d γ ) B ( B → X s γ ) = (cid:16) . +0 . − . (cid:12)(cid:12)(cid:12) mcmb ± . CKM ± . param . ± . scale (cid:17) × − . (19)These predictions are of NLL order. They are fully consistent with previousresults (91). A good part of the uncertainties cancel out in the ratio. The errorsare dominated by CKM uncertainties, and thus the measurement of B ( B → X d γ )constrains the CKM parameters. This measurement is also of specific interestwith respect to NP, because its CKM suppression by the factor | V td | / | V ts | inthe SM may not hold in extended models. Direct CP Asymmetry
Other important observables are the direct CP asymmetries ( q = s, d ), whose sign is always defined in terms of b − b , or A CP ( B → X q γ ) ≡ Γ( B → X q γ ) − Γ( B → X q γ )Γ( B → X q γ ) + Γ( B → X q γ ) . (20)As first noted in Ref. (174), the SM predictions are almost independent from thephoton energy cut-off and, for E γ > . A CP ( B → X s γ ) = (cid:16) . +0 . − . (cid:12)(cid:12)(cid:12) mcmb ± . CKM +0 . − . (cid:12)(cid:12)(cid:12) scale (cid:17) × − , (21) adiative and Electroweak Penguin Decays of B Mesons A CP ( B → X d γ ) = (cid:16) − . +2 . − . (cid:12)(cid:12)(cid:12) mcmb ± . CKM +2 . − . (cid:12)(cid:12)(cid:12) scale (cid:17) × − . (22)The two CP asymmetries are connected by the relative CKM factor λ [(1 − ρ ) + η ]. The small SM prediction for the CP asymmetry in the decay B → X s γ is aresult of three suppression factors: (a) α s to have a strong phase; (b) CKMsuppression of order λ ; and (c) GIM suppression of order ( m c /m b ) , whichreflects that in the limit m c = m u , any CP asymmetry in the SM would vanish.On the basis of CKM unitarity, one can derive the following U -spin relationbetween the un-normalized CP asymmetries (175): h Γ( B → X s γ ) − Γ( B → X s γ ) i + h Γ( B → X d γ ) − Γ( B → X d γ ) i = 0 (23) U -spin breaking effects can be estimated within the HME (even beyond the par-tonic level), so one arrives at the following prediction for the total (or untagged) B → X s + d γ asymmetry (176, 177): | ∆ B ( B → X s γ ) + ∆ B ( B → X d γ ) | ∼ · − . (24)Because this null test is based on the CKM unitarity, it represents a clear testfor new CP phases beyond the CKM phase (176, 177). NP sensitivities of direct CP asymmetries have been analyzed (174, 55). Inclusive B → X s ℓ + ℓ − The decay B → X s ℓ + ℓ − is particularly attractivebecause it offers several kinematic observables. The angular decomposition of thedecay rate provides three independent observables, H T , H A and H L , from whichone can extract the short-distance electroweak Wilson coefficients that test forNP (178): d Γ dq dz = 38 [(1 + z ) H T ( q ) + 2(1 − z ) H L ( q ) + 2 zH A ( q )] . (25)Here z = cos θ ℓ , θ ℓ is the angle between the negatively charged lepton and the B meson in the center-of-mass frame of the dilepton system, and q is the dilep-ton mass squared. H A is equivalent to the forward-backward asymmetry, andthe dilepton-mass spectrum is given by H T + H L . The observables depend onthe Wilson coefficients C , C and C in the SM. The present measurementsof the B → X s ℓ + ℓ − already favor the SM-sign of the coefficient C , which isundetermined by the B → X s γ mode (179).As discussed above, these observables are dominated by perturbative contribu-tions in the perturbative low- and high- q windows which are below (1 GeV . ) the cc resonances, respectively. Thepresent predictions are based on the perturbative calculations to NNLL preci-sion in QCD and to NLL precision in QED (see Section 2.2). For the branchingfraction in the low- q region one arrives at (89) B ( B → X s ℓ + ℓ − ) low = (cid:26) (1 . ± . × − ( ℓ = µ )(1 . ± . × − ( ℓ = e ) , (26)and for the high- q region, one arrives at (90) B ( B → X s ℓ + ℓ − ) high = ( . × − × (1 +0 . − . ) ( ℓ = µ )2 . × − × (1 +0 . − . ) ( ℓ = e ) . (27)0 Tobias Hurth and Mikihiko Nakao
As suggested in Ref. (108), normalizing the B → X s ℓ + ℓ − decay rate in the high- q region to the semileptonic B → X u ℓν decay rate with the same q cut (Eq. 14),significantly reduces the nonperturbative uncertainties (90): R (ˆ q = 14 . ) = (cid:26) . × − × (1 ± .
13) ( ℓ = µ )1 . × − × (1 ± .
16) ( ℓ = e ) . (28)The value of q for which the forward-backward asymmetry vanishes,( q )[ X s ℓ + ℓ − ] = (cid:26) (3 . ± .
12) GeV ( ℓ = µ )(3 . ± .
11) GeV ( ℓ = e ) , (29)is one of the most precise predictions in flavor physics and also determines therelative sign and magnitude of the coefficients C and C (90). However, unknownsubleading nonperturbative corrections of order O ( α s Λ /m b ), which are estimatedto give an additional uncertainty of order 5%, have to be added in all observablesof the B → X s ℓ + ℓ − mode (see Section 2.3).In all predictions, it is assumed that there is no cut in the hadronic massregion (see Section 2.4). Furthermore, after including the NLL QED matrixelements, the electron and muon channels receive different contributions due toterms involving ln( m b /m ℓ ) (see Section 2.2). This is the only source of thedifference between these two channels. All collinear photons are assumed to beincluded in the X s system, and the dilepton invariant mass does not contain anyphotons; in other words, q = ( p ℓ + + p ℓ − ) . Present experimental settings atthe B factories are different, and therefore the theoretical predictions have to bemodified (87).This difference in the settings also means that deviations from the SM predic-tion ( R SM X s = 1) in the muon-electron ratio R X s = Γ( B → X s µ + µ − ) [ q a , q b ] / Γ( B → X s e + e − ) [ q a , q b ] (30)can result from a different treatment of collinear photons in the two modes. Thisratio is interesting because it is sensitive to the neutral Higgs boson of two-Higgs-doublet models at large tan β (180,181), which is also valid in corresponding ratios R K ( ∗ ) of exclusive modes: In the SM, one finds R K = 1, as well as R K ∗ = 0 . q , including M e + e − < m µ . The exclusive penguin modes offer a larger variety of experimentally accessibleobservables than do the inclusive ones, but the nonperturbative uncertainties inthe theoretical predictions are in general sizable. B → K ∗ γ and B → ργ The large hadronic uncertainties, which arise fromthe nonperturbative input of the QCDF formula and from our limited knowledgeof power corrections, do not allow precise predictions of the branching fractions ofexclusive modes. However, within ratios of exclusive modes such as asymmetries,parts of the uncertainties cancel out and one may hope for higher precision.The ratio R th ( ργ/K ∗ γ ) [and similarly R th ( ωγ/K ∗ γ )] is given by (148, 149, 150,151). R th ( ργ/K ∗ γ ) = B th ( B → ργ ) B th ( B → K ∗ γ ) = S ρ (cid:12)(cid:12)(cid:12)(cid:12) V td V ts (cid:12)(cid:12)(cid:12)(cid:12) ( M B − m ρ ) ( M B − m K ∗ ) ζ [1 + ∆ R ( ρ/K ∗ )] , (31) adiative and Electroweak Penguin Decays of B Mesons m ρ is the mass of the ρ meson; ζ is the ratio of the transition form factors, ζ = T ρ (0) /T K ∗ (0); and S ρ = 1 and 1 / ρ ± and ρ mesons, respectively.The quantity (1 + ∆ R ) entails the explicit O ( α s ) corrections as well as the power-suppressed contributions. These functions also depend on CKM parameters,namely φ ≡ α = arg( − V cb V ∗ cb /V td V ∗ tb ) and R ut = | V ud V ∗ ub /V td V ∗ tb | , and one findsnumerically (150) that∆ R ( ρ ± /K ∗± ) = n − R ut cos φ [0 . +0 . − . ] + R ut [0 . +0 . − . ] o , (32)and ∆ R ( ρ /K ∗ ) = n − R ut cos φ [ − . +0 . − . ] + R ut [0 . +0 . − . ] o . (33)These results are consistent with the predictions given in Refs. (148, 149, 151).Obviously, the neutral mode is better suited for the determination of | V td /V ts | than is the charged mode, in which the function ∆ R is dominated by the weak-annihilation contribution, which leads to a larger error. The most recent deter-mination of the ratio ζ = T ρ (0) /T K ∗ (0) within the light-cone QCD sum ruleapproach (182), 1 /ζ = 1 . ± .
09, leads to the determination of | V td /V ts | viaEq. (31) (see Section 5.4). However, the experimental data on the branchingfractions of B → K ∗ γ and B → ργ calls for a larger error on ζ , if one assumesno large power corrections beyond the known annihilation terms (150). Isospin Asymmetry in Radiative Decays
Another important observableis the isospin breaking ratio given by∆ ( B → K ∗ γ ) = Γ( B → K ∗ γ ) − Γ( B + → K ∗ + γ )Γ( B → K ∗ γ ) + Γ( B + → K ∗ + γ ) , (34)where the partial decay rates are CP -averaged. In the SM spectator-dependenteffects enter only at the order Λ /m b , whereas isospin-breaking in the form fac-tors is expected to be a negligible effect. Therefore, the SM prediction is assmall as O (5%) (145, 148, 149, 150, 151). The ratio is especially sensitive to NPeffects in the penguin sector, namely to the ratio of the two effective couplings C /C . The analogous isospin ratio in the ρ sector strongly depends on CKMparameters (150):∆( ργ ) = Γ( B + → ρ + γ )2Γ( B → ρ γ ) − − . +5 . − . | CKM+5 . − . | had ) × − . (35)The hadronic error is due mainly to the weak-annihilation contribution, to whicha 50% error is assigned. CP asymmetries in Radiative Decays In the CP asymmetries, the un-certainties due to form factors cancel out to a large extent. But both the scaledependence and the dependence on the charm quark of the next-to-leading-orderpredictions are rather large because the CP asymmetries arise at O ( α s ) only. Al-though the direct CP asymmetry in B → K ∗ γ is doubly Cabibbo suppressed andexpected to be very small within the QCDF/SCET approach, one finds O ( − CP asymmetries in the B → ργ mode (131, 148, 150).Because the weak-annihilation contribution does not contribute significantly here,the neutral and charged modes are of similar sizes (150): A CP ( B → ρ γ ) = ( − . +1 . − . | CKM+3 . − . | had ) % (36)2 Tobias Hurth and Mikihiko Nakao and A CP ( B − → ρ − γ ) = ( − . +1 . − . | CKM+2 . − . | had ) % . (37)Finally, we reiterate that all predictions of exclusive observables within the QCDF/SCET approach may receive further uncertainties due to the unknown powercorrections. This possibility might be especially important in the case of CP asymmetries.The time-dependent CP asymmetry is given by two parameters, S CP and A CP : A CP ( B → f ; ∆ t ) = S CP sin(∆ m ∆ t ) + A CP cos(∆ m ∆ t ) , (38)where A CP represents the size of the direct CP asymmetry discussed above. Inhadronic decay modes such as B → J/ψK S , a large value of S CP due to the angle φ ≡ β = − arg( V td V ∗ tb /V ud V ∗ ub ) of the unitarity triangle has been established, anda similarly large CP asymmetry is expected for hadronic penguin decays. Thisasymmetry is suppressed in radiative penguin decays because the photon helicitiesare opposite between those from B and B decays under the left-handed currentof SM weak decays, and they do not interfere in the limit of massless quarks.This finding implies a suppression factor of m s /m b in the leading contribution to S CP that is induced by the electromagnetic dipole operator O : S SM CP = − sin 2 φ m s m b [2 + O ( α s )] + S SM ,sγg (39)However, there are also additional contributions, S SM ,sγg induced by the process b → sγg via operators other than O (183,184). These corrections are not helicity-suppressed but are power-suppressed. A conservative dimensional estimate of thecontribution from a nonlocal SCET operator series leads to |S SM ,sγg | ≈ .
06 (183,184), whereas within a QCD sum rule calculation, the contribution due to soft-gluon emission is estimated to be S SM ,sγg = − . ± .
01 (185, 151) whichleads to S SM CP = − . ± . +0 − . . The QCD sum rule estimates of powercorrections, namely long-distance contributions that arise from photon and soft-gluon emission from quark loops (151), lead to analogous results for the otherradiative decay modes, such as B → ργ (151). If a large value of S CP beyondthe SM prediction is observed, it will signal a new right-handed current beyondthe SM. B → K ∗ ℓ + ℓ − The isospin asymmetry in the mode B → K ∗ ℓ + ℓ − , as in theradiative mode, is a subleading Λ /m b effect, but the dominant isospin-breakingeffects can be calculated perturbatively, whereas other Λ /m b corrections are sim-ply estimated. Thus, the exact uncertainty is difficult to estimate due to unknownpower corrections, but the observable may still be useful in the NP search becauseof its high sensitivity to specific Wilson coefficients (186).The decay B → K ∗ ℓ + ℓ − (with K ∗ → K − π + on the mass shell) is completelydescribed by four independent kinematic variables: the lepton-pair invariant masssquared, q , and the three angles θ ℓ , θ K , and φ (for their precise definitions, see The symbol C CP = −A CP is also often used. This does not necessarily contradict a larger time-dependent CP asymmetry of approxi-mately 10% within the inclusive mode found in Ref. (183), because the SCET estimate (183,184)shows that the expansion parameter is Λ /Q . Here Q is the kinetic energy of the hadronic part.There is no contribution at leading order. Thus, the effect is expected to be larger for largerinvariant hadronic mass. The K ∗ mode must have the smallest effect, below the average 10%. adiative and Electroweak Penguin Decays of B Mesons d Γ B dq dθ ℓ dθ K dφ = 932 π I ( q , θ ℓ , θ K , φ ) sin θ ℓ sin θ K . (40)By integrating two of the angles, one finds d Γ ′ dθ K = 3Γ ′ θ K (cid:16) F L cos θ K + (1 − F L ) sin θ K (cid:17) , (41)and d Γ ′ dθ ℓ = Γ ′ (cid:18) F L sin θ ℓ + 38 (1 − F L )(1 + cos θ ℓ ) + A F B cos θ ℓ (cid:19) sin θ ℓ . (42)The observables appear linearly in the expressions so the fits can be performedon data binned in q . The fraction of longitudinal polarization F L from the kaonangular distribution and the forward-backward asymmetry A F B from the leptonangular distribution are accessible this way. The latter observable is defined asfollows ( θ ℓ is defined below Eq. 25): A F B ( q ) ≡ d Γ /dq Z d (cos θ ℓ ) d Γ dq d cos θ ℓ − Z − d (cos θ ℓ ) d Γ dq d cos θ ℓ ! . (43)The hadronic uncertainties of these two differential observables are large. How-ever, the value of the dilepton invariant mass q , for which the differential forward-backward asymmetry vanishes, can be predicted in quite a clean way. In theQCDF approach at leading order in Λ /m b , the value of q is free from hadronicuncertainties at order α s . A dependence on the soft form factor and on thelight-cone wave functions of the B and K ∗ mesons appears only at order α s . Atnext-to-leading order one finds (150): q [ K ∗ ℓ + ℓ − ] = 4 . +0 . − . GeV , q [ K ∗ + ℓ + ℓ − ] = 4 . +0 . − . GeV . (44)The small difference is due to isospin-breaking power corrections. However, an un-certainty due to unknown power corrections should be readded to the theoreticalerror bars. The zero is highly sensitive to the ratio of the two Wilson coefficients C and C . Thus, such a measurement would have a huge phenomenologicalimpact.In the near future, a full angular analysis based on the four-fold differentialdecay rate in Eq. 40 will become possible. Such rich information would allowfor the design of observables with specific NP sensitivity and reduced hadronicuncertainties (152, 187). These observables would be constructed in such a waythat the soft form factor dependence would cancel out at leading order for alldilepton masses, and they would have much higher sensitivity to new right-handedcurrents than would observables that are already accessible via the projectionfits (189,152,187). In these optimized observables, the unknown Λ /m b correctionswould be the source of the largest uncertainty. Further detailed NP analyses ofsuch angular observables have been presented in Refs. (191, 190). A full angularanalysis provides high sensitivity to various Wilson coefficients, but the sensitivityto new weak phases is restricted (192, 187).4 Tobias Hurth and Mikihiko Nakao
The huge samples of B meson decays collected by Belle and BaBar have madeit possible to fully explore the radiative penguin decays b → sγ and b → dγ , aswell as the electroweak penguin decays b → sℓ + ℓ − . B → X s γ Branching Fraction
An experimental challenge is how to lower the minimum photon energy to 1 . B factories, minimum photonenergy of 2.0 GeV was required in the measurement by CLEO (193). BaBarhas a minimum photon-energy requirement of 1.9 GeV based on 89 million BB pairs (194), whereas Belle first reported the result of 1.8 GeV with 152 million BB (195). Belle recently lowered the limit to 1.7 GeV by using 657 million BB pairs (196) (Fig. 5). Belle measured the branching fraction to be B ( B → X s γ ) =(345 ± ± × − for E γ > . ± ± ± × − for E γ > . B -reco technique have been used byBaBar with a minimum photon-energy requirement of 1.9 GeV (197, 198), whichcorresponds to a maximum recoil mass requirement of 2.8 GeV. Belle also madea sum-of-exclusive measurement using a very early data set (199).To calculate the average branching fraction based on the same phase spaceand to compare it with theory predictions, the Heavy Flavor Averaging Group(HFAG) (200) has made an extrapolation of the branching fraction to the sameminimum photon energy of 1 . . ± .
004 (0 . ± . B ( B → X s γ ) = (352 ± ± × − , (45)where the first error is statistical and systematic combined, and the second is dueto the extrapolation. The result is in agreement with the SM prediction given inEq. 17, and it provides stringent constraints on NP, as discussed in Section 4.1. b → sγ Processes
The recoil system of B → X s γ below 1.1 GeV is dominated by the K ∗ resonance,as the spin-0 state is forbidden. Above 1.1 GeV, X s is a mixture of various reso-nant and nonresonant states and therefore can usually be modeled as a continuumspectrum in the inclusive B → X s γ analysis. The K ∗ signal makes it possible touse the channel for various studies, as discussed below. The B → K ∗ γ branchingfractions have been measured precisely by Belle (201) and BaBar (154) and havebeen averaged by HFAG to be B ( B → K ∗ γ ) = (43 . ± . × − B ( B + → K ∗ + γ ) = (42 . ± . × − , (46)which corresponds to approximately 12% of the total B → X s γ branching frac-tion. The SM predictions for the branching fraction have been calculated bymany groups. They are consistent with the measured values but have very largeerrors of 30% to 50%, which arise mainly from the uncertainty of the B → K ∗ form factor (130, 131). adiative and Electroweak Penguin Decays of B Mesons X s system has also been explored. Sofar, B → K ∗ (1430) γ (202, 203) and B → K (1270) γ (204) have been measured,but other decay channels such as B → K (1400) γ seem to have small branchingfractions and have not yet been observed. In addition, many multi-body finalstates have been measured, such as B → Kππγ (205, 202) (including B → K ∗ πγ and B → Kργ ), B → Kφγ (206, 207), B → Kηγ (208, 209), B → Kη ′ γ (210),and B → Λ pγ (211). CP and Isospin Asymmetries in b → sγ Processes
The measurement of the inclusive direct CP asymmetry (Eq. 20) was performedby use of the sum-of-exclusive method to tag the flavor of the B candidate. For B ( B ), only the self-tagging modes with a K + ( K − ) were used. The measuredasymmetry was corrected for a small dilution due to the doubly misidentified pairof a charged kaon and a pion. The results, based on 152 and 383 million BB samples by Belle and BaBar, are 0 . ± . ± .
030 (212) and − . ± . ± .
014 (213), respectively, and have been averaged by HFAG to be A CP ( B → X s γ ) = − . ± . . (47)This is consistent with null asymmetry, and the size of the error is still much largerthan the SM precision (Eq. 21). Assuming the systematic error can be reducedalong with the statistical error, a data set two orders of magnitude larger wouldbe more sensitive to NP, although still insufficient to measure the small A CP predicted by the SM.In the exclusive B → K ∗ γ channel, the direct CP asymmetry is also small buthas less-understood theoretical uncertainties. Experimentally it can be measuredmore precisely. The current HFAG average is A CP ( B → K ∗ γ ) = − . ± . , (48)which is also consistent with null asymmetry.In the fully inclusive measurement, flavor information is not available for thesignal side, but it can be obtained from the charge of the lepton in the event if theother B decays into a semileptonic final state. In this case, it is not possible todiscriminate B → X d γ from B → X s γ and the measured asymmetry correspondsto a combined one. This combined asymmetry (note the different normalizationin comparison with Eq. 24) has been measured by BaBar to be A CP ( B → X s + d γ ) = − . ± . ± . , (49)which is consistent with null asymmetry but has a much larger error than do theother two asymmetry measurements.The measurement of the isospin asymmetry (Eq. 34) is another way to utilizethe reconstructed B → K ∗ γ events. Here, the measured branching fractions arecorrected by the lifetime ratio: τ B + /τ B = 1 . ± .
009 (214). Usually B decaybranching fractions are quoted based on the assumption of B (Υ(4 S ) → B + B − ) = B (Υ(4 S ) → B B ) = 0 .
5, but the isospin-asymmetry measurement is alreadyprecise enough to be affected by the difference in these branching fractions. Bellemeasures ∆ ( B → K ∗ γ ) = 0 . ± . ± .
026 without this correction, whereasBaBar measures ∆ ( B → K ∗ γ ) = 0 . ± . ± .
022 using B (Υ(4 S ) → B + B − ) = 0 . ± .
006 and B (Υ(4 S ) → B B ) = 0 . ± .
006 (214). After6
Tobias Hurth and Mikihiko Nakao scaling the Belle result and including the CLEO result using the aforementionedlifetime and production ratios, the na¨ıve world average is∆ ( B → K ∗ γ ) = 0 . ± . . (50)This average is in agreement with the SM expectation.A similar isospin asymmetry can be also measured for the inclusive B → X s γ decay by use of the sum-of-exclusive method. The result by BaBar is∆ ( B → X s γ ) = − . ± . ± . ± . , (51)which is consistent with null asymmetry but is not yet as precise as that for B → K ∗ γ .The measurement of the time-dependent CP asymmetry (Eq. 38) for b → sγ faces two experimental challenges. First, the modes and statistics that canbe used for time-dependent CP asymmetry measurements are rather limited.Although the B → K ∗ γ branching fraction is not very small, only 1/9 of theevents that decay into the K S ( → π + π − ) π γ final state can be used. Second,the B meson decay vertex position has to be extrapolated from the displaced K S → π + π − vertex and the K S momentum vector. Therefore, the K S decaysinside the vertex detector volume (55% in Belle, 68% in BaBar) and the resultingvertex resolution is somewhat degraded.Because any B → P Q γ final states [where P and Q are CP eigenstates (215)]can be used, the B → K S π γ events with M K S π up to 1 . K ∗ (1430) were measured by both Belle (216) and BaBar (217) and have beenaveraged by HFAG as S CP ( B → K S π γ ) = − . ± . , (52)in which the B → K ∗ γ contribution gives S CP ( B → K ∗ γ ) = − . ± . S CP ( B → K S ηγ ) = − . +0 . − . ± .
12 (209), and Belle has measured S CP ( B → K S ρ γ ) = 0 . ± . +0 . − . (218).The latter is slightly diluted by the B → K ∗ + π − γ events, by a factor whichwas measured to be 0 . +0 . − . , but is free from the restriction of K S vertexingand has a statistical error comparable in size to that of the B → K S π γ mode.Currently all results are compatible with null asymmetry with errors that are stillnot small enough to provide nontrivial constraints on right-handed currents, butthis observable will be one of the best ways to search for NP in future experiments. b → dγ Processes
Three exclusive b → dγ decay modes are considered to be the easiest modes tostudy the b → dγ process: B + → ρ + γ , B → ρ γ and B → ωγ . Although thesemodes have been searched for since the beginning of Belle and BaBar, only inthe later stage of these experiments were measurements of the B → ργ modesestablished. This is partly because of the large B → K ∗ γ background and partlydue to the huge continuum background, which is more severe for modes without akaon in the final state. Therefore, large statistics and good particle identificationare essential; Belle has the advantage in the former, whereas BaBar leads in thelatter.To gain statistics, these three modes have been combined by assuming theirna¨ıve quark contents, through the use of Γ( B + → ρ + γ ) = 2Γ( B → ρ γ ) = adiative and Electroweak Penguin Decays of B Mesons B → ωγ ). In the latest measurements by both Belle (219) and BaBar (220),the B → ρ γ mode was measured with more than 5 σ significance and B + → ρ + γ with more than 3 σ significance, whereas B → ωγ remains unestablished withsignificance less than 3 σ . Using the symbol B → ( ρ, ω ) γ for the combined resultsthat are adjusted for the B + → ρ + γ mode, the averaged branching fraction byHFAG becomes B ( B → ( ρ, ω ) γ ) = (1 . +0 . − . ) × − . (53)The results are consistent with the SM predictions. However, these predictionsare affected by form factor uncertainties and do not have effective predictionpower for NP.A more effective way to use these results is to combine them with the B → K ∗ γ measurements to determine | V td /V ts | . Using Eq. 31, Belle and BaBar reportedthe value of | V td /V ts | to be 0 . +0 . − . ± .
015 and 0 . +0 . − .
024 +0 . − . , respectively,where the errors are experimental and theoretical. These results from the penguindiagrams are in agreement with the determination from the box diagrams usingthe ratio of the B and B s mixing parameters ∆ m d / ∆ m s , where ∆ m d is measuredat the B factories and ∆ m s at the Tevatron. The results are also in agreementwith the more indirect determination by the unitarity triangle fit from otherobservables. This is a nontrivial test of the CKM scheme. However, althoughthe experimental errors are still larger than the theoretical errors, the size of thetheoretical error is unlikely to be reduced.A possible way to improve this situation utilizes the inclusive B → X d γ mea-surement with the sum-of-exclusive method. BaBar has reconstructed the X d γ system in seven final states ( π + π − γ , π + π γ , π + π − π + γ , π + π − π γ , π + π − π + π − γ , π + π − π + π γ , π + ηγ ) (221) in the mass range 0 . < M X d < . X s γ modes were also measured in thecorresponding seven final states in the same mass range, where the first π + wasreplaced with K + . The ratio of the two inclusive branching fractions is B ( B → X d γ ) B ( B → X s γ ) = 0 . ± . ± . , (54)which is converted to | V td /V ts | = 0 . ± . ± . CP asymmetry for B → ργ can be as large as ∼ −
10% in theSM, whereas the time-dependent CP asymmetry is doubly suppressed due to thephoton helicity and the cancellation of the CKM element V td . The latter appearsin the mixing and in the b → d penguin decay. However, the ρ → π + π − decayprovides clear vertex information for B → ρ γ . Both CP asymmetries havebeen measured by Belle (219, 222) as A CP ( B + → ρ + γ ) = − . ± . ± . , A CP ( B → ρ γ ) = − . ± . ± . , and S CP ( B → ρ γ ) = − . ± . ± . . (55)So far the results are consistent with null asymmetry. A nonzero direct CP violation may be measured earlier in B → ργ than in B → K ∗ γ .The isospin asymmetry (Eq. 35) is also expected to be as large as ∼ − B → ργ . Belle measures ∆( ργ ) = − . +0 . − .
19 +0 . − . (219), and BaBar mea-8 Tobias Hurth and Mikihiko Nakao sures ∆( ργ ) = − . +0 . − . ± .
10 (220); both measurements show a large isospinasymmetry. The average by HFAG is∆( ργ ) = − . +0 . − . . (56)A significant nonzero isospin asymmetry could indicate NP. B → K ( ∗ ) ℓ + ℓ − Branching Fraction
Despite their small branching fractions, the exclusive decay channels B → K ( ∗ ) ℓ + ℓ − have been measured efficiently with small background at Belle and BaBar, giventhat their final states are the same as those of B → J/ψK ( ∗ ) for which the B factories were designed. Here, K ( ∗ ) is one of K + , K S , K ∗ + and K ∗ , and ℓ + ℓ − is either e + e − or µ + µ − .Electrons are identified by their energy deposit through an electromagneticshower in the calorimeter. The minimum momentum is required to be greaterthan 0.4 GeV by Belle or 0.5 GeV by BaBar. The momentum of the bremsstrahlungphotons that may be emitted by the electrons are added to their momenta if theyare found near the electron direction. Muons have to reach and penetrate intothe outer muon detectors and the minimum momentum is required to be 0.7 GeVby Belle or 1.0 GeV by BaBar. The dilepton mass regions around J/ψ and ψ (2 S )are vetoed.The branching fractions, averaged over the lepton and kaon flavors and inte-grated over the dilepton masses, assuming the SM distribution over the vetoed J/ψ and ψ (2 S ), were measured by Belle (223) and BaBar (224) and have beenaveraged by HFAG as B ( B → Kℓ + ℓ − ) = (4 . ± . × − , B ( B → K ∗ ℓ + ℓ − ) = (10 . +1 . − . ) × − . (57)The results are consistent with SM expectations. At present, the irreducibleform factor uncertainty in the SM calculations prevents these results from placingmeaningful constraints on NP.A small subset of these combinations, B + → K + µ + µ − and B → K ∗ µ + µ − ,can be efficiently measured at hadron colliders. CDF has reported the mostprecise measurements of these modes (225). B → K ( ∗ ) ℓ + ℓ − Asymmetries and Angular Distributions
The direct CP and isospin asymmetries in B → K ( ∗ ) ℓ + ℓ − are also useful inthe search for NP. The direct CP asymmetries are consistent with null values, A CP ( B → K + ℓ + ℓ − ) = − . ± .
09 and A CP ( B → K ∗ ℓ + ℓ − ) = − . ± .
08 asaveraged by HFAG. However, nonzero negative isospin asymmetries in the small q -region of B → Kℓ + ℓ − (3 . σ ) and B → K ∗ ℓ + ℓ − (2 . σ ) have been reported byBaBar (3 . σ when combined). The corresponding isospin asymmetries by Belleare 1 . σ and 1 . σ from zero, and are consistent with both BaBar’s results and nullasymmetry. The isospin asymmetry combined for B → K ( ∗ ) ℓ + ℓ − and averagedby HFAG is A K ( ∗ ) I = − . ± . . (58)The SM prediction is essentially zero at this level of statistics (see Section 4.2). adiative and Electroweak Penguin Decays of B Mesons B → K ( ∗ ) ℓ + ℓ − (Eq. 30) are also measured by bothBelle and BaBar. Results are consistent with the SM, and their na¨ıve averagesare R K = 1 . ± .
18 and R K ∗ = 0 . ± . B → K ∗ ℓ + ℓ − → Kπℓ + ℓ − allows ex-traction of further information from the angular distributions of the final-stateparticles. The most interesting observables are the fraction of longitudinal polar-ization F L from the kaon angular distribution (Eq. 41) and the forward-backwardasymmetry A F B from the lepton angular distribution (Eq. 42). Belle has mea-sured F L and A F B in six bins of q (223), whereas BaBar has done so in twobins (226). Current statistics are not enough to tell whether there is a zero-crossing point at low q , although the results favor the case with no crossing,for which the sign of the Wilson coefficient C is flipped. Both results have pos-itive A F B for high q (Fig. 6), which sets nontrivial constraints on the Wilsoncoefficients. CDF has also measured F L and A F B in the same six bins as Bellefor B → K ∗ µ + µ − events (225). The results are in agreement with Belle andBaBar. B → X s ℓ + ℓ − Branching Fraction
The inclusive B → X s ℓ + ℓ − branching fraction has been measured by Belle andBaBar using the sum-of-exclusive technique. The X s system includes final stateswith one kaon and up to four (two) pions that have masses up to 2.0 (1.8) GeV forthe result by Belle (BaBar). Belle recently announced a preliminary result basedon 657 million BB (227), and BaBar’s result is based on 89 million BB (228).In Belle’s new analysis, partial branching fractions are measured in bins of the X s mass, and then the total branching fraction is calculated as their sum. Thismethod reduces the large systematic error observed in previous studies that arosefrom the strong X s mass dependence of the efficiency and the unknown fractionsof exclusive channels B → K ( ∗ ) ℓ + ℓ − . The measurement is still dominated bythe statistical error and will be more precise in the future. Belle and BaBarreported branching fractions as B ( B → X s ℓ + ℓ − ) = (3 . ± . +0 . − . ) × − and B ( B → X s ℓ + ℓ − ) = (5 . ± . ± . ± . × − , respectively, which were averagedby HFAG as B ( B → X s ℓ + ℓ − ) = (3 . +0 . − . ) × − , (59)integrated over the entire subset of phase space with q > . J/ψ and ψ (2 S ) regions. The results are in good agreement with the SMprediction. They strongly disfavor the case with the flipped sign of C (179). Remarkably, the B factories have measured all the observables within the radia-tive and electroweak penguin decays at values that are consistent with the SMpredictions. These measurements rule out O (1) corrections to the SM and iden-tify the CKM theory as the dominant effect for flavor violation as well as for CP violation. The success of the simple CKM theory of CP violation was honoredwith the Nobel Prize in Physics in 2008. Theoretical tools and precision havesignificantly advanced during the past decade, and we are ready to challenge theSM if a clear deviation is found or to discriminate different NP scenarios if directevidence is found at the LHC.0 Tobias Hurth and Mikihiko Nakao
Also, the future offers great experimental opportunities in flavor physics. LHCbhas finally started taking data and promises to overwhelm many B factory results,and ATLAS and CMS will also contribute to flavor physics. In the radiative andelectroweak penguin decays, the most promising measurements are the angularanalysis of B → K ∗ µ + µ − and the analysis of time-dependent CP asymmetryin B s → φγ ; the latter measurement cannot be performed at the B factories dueto the fast B s oscillation. However, the theoretically clean inclusive modes andmany modes involving neutral particles like the π can be pursued only at the e + e − B factories. Two proposed super- B factories, Belle II at KEK and SuperBin Italy, would accumulate two-orders-of-magnitude-larger data samples. Suchdata would push experimental precision to its limit.Theoretical and experimental techniques are ready for such large data samples.The results provided by LHCb and the next-generation e + e − B factories areeagerly awaited, as they may be the key to identifying physics beyond the SM. ACKNOWLEDGMENTS
We thank Christoph Greub, Colin Jessop, and Kurtis Nishimura for their carefulreading of the manuscript, and Thorsten Feldmann, Matthias Neubert, and GilPaz for comments. T.H. thanks the CERN Theory Group for its hospitalityduring his visits to CERN.
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XXIV International Symposium on LeptonPhoton Interactions (2009)8
Tobias Hurth and Mikihiko Nakao
Phys. Rev. Lett. adiative and Electroweak Penguin Decays of B Mesons W − tu, db u, dsγ (b) χ − ˜ tu, db u, dsγ (c) b Wt sl l Figure 1: Examples of radiative penguin decay diagrams (a) in the StandardModel and (b) beyond. (c) A penguin. µ b [GeV] µ c [GeV]Figure 2: Renormalization-scale dependence of B ( B → X s γ ) in units 10 − atleading log (dotted lines), next-to-leading log (dashed lines) and next-to-next-to-leading log (solid lines). The plots describe the dependence on (left) the the low-energy scale µ b and (right) the charm mass renormalization scale µ c , from (58). d Γ/ dE γ γ (GeV) T O Y
S P E C T R U M s d B ( B → X s ℓ + ℓ − ) / d s [ − ] Figure 3: Spectra in inclusive modes: (left) Cut in the photon-energy spectrumin B → X s γ . (right) Differential B → X s ℓ + ℓ − branching fraction as a functionof s = q /m b ≡ m ℓ + ℓ − /m b , including the effect of charm resonances in theKr¨uger-Sehgal method (solid line). For comparison, the dashed curve shows thesame quantity obtained within a purely partonic calculation at next-to-leading-log precision, from Ref. (113).0 Tobias Hurth and Mikihiko Nakao ) (GeV/c ES M E ve n t s / M e V / c - π + K ) (GeV/c ES M E ve n t s / M e V / c E (GeV) ∆ -0.3 -0.2 -0.1 0 0.1 0.2 0.3 E ve n t s / M e V - π + K E (GeV) ∆ -0.3 -0.2 -0.1 0 0.1 0.2 0.3 E ve n t s / M e V Figure 4: Example of M ES ( M bc ) and ∆ E for B → K ∗ γ by BaBar (fromRef. (154)). [GeV] γ c.m.s E1.5 2 2.5 3 3.5 4 P ho t on s / M e V -4000-20000200040006000 a) [GeV] γ c.m.s E1.5 2 2.5 3 3.5 4 P ho t on s / M e V -1000-50005001000 b) [GeV] γ c.m.s E1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 P ho t on s / M e V -30000-20000-100000100002000030000 c) Figure 5: Photon-energy spectrum for B → X s γ , as measured by Belle (a) with-out lepton tag, (b) with a lepton tag, and (c) their average from Ref. (196). F L A F B q (GeV /c ) A I -1010 2 4 6 8 10 12 14 16 18 20 F B A −0.6−0.4−0.200.20.40.60.811.2 (a) ( S ) ψ ψ J/ ] /c [GeV q L F (b) ( S ) ψ ψ J/ Figure 6: Longitudinal polarization fraction, forward-backward asymmetry andisospin asymmetry of B → K ∗ ℓ + ℓ −−