Inclusive \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \) : complete angular analysis and a thorough study of collinear photons

Journal of High Energy Physics, Jun 2015

We investigate logarithmically enhanced electromagnetic corrections of all angular observables in inclusive \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \). We present analytical results, which are supplemented by a dedicated Monte Carlo study on the treatment of collinear photons in order to determine the size of the electromagnetic logarithms. We then give the Standard Model predictions of all observables, considering all available NNLO QCD, NLO QED and power corrections, and investigate their sensitivity to New Physics. Since the structure of the double differential decay rate is modified in the presence of QED corrections, we also propose new observables which vanish if only QCD corrections are taken into account. Moreover, we study the experimental sensitivity to these new observables at Belle II.

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Inclusive \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \) : complete angular analysis and a thorough study of collinear photons

Received: March and a thorough study of collinear photons Tobias Huber 0 1 2 Tobias Hurth 0 1 2 3 Enrico Lunghi 0 1 2 4 0 Bloomington , IN 47405 , U.S.A 1 Johannes Gutenberg University , D-55099 Mainz , Germany 2 Universit ̈at Siegen , D-57068 Siegen , Germany 3 PRISMA Cluster of Excellence and Institute for Physics , THEP 4 Physics Department, Indiana University We investigate logarithmically enhanced electromagnetic corrections of all angular observables in inclusive B¯ supplemented by a dedicated Monte Carlo study on the treatment of collinear photons in order to determine the size of the electromagnetic logarithms. We then give the Standard Model predictions of all observables, considering all available NNLO QCD, NLO QED and power corrections, and investigate their sensitivity to New Physics. Since the structure of the double differential decay rate is modified in the presence of QED corrections, we also propose new observables which vanish if only QCD corrections are taken into account. Moreover, we study the experimental sensitivity to these new observables at Belle II. aTheoretische Physik 1; Naturwissenschaftlich-Technische Fakult¨at 1 Introduction 2 Definition of the observables 3 Log-enhanced QED corrections to the double differential decay rate 5.1 5.2 5.3 Master formulas for the observables 5 Phenomenological results Branching ratio, low-q2 region Branching ratio, high-q2 region 5.6 The ratio R(s0) 6 New physics sensitivities 7 On the connection between theory and experiments Various experimental settings Validation Monte Carlo estimate of QED corrections to HT and HL 8 Conclusion A QED and QCD functions A.1 QED functions for the double differential rate A.2 Functions for the QCD corrections to the HI A.3 Functions for the QED corrections to the HI B New physics formulas Introduction By now the LHC experiment has not discovered any new degrees of freedom beyond the Standard Model (SM). In particular, the measurements of the LHCb experiment and the B-physics experiments of ATLAS and CMS have confirmed the simple Cabibbo-KobayashiMaskawa (CKM) theory of the SM [1–3]. This corresponds to the general result of the B-factories [4, 5] and of the Tevatron B-physics experiments [6, 7] which have not indicated any sizable discrepancy from SM predictions in the B-meson sector (for reviews see However, recently the first measurement of new angular observables in the exclusive tainties it is not clear if this anomaly is a first sign for new physics beyond the SM, or a consequence of hadronic power corrections; but of course, it could turn out to just be a staeagerly awaited to clarify the situation. More recently, another slight discrepancy occurred. is affected by unknown power corrections, the ratio RK is theoretically rather clean. This might be a sign for lepton non-universality (see e.g. refs. [22–31]). clean modes of the indirect search for new physics via flavour observables (for a review and updates see refs. [32–34]); especially it allows for a nontrivial crosscheck of the recent LHCb data on the exclusive mode [18, 35]. The observables within this inclusive mode are dominated by perturbative contributions if the cc¯ resonances that show up as large peaks in the dilepton invariant mass spectrum are removed by appropriate kinematic cuts — leading to so-called ‘perturbative di-lepton invariant mass windows’, namely the low di-lepton mass region 1 GeV2 < s = q 2 = m`2` < 6 GeV2, and also the high dilepton mass region with 2 > 14.4 GeV2 (or q2 > 14.2 GeV2). In these regions a theoretical precision of order 10% is in principle possible. By now the branching fraction has been measured by Belle and BaBar using the sumof-exclusive technique only. The latest published measurement of Belle [36] is based on a sample of 152 × 106 BB¯ events only, which corresponds to less than 30% of the dataset available at the end of the Belle experiment. Babar has just recently presented an analysis based on the whole dataset of Babar using a sample of 471 × 106 BB¯ events [37] which updated the former analysis of 2004 [38]. In the low- and high-dilepton invariant mass region the weighted averages of the experimental results read B(B¯ → Xs`+`−)leoxwp = (1.58 ± 0.37) × 10−6 , B(B¯ → Xs`+`−)ehxigph = (0.48 ± 0.10) × 10−6 . All the measurements are still dominated by the statistical error. The expectation is that In addition, Belle has presented a first measurement of the forward-backward asymmetry [39] and Babar a measurement of the CP violation in this channel [37]. The super flavour factory Belle II at KEK will accumulate two orders of magnitude larger data samples [40]. Such data will push experimental precision to its limit. This is the main motivation for the present study to decrease the theoretical uncertainties accordingly. The theoretical precision has already reached a highly sophisticated level. Let us briefly review the previous analyses. contributions are calculated up to NNLL precision. The complete NLL QCD contributions have been presented [41, 42]. For the NNLL calculation, many components components for the NNLL QCD precision have been calculated in refs. [43–55]. smin = qm2in/mb2 [53]. • If only the leading operator of the electroweak hamiltonian is considered, one is led to a local operator product expansion (OPE). In this case, the leading hadronic power already been analysed [56–61]. Power correction that scale with 1/mc2 [62] have also been considered. They can be calculated quite analogously to those in the decay Such analysis goes beyond the local OPE. An additional uncertainty of ±5% has been In the high-q2 region, one encounters the breakdown of the heavy-mass expansion (HME) at the end point of the dilepton mass spectrum: whereas the partonic contribution vanishes, the 1/mb2 and 1/mb3 corrections tend towards a finite, non-zero value. decay, no partial all-order resummation into a shape-function is possible. However, for the integrated high-q2 spectrum an effective expansion is found in inverse powless rapidly, and the convergence behaviour depends on the lower dilepton-mass cut The large theoretical uncertainties could be significantly reduced by normalizing the R(s0) = For example, the uncertainty due to the dominating 1/mb3 term could be reduced from 19% to 9% [66]. independent kinematical quantities. A hadronic invariant-mass cut is imposed in the experiments. The high-dilepton-mass region is not affected by this cut, but in the the relevance of the shape function. A recent analysis in soft-collinear effective theory tion in the dilepton-mass spectrum can be accurately computed. Nevertheless, the effects of subleading shape functions lead to an additional uncertainty of 5% [67, 68]. A more recent analysis [69] estimates the uncertainties due to subleading shape functions more conservatively. By scanning over a range of models of these functions, one finds corrections in the rates relative to the leading-order result to be between decrease such uncertainties significantly by constraining both the leading and subperforming the matching from QCD onto SCET at NNLO, and a prediction of the zero of the forward-backward asymmetry in this semi-inclusive channel was provided. • As already discussed, the cc¯ resonances can be removed by making appropriate kinematic cuts in the invariant mass spectrum. However, nonperturbative contributions away from the resonances within the perturbative windows are also important. In the KS approach [72, 73] one absorbs factorizable long-distance charm rescattering effects (in which the B¯ → Xscc¯r transition can be factorized into the product of s¯b and cc¯ color-singlet currents) into the matrix element of the leading semileptonic operator O9. Following the inclusion of nonperturbative corrections scaling with 1/mc2, the KS approach avoids double-counting. For the integrated branching fractions one due to the 1/mb corrections. • The integrated branching fraction is dominated by this resonance background which exceeds the nonresonant charm-loop contribution by two orders of magnitude. This feature should not be misinterpreted as a striking failure of global parton-hadron duality [74], which postulates that the sum over the hadronic final states, including resonances, should be well approximated by a quark-level calculation [75]. Crucially, of a phase-space integral over the absolute square of a correlator. For such a quantity global quark-hadron duality is not expected to hold. Nevertheless, local quark-hadron duality (which, of course, also implies global duality) may be reestablished by resumming Coulomb-like interactions [74]. • Also electromagnetic perturbative corrections were calculated: NLL quantum electrodynamics (QED) two-loop corrections to the Wilson coefficients are of O(2%) [55]. In the QED one-loop corrections to matrix elements, large collinear logarithms of the form log(mb2/m`2) survive integration over phase space if only a restricted part of the dilepton mass spectrum is considered. These collinear logarithms add another contribution of order +2% in the low-q2 region of the dilepton mass spectrum in Based on all these scientific efforts of various groups, the latest theoretical predictions have been presented in ref. [66]. In the present manuscript, we make the effort to provide all missing relevant perturwell-known, the angular decomposition of this inclusive decay rate provides three independent observables, HT , HA, HL from which one can extract the short-distance electroweak Wilson coefficients that test for possible new physics [77]: (1 + z2)HT (q2) + 2(1 − z2)HL(q2) + 2zHA(q2) . di-lepton rest frame, HA is equivalent to the forward-backward asymmetry [78], and the q2 spectrum is given by HT + HL. The observables dominantly depend on the effective Wilson coefficients corresponding to the operators O7, O9, and O10. The paper is organized as follows. In section 2 we define the observables which we consider in the present analysis. In section 3 the derivation of the log-enhanced terms is presented. Master formulae for our observables are given in section 4, our phenomenological results in section 5. We briefly discuss the new physics sensitivity of our observables in section 6. Finally we explore the precise connection between experimental and theoretical quantities using Monte Carlo techniques in section 7. The latter analysis updates, and in parts supersedes, our previous statements in ref. [79]. We conclude in section 8. In the appendices we collect various functions that arise in the computation of QED and QCD corrections to the observables (appendix A), as well as formulas that parametrise the observables in terms of ratios of high-scale Wilson coefficients (appendix B). Definition of the observables The z dependence of the double differential decay distribution presented in eq. (1.4) is exact to all orders in QCD because it is controlled by the square of the leptonic current. The inclusion of QED bremsstrahlung modifies the simple second order polynomial structure and replaces it with a complicated analytical z dependence (see eqs. (3.28)–(3.33)). In particular this implies that, as long as QED effects are observably large, a simple fit to a quadratic polynomial will introduce non-negligible distorsions in the comparison between theory and experiment. In this section we explain the procedure that we adopt to construct various q2 differential distributions and suggest that experimental analyses follow the same prescriptions. The extraction of multiple differential distributions from eq. (1.4) is phenomenologically important because the various observables have different functional dependence on the Wilson coefficients. For instance, at next-to-leading order in QCD and without including any QED effect, the three HI defined in eq. (1.4) are given by [77]: HT (q2) = HL(q2) = HA(q2) = 2 G2F mb5 |Vt∗sVtb| (1 − sˆ)2h C9 + 2 C7 2 G2F mb5 |Vt∗sVtb| (−4sˆ) (1 − sˆ)2 Re C10 C9 + and of the forward-backward asymmetry dAFB/dq2: dq2 ≡ dq2 ≡ −1 dq2dz −1 dq2dz with the understanding that AFB does not coincide with the coefficient of the linear term We extract other single-differential distributions by projecting the double-differential rate onto various Legendre polynomials, Pn(z). These polynomials are orthogonal in the connection with the existing literature we choose the first two projections in such a way to reproduce HT and HL in the limit of no QED radiation. For the higher order terms we simply adopt the corresponding Legendre polynomials. The observables are defined as and the weights we use are: HI (q2) = −1 dq2dz WT = WL = WA = P0(z) − 3 W3 = P3(z) , W4 = P4(z) , The unnormalized (defined in eq. (2.5)) forward-backward asymmetry receives contributions form all odd powers of z in the Taylor expansion of the double differential rate and is given by In the literature the normalized differential and integrated forward-backward asymmetries are often considered (see for instance ref. [66]): dq2 HT (q2) + HL(q2) dq2 ≡ AFB[qm2, qM2 ] ≡ −1 −1 −1 −1 dAFB = 4 HT (q2) + HL(q2) , ℓ− p− ≡ p2 p+ ≡ x¯p1 ℓ− from the effective weak Hamiltonian. The arrows indicate momentum rather than fermion flow. x denotes the momentum fraction of the collinear photon. The new observables H3 and H4 (obtained by employing the weights W3 and W4) vanish exactly in the limit of no QED radiation but are still potentially important for phenomenology because of their non trivial dependence on the Wilson coefficients. find that projections with even higher Legendre polynomials are suppressed and will not be considered further. Note that the expected statistical experimental uncertainties (at a given luminosity) are well understood in the total width (HT + HL) and forward-backward asymmetry (3/4 HA) cases. On the other hand, HT , HL, H3 and H4 are obtained by projecting the double differential rate with weights that (especially for W3 and W4) are essentially arbitrary. As a consequence a simple rescaling of these weights implies a corresponding rescaling of the central values we find. In section 6 we show how to use the squared weights (WI2) to assess the expected Belle II reach for each of these observables. The experimental procedure that we recommend is to use the weights WI to extract single-differential distributions and to refrain from attempting polynomial fits to the data. Log-enhanced QED corrections to the double differential decay rate In this section we work out the formulas for the logarithmically enhanced electromagnetic coefficients of the effective weak Hamiltonian are the same as in [66, 76]. The kinematics can be inferred from figure 1. Let us first consider the case without photon radiation. The momenta of the quarks Moreover, we define sij ≡ i ∈ {1, 2, s, b} . y1 ≡ y2 ≡ decaying b-quark. From momentum conservation and by treating all final-state particles s1s = 1 − y2 , s2s = 1 − y1 , s1b = y1 , s2b = y2 . ssb = 1 − s12 , the b-quark and the positively charged lepton in the centre-of-mass system (c.m.s.) of the final-state lepton pair. Hence z = y2 − y1 1 − s12 where all primed momenta are taken in the c.m.s. of the final-state lepton pair. It turns out that z is simply given by [57] At this point we stress that the l.h.s. of this equation is evaluated in the lepton c.m.s., whereas its r.h.s. is evaluated in the rest-frame of the decaying b-quark. The connection We now switch on QED and consider the radiation of a collinear photon off a lepton leg as shown in figure 1. The momentum of the positively (negatively) charged lepton is lepton radiates the photon (left panel of figure 1), its momentum p+ after radiation is and hence we have p panel of figure 1), we obviously have p − = x¯p2 and p+ = p1. In analogy to eq. (3.2), ± is the zero-component of p±, again evaluated in the rest-frame of the decaying b-quark. We will also need the definition As already discussed in refs. [66, 76], the logarithmically enhanced contributions stem± ≡ s+− ≡ x¯ s12 . ming from collinear photon radiation are evaluated by invariant s12 = (p+ + p the triple invariant, where the formulae look exactly the same as in the case without QED, since we can lump the lepton and the collinear photon. We therefore arrive at P F = G2F mb|VtbVt∗s|2 , [1 + (1 − x)2] The squared matrix elements |A|2 for the different operators read f2(s12) = |A|727 (s12, y1, y2) = 8mb4 [(1 − y2) y1 + (1 − y1) y2] , |A|729 (s12, y1, y2) = 4mb4 (1 − s12) , 2 |A|710 (s12, y1, y2) = 4mb4 (y1 − y2) , 2 |A|910 (s12, y1, y2) = 2mb4 s12 (y1 − y2) , |A|229 (s12, y1, y2) = αee f2(s12) |A|929 (s12, y1, y2) , |A|227 (s12, y1, y2) = αee f2(s12) |A|729 (s12, y1, y2) , |A|222 (s12, y1, y2) = αee2 |f2(s12)|2 |A|929 (s12, y1, y2) , |A|210 (s12, y1, y2) = αee f2(s12) |A|910 (s12, y1, y2) . The function f2(s12) denotes the one-loop matrix element of P2 and is given by − 9 (2 + yc)p|1 − yc| 1 + √ 1 −  2 arctan √ 1 − yc 1 − yc yc − 1 when yc ≥ 1 , are complex. However, after taking into account the Wilson coefficients and adding the appropriate complex conjugate expression, the double differential rate turns out to be real, see eq. (3.28). and changing variables according to eq. (3.5) we arrive at 1 − s12 z, 1 − s12 z 1 − s12 θ(1 − z) θ(1 + z) θ(s12) θ(1 − s12) . The factor of two stems from the fact that both diagrams in figure 1 are relevant. Note once all expressions on its r.h.s. are plugged in. We now turn our attention to the more complicated case of the double invariant p~1 + x¯p~2 −→ 0. z = x¯y2 − y1 (y1 + x¯y2)2 − 4x¯s12 Again, the primed momenta are evaluated in the lepton c.m.s., whereas the r.h.s. of the equation is evaluated in the rest-frame of the b-quark. The differential decay width reads favour of z according to eq. (3.16). This transformation reads y2(±)(z) = x2(1 − z2) + 4x¯ It turns out that this in an injective mapping only for s12 < x¯. For s12 > x¯ we have to subdivide the y2-interval into two pieces, so that we get a total of three contributions. ds+− dz = P F ds+− dz = ±P F Z 1−√s+− ∂∂z y2(+)(z) h |A|2 (s12, 1 + s12 − y2, y2)i ×θ(1 − z) θ(1 + z) θ(s+−) θ(1 − s+−) , Z x− 1−√s+− ∂∂z y2(±)(z) h |A|2 (s12, 1+s12 −y2, y2)i y2 = y2(+)(z) s12 = s+−/x¯ y2 = y2(±)(z) s12 = s+−/x¯ Once the photon is radiated off `+, we apply very similar steps. As can be seen from is determined by x± = 1 − s+− 1 ∓ p(1 − z2)s+− z = y2 − x¯y1 (x¯y1 + y2)2 − 4x¯s12 favour of z according to eq. (3.23). This transformation reads y1(±)(z) = x2(1 − z2) + 4x¯ As mentioned before, this is an injective mapping only for s12 < x¯. For s12 > x¯ we have to subdivide the y1-interval into two pieces, so that in this case we also get a total of three ds+− dz = −P F ds+− dz = ∓P F Z 1−√s+− ∂∂z y1(−)(z) h |A|2 (s12, y1, 1 + s12 − y1)i ×θ(1 − z) θ(1 + z) θ(s+−) θ(1 − s+−) , Z x− 1−√s+− ∂∂z y1(∓)(z) h |A|2 (s12, y1, 1 + s12 − y1)i The total contribution in case of the double invariant is now obtained by ds+− dz ds+− dz ds+− dz into eq. (3.8). This leads us to the following expression for the logarithmically enhanced collinear decay width +Re [C9C1∗0] ξ9(e1m0)(s, z) + αee2 Re h(C2 + CF C1) C7eff ∗ ξ2(e7m)(s, z) where we assumed that the Wilson coefficients C1 and C2 are real, and we neglected 64 p1(s, z) √s ln q s 1−z2 − 1−z2 − 1 (z2 − 1)3 √s + z2 − 1 s(1−z2) − s(z2 − 1)3 (s (z2 − 1) + 1)3/2 64 z p2(s, z) ln z1−+z1 s(z2 − 1)3 16 p4(s, z) ln(s) s(z2 − 1)3 3s (z2 − 1)2 (s (z2 − 1) + 1) − 16(s − 1)2 p6(s, z) ln √2(1−s) 1−z2 y1 = y1(−)(z) s12 = s+−/x¯ y1 = y1(∓)(z) s12 = s+−/x¯ (z2 − 1)4 8 s3/2 p9(s, z) ln 1−z2 − (z2 − 1)4 1−z2 − 1 (z2 − 1)4 (s + z2 − 1)5/2 +4(s − 1)2 sz2 + s − z2 + 1 ln 2(1 − s) 1 − z2 32 p12(s, z) ln(s) (z2 − 1)3 64√s p14(s, z) ln 3(z2 − 1)3 (s + z2 − 1)2 1−z2 − 1−z2 − 1 (z2 − 1)3 (s + z2 − 1)3/2 64 z p11(s, z) ln z1−+z1 (z2 − 1)3 8 p13(s, z) − (z2 − 1)2 (s + z2 − 1) +32 (s − 1)2 ln 64 p15(s, z) sign(z) ln 32 s z p19(s, z) ln(s) (z2 − 1)4 4s (√s−1)2 z p21(s, z) (z2 − 1)3 (s+z2 −1)2 − −16(s − 1)2 s z ln 1 − (z2 − 1)2 ps (z2 − 1) + 1 (√s+1)√1−z2 32 z p17(s, z) ln 12 (√s + 1) √1 − z2 (z2 − 1)3 (s + z2 − 1)3/2 64 s z 9sz2 + 7s + 4z2 − 4 ln(s) (z2 − 1)3 (z2 − 1)3 2(1 − s) 1 − z2 64 p15(s, z) ln s(1−z2) − (z2 − 1)2 ps (z2 − 1) + 1 −√s(z2−1)−√z2√s(z2−1)+1+1 (√s+1)√1−z2 8 (√s − 1)2 z p18(s, z) (z2 − 1)2 (s + z2 − 1) − 32(s − 1)2 z ln 1 − 16 s z p20(s, z) ln 12 (√s + 1) √1 − z2 (z2 − 1)4 (√s+1)√1−z2 (z2 − 1)4 (s + z2 − 1)5/2 The pi(s, z) are polynomials in s and z and are given in appendix A. In case of negative or complex arguments, the logarithms and square-roots are defined as z = p|z| ei/2 arg(z) , ln(z) = ln|z| + i arg(z) , squared matrix elements (see eq. (3.12)) are complicated functions of s12. We therefore refrain from presenting their explicit expressions. They can easily be computed numerically by applying the steps outlined above. subsequently integrating over z. After proper normalisation one obtains the functions to introduce the variable z we performed the calculation entirely in terms of the rescaled energies yi. Moreover, there was more freedom in choosing the order of integrations since we These two simplifications led to significantly simpler variable substitutions and shorter expressions. With the ability to reproduce them by the more complicated calculation can therefore be regarded as a non-trivial cross-check. Master formulas for the observables We start again from the double differential decay width dz dq2 = (1 + z2)HT (q2) + 2zHA(q2) + 2(1 − z2)HL(q2) , di-lepton rest frame. This formula is modified once QED corrections are taken into account (see sections 2 and 3) due to the appearance of higher powers of z. As stated in section 2, lepton forward-backward asymmetry; the q2-spectrum is given by HT + HL, C = dq2 = dAFB = −1 −1 dz dq2 = HT (q2) + HL(q2) , dz dq2 sign(z) = HI (q2) = G2F mb5,pole |Vt∗sVtb|2 Φ`I`(sˆ), Moreover, we normalise the observables to the inclusive semi-leptonic corrections), and also use the ratio [55, 80] Consequently, our expression of the normalised angular observables HI reads ϕ(1) = 3 − 3 ϕ(2) = nh − + O(αes3, κ2, αsκ, αesΛ2/mb2, Λ3/mb3) , e 2mb2 − 2mb2 − 27 Φ`I`(sˆ) = X Re hCieff (μb) Cjeff∗(μb) HiIj(μb, sˆ)i , i≤j − 27 As explained in detail in [76], a consistent perturbative expansion in inclusive ¯ B → Xs`+`− in the presence of QED corrections is done in αes = αs(μb)/(4π) and also exist QED corrections at O(αsκ) which could be computed in principle. However, e infrared safe observable with respect to collinear photon radiation. We therefore neglect this contribution, but include it lateron in the quantity R(s0), where QED logs will be present in the normalisation. of heavy and light quark flavours, respectively, and β(5) = 23/3 is the one-loop QCD 0 terms represent the matrix element of the kinetic energy and magnetic moment operator, respectively, and are defined as λ2 = −hB|h¯iσμνGμνh|Bi/(12MB) ≈ 4 (MB2 ∗ − MB2 ) . of products of the low-scale Wilson coefficients and various functions arising from the matrix elements, and (16) in [76]. Their low-scale Wilson coefficients are also given explicitly (analytically HiIj =  N=7,9,10 N=7,9,10 |MiN |2 SNIN + Re(Mi7Mi9∗) S7I9 + ΔHiIi , for i = j , 2MiN MjN∗ SNIN + Mi7Mj9∗ + Mi9Mj7∗ For I = A the formula is simpler, HiAj =   N=7,9 MiN Mj10∗ + Mi10MjN∗ The coefficients MiA are listed in table 6 of [76]. The building blocks SNIM have the following SNM = σNM (sˆ) n1 + 8 αes ωN(1M),I (sˆ) + 16 αes2 ω(2) I I From (4.11) and (4.12) we see that the possible combinations of indices are N M NM (sˆ) read for i = j , to appendix A. NM,I (sˆ) can be extracted from [50] and have already NM,I (sˆ) have so far only been available for the q2-spectrum [84–87], but not for the double differential rate. Due to a recent calculation in QCD [88], they can be extracted for N M = 99, 1010 and I = T, L as well as for the authors of [88, 89] and we can therefore present them here for the first time. All NM,I (sˆ) are rather lengthy and we therefore relegate their explicit expressions corrections can be obtained from [57] (see also [56, 59]) and were previously computed in [77]. We confirm their expressions, (1 − sˆ)(5sˆ + 3) , (sˆ − 1)(3sˆ + 13) , 3sˆ2 + 2sˆ + 3 , 3sˆ2 + 2sˆ − 9 , χ2,99(sˆ) = sˆ 15sˆ2 − 14sˆ − 5 , T 2 −17sˆ2 + 10sˆ + 3 , χ2,710(sˆ) = −4 9sˆ2 − 10sˆ − 7 , A χ2,910(sˆ) = −2sˆ 15sˆ2 − 14sˆ − 9 . A Here the contributions biIj represent finite bremsstrahlung corrections that appear at NNLO. only include them for these two cases, but not for HT and HL separately. This is still an excellent approximation since the effect of finite bremsstrahlung corrections is very small anyway. The explicit formulas can be found in [48, 51] and will therefore not be repeated. differential rate can be inferred from that paper. One obtains c2Tj = −αesκ 98mλ2c2 (1 − sˆ)2(1 + 3sˆ) F (r) c1Tj = − 61 c2Tj , for j 6= 1, 2 , c2T2 = −αesκ 98mλ2c2 (1 − sˆ)2(1 + 3sˆ) F (r) c1T1 = +αsκ 247λm2c2 (1 − sˆ)2(1 + 3sˆ) F (r) e c1T2 = −αesκ 98mλ2c2 (1 − sˆ)2(1 + 3sˆ) F ∗(r) M27∗ + M29∗ , M17∗ + M19∗ , Mj9∗ , c2Lj = −αesκ 98mλ2c2 (1 − sˆ)2(3 − sˆ) F (r) Mj7∗ + c1Lj = − 61 c2Lj , for j 6= 1, 2 , Mj7∗ + Mj9∗ , for j 6= 1, 2 , − 6 M27∗ + M29∗ for j 6= 1, 2 , c2L2 = −αesκ 98mλ2c2 (1 − sˆ)2(3 − sˆ) F (r) M27∗ + c1L1 = +αsκ 247λm2c2 (1 − sˆ)2(3 − sˆ) F (r) M17∗ + e c1L2 = −αesκ 98mλ2c2 (1 − sˆ)2(3 − sˆ) F ∗(r) M29∗ , M19∗ , − 6 M27∗ + M29∗ we also include factorisable non-perturbative charm contributions which we implement by means of the Kru¨ger-Sehgal approach [72, 73]. We elaborated extensively on this approach and also the formulas by means of which these corrections are taken into account in ref. [66]. Given their length we do not repeat these formulas here but refer the inclined reader to refs. [66, 72, 73] for all necessary details. Finally, the coefficients eiIj collect the ln(mb2/m`2)-enhanced electromagnetic corrections which we calculated in section 3 for the double differential rate. Their contribution to the HI can be derived from (3.28) by applying the projections given in section 2. One finds eI11 = eI12 = eI1j = for j = 7, 9 , for I = T, L, while for I = A one gets e910 = 8 αsκ σ9A10(sˆ) ω9(e1m0,)A(sˆ) , A We consider the observables HI (or equivalently HI ) in the low-q2 region only, because their sensitivity to New Physics is highest in this region [77]. Besides, there are two more of the forward-backward asymmetry, which we extract numerically from HA by means of the formulas given above. Moreover, there is the branching ratio. In principle, it can be obtained by taking the sum of HT and HL. Its master formula has already been given in [76]. We therefore only highlight two small pieces which are available for the branching ratio only, but not for HT and HL individually. These are only the finite bremsstrahlung In the high-q2 region we consider two observables. The first one is the branching ratio, where we include the same terms as in the low-q2 region. As far as QED corrections are obtained from a numerical fit. To take into account our most recent input parameters (see table 1), we re-did the fits and collected the results in appendix A. In addition, the twoloop QCD matrix element functions F17,2(sˆ) and F19,2(sˆ), which were originally computed in [53], were given explicitly only in [54]. We implement these formulas in our numerical code. Moreover, non-perturbative 1/mb3 corrections become sizable in the high-sˆ region. They were originally computed in [60] and we implement the formulas of refs. [60, 61]. The second observable is the ratio R(s0) which we have already mentioned in the introduction. uncertainties that stem from poorly known parameters in the 1/mb2 and 1/mb3 powercorrections can be significantly reduced, as we will see in our numerical analysis in section 5. In terms of our perturbative quantities, it reads R(s0) = = 4 Vt∗sVtb 2 Rsˆ10 dsˆ Φ``(sˆ) We would rather and which was absent in [66]. Once the integration over sˆ is restricted to the high-q2 Φu(sˆ) αesκ = 8 αsκ (1 − sˆ)2 (1 + 2sˆ) ω9(e9m)(sˆ) e Let us conclude this section by a few remarks on the renormalisation schemes for the that are present in the definition of sˆ and in several loop functions suffer from renormalon ambiguities [90, 91]. We therefore convert them analytically to short-distance schemes (1S and MS, respectively) before any numerical evaluation of the observables is carried out. In the mass of the top quark is concerned we take the pole mass as input and convert it to 1Note that we use a different pre-factor here. q2 ∈ [1, 6] GeV2 q2 ∈ [1, 3.5] GeV2 the integrated observable and its QED correction normalized to the total low-q2 branching ratio From inspection of the left plot in figure 12 we see that, in the low-q2 region HT is much smaller than HL. We can understand the origin of this effect by looking at the ratio HT /HL at leading order: C120 + (C9 + 2C7)2 The suppression comes from the small 2sˆ . 1 factor and from the accidental strong cancellation between C9 and 2C7/sˆ at low sˆ (in fact, the combination C9 + 2C7/sˆ vanishes for C9 + 2C7/sˆ > C9 + 2C7 and the integrated HT and HL observables at low-q2 would assume very similar values. In table 4 we present the results we obtain by integrating the Monte Carlo generated b → s`` histograms. For each bin ([s1, s2]) and for each observable O (HT + HL, HT and HL) we show the total integrated observable (Rss12 O/ R16(HT + HL)), the total amongst the three observables (with the effect on HT being only slightly larger) and that the suppression of HT with respect to HL is responsible for very large relative effects in the 30–50% range. Finally we must point out that the numerical estimates presented in table 4 are affected by sizable uncertainties that are hard to quantify and that only the analytical results presented in table 2 should be utilized. The Monte Carlo study was nevertheless extremely valuable to build confidence in our study. search for new physics via quark flavour observables. It is theoretically clean, while the exclusive mode is affected by unknown power corrections. Thus, besides allowing for a nontrivial check of the recent LHCb data on the exclusive mode, it contains complementary information both in Standard Model predictions and in pinning down new physics. It is therefore a precious channel to be measured at Belle II, and might be accessible even In the present article we perform a complete angular analysis of the inclusive decay able to date. We confirm the findings of ref. [77] that a separation of the double differential decay width into three observables HT,A,L(q2), as well as subdivision of the low-q2 region into two bins (see also [66]), provides significantly more information than the branching ratio or forward-backward asymmetry in the entire low-q2 region alone. We compute logarithmically enhanced QED corrections to these observables and find double differential decay width in the absence of QED corrections. We therefore propose to project out HT,A,L(q2) using weight functions, and argue that the Legendre polynomials Pn(z) are the optimal choice for the latter. Besides reproducing HT (q2) and HL(q2) in the absence of QED radiation, they allow to construct observables H3,4(q2) (eq. (2.6)) that vanish if only QCD corrections are taken into account, and are therefore particular sensitive to QED effects. In view of the benefits of the Legendre weight functions we urgently recommend the experiments to use the weights (2.6) to extract single-differential distributions, and to refrain from attempting polynomial fits to the data. The absolute values of the QED effects that we compute are natural in size. However, due to the phase-space and Wilson coefficient suppression of HT (q2) the relative size of the QED corrections is large in this observable. We argue carefully that this does clearly not indicate a breakdown of perturbation theory. On the contrary, we can benefit from the fact that QED corrections lift the smallness of HT (q2) to a certain extent, which makes it an observable that is particular sensitive to QED radiation. To supplement our calculation we carry out a dedicated Monte Carlo study, whose main purpose is three-fold. First, we investigate how the electromagnetic logarithms are treated correctly in the presence of angular and energy cuts. We find that our analytical predictions can be directly applied, with the exception of the electron channel at BaBar, where our numbers have to be modified according to eqs. (7.1) and (7.2). Second, the size of the QED corrections, in particular their large relative size in HT (q2), are confirmed by the Monte Carlo (cf. tables 2 and 4). Last but not least, it consitutes also a validation of PHOTOS, which is used by experiments to estimate QED effects in the calculation of efficiencies. We update the Standard Model predictions for all angular observables integrated over two bins in the low-q2 region. The branching ratio and the observable R(s0) are also evaluated in the high-q2 region. Moreover, we provide our prediction for the zero crossing of the forward-backward asymmetry (or, equivalently, HA). The parametric and perturbative branching ratio, where the relative errors are much larger. In the former case the reason is the zero crossing of HA which entails a cancellation between the central values of the two bins in the low-q2 region. In the latter case we suffer from poorly known hadronic same cut in q2 [61]. parameters in the 1/mb2,3 power-corrections, a drawback that is circumvented in the ratio independent way. We give all observables in terms of ratios R7,8,9,10 of high-scale Wilson coefficients, which we assume to be altered by the new interactions. We also study cortightest constraints. On the other hand, if deviations from the Standard Model are seen, all observables become crucial to pin down the structure of new physics. ferent from unity, one might wonder whether this sign of lepton non-universality could be traced back to logarithmically enhanced QED corrections. LHCb uses the PHOTOS Monte Carlo to eliminate the impact of collinear photon emissions from the final state electrons. Therefore, the corrections calculated in this paper do not seem to apply to the ratio RK . Given that the agreement between PHOTOS and our analytical calculations is not perfect (see e.g. tables 2 and 4), it would be advisable to correct for photon radiation using data-driven methods that do not rely on PHOTOS. Acknowledgments We would like to thank Javier Virto for useful discussions, and Kevin Flood, Chris Schilling and Owen Long for logistic and technical support that allowed the Monte Carlo study presented in section 7. We are indebted to Mathias Brucherseifer, Fabrizio Caola, and Kirill studies [88, 89]. T. Huber acknowledges support from Deutsche Forschungsgemeinschaft within research unit FOR 1873 (QFET). T. Hurth thanks the CERN theory group for its hospitality during his regular visits to CERN. All authors are grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality at the Capri-Institute in May 2014, where part of this work was done. QED and QCD functions QED functions for the double differential rate QED corrections to the double differential rate in eq. (3.28). p3(s, z) = s 3 z2 − 1 + s2 z2 − 1 +s 6z6 + 37z4 − 36z2 − 7 + 5z4 + 24z2 + 3 , p5(s, z) = s4 13z8 − 56z6 + 210z4 − 112z2 − 55 −15z8 + 31z6 − 127z4 + 149z2 + 154 +3s2 5z8 − 9z6 + 55z4 − 31z2 − 84 +s −13z8 + 65z6 − 285z4 + 355z2 + 262 −13z6 + 37z4 − 299z2 − 109 , p6(s, z) = s z2 − 1 − z2 − 1 , p8(s, z) = s2 −z10 + 3z8 + 32z6 + 364z4 + 289z2 + 17 − 19z8 + 106z6 + 102z4 − 173z2 − 19 +2 −z8 + 7z6 − 9z4 + z2 + 2 , +s2 z2 − 1 +s z2 − 1 2 46z6 + 889z4 + 1030z2 + 87 3 256z4 + 483z2 + 51 + z2 − 1 −26z10 + 173z8 − 2504z6 − 2098z4 + 7690z2 + 989 −13z12 + 122z10 − 1190z8 + 830z6 + 8809z4 − 7288z2 − 1270 +s2 z2 − 1 −s z2 − 1 2 15z8 − 18z6 + 397z4 + 3716z2 + 706 3 15z6 + 19z4 − 403z2 − 143 + z2 − 1 4 13z4 − 22z2 + 1 , p11(s, z) = s2 5z2 + 3 + z2 − 1 , p12(s, z) = s2 z6 − 6z4 − 9z2 − 2 − s z2 − 1 + z4 − 1 , −s 2z6 + 17z4 − 24z2 + 5 + z2 − 1 +s z2 − 1 2 7z2 − 1 − z2 − 1 p15(s, z) = s z2 − 1 + z2 + 1 , +s z2 − 1 2 19z2 + 5 + z2 − 1 −2 s z2 − 5 z2 − 1 + z2 − 1 + z2 − 1 2 z4 − 2z2 + 5 , 2 z4 − 27z2 − 34 +s z2 − 1 + z2 − 1 3 3z4 − 10z2 − 33 , 2 42z6 + 717z4 + 742z2 + 63 + 5s z2 − 1 +s2 z2 − 1 + z2 − 1 3(sˆ− 1)2 3(sˆ− 1)2 ω7(19),T(sˆ) = − 34 log μb mb − ω9(19),T(sˆ) = (√sˆ+ 1)2(8sˆ3/2 (5sˆ+ 1)log(1 − sˆ) + sˆ(3sˆ+ 1)log(sˆ) + 6(sˆ− 1)2 − 15sˆ+ 4 sˆ− 5)Li2(1 − sˆ) 6(sˆ− 1)2√sˆ 36(sˆ− 1)2√sˆ 2(sˆ2 − 12sˆ− 5)Li2(1 − √sˆ) 3(sˆ− 1)2√sˆ 3(sˆ− 1)2 3(sˆ− 1)2 6(sˆ− 1)2sˆ A.2 Functions for the QCD corrections to the HI The one-loop QCD functions [50, 77] can be computed analytically, ω7(17),T(sˆ) = − 38 log mμbb − (√sˆ+ 1)2(sˆ3/2 − 10sˆ+ 13√sˆ− 8)Li2(1 − sˆ) 6(sˆ− 1)2 + 5sˆ3 − 54sˆ2 + 57sˆ− 8 18(sˆ− 1)2 3(sˆ− 1)2 36(sˆ− 1)2 sˆ(5sˆ+ 1)log(sˆ) + 3(sˆ− 1)2 − log(1 − sˆ) + 3(sˆ− 1)2 2√sˆ(sˆ+ 3)Li2(1 − √sˆ) π2(16sˆ+ 29√sˆ+ 19)(√sˆ− 1)2 36(sˆ− 1)2 sˆ2 + 18sˆ− 19 (2sˆ+ 1)log(1 − sˆ) + 2 log(1 − sˆ)log(sˆ), 3sˆ 3 √sˆ)log(sˆ) 3(sˆ− 1)2 3(sˆ− 1)2 3(sˆ− 1)2 3(sˆ − 1)2 3(sˆ − 1)2 (2sˆ3 − 11sˆ2 + 10sˆ − 1) log(1 − sˆ) 2sˆ(2sˆ − 5) log(1 − 3(sˆ − 1)2sˆ ( sˆ + 1)2(4sˆ3/2 − 7sˆ + 2 sˆ − 3)Li2(1 − sˆ) 3(sˆ − 1)2 √sˆ) log(sˆ) − 3 9sˆ2 −38sˆ+29 6(sˆ − 1)2 7sˆ2 − 2sˆ − 5 6(sˆ − 1)2 (sˆ − 7)sˆ log(sˆ) 3(sˆ − 1)2 ( sˆ + 1)2(sˆ3/2 4(sˆ2 − 6sˆ − 3)Li2(1 − 3(sˆ − 1)2√sˆ 3(sˆ − 1)2sˆ √sˆ) (sˆ3 − 3sˆ + 2) log(1 − sˆ) 2(sˆ2 − 3sˆ − 3) log(sˆ) π2(8sˆ3/2 + 13sˆ + 2 sˆ + 9)(√sˆ − 1)2 √ 18(sˆ − 1)2√sˆ 4√sˆ(sˆ + 3)Li2(1 − √sˆ) 3(sˆ − 1)2 18(sˆ−1)2 log(1 − sˆ) log(sˆ) , − 8sˆ + 3 sˆ − 4)Li2(1 − sˆ) 3(sˆ − 1)2 log(1 − sˆ) log(sˆ) , ( sˆ + 1)2(4sˆ−9 sˆ+3)Li2(1−sˆ) 3(sˆ−1)2 (2sˆ + 1) log(1 − sˆ) 4√sˆ(sˆ2 − 12sˆ − 5)Li2(1 − √sˆ) 3(sˆ − 1)2 3(sˆ − 1)2√sˆ 3(sˆ − 1)2 for five active flavours. sˆ + 8)(√sˆ − 1)2 18(sˆ − 1)2 4sˆ3 − 51sˆ2 + 42sˆ + 5 6(sˆ − 1)2 3(sˆ − 1)2 log(1 − sˆ) log(sˆ) . − log(1 − sˆ) The two-loop QCD functions [88, 89] are obtained from least-squares fits and are also valid for all q2. The necessary data was kindly provided by the authors of [88, 89]. ω9(19),T (sˆ) + 54.919(1 − sˆ)4 − 136.374(1 − sˆ)3 + 119.344(1 − sˆ)2 − 15.6175(1 − sˆ) − 31.1706 , ω9(11)0,A(sˆ) + 74.3717(1 − sˆ)4 − 183.885(1 − sˆ)3 + 158.739(1 − sˆ)2 − 29.0124(1 − sˆ) − 30.8056 , ω9(19),L(sˆ) − 5.95974(1 − s)3 + 11.7493(1 − s)2 + 12.2293(1 − s) − 38.6457 . Functions for the QED corrections to the HI The following functions are again obtained by least-squares fits. They are valid in the low-q2 region (1 GeV2 < q2 < 6 GeV2) only. −0.768521 − 80.8068 sˆ4 + 70.0821 sˆ3 − 21.2787 sˆ2 + 2.9335 sˆ − 0.0180809 sˆ 19.063 + 2158.03sˆ4 − 2062.92sˆ3 + 830.53sˆ2 − 186.12sˆ + 0.324236 sˆ 8(1 − sˆ)2 −6.03641 − 896.643sˆ4 + 807.349sˆ3 − 278.559sˆ2 + 47.6636sˆ − 0.190701 sˆ 21.5291 + 3044.94sˆ4 − 2563.05sˆ3 + 874.074sˆ2 − 175.874sˆ + 0.121398 sˆ 2.2596 + 157.984 sˆ4 − 141.281 sˆ3 + 52.8914 sˆ2 − 13.5377 sˆ + 0.0284049 sˆ 8(1 − sˆ)2 4(1 − sˆ)2 2sˆ(1 − sˆ)2 (1 − sˆ)2 4(1 − sˆ)2 8(1 − sˆ)2 8(1 − sˆ)2 4(1 − sˆ)2 −8.01684 − 1121.13sˆ4 + 882.711sˆ3 − 280.866sˆ2 + 54.1943sˆ − 0.128988 sˆ 4(1 − sˆ)2 + i −2.14058 − 588.771sˆ4 + 483.997sˆ3 − 124.579sˆ2 + 12.3282sˆ + 0.0145059 sˆ 4.54727+330.182sˆ4 −258.194sˆ3 +79.8713sˆ2 −19.6855sˆ+ 0.0371348 sˆ 2sˆ(1 − sˆ)2 2sˆ(1 − sˆ)2 −2.27221 − 298.369sˆ4 + 224.662sˆ3 − 65.1375sˆ2 + 11.5686sˆ − 0.0233098 sˆ (1 − sˆ)2 + i −0.666157 − 120.303sˆ4 + 109.315sˆ3 − 28.2734sˆ2 + 2.44527sˆ + 0.00279781 sˆ (1 − sˆ)2 1 − 5 sˆ 1 − sˆ 7 − 16 √sˆ + 9 sˆ 4 (1 − sˆ) ln(1 − √sˆ) − + ln(1 − √sˆ) + 5 − 16 sˆ + 11 sˆ 4 (1 − sˆ) (1 − 3 sˆ) ln sˆ (1 − sˆ) 2sˆ(1 − sˆ)2 9sˆ(1 − sˆ)2 (1 − sˆ)2 9(1 − sˆ)2 −1.71832 − 234.11sˆ4 + 162.126sˆ3 − 37.2361sˆ2 + 6.29949sˆ − 0.00810233 sˆ 8(224.662sˆ3 −2.27221−298.369sˆ4 −65.1375sˆ2 +11.5686sˆ− 0.0233098 ) sˆ 1 − sˆ − (1 − sˆ) 24sˆ(1 − sˆ)2 7.98625 + 238.507 (sˆ − a) − 766.869 (sˆ − a)2 24sˆ(1 − sˆ)2 with a = (4mc2/mb2)2. squares fit (for fixed values of mb and mc) read The respective high-q2 functions for the branching ratio that are obtained by a least8(1 − sˆ)2(1 + 2sˆ) 8(1 − sˆ)2(1 + 2sˆ) 96(1 − sˆ)2 9(1 − sˆ)2(1 + 2sˆ) New physics formulas + 0.204994 |R7|2 + 0.00230146 |R8|2 + 0.244813 |R9|2 + 1.74294 |R10|2 + 0.632156 × 10−7 , + 0.0631028 |R7|2 + 0.000727107 |R8|2 + 0.273706 |R9|2 + 1.96638 |R10|2 + 0.257773 × 10−7 , + 0.268097 |R7|2 + 0.00302857 |R8|2 + 0.518519 |R9|2 + 3.70932 |R10|2 + 0.889929 × 10−7 , + 0.206371 |R7|2 + 0.00230943 |R8|2 + 0.179467 |R9|2 + 1.28881 |R10|2 + 0.436438 × 10−7 , − 0.0027675 |R9|2 − 0.0192329 |R10|2 − 0.0115297 × 10−7 , + 0.0650616 |R7|2 + 0.000738436 |R8|2 + 0.239011 |R9|2 + 1.72527 |R10|2 + 0.123204 × 10−7 , + 0.271433 |R7|2 + 0.00304786 |R8|2 + 0.418478 |R9|2 + 3.01408 |R10|2 + 0.559642 × 10−7 , − 0.00174093 |R9|2 − 0.0120987 |R10|2 + 0.0121072 × 10−7 , − 0.00174093 |R9|2 − 0.0120987 |R10|2 + 0.0131242 × 10−7 , − 0.0027675 |R9|2 − 0.0192329 |R10|2 − 0.0113078 × 10−7 , − 0.00450843 |R9|2 − 0.0313316 |R10|2 + 0.00181642 × 10−7 , × 10−9 , (B.13) × 10−9 , (B.14) × 10−9 , (B.15) × 10−9 , (B.16) × 10−9 , × 10−9 , (B.18) + 0.482688 |R9|2 + 0.0892516 |R7|2 + 0.00051617 |R8|2 + 3.35446 |R10|2 + 1.6742 × 10−9 , + 0.662601 |R9|2 + 0.112159 |R7|2 + 0.00064865 |R8|2 + 4.60478 |R10|2 + 2.20357 × 10−9 , + 0.200654 |R9|2 + 0.0371021 |R7|2 + 0.000214573 |R8|2 + 1.39446 |R10|2 + 0.697498 × 10−9 , + 0.0747905 |R9|2 + 0.00952261 |R7|2 + 0.0000550722 |R8|2 + 0.51976 |R10|2 + 0.22061 × 10−9 , + 0.275445 |R9|2 + 0.0466247 |R7|2 + 0.000269645 |R8|2 + 1.91422 |R10|2 + 0.918108 × 10−9 , + 0.00589466 |R7|2 + 0.000128527 |R8|2 + 0.575967 |R9|2 + 4.20578 |R10|2 + 0.806915 × 10−7 , + 0.00631092 |R7|2 + 0.0000975709 |R8|2 + 0.439598 |R9|2 + 3.20293 |R10|2 + 0.701014 × 10−7 , + 0.0122056 |R7|2 + 0.000226098 |R8|2 + 1.01556 |R9|2 + 7.40871 |R10|2 + 1.50793 × 10−7 , + 0.00897313 |R7|2 + 0.000146331 |R8|2 + 0.609248 |R9|2 + 4.43707 |R10|2 + 0.914888 × 10−7 , + 0.00748476 |R7|2 + 0.00010436 |R8|2 + 0.466941 |R9|2 + 3.39295 |R10|2 + 0.791074 × 10−7 , + 0.0164579 |R7|2 + 0.000250691 |R8|2 + 1.07619 |R9|2 + 7.83003 |R10|2 + 1.70596 × 10−7 , + 0.210889 |R7|2 + 0.0028916 |R8|2 + 0.813297 |R9|2 + 5.94874 |R10|2 + 1.46402 × 10−7 , + 0.0694138 |R7|2 + 0.000881518 |R8|2 + 0.714084 |R9|2 + 5.16931 |R10|2 + 0.985134 × 10−7 , + 0.280302 |R7|2 + 0.00377311 |R8|2 + 1.52738 |R9|2 + 11.1181 |R10|2 + 2.44915 × 10−7 , + 0.215344 |R7|2 + 0.00291736 |R8|2 + 0.78123 |R9|2 + 5.7259 |R10|2 + 1.3762 × 10−7 , + 5.11822 |R10|2 + 0.940534 × 10−7 , + 0.287891 |R7|2 + 0.003817 |R8|2 + 1.48796 |R9|2 + 10.8441 |R10|2 + 2.31673 × 10−7 , + 0.00287361 |R7|2 + 0.0000373632 |R8|2 + 0.211548 |R9|2 + 1.50748 |R10|2 + 0.200589 × 10−7 , + 0.00370104 |R7|2 + 0.0000421485 |R8|2 + 0.234333 |R9|2 + 1.66583 |R10|2 + 0.292268 × 10−7 , + 0.00293717 |R7|2 + 0.0000444449 |R8|2 + 0.228597 |R9|2 + 1.6322 |R10|2 + 0.174573 × 10−3 , + 0.00393316 |R7|2 + 0.000050205 |R8|2 + 0.256024 |R9|2 + 1.82281 |R10|2 + 0.266662 × 10−3 . Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [3] https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsBPH. [4] http://www.slac.stanford.edu/BFROOT/. [5] http://belle.kek.jp/. [7] http://www-d0.fnal.gov/Run2Physics/WWW/results/b.htm. model, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355 [arXiv:1002.0900] [INSPIRE]. Nucl. Part. Sci. 60 (2010) 645 [arXiv:1005.1224] [INSPIRE]. Rev. Mod. Phys. 85 (2013) 795 [arXiv:1211.6453] [INSPIRE]. Phys. Rev. D 88 (2013) 074002 [arXiv:1307.5683] [INSPIRE]. [14] F. Beaujean, C. Bobeth and D. van Dyk, Comprehensive bayesian analysis of rare (semi)leptonic and radiative B decays, Eur. Phys. J. C 74 (2014) 2897 [arXiv:1310.2478] to QCD and back, Phys. Rev. D 89 (2014) 074014 [arXiv:1308.4379] [INSPIRE]. [arXiv:1312.5267] [INSPIRE]. anomaly, Phys. Rev. D 89 (2014) 015005 [arXiv:1308.1959] [INSPIRE]. in b → s`+`−?, arXiv:1406.0566 [INSPIRE]. Lett. 113 (2014) 151601 [arXiv:1406.6482] [INSPIRE]. shape of new physics in rare B decays, Phys. Rev. Lett. 113 (2014) 241802 [arXiv:1407.7044] [INSPIRE]. opportunities, Phys. Rev. D 90 (2014) 054014 [arXiv:1408.1627] [INSPIRE]. meson decays, JHEP 12 (2014) 131 [arXiv:1408.4097] [INSPIRE]. the LHCb and CMS within a unified framework, JHEP 02 (2015) 142 [arXiv:1409.0882] of LHCb measurements and future prospects, October 14–16, CERN, Switzerland (2014). non-universality, JHEP 12 (2014) 053 [arXiv:1410.4545] [INSPIRE]. Lett. 114 (2015) 091801 [arXiv:1411.0565] [INSPIRE]. arXiv:1411.3161 [INSPIRE]. 055 [arXiv:1411.4773] [INSPIRE]. (2014) 211802 [arXiv:1312.5364] [INSPIRE]. [hep-ex/0404006] [INSPIRE]. corrections, Nucl. Phys. B 393 (1993) 23 [Erratum ibid. B 439 (1995) 461] [INSPIRE]. in the NDR and HV schemes, Phys. Rev. D 52 (1995) 186 [hep-ph/9501281] [INSPIRE]. rare semileptonic B decays up to three loops, Nucl. Phys. B 673 (2003) 238 [hep-ph/0306079] [INSPIRE]. NNLO in QCD, Nucl. Phys. B 713 (2005) 291 [hep-ph/0411071] [INSPIRE]. [hep-ph/0109140] [INSPIRE]. Phys. Lett. A 19 (2004) 603 [hep-ph/0311187] [INSPIRE]. [hep-ph/0312128] [INSPIRE]. [hep-ph/9609449] [INSPIRE]. invariant mass, Nucl. Phys. B 525 (1998) 333 [hep-ph/9801456] [INSPIRE]. Phys. Lett. B 653 (2007) 404 [arXiv:0707.1694] [INSPIRE]. decays, Nucl. Phys. B 511 (1998) 594 [hep-ph/9705253] [INSPIRE]. spectrum, JHEP 07 (2000) 022 [hep-ph/0006068] [INSPIRE]. decays, Phys. Rev. D 64 (2001) 113004 [hep-ph/0107074] [INSPIRE]. [arXiv:0712.3009] [INSPIRE]. Phys. Rev. D 74 (2006) 014005 [hep-ph/0511334] [INSPIRE]. observables, Phys. Rev. D 79 (2009) 114021 [arXiv:0812.0001] [INSPIRE]. (1997) 2799 [hep-ph/9608361] [INSPIRE]. quark-hadron duality, Eur. Phys. J. C 61 (2009) 439 [arXiv:0902.4446] [INSPIRE]. (1976) 1958 [INSPIRE]. Nucl. Phys. B 740 (2006) 105 [hep-ph/0512066] [INSPIRE]. B¯ → Xs`+`−, arXiv:0807.1940 [INSPIRE]. 338 [hep-ph/0104034] [INSPIRE]. Phys. Rev. D 89 (2014) 014022 [arXiv:1307.4551] [INSPIRE]. mediated by W , Nucl. Phys. B 196 (1982) 83 [INSPIRE]. rate, Phys. Lett. B 454 (1999) 353 [hep-ph/9903226] [INSPIRE]. Phys. Rev. Lett. 88 (2002) 131801 [hep-ph/0112264] [INSPIRE]. two loop accuracy, Phys. Rev. Lett. 93 (2004) 062001 [hep-ph/0403221] [INSPIRE]. calculations for heavy-to-light decays, Phys. Rev. D 71 (2005) 054004 [hep-ph/0503039] inclusive decays, Phys. Lett. B 721 (2013) 107 [arXiv:1302.0444] [INSPIRE]. [92] A.H. Hoang, Bottom quark mass from Upsilon mesons: charm mass effects, running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun. neutrino anti-neutrino for arbitrary Higgs boson mass, Phys. Rev. D 57 (1998) 216 [95] CKMfitter Group collaboration, J. Charles et al., CP violation and the CKM matrix: assessing the impact of the asymmetric B factories, Eur. Phys. J. C 41 (2005) 1 [96] Heavy Flavor Averaging Group collaboration, Y. Amhis et al., Averages of B-hadron, [101] A. Ali, E. Lunghi, C. Greub and G. Hiller, Improved model independent analysis of semileptonic and radiative rare B decays, Phys. Rev. D 66 (2002) 034002 [104] C. Bobeth, M. Bona, A.J. Buras, T. Ewerth, M. Pierini et al., Upper bounds on rare K and B decays from minimal flavor violation, Nucl. Phys. B 726 (2005) 252 [hep-ph/0505110] [106] T. Hurth and F. Mahmoudi, The minimal flavour violation benchmark in view of the latest Rev. D 74 (2006) 034016 [hep-ph/0605076] [INSPIRE]. 55 (1997) 5549 [hep-ph/9610323] [INSPIRE]. 713 (2005) 522 [hep-ph/0409293] [INSPIRE]. Comput. Phys. Commun. 82 (1994) 74 [INSPIRE]. corrections in decays, Comput. Phys. Commun. 66 (1991) 115 [INSPIRE]. Version 2.0, Comput. Phys. Commun. 79 (1994) 291 [INSPIRE]. [1] LHCb collaboration , http://lhcb.web.cern.ch/lhcb/. [2] https://twiki.cern.ch/twiki/bin/view/AtlasPublic/BPhysPublicResults. [8] G. Isidori , Y. Nir and G. Perez , Flavor physics constraints for physics beyond the standard [9] T. Hurth and M. Nakao , Radiative and electroweak penguin decays of B mesons , Ann. Rev. [10] T. Hurth and F. Mahmoudi , Colloquium: new physics search with flavor in the LHC era, [11] LHCb collaboration, Measurement of form-factor-independent observables in the decay [12] S. Descotes-Genon , J. Matias and J. Virto , Understanding the B → K ∗ μ+μ− anomaly, [13] W. Altmannshofer and D.M. Straub , New physics in B → K∗μμ?, Eur. Phys. J. C 73 [15] C. Hambrock , G. Hiller , S. Schacht and R. Zwicky , B → K? form factors from flavor data [17] A.J. Buras , F. De Fazio and J. Girrbach , 331 models facing new b → sμ+μ− data , JHEP 02 [18] T. Hurth and F. Mahmoudi , On the LHCb anomaly in B → K∗`+`−, JHEP 04 ( 2014 ) 097 [19] F. Mahmoudi , S. Neshatpour and J. Virto , B → K∗ μ+μ− optimised observables in the MSSM , Eur. Phys. J. C 74 ( 2014 ) 2927 [arXiv:1401.2145] [INSPIRE]. [20] J. Lyon and R. Zwicky , Resonances gone topsy turvy - The charm of QCD or new physics [21] LHCb collaboration , Test of lepton universality using B+ → K+`+`− decays, Phys. Rev. [22] R. Alonso , B. Grinstein and J. Martin Camalich , SU( 2 ) × U(1) gauge invariance and the [23] G. Hiller and M. Schmaltz , RK and future b → s`` physics beyond the standard model [24] D. Ghosh , M. Nardecchia and S.A. Renner , Hint of lepton flavour non-universality in B [25] S. Biswas , D. Chowdhury , S. Han and S.J. Lee , Explaining the lepton non-universality at [26] D. Straub , New Physics models facing b → s data , talk given at the workshop Implications [27] T. Hurth , F. Mahmoudi and S. Neshatpour , Global fits to b → s`` data and signs for lepton [28] S.L. Glashow , D. Guadagnoli and K. Lane , Lepton flavor violation in B decays? , Phys. Rev. [29] W. Altmannshofer and D.M. Straub , New physics in b → s transitions after LHC run 1, [30] G. Hiller and M. Schmaltz , Diagnosing lepton-nonuniversality in b → s``, JHEP 02 ( 2015 ) [31] B. Bhattacharya , A. Datta , D. London and S. Shivashankara , Simultaneous explanation of the RK and R(D(∗)) puzzles, Phys. Lett . B 742 ( 2015 ) 370 [arXiv:1412.7164] [INSPIRE]. [32] M. Misiak , Perturbative contributions to rare B-meson decays, PoS(DIS2014)198 [INSPIRE]. [33] M. Misiak , Rare B-meson decays, arXiv:1112 .5978 [INSPIRE]. [91] A.H. Hoang , Z. Ligeti and A.V. Manohar , B decays in the Upsilon expansion , Phys. Rev . D [93] K.G. Chetyrkin , J.H. Kuhn and M. Steinhauser , RunDec: a Mathematica package for [97] C. Schwanda , Determination of | Vcb| from inclusive decays B → Xc`ν using a global fit , [98] C.W. Bauer , Z. Ligeti , M. Luke , A.V. Manohar and M. Trott , Global analysis of inclusive B decays , Phys. Rev . D 70 ( 2004 ) 094017 [hep-ph/0408002] [INSPIRE]. [99] Particle Data Group collaboration , K.A. Olive et al., Review of particle physics, Chin. [100] M. Beneke , T. Feldmann and D. Seidel , Systematic approach to exclusive B → V `+`−, V γ [102] Y.G. Kim , P. Ko and J.S. Lee , Possible new physics signals in b → sγ and b → s`+`−, Nucl. [105] T. Hurth , G. Isidori , J.F. Kamenik and F. Mescia , Constraints on new physics in MFV [107] S. Schilling , C. Greub , N. Salzmann and B. Toedtli , QCD corrections to the Wilson coefficients C(9) and C(10) in two-Higgs doublet models , Phys. Lett. B 616 (2005) 93 [108] Z.-j. Xiao and L.-x . Lu, B → Xs` +`− decay in a top quark two-Higgs-doublet model , Phys. [109] S. Bertolini , F. Borzumati , A. Masiero and G. Ridolfi , Effects of supergravity induced electroweak breaking on rare B decays and mixings, Nucl . Phys . B 353 ( 1991 ) 591 [INSPIRE]. [110] P.L. Cho , M. Misiak and D. Wyler , K(L) → π0e+e− and B → Xsl+l− decay in the MSSM, Phys. Rev. D 54 ( 1996 ) 3329 [hep-ph/9601360] [INSPIRE]. [111] T. Goto , Y. Okada , Y. Shimizu and M. Tanaka , b → s lepton anti-lepton in the minimal supergravity model , Phys. Rev. D 55 ( 1997 ) 4273 [Erratum ibid . D 66 ( 2002 ) 019901] [112] J.L. Hewett and J.D. Wells , Searching for supersymmetry in rare B decays , Phys. Rev . D [113] C.-S. Huang , W. Liao and Q.-S. Yan , The Promising process to distinguish supersymmetric models with large tan β from the standard model: B → Xsμ+μ− , Phys. Rev. D 59 ( 1999 ) [114] E. Lunghi , A. Masiero , I. Scimemi and L. Silvestrini , B → Xsl+l− decays in supersymmetry, Nucl. Phys . B 568 ( 2000 ) 120 [hep-ph/9906286] [INSPIRE]. [115] C. Bobeth , A.J. Buras and T. Ewerth , B¯ → Xs`+`− in the MSSM at NNLO, Nucl . Phys . B [116] G. Cowan , http://www.pp. rhul.ac .uk/∼cowan/stat/notes/weights.pdf [117] T. Sj ¨ostrand, High-energy physics event generation with PYTHIA 5.7 and JETSET 7 . 4 , [118] A. Ishikawa and M. Nakao , private communication. [119] S. Playfer , G. Eigen and K. Flood , private communication. [120] D.J. Lange , The EvtGen particle decay simulation package , Nucl. Instrum. Meth. A 462 [121] E. Barberio , B. van Eijk and Z. Was , PHOTOS: a universal Monte Carlo for QED radiative [122] E. Barberio and Z. Was , PHOTOS: a universal Monte Carlo for QED radiative corrections .


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Tobias Huber, Tobias Hurth, Enrico Lunghi. Inclusive \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \) : complete angular analysis and a thorough study of collinear photons, Journal of High Energy Physics, 2015, 176, DOI: 10.1007/JHEP06(2015)176