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February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial Chapter 1 5 1 Inclusive semileptonic B decays and |V | 0 cb 2 In memoriam Kolya Uraltsev b e F Paolo Gambino 3 Università di Torino, Dipartimento di Fisica, and INFN, Torino 1 Via Giuria 1, I-10125 Torino, Italy [email protected] ] h p ThemagnitudeoftheCKMmatrixelementsV canbeextractedfrominclusive cb - semileptonic B decays in a model independent way pioneered by Kolya Uraltsev p and collaborators. I review here the present status and latest developments in e h this field. [ 2 1. A semileptonic collaboration v 4 1 MycontinuinginvolvementwithsemileptonicB decaysismostlyduetoafortuitous 3 encounterwithKolyaUraltsevin2002. AgroupofexperimentalistsoftheDELPHI 0 Collaboration,amongwhomMarcoBattagliaandAchilleStocchi,hadembarkedin 0 ananalysisofsemileptonicmomentsandaskedKolyaandmetohelpthemout. He . 1 wastheexpert,Iwasanoviceinthefieldandthingshavestayedthatwayforalong 0 5 time thereafter. The joint paper that appeared later that year [1] contained one of 1 the first fits to semileptonic data to extract |V |, the masses of the heavy quarks cb : v and some non-perturbative parameters, and it used Kolya’s proposal to avoid any i 1/m expansion [2]. The next step for us was to compute the moments with a cut X c on the lepton energy [3], as measured by Cleo and Babar [4, 5]. Impressed as I was r a by Kolya’s deep physical insight and enthusiasm, I was glad that he asked me to continue our collaboration. Kolya thought that a global fit should be performed by experimentalists, but as theoretical issues kept arising he tirelessly discussed with them every single detail; the BaBar fit [6], where the kinetic scheme analysis of [1] was extended to the BaBar dataset, and the global fit of [7] owe very much to his determination. Kolya’s patience in explaining was unlimited and admirable: countless times I took advantage of it and learned from him. Our semileptonic collaboration later covered perturbative corrections [8], the extraction of |V | [9], and a reassessment ub of the zero-recoil heavy quark sum rule [10, 11]. It was during one of his visits to Turin that he suffered a first heart attack, but it did not take long before he was back to his usual dynamism. Working with Kolya was sometimes complicated, 1 February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial 2 Paolo Gambino but it was invariably rewarding. He was stubborn and we could passionately argue about a single point for hours. For him discussion, even heated discussion, was an essentialpartofdoingphysics. IwillforevermissKolya’spassionateloveofphysics and his total dedication to science. They were the marks of a noble soul, a kind and discreet friend. In the following I will review the present status of the inclusive B → X (cid:96)ν¯ c decays, the subject of most of my work with Kolya, who was a pioneer of the field. Semileptonic B decays allow for a precise determination of the magnitude of the CKM matrix elements V and V , which are in turn crucial ingredients in the cb ub analysisofCPviolationinthequarksectorandinthepredictionofflavour-changing neutral current transitions. In the case of inclusive decays, the Operator Product Expansion (OPE) allows us to describe the relevant non-perturbative physics in termsofafinitenumberofnon-perturbativeparametersthatcanbeextractedfrom experiment, while in the case of exclusive decays like B → D(∗)(cid:96)ν¯ or B → π(cid:96)ν¯ the form factors have to be computed by non-perturbative methods, e.g. on the lattice. Presently, the most precise determinations of |V | (the inclusive one [12] cb andtheonebasedonB →D∗(cid:96)ν atzerorecoilandalatticecalculationoftheform- factor [13]) show a ∼ 3σ discrepancy that does not seem to admit a new physics explanation, as I will explain later on. A similar discrepancy between the inclusive and exclusive determinations occurs in the case of |V | [14]. It is a pity that Kolya ub will not witness how things eventually settle. 2. The framework Our understanding of inclusive semileptonic B decays is based on a simple idea: since inclusive decays sum over all possible hadronic final states, the quark in the finalstatehadronizeswithunitprobabilityandthetransitionamplitudeissensitive only to the long-distance dynamics of the initial B meson. Thanks to the large hi- erarchybetweenthetypicalenergyrelease,ofO(m ),andthehadronicscaleΛ , b QCD and to asymptotic freedom, any residual sensitivity to non-perturbative effects is suppressed by powers of Λ /m . QCD b The OPE allows us to express the nonperturbative physics in terms of B meson matrixelementsoflocaloperatorsofdimensiond≥5, whiletheWilsoncoefficients can be expressed as a perturbative series in α [15–19]. The OPE disentangles the s physics associated with soft scales of order Λ (parameterized by the matrix QCD elements of the local operators) from that associated with hard scales ∼m , which b determine the Wilson coefficients. The total semileptonic width and the moments ofthekinematicdistributionsarethereforedoubleexpansionsinα andΛ /m , s QCD b with a leading term that is given by the free b quark decay. Quite importantly, the power corrections start at O(Λ2 /m2) and are comparatively suppressed. At QCD b higher orders in the OPE, terms suppressed by powers of m also appear, starting c with O(Λ3 /m3 × Λ2 /m2) [20]. For instance, the expansion for the total QCD b QCD c February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial Inclusive semileptonic B secays 3 semileptonic width is (cid:104) α (m ) (cid:16)α (cid:17)2 (cid:16)α (cid:17)2 Γ =Γ 1+a(1) s b +a(2,β0)β s +a(2) s sl 0 π 0 π π (cid:18) 1 α (cid:19) µ2 (cid:16) α (cid:17)µ2(m ) + − +p(1) s π + g(0)+g(1) s G b 2 π m2 π m2 b b ρ3 ρ3 (cid:21) +d(0) D −g(0) LS +higher orders , (1) m3 m3 b b where Γ = A |V2|G2 m5(1−8ρ+8ρ3−ρ4−12ρ2lnρ)/192π3 is the tree level 0 ew cb F b free quark decay width, ρ = m2/m2, and A (cid:39) 1.014 the leading electroweak c b ew correction. I have split the α2 coefficient into a BLM piece proportional to β = s 0 11 − 2/3n and a remainder. The expansions for the moments have the same f structure. The relevant parameters in the double series of Eq.(1) are the heavy quark masses m and m , the strong coupling α , and the B meson expectation values of b c s localoperatorsofdimension5and6,denotedbyµ2,µ2,ρ3 ,ρ3 . Asthereareonly π G D LS two dimension five operators, two matrix elements appear at O(1/m2): b 1 µ2(µ)= (cid:104)B|¯b (cid:126)π2b |B(cid:105) , (2) π 2M v v µ B 1 i µ2(µ)= (cid:104)B|¯b σ Gµνb |B(cid:105) (3) G 2M v2 µν v µ B where (cid:126)π = −iD(cid:126), Dµ is the covariant derivative, bv(x) = e−imbv·xb(x) is the b field deprived of its high-frequency modes, and Gµν the gluon field tensor. The matrix element of the kinetic operator, µ2, is naturally associated with the average kinetic π energy of the b quark in the B meson, while that of the chromomagnetic operator, µ2, is related to the B∗-B hyperfine mass splitting. They generally depend on G a cutoff µ = O(1GeV) chosen to separate soft and hard physics. The cutoff can be implemented in different ways. In the kinetic scheme [21, 22], a Wilson cutoff on the gluon momentum is employed in the b quark rest frame: all soft gluon contributions are attributed to the expectation values of the higher dimensional operators,whilehardgluonswithmomentum|(cid:126)k|>µcontributetotheperturbative correctionstotheWilsoncoefficients. MostcurrentapplicationsoftheOPEinvolve O(1/m3) effects [23] as well, parameterized in terms of two additional parameters, b generallydenotedbyρ3 andρ3 [22]. AlloftheOPEparametersdescribeuniversal D LS properties of the B meson or of the quarks and are useful in several applications. The interesting quantities to be measured are the total rate and some global shape parameters, such as the mean and variance of the lepton energy spectrum or of the hadronic invariant mass distribution. As most experiments can detect the leptons only above a certain threshold in energy, the lepton energy moments are defined as 1 (cid:90) dΓ (cid:104)En(cid:105)= En dE , (4) (cid:96) Γ (cid:96) dE (cid:96) E(cid:96)>Ecut E(cid:96)>Ecut (cid:96) February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial 4 Paolo Gambino where E is the lepton energy in B → X (cid:96)ν, Γ is the semileptonic width (cid:96) c E(cid:96)>Ecut above the energy threshold E and dΓ/dE is the differential semileptonic width cut (cid:96) as a function of E . The hadronic mass moments are (cid:96) 1 (cid:90) dΓ (cid:104)m2n(cid:105)= m2n dm2 . (5) X Γ X dm2 X E(cid:96)>Ecut E(cid:96)>Ecut X Here, dΓ/dm2 is the differential width as a function of the squared mass of the X hadronic system X. For both types of moments, n is the order of the moment. For n>1, the moments can also be defined relative to (cid:104)E (cid:105) and (cid:104)m2 (cid:105), respectively, in (cid:96) X which case they are called central moments: (cid:96) (E )=(cid:104)E (cid:105) , (cid:96) (E )=(cid:104)(E −(cid:104)E (cid:105))2,3(cid:105) ; (6) 1 cut (cid:96) E(cid:96)>Ecut 2,3 cut (cid:96) (cid:96) E(cid:96)>Ecut h (E )=(cid:104)M2(cid:105) , h (E )=(cid:104)(M2 −(cid:104)M2(cid:105))2,3(cid:105) . (7) 1 cut X E(cid:96)>Ecut 2,3 cut X X E(cid:96)>Ecut Since the physical information that can be extracted from the first three linear moments is highly correlated, it is more convenient to study the central moments (cid:96) and h , which correspond to the mean, variance, and asymmetry of the lepton i i energy and invariant mass distributions. The OPE cannot be expected to converge in regions of phase space where the momentum of the final hadronic state is O(Λ ) and where perturbation theory QCD has singularities. This is because what actually controls the expansion is not m b buttheenergyrelease,whichisO(Λ )inthosecases. TheOPEisthereforevalid QCD only for sufficiently inclusive measurements and in general cannot describe differential distributions. The lepton energy moments can be measured very precisely, while the hadronic mass central moments are directly sensitive to higher dimensional matrix elements such as µ2 and ρ3 . The leptonic and hadronic moments, π D which are independent of |V |, give us constraints on the quark masses and on cb the non-perturbative OPE matrix elements, which can then be used, together with additional information, in the total semileptonic width to extract |V |. cb 3. Higher order effects The reliability of the inclusive method depends on our ability to control the higher order contributions in the double series and to constrain quark-hadron duality violation, i.e. effects beyond the OPE, which we know to exist but expect to be rather suppressed in semileptonic decays. The calculation of higher order effects allows us to verify the convergence of the double series and to reduce and properly estimate the residual theoretical uncertainty. Duality violation, see [24] for a review, is related to the analytic continuation of the OPE to Minkowski space-time. It can be constrained a posteriori, considering how well the OPE predictions fit the experimental data. This in turn depends on precise measurements and precise OPE predictions. As the experimental accuracy reached at the B factories is already February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial Inclusive semileptonic B secays 5 better than the theoretical accuracy for most of the measured moments and will further improve at Belle-II, efforts to improve the latter are strongly motivated. The main ingredients for an accurate analysis of the experimental data on the moments and the subsequent extraction of |V | have been known for some time. cb Letusconsiderfirstthepurelyperturbativecontributions. TheO(α )perturbative s corrections to various kinematic distributions and to the rate have been computed longago. Inparticular,thecompleteO(α )andO(α2β )correctionstothecharged s s 0 leptonic spectrum have been first calculated in [25–27] and [28]. The so-called BLM corrections [29], of O(α2β ), are related to the running of the strong coupling s 0 inside the loops and are usually the dominant source of two-loop corrections in B decays. ThefirstO(α )calculationsofthehadronicspectraappearedin[30–32]and s were later completed in [8, 33, 34], while the O(α2β ) contributions were studied s 0 in [8, 31, 32, 34]. The triple differential distribution was first computed at O(α ) s in [8, 33]; its O(αnβn−1) corrections can be found in [8]. s 0 The complete two-loop perturbative corrections to the width and moments of the lepton energy and hadronic mass distributions have been computed in [35– 37] by both numerical and analytic methods. The kinetic scheme implementation for actual observables can be found in [38]. In general, using α (m ) in the one- s b loop result and adopting the on-shell scheme for the quark masses, the non-BLM correctionsamounttoabout−20%ofthetwo-loopBLMcorrectionsandgivesmall contributionstonormalizedmoments. Inthekineticschemewithcutoffµ=1GeV, the perturbative expansion of the total width is α (m ) (cid:16)α (cid:17)2 Γ[B¯ →X eν¯]∝1−0.96 s b −0.48β s c π 0 π (cid:16)α (cid:17)2 +0.82 s +O(α3)≈0.916 (8) π s HigherorderBLMcorrectionsofO(αnβn−1)tothewidtharealsoknown[8,39]and s 0 can be resummed in the kinetic scheme: the resummed BLM result is numerically very close to that of from NNLO calculations [39]. The residual perturbative error in the total width is about 1%. Inthenormalizedleptonicmomentstheperturbativecorrectionscanceltoalarge extent,independentlyofthemassscheme,becausehardgluonemissioniscompara- tivelysuppressed. Thispatternofcancelations, crucialforacorrectestimateofthe theoretical uncertainties, is confirmed by the complete O(α2) calculation, although s thenumericalprecisionoftheavailableresultsisnotsufficienttoimprovetheoverall accuracy for the higher central leptonic moments [38]. The non-BLM corrections turnouttobemoreimportantforthehadronicmoments. Eventhoughitimproves theoveralltheoreticaluncertaintyonlymoderately,thecompleteNNLOcalculation leadstothemeaningfulinclusionofprecisemassconstraintsinvariousperturbative schemes. Thecoefficientsofthenon-perturbativecorrectionsofO(Λn /mn)inthedou- QCD b ble series are Wilson coefficients of power-suppressed local operators and can be February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial 6 Paolo Gambino Fig. 1. One-loop diagrams contributing to the current correlator. The background gluon can be attachedwhereveracrossismarked. computed perturbatively. The calculation of the O(α Λ2 /m2) corrections has s QCD b been recently completed. The O(α ) corrections to the coefficient of µ2 have been s π computed numerically in [40] and analytically in [41]. They can be also obtained from the parton level O(α ) result using reparameterization invariance (RI) rela- s tions [16, 42, 43]. In fact, these RI relations have represented a useful check for the calculation of the remaining O(α Λ2 /m2) corrections, those proportional to µ2, s QCD b G which was completed in [44]. The calculation consists in matching the one-loop diagrams in Fig. 1, representing the correlator of two axial-vector currents computed in an expansion around the mass-shell of the b quark, onto local HQET operators. A recent independent calculation [45] of the semileptonic width at m = 0 c seems to be in agreement with the m → 0 limit of [44]. Refs. [41, 44] provide c analytic results for the O(α Λ2 /m2) corrections to the three relevant structure s QCD b functionsandhencetothetripledifferentialsemileptonicB decaywidth. Themost general moment have now been computed to this order and employed to improve the precision of the fits to |V | [12]. cb Numerically,usingfortheheavyquarkon-shellmassesthevaluesm =4.6GeV b and m =1.15 GeV, the total semileptonic width reads c (cid:20)(cid:16) α (cid:17)(cid:18) µ2 (cid:19) (cid:16) α (cid:17)µ2(m )(cid:21) Γ =Γ 1−1.78 s 1− π − 1.94+2.42 s G b , B→Xc(cid:96)ν 0 π 2m2 π m2 b b where Γ is the tree level width and we have omitted higher order terms of O(α2) 0 s and O(1/m3). The coefficient of µ2 is fixed by RI (or equivalently, by Lorentz b π invariance) at all orders. The parameter µ2 is renormalized at the scale m . It is G b advisable to evaluate the QCD coupling constant at a scale lower than m . If we b adopt α =0.25 the O(α ) correction increases the µ2 coefficient by about 7%. In s s G the kinetic scheme with cutoff µ=1GeV and for the same values of the masses the February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial Inclusive semileptonic B secays 7 0.30 0.25 0.20 0.15 0.10 0.05 2 4 6 8 10 Μ(cid:72)GeV(cid:76) Fig. 2. Relative NLO correction to the µ2 coefficients in the width (blue), first (red) and second G central(yellow)leptonicmomentsasafunctionoftherenormalizationscaleofµ2. G width becomes (cid:20) α (cid:18)1 α (cid:19) µ2 (cid:16) α (cid:17)µ2(m )(cid:21) Γ =Γ 1−0.96 s − −0.99 s π − 1.94+3.46 s G b , B→Xc(cid:96)ν 0 π 2 π m2 π m2 b b where the NLO corrections to the coefficients of µ2,µ2 are both close to 15% but π G have different signs. Overall, the O(α Λ2 /m2) contributions decrease the total s QCD b width by about 0.3%. However, NLO corrections also modify the coefficients of µ2,µ2 inthemomentswhicharefittedtoextractthenon-perturbativeparameters, π G and will ultimately shift the values of the OPE parameters to be employed in the width. Therefore, in order to quantify the eventual numerical impact of the new corrections on the semileptonic width and on |V |, a new global fit has to be per- cb formed. The size of the O(α µ2/m2) corrections depends on the renormalization s G b scale µ of the chromomagnetic operator. This is illustrated in Fig.2, where the size of the NLO correction relative to the tree level results is shown for the width and thefirsttwoleptoniccentralmomentsatdifferentvaluesofµ. TheNLOcorrections are quite small for µ ≈ 2GeV and, as expected, increase with µ. For µ>∼mb the runningofµ2 appearstodominatetheNLOcorrections. Inviewoftheimportance G of O(1/m3) corrections, if a theoretical precision of 1% in the decay rate is to be b reached, the O(α /m3) effects need to be calculated. s b As to the higher power corrections, the O(1/m4) and O(1/m5) effects were b Q computed in [46]. The main problem here is the proliferation of non-perturbative parameters: as many as nine new expectation values appear at O(1/m4) and more b at the next order. Because they cannot all be extracted from experiment, in [46] they have been estimated in the ground state saturation approximation, thus re- ducing them to products of the known O(1/m2,3) parameters, see also [47]. In b this approximation, the total O(1/m4,5) correction to the width is about +1.3%. Q The O(1/m5) effects are dominated by O(1/m3m2) intrinsic charm contributions, Q b c amountingto+0.7%[20]. Theneteffecton|V |alsodependsonthecorrectionsto cb February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial 8 Paolo Gambino the moments. Ref. [46] estimate that the overall effect on |V | is a 0.4% increase. cb While this sets the scale of higher order power corrections, it is as yet unclear how much the result depends on the assumptions made for the expectation values. A new preliminary global fit [48] performed using different ansatz for the new non- perturbative parameters seems to confirm that these corrections lead to a small shift in |V |. cb Two implementations of the OPE calculation have been employed in global analyses; they are based either on the kinetic scheme [3, 21, 22, 39] or on the 1S mass scheme for the b quark mass [49, 50]. They both include power corrections up to and including O(1/m3) and perturbative corrections of O(α2β ). Beside b s 0 differing in the perturbative scheme adopted, the global fits may include a different choiceofexperimentaldata,employspecificassumptions,orestimatethetheoretical uncertainties in different ways. Recently, the kinetic scheme implementation has beenupgradedtoincludefirstthecompleteO(α2)[38]andlaterthe(α Λ2/m2)[12] s s b contributions. 4. |V | and the fit to semileptonic moments cb The OPE parameters can be constrained by various moments of the lepton energy and hadron mass distributions of B → X (cid:96)ν that have been measured with good c accuracy at the B-factories, as well as at CLEO, DELPHI, CDF [6, 51–56]. The totalsemileptonicwidthcanthenbeemployedtoextract|V |. Thesituationisless cb favorable in the case of |V |, where the total rate is much more difficult to access ub experimentally because of the background from B → X (cid:96)ν, but the results of the c semileptonicfitsarecrucialalsointhatcase. Thisstrategyhasbeenrathersuccess- ful and has allowed for a ∼2% determination of V and for a ∼5% determination cb of V from inclusive decays [14, 82]. ub The first few moments of the charged lepton energy spectrum in B → X (cid:96)ν c decays are experimentally measured with high precision — better than 0.2% in the case of the first moment. At the B-factories a lower cut on the lepton energy, E ≥ (cid:96) E , is applied to suppress the background. Experiments measure the moments at cut differentvaluesofE ,whichprovidesadditionalinformationasthecutdependence cut isalsoafunctionoftheOPEparameters. Therelevantquantitiesaretherefore(cid:96) , 1,2,3 h , as well as the ratio R∗ between the rate with and without a cut 1,2,3 (cid:82)EmaxdE dΓ R∗(E )= Ecut (cid:96) dE(cid:96) . (9) cut (cid:82)EmaxdE dΓ 0 (cid:96) dE(cid:96) This quantity is needed to relate the actual measurement of the rate with a cut to thetotalrate,fromwhichoneconventionallyextracts|V |. Alloftheseobservables cb February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial Inclusive semileptonic B secays 9 43.5 4.80 4.75 43.0 4.70 (cid:76)V 42.5 kin(cid:72)m1Geb4.65 3(cid:200)(cid:200)10Vcb42.0 4.60 4.55 41.5 4.50 41.0 1.0 1.1 1.2 1.3 4.50 4.55 4.60 4.65 4.70 4.75 4.80 mcMS(cid:72)3GeV(cid:76) mbkin(cid:72)GeV(cid:76) Fig.3. Two-dimensionalprojectionsofthefitsperformedwithdifferentassumptionsforthetheoretical correlations. The orange, magenta, blue, light blue 1-sigma regions correspond to the four scenarios consideredin[58]. Theblackcontoursshowthesameregionswhenthemc constraintofRef.[59]is employed. canbeexpressedasdoubleexpansionsinα andinversepowersofm ,schematically s b α (µ) (cid:16)α (cid:17)2 (cid:18) α (µ) (cid:19) µ2 M =M(0)+ s M(1)+ s M(2)+ M(π,0)+ s M(π,1) π i i π i π i i π i m2 b (cid:18) α (µ) (cid:19) µ2 ρ3 ρ3 + M(G,0)+ s M(G,1) G +M(D) D +M(LS) LS +... (10) i π i m2 i m3 i m3 b b b whereallthecoefficientsM(j) dependonm ,m ,E ,andonvariousrenormaliza- i c b cut tion scales. The dots represent missing terms of O(α3), O(α2/m2), O(α /m3), and s s b s b O(1/m4), which are either unknown or not yet included in the latest analysis [12]. b It is worth stressing that according to the adopted definition the OPE parameters µ2, ... are matrix elements of local operators evaluated in the physical B meson, π i.e. without taking the infinite mass limit. The semileptonic moments are sensitive to a specific linear combination of m c and m , ≈ m −0.8m [57], see Fig. 3, which is close to the one needed for the b b c extraction of |V |, but they cannot resolve the individual masses with good accu- cb racy. Itisimportanttochecktheconsistencyoftheconstraintsonm andm from c b semileptonic moments with precise determinations of these quark masses, as a step in the effort to improve our theoretical description of inclusive semileptonic decays. Moreover, the inclusion of these constraints in the semileptonic fits improves the accuracy of the |V | and |V | determinations. The heavy quark masses and the ub cb non-perturbative parameters obtained from the fits are also relevant for a precise calculation of other inclusive decay rates such as that of B →X γ [58]. s In the past, the first two moments of the photon energy in B →X γ have gen- s February16,2015 1:52 WorldScientificReviewVolume-9.75inx6.5in memorial 10 Paolo Gambino erally been employed to improve the accuracy of the fit. Indeed, the first moment corresponds to a determination of m . However, in recent years rather precise de- b terminations of the heavy quark masses (e+e− sum rules, lattice QCD etc.) have becomeavailable,basedoncompletelydifferentmethods,seee.g.[59–68]and[69,70] for reviews. The charm mass determinations have a smaller absolute uncertainty and appear quite consistent with each other, providing a good external constraint forthesemileptonicfits. Radiativemomentsremaininterestingintheirownrespect, but they are not competitive with the charm mass determinations. Moreover, experiments place a lower cut on the photon energy, which introduces a sensitivity to theFermimotion oftheb-quarkinsidethe B mesonand tendstodisruptthe OPE. One can still resum the higher-order terms into a non-local distribution function and parameterize it assuming different functional forms [71–73], but the parame- terization will depend on m ,µ2 etc., namely the same parameters one wants to b π extract. Another serious problem is that only the leading operator contributing to inclusive radiative decays can be described by an OPE. Therefore, radiative moments are in principle subject to additional O(Λ /m ) effects, which have not QCD b yetbeenestimated[74]. Forallthesereasonsthemostrecentanalyses[12,58]have relied solely on charm and possibly bottom mass determinations. TheglobalfitsofRefs.[12,58]areperformedinthekineticschemewithacutoff µ = 1GeV and follow the implementation described in [3, 38]. The two fits only differ in the inclusion of O(α2/m2) corrections and in the consequent reduction s b of theoretical uncertainties. In order to use the high precision m determinations c withoutintroducingadditionaltheoreticaluncertaintyduetothemassschemecon- version, it is convenient to employ the MS scheme for the charm mass, denoted by m (µ ), and to choose a normalization scale µ well above m , e.g. 3 GeV. c c c c The experimental data for the moments are fitted to the theoretical expressions inordertoconstrainthenon-perturbativeparametersandtheheavyquarkmasses. 43measurementsareincluded,see[58]forthelist. Thechromomagneticexpectation value µ2 is also constrained by the hyperfine splitting G 2µ2 (cid:18)α µ2 1 (cid:19) M −M = G +O s G, . B∗ B 3m m m2 b b b Unfortunately, little is known of the power corrections to the above relation and onlyaloosebound[75]canbeset,see[11]forarecentdiscussion. Forwhatconcerns ρ3 , it is somewhat constrained by the heavy quark sum rules [75]. Refs. [12, 58] LS use the constraints µ2 =(0.35±0.07)GeV2, ρ3 =(−0.15±0.10)GeV3. (11) G LS It should be stressed that ρ3 plays a minor role in the fits because its coefficients LS are generally suppressed with respect to the other parameters. Itisinterestingtonotethatthefitwithouttheoreticaluncertaintiesisnotgood, with χ2/dof ∼ 2, corresponding to a very small p-value and driven by a strong tension (∼ 3.5σ) between the constraints in Eq. (11) and the measured moments.