Subleading-power corrections to the radiative leptonic B → γℓν decay in QCD

Journal of High Energy Physics, May 2018

Abstract Applying the method of light-cone sum rules with photon distribution amplitudes, we compute the subleading-power correction to the radiative leptonic B → γℓν decay from the twist-two hadronic photon contribution at next-to-leading order in QCD; and further evaluate the higher-twist “resolved photon” corrections at leading order in α s , up to twist-four accuracy. QCD factorization for the vacuum-to-photon correlation function with an interpolating current for the B-meson is established explicitly at leading power in Λ/m b employing the evanescent operator approach. Resummation of the parametrically large logarithms of m b 2 /Λ2 entering the hard function of the leading-twist factorization formula is achieved by solving the QCD evolution equation for the light-ray tensor operator at two loops. The leading-twist hadronic photon effect turns out to preserve the symmetry relation between the two B → γ form factors due to the helicity conservation, however, the higher-twist hadronic photon corrections can yield symmetry-breaking effect already at tree level in QCD. Using the conformal expansion of photon distribution amplitudes with the non-perturbative parameters estimated from QCD sum rules, the twist-two hadronic photon contribution can give rise to approximately 30% correction to the leading-power “direct photon” effect computed from the perturbative QCD factorization approach. In contrast, the subleading-power corrections from the higher-twist two-particle and three-particle photon distribution amplitudes are estimated to be of \( \mathcal{O}\left(3\sim 5\%\right) \) with the light-cone sum rule approach. We further predict the partial branching fractions of B → γℓν with a photon-energy cut E γ ≥ Ecut, which are of interest for determining the inverse moment of the leading-twist B-meson distribution amplitude thanks to the forthcoming high-luminosity Belle II experiment at KEK.

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Subleading-power corrections to the radiative leptonic B → γℓν decay in QCD

Received: March Subleading-power corrections to the radiative leptonic Yu-Ming Wang 0 1 2 3 5 Yue-Long Shen 0 1 2 4 0 Songling Road 238 , Qingdao, 266100 Shandong , P.R. China 1 Boltzmanngasse 5 , 1090 Vienna , Austria 2 Weijin Road 94 , 300071 Tianjin , China 3 Fakultat fur Physik, Universitat Wien 4 College of Information Science and Engineering, Ocean University of China 5 School of Physics, Nankai University Applying the method of light-cone sum rules with photon distribution amplitudes, we compute the subleading-power correction to the radiative leptonic B ! decay from the twist-two hadronic photon contribution at next-to-leading order in QCD; and further evaluate the higher-twist \resolved photon" corrections at leading order in s up to twist-four accuracy. QCD factorization for the vacuum-to-photon correlation function with an interpolating current for the B-meson is established explicitly at leading power b =mb employing the evanescent operator approach. Resummation of the parametrically large logarithms of m2= 2 entering the hard function of the leading-twist factorization formula is achieved by solving the QCD evolution equation for the light-ray tensor operator at two loops. The leading-twist hadronic photon e ect turns out to preserve the symmetry relation between the two B ! form factors due to the helicity conservation, however, the higher-twist hadronic photon corrections can yield symmetry-breaking e ect already at tree level in QCD. Using the conformal expansion of photon distribution amplitudes with the non-perturbative parameters estimated from QCD sum rules, the twist-two hadronic photon contribution can give rise to approximately 30% correction to the leading-power \direct photon" e ect computed from the perturbative QCD factorization approach. In contrast, the subleading-power corrections from the higher-twist two-particle and three-particle photon distribution amplitudes are estimated to be of O(3 approach. We further predict the partial branching fractions of B ! Heavy Quark Physics; Perturbative QCD - B energy cut E Ecut, which are of interest for determining the inverse moment of the leading-twist B-meson distribution amplitude thanks to the forthcoming high-luminosity Belle II experiment at KEK. Keywords: Heavy Quark Physics, Perturbative QCD 1 Introduction 2 3 4 5 6 A Spectral representations B Higher-twist photon DAs 1 Introduction Theoretical overview of B ! ` decay Leading-twist hadronic photon correction in QCD The twist-two hadronic photon correction at tree level The twist-two hadronic photon correction at one loop Higher-twist hadronic photon corrections in QCD 3.1 3.2 4.1 4.2 5.1 5.2 Numerical analysis Conclusion Higher-twist two-particle corrections Higher-twist three-particle corrections Theory inputs Predictions for the B ! ` form factors the LHC phenomenologically. In these respects, the radiative leptonic decay B ! with an energetic photon in the nal state is widely believed to provide a clean probe of the strong interaction dynamics of a heavy quark system and to put stringent constraints on the inverse moment of the leading-twist B-meson distribution amplitude (DA). Factorization properties of B ! ` have been investigated extensively at leading power in =mb with distinct QCD techniques [1, 2] and with the soft-collinear e ective theory (SCET) [3{5] which established the corresponding QCD factorization formula to all orders in perturbation theory. Subleading-power corrections to the B ! ` transition form factors were discussed in QCD factorization at tree level [6], where the symmetry-preserving form factor (E ) was introduced to parameterize the non-local SCET matrix element without integrating { 1 { out the hard-collinear scale. Systematic studies on the higher-power terms of the radiative leptonic B-meson decay amplitude in the heavy quark expansion are, however, still absent in the framework of SCET beyond the leading-order in s . Applying the dispersion relations and the parton-hadron duality, an alternative approach without identifying manifest structures of the subleading-power e ective operators was proposed [7] to estimate the power suppressed soft contributions at tree level and was further extended [8] to compute the soft-overlap contribution at next-to-leading-order (NLO) in QCD. Consequently, there will be a price to pay for the dispersion approach when taking into account the hadronic photon corrections and the end-point contributions (the so-called Feynman mechanism) by implementing the non-perturbative modi cations of the QCD spectral densities, as two additional non-perturbative parameters (vector meson mass m and e ective threshold parameter s0) are introduced when compared to the direct QCD calculation. It is then evident that evaluating the higher-power terms in the expansion of =mb individually with direct QCD approaches is of particular interest to deepen our understanding of perturbative QCD factorization for hard exclusive reactions. The major objective of this paper is to perform QCD calculations of the subleadingpower corrections induced by the hadronic component of the energetic photon at NLO in the strong coupling constant. QCD factorization formula for the two-particle hadronic ! photon correction to the B ` amplitude was demonstrated to be invalidated by the rapidity divergence in the convolution integral of the hard scattering kernel with the light-cone DAs of the B-meson and of the photon [5]. Employing the technique of lightcone sum rules (LCSR) with the two-particle photon DAs, the power suppressed \resolved photon" contribution was computed at twist-four accuracy and at leading-order (LO) in s [9{11], and was further updated [12] by including the NLO correction to the leadingtwist hadronic photon DA contribution and by calculating the higher-twist correction from the three-particle photon DAs at tree level. However, QCD factorization for the vacuum-tophoton correlation function with an interpolating current for the B-meson is not explicitly demonstrated with the operator-product-expansion (OPE) technique at one loop in [12], where the renormalization scheme dependence of 5 for the QCD amplitude in dimensional regularization was not addressed in any detail. It is therefore necessary to perform an independent calculation of the twist-two hadronic photon correction to the B ! ` form factors at NLO in s by compensating the above-mentioned gaps. To this end, we will apply the standard perturbative matching procedure including the evanescent SCET operators to establish QCD factorization formulae for the vacuum-to-B-meson correlation function with the Dirac matrix 5 de ned in naive dimensional regularization (NDR) (see [13, 14] for an overview, and [15] for a discussion in the context of the pion-photon transition form factor). The presentation is organized as follows. We rst summarize the theoretical status on QCD calculations of the B ! ` form factors with di erent techniques based upon the heavy quark expansion and discuss the origin of subleading-power corrections in section 2. To construct the sum rules for the leading-twist hadronic photon correction, we then establish QCD factorization for the correlation function de ned with an interpolating current for the B-meson and with the weak transition current [u (1 5) b] in section 3, where the master formula of the hard matching coe cient entering the factorization formula at one { 2 { loop will be derived with the implementation of the infrared (IR) subtraction including the evanescent SCET operator. With the aid of the evolution equation of the twist-two photon DA at two loops, summation of the parametrically large logarithms of m2= 2 in the hard function will be further preformed at next-to-leading-logarithmic (NLL) accuracy applying the momentum-space renormalization group (RG) approach. The NLL resummation improved LCSR for the twist-two hadronic correction to the B ! form factors will be also presented here, taking advantage of the dispersion relation technique and the partonhadron duality ansatz. The subleading-power corrections to the B ! ` decay amplitude from both the two-particle and three-particle higher-twist photon DAs displayed in [16] will b be computed with the LCSR approach at tree level in section 4, where a comparison of our results with that obtained in [11, 12] will be also presented. Phenomenological impacts of the various subleading-power corrections with the non-perturbative parameters of the photon DAs determined from QCD sum rules [17] will be explored in section 5, including the dependence of the partial branching fractions of B ! ` , with the phase-space cut of the photon energy, on the inverse moment B. A summary of our main observations and future perspectives will be presented in section 6. We further collect spectral representations of the convolution integrals entering the leading-twist factorization formulae for the vacuumto-photon correlation function at one-loop accuracy and the operator-level de nitions of the higher-twist photon DAs up to the twist-four in appendices A and B, respectively. 2 Theoretical overview of B ` decay ! The radiative leptonic B ! ` decay amplitude is de ned by the following matrix element (p) `(p`) (p ) ` (1 5) [u (1 5) b] B (pB) : A(B into account by the rede nition of the axial form factor FA(n p). At leading power in =mb the QCD factorization formula for the B ! form factors can be readily derived with the SCET technique [3, 4] FV; LP(n p) = FA; LP(n p) = n p Qu mB f~B( ) C?(n p; ) Z 1 0 d! B+(!; ) ! J?(n p; !; ) : (2.4) { 3 { The hard function C ? arises from matching the QCD weak current u the corresponding SCET current and the one-loop expression is given by [18, 19] 5) b onto C ? = 1 4 s CF 2 ln2 + 3r 1 2 r n p 2 12 ln r + ? entering the SCET factorization formula (2.4) reads [3, 4, 8] HJEP05(218)4 J ? = 1 + s CF 4 ln2 2 n p (! n p) 2 6 1 + O( s2) : Setting as a hard-collinear scale of order p of the parametrically large logarithms in the hard function yields mb and performing the NLL resummation FV; LP(n p) = FA; LP(n p) = Qu mB n p B( ) h U2(n p; h2; ) f~B( h2)i [U1(n p; h1; ) C?(n p; h1)] 1 + de ned in [6] and the manifest expressions of the evolution functions U1 and U2 can be The subleading-power corrections from photon radiation o the heavy quark and from higher-twist B-meson DAs were addressed [6] by computing the two diagrams for the W amplitude in QCD. Since the factorization property of the non-local subleading-power correction from photon radiation o the light quark has not been explored yet, we will only focus on the local subleading-power contribution to the B ! As discussed in [5] the subleading-power contribution can be further generated by the e ective matrix element h (p)jOjB (pB)i with the SCET operator O [qs hv]s [ ]c containing no photon eld, due to the unsuppressed interactions of photons with any numbers of collinear quark and gluon elds. The collinear matrix element h (p)j[ ]cj0i de nes the photon DAs on the light cone, making the photon behave in analogy to an energetic vector meson. Consequently, these terms are also referred to as the \hadronic (resolved) photon" contributions in di erent contexts. QCD calculations of such power suppressed corrections to the B ! DAs accuracy, with the LCSR approach in the following. ` decay form factors will be carried out, to the twist-four photon { 4 { (2.5) (2.6) 1 ; (2.7) To obtain the sum rules for the form factors FV (n p) and FA(n p), we construct the vacuum-to-photon correlation function with an interpolating current for the B-meson Z 5) b(x); b(0) 5 u(0)gj0i ; (3.1) where q = p` + p refers to the four-momentum of the lepton-neutrino pair. QCD factorization for the correlation function (3.1) can be demonstrated with the technique of the light-cone OPE at (p + q)2 following power counting scheme mb2 and q 2 b m2. For de niteness, we will employ the The twist-two hadronic photon correction at tree level QCD factorization for the twist-two contribution to the correlation function (3.1) can be justi ed by investigating the QCD amplitude d4x ei q x hq(z p) q(z p)jT fu(x) ? (1 5) b(x); b(0) 5 u(0)gj0i ; (3.3) where z indicates the momentum fraction carried by the collinear quark and z 1 Evaluating the tree diagram displayed in gure 1 leads to F (0)(p; q) = = i i 2 z(p + q)2 + z q2 n q n q where the convolution integral of z0 is represented by an asterisk. hOA; (z; z0)i(0) indicates the partonic matrix element of the SCET operator OA; at tree level hOA; (z; z0)i = hq(z p) q(z p)jOA; (z0)j0i = (z p) ? 6 n (1 + 5) (z p) (z z0) + O( s); where the general de nition of the collinear operator in moment space reads Z 0 { 5 { Oj; (z0) = n p Z 2 j = ? 6 n (1 + 5) : d e i z0 n p ( n) Wc( n; 0) j (0) ; The collinear Wilson line with the convention of the covariant derivative D Wc( n; 0) = P Exp i gs d n Ac( n) : (3.4) (3.5) (3.6) (3.7) To establish the hard-collinear factorization for the QCD amplitude (3.3), we further decompose the SCET operator OA; into the light-ray operators de ning the photon DAs displayed in [16] with OA; = O1; + O2; + OE; ; 1 = ? 6 n ; 2 = n 2 ; E = Expanding the operator matching equation including the evanescent operator we can readily derive the tree-level factorization formula for the correlation function (3.1) p + q i p u u¯ (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) at the LO in the strong coupling constant, gives rise to X i (p; q) = i (z ; ) Employing the de nition of the B-meson decay constant in QCD hB (pB)jb 5 uj0i = i fB m2B ; we can derive the hadronic dispersion relation of (3.1) as follows where s0 is the e ective threshold of the B-meson channel. The tree-level LCSR for the B ! ` form factors can be obtained by matching the factorization formula (3.13) and (3.15) with the aid of the parton-hadron duality approximation and the Borel transformation HJEP05(218)4 fB mB q2). With the power counting scheme for the threshold parameter and the Borel mass entering the sum rules (3.16) s0 mb2 M 2 O(mb ) ; z0 =mb ; the heavy-quark scaling of the hadronic photon correction at leading twist can be established i (3.15) (z; ) (3.16) (3.17) (3.18) (3.19) FV2;PpLhToton FA2P;pLhToton O mb which is indeed suppressed by a factor of =mb compared with the direct photon contribution (see [5] for more details) * (p) qs A6?( ) 1 i n D s 6 n 2 (1 5) hv B (pB) mb 1=2 : 3.2 The twist-two hadronic photon correction at one loop In this subsection we will proceed to derive the NLO sum rules for the twist-two hadronic photon correction to the B ! form factors and to perform resummation of the large b logarithms of m2= 2 in the hard function at NLL accuracy. To this end, we will need to demonstrate QCD factorization for the vacuum-to-photon correlation function (3.1) at one loop, applying the technique of the light-cone OPE. For the sake of determining the NLO matching coe cients entering the factorization formulae of (p; q), we will rst evaluate the one-loop diagrams for the QCD matrix element F (p; q) displayed in gure 2. The one-loop QCD correction to the weak vertex diagram shown in gure 2(a) can be readily computed as F (1;w)eak = gs2CF z(p+q)2 +zq2 u(zp) (z 6p+ 6 l) dDl 1 (2 )D [(zp+l)2 +i0][(zp+q +l)2 5)(z 6p+ 6q+ 6 l+mb) (z 6p+ 6 l+mb) 5v(zp) ; (3.20) { 7 { + 3=2 ; O (a) (b) (c) (d) where the external partons are already taken to be on the mass-shell due to the insensitivity of the hard matching coe cients on the IR physics. With the power counting scheme speci ed in (3.2), one can identify the leading-power contributions of the scalar integral HJEP05(218)4 I1 = Z dDl 1 (2 )D [(z p + l)2 + i0][(z p + q + l)2 mb2 + i0][l2 + i0] ; (3.21) from the hard and collinear regions as expected. Applying the method of regions [20], the collinear contribution of I1 vanishes in dimensional regularization due to the resulting scaleless integral. The collinear contribution of I1 may not vanish with a di erent regularization scheme, however, it will be always cancelled by the corresponding IR subtraction term. Reducing the Dirac algebra of F (1;w)eak with the NDR scheme of the Dirac matrix 5 and preforming the loop-momentum integration leads to F (1;w);ehak = + + where r1 = (z p + q)2=mb2 and r2 = q2=mb2. displayed in gure 2(b) can be written as F (1;B) = dDl (z 6p+ 6q+ 6 l+mb) 5 (z 6p 6 l) 1 (2 )D [(z p l)2 +i0][(z p+q +l)2 which again depends on the precise prescription of 5 in the complex D-dimensional space. It is straightforward to verify that the leading-power contributions to the B-meson vertex diagram also arise from the hard and collinear regions. Evaluating the hard contribution { 8 { to F (1;B) with the method of regions in the NDR scheme of 5 yields F (1;B);h = 1 r1 1 r3 (1 r3) Li2 1 1 r1 1 r3 1 1 +ln 2 m2 b +(3 r1 r3 1) 1 r3 1 3 r1 +3 F (0) ; with r3 = (p + q)2=mb2. gure 2(c) can be computed as gure 2(d) F (1;b)ox = dDl The self-energy correction to the intermediate bottom-quark propagator displayed in F (1;w)fc = Furthermore, the wave function renormalization of the external quarks will be cancelled precisely by the corresponding collinear subtraction term and hence will not contribute to the perturbative matching coe cients. Now we turn to compute the one-loop correction to the box diagram displayed in (2 )D [(z p+l)2 +i0][(z p+q +l)2 mb2 +i0][(u p l)2 +i0][l2 +i0] dDl (2 )D [(z p+l)2 +i0][(z p+q +l)2 mb2 +i0][(u p l)2 +i0][l2 +i0] (D 4) D D 4 l2 + 2 ? n l n q n l n (u p+q)+l2 ; (3.26) where the reduction of the Dirac algebra is achieved with the NDR scheme of 5 in the second step and l2 ? g? l l . Performing the loop-momentum integration we nd that the one-loop box diagram only contributes at O( ), vanishing in four dimensional space. Such observation is in analogy to the hard-collinear factorization for the hadronic photon correction to the pion-photon form factor at leading-twist accuracy [15]. Adding up di erent pieces together, we obtain the one-loop QCD correction to the four-point QCD matrix element as follows F ( 1 )(p; q) = T A(1;)hard(z0; (p + q)2; q2) hOA; (z; z0)i(0) + : : : = X i=1;2;E Ti(;1h)ard(z0; (p + q)2; q2) hOi; (z; z0)i(0) + : : : ; (3.27) { 9 { where the explicit expression of the NLO hard amplitude is given by Ti(;1h)ard NDR = 1 r1 + 1 r3 3 1 r1 1 r1 1 r3 1 +ln 2 m2 b + + + + + 1 r1 + 4 r2 +2 ln(1 r2)+ 3 1 r1 15 2 Ci(;0h)ard ; ln2(1 r1) 6 r3 where the parameter z in the de nition of r1 should be apparently understood as z0. We are now in a position to derive the master formulae for the hard functions C1;2(z0; (p + q)2; q2) by implementing the ultraviolet (UV) renormalization and the IR subtraction. Expanding the operator matching condition (3.10) at O( s) gives rise to X Ti( 1 )(z0; (p+q)2; q2) hOi; (z; z0)i(0) = X hCi( 1 )(z0; (p+q)2; q2) hOi; (z; z0)i(0) +Ci(0)(z0; (p+q)2; q2) hOi; (z; z0)i( 1 )i : (3.29) The UV renormalized one-loop SCET matrix elements hOi; i ( 1 ) can be further written hOi; i ( 1 ) = X hMi(j1;)b;aRre + Zi(j1)i hOj; i (0) ; j i i as [21] where Mi(j1;)b;aRre are the bare matrix elements dependent on the IR regularization scheme and Zi(j1) are the UV renormalization constants at one loop. When both UV and IR divergences are coped with dimensional regularization, the bare SCET matrix elements vanish due to the resulting scaleless integrals from the corresponding one-loop diagrams. Comparing the coe cients of hOi; i (0) (i = 1; 2) on both sides of (3.29) with the aid of (3.30) yields The SCET operators O1; and O2; do not mix into each other, which can be veri ed explicitly by computing the one-loop correction to the two SCET matrix elements hOi; i ( 1 ) = Zi(i1) hOi; i (0) ; with i = 1; 2 : The collinear subtraction term Zi(i1) hOi; i pseudoscalar current b 5 u will remove the divergent terms of the NLO QCD amplitude Ti( 1 ) (0) and the UV renormalization of the QCD (a) (b) (c) to guarantee that the perturbative matching coe cients entering the factorization formulae of the correlation function (3.1) are free of singularities and are entirely from the hardscale dynamics of . We further turn to determine the IR subtraction term ZE(1i) (i = 1; 2) originated from the renormalization mixing of the evanescenet operators OE; into the physical SCET operators O1; and O2; . As discussed in [21{23], the renormalization constants ZE(1i) (i = 1; 2) will be determined by implementing the prescription that the IR nite matrix element of the evanescent operator OE; vanishes, when applying dimensional regularization only to the UV divergences and regularizing the IR singularities with any other scheme di erent from the dimensions of spacetime. In accordance with (3.30) this amounts to HJEP05(218)4 ZE(1i) = M E(1i);;boare : Inserting (3.33) into (3.31) leads to the following master formula where Ti(;1h)a;rrdeg is the regularized terms of the NLO hard contribution to the QCD matrix element F as presented in (3.28) and i = 1; 2. hOE; i We proceed to compute the one-loop matrix element of the evanescent SCET operator ( 1 ) by evaluating the e ective diagrams shown in gure 3. Employing the SCET Feynman rules, we nd that only the diagram (a) with a collinear-gluon exchange between two collinear quarks could give rise to a non-trivial contribution to M E(1i);;boare. Evaluating this one-loop SCET diagram explicitly yields hOE; (z; z0)i( 1 ) i gs2 CF Z dDl 1 (2 )D [(z p + l)2 + i0][(l z p)2 + i0][l2 + i0] which only generates a non-vanishing contribution proportional to the SCET matrix element hOE; i (0) at O( ) with the NDR scheme of 5. Explicitly, we obtain C(0) E M E(1i);;boare = 0 ; with i = 1; 2 ; from which the one-loop hard matching coe cients can be written as C( 1 ) = Ti(;1h)a;rrdeg : i (3.33) (3.34) (3.35) (3.36) (3.37) Now we are ready to demonstrate the factorization-scale independence of the factorization formula for the vacuum-to-photon correlation function (3.1) To this end, we need to make use of the evolution equation for the leading-twist photon DA The explicit expression of the one-loop evolution kernel Ve0 is given by [26, 27] Ve0(z; z0) = 2 CF (z z0) + (z0 z) CF (z z0) ; (3.42) z z0 z0 1 z + where the plus function is de ned as It is then straightforward to write down f (z; z0) + = f (z; z0) (z z0) dt f (t; z0) : d mb( ) d ln z z0 z = 1 z0 0 X n=0 with the renormalization kernel Ve (z; z0) expanded perturbatively in QCD and the RG equation for the bottom-quark mass [24, 25] Ve (z; z0) = 4 s n+1 Ven(z; z0) ; X n=0 s( ) n+1 4 m(n) ; m(0) = 6 CF : 0 i (3.38) (3.40) (3.41) (3.43) (3.44) d dependence at one loop arises from the UV renormalization of the pseudoscalar QCD current de ning the correlation function (3.1). Distinguishing the renormalization scale , due to the non-conservation of the pseudoscalar current in QCD, from the factorization scale governing the RG evolution in SCET, we are led to conclude that the factorization formula (3.38) of (p; q) is indeed independent of the scale at one-loop accuracy. According to the QCD factorization formula (3.38) for the correlation function (p; q), we cannot avoid the parametrically large logarithms of O(ln (mb2= 2)) by adopting a universal scale in the hard matching coe cient and in the photon DA. We will perform resummmation of the above-mentioned large logarithms at NLL accuracy by applying the two-loop RG equation of the twist-two photon DA and by setting the factorization scale as mb. The NLO evolution kernel Ve1 in QCD can be decomposed as follows [28{30] Ve1(z; z0) = 2 Nf CF VeN (z; z0) + CF CA VeG(z; z0) + CF2 VeF (z; z0) ; (3.45) where the explicit expressions of the evolution functions can be found in [29]. Symmetry properties of the RG evolution equation (3.39) imply the series expansion of the leadingtwist photon DA in terms of the Gegenbauer polynomials 1 n=0 (z; ) = 6 z z X an( ) Cn3=2(2z (0)(s; q2) = s b s q mb2) : The two-loop evolution of the Gegenbauer moment an( ) can then be obtained as follows ( ) hqqi( ) an( ) = ETN;LnO( ; 0) ( 0) hqqi( 0) an( 0) + s( ) X where k; n = 0; 2; 4; : : : and the explicit expressions of the RG functions ET(N;n)LO and the o -diagonal mixing coe cients can be found in appendix A of [15]. In contrast to the LO evolution in QCD, the Gegenbauer coe cients an( ) do not renormalize multiplicatively at NLO accuracy. Inserting (3.46) and (3.47) into the NLO factorization formula (3.38) gives rise to the NLL resummation improved expression (p; q) = gem Qu n p ( ) hqqi( ) where the perturbative function Kn((p + q)2; q2) is determined by Kn = dz hCi(0)(z; (p + q)2; q2) + Ci( 1 )(z; (p + q)2; q2)i h6 z z Cn3=2(2 z i 1) : To construct the sum rules for the twist-two hadronic photon correction to the B ! ` form factors, we need to derive the dispersion representation for the NLL factorization formula (3.48). Applying the spectral representations of the convolution integrals collected in appendix A, we can readily obtain i 2 gem Qu n p n q ( ) hqqi( ) (p) hg? i n v i (p + q)2 i0 where the LO spectral function (0)(s; q2) is given by (3.48) (3.49) (3.50) (3.51) The resulting NLO spectral function ( 1 )(s; q2) is rather involved and can be written as ( 1 )(s; q2) = ( 2) ln 2 m2 b 1 0 dz ln + ln + + (z r3 +z r2 1) 0 (z; )+ z z (z; ) (r3 1) z r2 1 z r3 +z r2 1 4 z 1 1 z r2 P z r3 +z r2 1 ; ; +2 ln(r3 1) 0 (z = 1; ) + r2 d 1 r2 dz r2(r2 2)(1+z)+2 z r2 (1 r2) zz 2 (1 3 z r2) 1 r3 r2 z r2 (1 z r2) r3 0 dz (r3 1) ; (z; ) z (z; ) 1 r2 r3 (3.52) HJEP05(218)4 where P indicates the principle-value prescription. Finally, the NLL sum rules for the hadronic photon correction to the B ! form factors at leading twist can be written as fB mB It is evident that the twist-two hadronic photon correction preserves the symmetry relation of the two form factors FV and FA at leading power in =mb. Higher-twist hadronic photon corrections in QCD In this section we will aim at computing the higher-twist hadronic photon corrections to s, up to the twist-four accuracy, from the LCSR approach. Following the discussion on a general classi cation of the photon DAs [16], we will need to take into account the subleading-power contributions arising from the light-cone matrix elements of both the two-body and three-body collinear operators. To achieve this goal, we rst demonstrate QCD factorization for the two-particle and three-particle highertwist contributions to the vacuum-to-photon correlation function (3.1) and then construct the tree-level sum rules for the form factors FV and FA following the standard strategy. 4.1 Higher-twist two-particle corrections Employing the light-cone expansion of the bottom-quark propagator and keeping the subleading-power contributions to the correlation function (3.1) leads to Z d4k Z (2 )4 i Z d4k Z (2 )4 d4x ei (q k) x k2 k m2 h (p)ju(x) (1 + 5) u(0)j0i d4x ei (q k) x k2 b mb b m2 h (p)ju(x) (1 5) u(0)j0i : (4.1) Making use of the de nitions of the higher-twist photon DAs displayed in appendix B, it is straightforward to write down i 4 gem Qq (p q) ( " 2AP;2HT((p+q)2; q2; z) [(z p+q)2 + 2AP;3HT((p+q)2; q2; z) # + 2VP;3HT((p+q)2; q2; z) # ) where the explicit expressions of the invariant functions 2VP(HAT); i (i = 2 ; 3) are given by 2PHT((p + q)2; q2; z) = 2 mb f3 ( ) (a)(z; ) V;2 2VP;3HT((p + q)2; q2; z) = 2 mb2 hqqi( ) A(z; ) ; 2AP;2HT((p + q)2; q2; z) = 4 mb f3 ( ) (v)(z; ) + A(z; ) 2AP;3HT((p + q)2; q2; z) = 2 mb2 hqqi( ) A(z; ) 2 h (z; ) : hqqi( ) A(z; ) ; 2 h (z; ) hqqi( ) ; (4.3) The two new functions (v)(z; ) and h (z; ) introduced in (4.3) are de ned by Z z 0 Z z 0 (v)(z; ) = 2 d (v)( ; ) ; h (z; ) = 4 d (z ) h ( ; ) : (4.4) The resulting LCSR for the two-particle higher-twist hadronic photon corrections to the B ! ` form factors can be further derived as follows FV2 P(AH)T; p;LhoLton(n p) m2B q2 exp = fB mB exp 1 q2 M 2 exp q2)2 exp q2)2 ds0 d h q 2 ds s 2 (mb2 d exp V (A);2 s0; q2; z = f1(s0; q2) 2PHT d dz 1 2 z 2VP(HAT);3(s0; q2; z) V (A);2 s; q2; z = f1(s; q2) 2PHT 2VP(HAT);3(s0; q2; z) i s z=f1(s0;q2) z=f1(s0;q2) z=f1(s;q2) ) ; (4.5) where we have de ned f1(s; q2) = (mb2 q2) to compactify the above expressions. Several comments on the tree-level sum rules for the higher-twist corrections to the form factors FV and FA presented in (4.5) are in order. It is evident from (4.3) that the higher-twist two-particle hadronic photon corrections can lead to the symmetry-breaking contributions to the B ! ` form factors already at tree level, in agreement with the observation made in [12]. However, it needs to be pointed out that the subleading-twist e ects do not always violate the symmetry relation of the two B ! form factors at leading power in =mb [8]. Applying the power-counting scheme for the threshold parameter (3.17) and the endpoint behaviours of the two-particle photon DAs (v)(z; ), (a)(z; ), A(z; ) and h (z; ), we can readily identify the heavy-quark scaling for the two-particle highertwist corrections FV2;PpHhTot;oLnL(n p) FA2P;pHhTot;oLnL(n p) mb 3=2 ; (4.6) B ! of the pion-photon form factor). which is of the same power as the twist-two hadronic photon contribution obtained in (3.18) and is suppressed by only one factor of =mb compared with the \direct" photon contribution (3.19). We are then led to conclude that there is generally no correspondence between the heavy-quark expansion and the twist expansion for the ` form factors in the LCSR approach (see [31] for a discussion in the context F (p; q) de ned in (3.3) at tree level. Higher-twist three-particle corrections We will proceed to compute the higher-twist three-particle hadronic photon corrections to ` form factors at tree level with the sum rule technique. Following the standard strategy, we rst compute the three-particle contribution to the four-point QCD amplitude F (p; q) (3.3) displayed in gure 4. Keeping the one-gluon part for the light-cone expansion of the bottom-quark propagator in the background gluon/photon eld [32, 33] and employing the de nitions of the three-particle photon DAs in appendix B, we obtain (p; q) i gem Qq (p q) Z 1 0 dv Z [D i ( 3AP;2((p+q)2; q2; i; v) [(( q +v g) p+q)2 where the integration measure is de ned as " # + i " Z 1 0 [(( q +v g) p+q)2 d q d q q q g) : (4.9) Z q 0 Z q 0 Implementing the continuum subtraction with the aid of the parton-hadron duality relation and preforming the Borel transformation in the variable (p + q)2 the desired sum rules for the three-particle hadronic photon corrections at tree level ! s gives rise to mb +mu fB mB exp = Qq hqqi( ) 0 m2B M 2 ( Z f1(s0;q2) FV3 P(A;L);Lphoton(n p) d q 3P V (A);2 s0; q2; q; q; g; v = (1 q f2( q; s; q2)) 1 b s M 2 + Z s0 m2 b mb2 q2 ds 0 Z f1(s;q2) Z 1 q d q V (A);2 s; q2; q; q; g; v = 3P exp s0 M 2 d g g f2( q;s;q2) f2( q; s; q2) g g; q; g; v = 1 The resulting expressions for the invariant functions 3VP(A);i (i = 2; 3) are given by 3VP;2 = where we have introduced the following notations (4.10) (4.11) (4.12) + + + 0 0 0 0 d q 1 d q f2( q;s0;q2) f2( q;s0;q2) Z f1(s0;q2) Z 1 q d h ds0 ds Z s0 m2 b exp 0 d2 h ds2 exp s0 M 2 s M 2 d dv 2 ( q +v g) V (A);3 s0; q2; q; q; g; v 3P d g s0 q 2 g 2 (mb2 q2) V (A);3 s0; q2; q; q; g; v 3P v=f2( q;s0;q2)= g (1 q f2( q; s0; q2)) Z f1(s;q2) Z 1 q d q d g s q 2 g 2 (mb2 q2) f2( q;s;q2) 3P V (A);3 s; q2; q; q; g; v i i v=f2( q;s0;q2)= g (1 q f2( q; s; q2)) v=f2( q;s;q2)= g ) ; where for brevity we have introduced the auxiliary function f2( q; s; q2) de ned by In accordance with the power counting scheme for the threshold parameter and the endpoint behaviours of the three-particle photon DAs entering the sum rules (4.12), we can deduce the heavy-quark scaling of the three-particle hadronic photon corrections FV3;Pp;hLoLton(n p) FA3P;p; hLoLton(n p) mb 5=2 ; which is suppressed by one factor of =mb compared with the higher-twist two-particle contributions to the B ! ` form factors at tree level as presented in (4.5). It remains interesting to verify whether the NLO QCD corrections to the three-particle hadronic photon contributions can give rise to a dynamically enhancement to remove the power-suppression mechanism of the LO contributions (see [34, 35] for a discussion in the context of the NLO sum rules for the B ! yet higher-order corrections for future work. form factors) and we will leave explicit QCD calculations of the Collecting the di erent pieces together, the resulting expressions for the B ! form factors including the subleading-power contributions from the tree-level b u ! amplitude in QCD and from the hadronic photon corrections can be written as ` W FV (n p) = FV; LP(n p)+FVL;CNLP(n p)+FV2;PpLhToton(n p)+FV2;PpHhTot;oLnL(n p)+FV3;Pp;hLoLton(n p) ; FA(n p) = FA; LP(n p)+FAL;CNLP(n p)+FA2P;pLhToton(n p)+FA2P;pHhTot;oLnL(n p)+FA3P;p;hLoLton(n p) + Q` fB v p ; where the last term proportional to the electric charge of the lepton comes from the rede nition of the axial-vector form factor as discussed in section 2. The detailed expressions of the individual terms displayed on the right-hand side of (4.15) are given by (2.7), (2.8), (3.53), (4.5) and (4.12), respectively. We mention in passing that the LCSR calculations of the hadronic photon corrections to the B ! ` decay form factors presented here su er from the systematic uncertainty due to the parton-hadron duality ansatz in the B-meson channel. Future development of the subleading-power contributions to the radiative leptonic B-meson decays in the framework of SCET including a proper treatment of the rapidity divergences will be in demand for a model-independent QCD analysis. Several comments on the general structure of the B ! ` form factors (4.15) are in order. Both the point-like (short-distance) and the hadronic (long-distance) photon contributions to the B ! ` amplitude were computed with same correlation function (3.1) in [12] (see also [9, 10]), where the leading-power point-like photon contribution was represented by the triangle quark diagrams. By contrast, we apply the QCD factorization approach for the evaluation of the short-distance photon effect and employ the LCSR method with the photon DAs for the computation of the (4.13) (4.14) subleading power hadronic photon corrections. Since both the short-distance and long-distance photon contributions can be de ned by hadronic matrix elements of the corresponding e ective operators in SCET [5], we are allowed to compute the different hadronic matrix elements contributing to the B ! ` amplitude with the aid of distinct QCD techniques, and such \hybrid" computation scheme as implemented in this work is free of the double-counting issue. However, it needs to be pointed out that an additional source of the systematic uncertainty could be introduced in the \hybrid" approach, due to the scheme dependence of separating the leading and sub-leading power contributions. The subleading power contributions to the B ! ` transition form factors were also computed from the dispersion approach [7, 8, 36] by investigating the B ! ` amplitude with the parton-hadron duality ansatz. In particular, the higher-twist corrections to the form factors due to the higher Fock states of the B-meson and to the transverse momentum of the light-quark in the valence state were computed at tree level [36], where the soft contributions from the twist- ve and -six B-meson DAs were also estimated in the factorization approximation. 5 Numerical analysis We are now ready to explore the phenomenological implications of the hadronic photon corrections to the B ! ` amplitude computed from the LCSR approach. To this end, we will proceed by specifying the nonperturbative models of the two-particle and threeparticle photon DAs, the rst inverse moment B( ) and the logarithmic moments 1( ) and 2( ) of the leading-twist B-meson DA, and by determining the Borel mass and the hadronic threshold parameter entering the sum rules for the subleading-power resolved photon contributions. Having at our disposal the theory predictions for the form factors FV and FA, we will further explore the opportunity of constraining the inverse moment B( ) taking advantage of the improved measurements at the Belle II experiment in the near future. 5.1 Theory inputs In analogy to the leading-twist photon DA, we employ the conformal expansion for the twist-three DAs de ned by the chiral-even light-cone matrix elements 5 2 V ( i; ) = 540 !V ( ) ( q A( i; ) = 360 q q g2 1 + " 3 64 (v)(z; ) = 5 3 2 1 + 15 !V ( ) 5 !A( ) 3 30 2 + 35 4 ; (a)(z; ) = 1 2 (5 2 1) 1 + 3) ; 3 16 (5.1) HJEP05(218)4 with = 2 z 1, and for the chiral-odd twist-four DAs A(z; ) = 40 z2 z2 3 ( ) 2 2 g (1 g (1 q) [4 7 ( q + q)] ; q) q q g ; q) q q g ; where the anomalous dimension matrix ! is given by [16, 37] ! = Due to the Ferrara-Grillo-Parisi-Gatto theorem [38], the twist-four parameters corresponding to the \P"-wave conformal spin satisfy the following relations ( ) +( ) + 1( ) 1+( ) (1 2 g)+ 2( ) (3 4 g) ; ( )+ +( ) + 1( )+ 1+( ) (1 2 g)+ 2( ) (3 4 g) ; de ned by the chiral-odd light-ray matrix elements. Here, we have truncated the conformal expansion of the photon light-cone DAs up to the next-to-leading conformal spin (i.e., ``P"-wave). The renormalization-scale dependence of the twist-three parameters can be written as f3 ( ) = s( ) s( 0) !A( ) !V ( ) + !A( ) ! f = 0 f3 ( 0) ; f = s( ) s( 0) != 0 3 CF + 3 CA ; 1( ) + 11 2( ) 2 2+( ) = (5.2) (5.3) (5.4) (5.5) ( 0) 2:1 1:0 0:2 0:2 +( 0) 0 1( 0) 0:4 0:4 0 0 renormalization scale 0 = 1:0 GeV. The scale evolution of the nonperturbative parameters at twist-four accuracy is given by 1( ) = 2 ( ) = s( ) s( 0) s( ) s( 0) s( ) s( 0) ( + qq)= 0 Q(3) qq = 0 + = 3 CA rqq = 3 CF ; Q(3) = CF ; 13 3 1( 0) ; + 2 ( 0) ; 5 3 CF ; ( ) = + 1 ( ) = s( ) s( 0) s( ) s( 0) ( qq)= 0 Q(5) qq = 0 = 4 CA 3 CF ; 11 2 Q( 1 ) = CA 3 CF ; Q(5) = 5 CA CF : 8 3 ( 0) ; + where the anomalous dimensions of these twist-four parameters at one loop are given by [16] we will take the interval the semileptonic B ! tion equation of B+(!; ) [41] Numerical values of the input parameters entering the photon DAs up to twist-four are collected in table 1, where we have assigned 100 % uncertainties for the estimates of the twist-four parameters from QCD sum rules [17]. The second Gegenbauer moment of the leading-twist photon DA will be further taken as a2( 0) = 0:07 0:07 as obtained in [16]. The magnetic susceptibility of the quark condensate (1 GeV) = (3:15 0:3) GeV 2 computed from the QCD sum rule approach including the O( s) corrections [16] and the quark condensate density hqqi(1 GeV) = will be also employed for the numerical estimates in the following. (246+2189 MeV)3 determined by the GMOR relation [39] The key quantity entering the leading-power factorization formula of the B ! ` form factors is the rst inverse moment of the B-meson DA B( ), whose determination has been discussed extensively in the context of exclusive B-meson decays with distinct QCD approaches (see [34, 40] for more discussions). To illustrate the phenomenological consequences of the subleading-power corrections from the hadronic photon contributions B(1 GeV) = 354+3380 MeV implied by the LCSR calculations of form factors with B-meson DAs on the light-cone [34]. The renormalization-scale dependence of B( ) at one loop can be determined from the evoluB( ) B( 0) = 1 + 4 s( 0) CF ln 0 2 2 ln 0 4 1( 0) + O( s2) ; (5.8) where the inverse-logarithmic moments n( 0) are de ned as [6] n( 0) = B( ) Z 1 d! 0 ! lnn 0 ! B+(!; 0) : (5.9) Numerically we will employ 1(1GeV) = 1:5 1 consistent with the NLO QCD sum rule calculation [42] and 2(1GeV) = 3 2 from [6]. Furthermore, the static B-meson decay constant f~B( ) will be expressed in terms of the QCD decay constant fB f~B( ) = fB 1 + and the determination fB = (192:0 be taken in the numerical analysis. Following the discussions presented in [6, 34], the hard scales h1 and h2 entering the leading-power factorization formula will be chosen as h1 = h2 2 [mb=2; 2 mb] around the default value mb and the factorization scale in (2.7) will be varied in the interval 1 GeV 2 GeV with the central value = 1:5 GeV. In contrast, the factorization scale entering the LCSR for the hadronic photon corrections will be taken as around the default choice mb. In addition, we adopt the numerical values of the bottom quark mass mb(mb) = 4:193+00::002323 GeV [44] in the MS scheme from non-relativistic sum rules. Finally, we turn to determine the Borel mass M 2 and the threshold parameter s0 in the LCSR for the hadronic photon contributions. Applying the standard strategies presented in [34] (see also [45] for a review) gives rise to following intervals 4:3) MeV from the FLAG Working Group [43] will s0 = (37:5 2:5) GeV2 ; M 2 = (18:0 3:0) GeV2 ; (5.11) which is consistent with the determinations from the LCSR of the B ! form factors [46]. 5.2 Predictions for the B ! ` form factors We are now in a position to explore the phenomenological signi cance of the hadronic photon corrections to the B ! ` form factors. To develop a better understanding of the heavy quark expansion for the bottom sector, we plot the photon-energy dependence of the leading-power contribution, the subleading-power local correction and the subleadingpower two-particle and three-particle hadronic photon e ects in gure 5. It is apparent that the twist-two hadronic photon contribution at NLL can generate sizeable destructive interference with the leading-power \direct photon" contribution: approximately O(30%) for n p 2 [3 GeV ; mB] with B( 0) = 354 MeV. However, both the two-particle highertwist and the three-particle hadronic photon contributions turn out to be numerically insigni cant at tree level and will only shift the leading-power prediction by an amount of O (3 FVL;CNLP at tree level displayed in (2.8) will give rise to O(3%) correction at n p = mB and 5)% for n p 2 [3 GeV ; mB]. Furthermore, the subleading-power local contribution O(10%) correction at n p = 3 GeV. On account of the observed pattern for the separate terms contributing to the B ! form factors numerically, we are led to conclude that the power suppressed contributions to the radiative leptonic B-meson decay are dominated by the leading-twist hadronic photon correction with the default theory inputs. form factor FV (2 E ) as displayed in (4.15) with the central values of theory inputs. The individual contributions correspond to the leading-power contribution at NLL computed from the QCD factorization approach (FVN;LLLP, black), the subleading-power local contribution at LO (FVL;CNLP, green), the two-particle leading-twist hadronic photon correction at NLL (FV2;PpLhTo;tNonLL, blue), the two-particle higher-twist hadronic photon correction at leading-logarithmic (LL) accuracy (FV2;PpHhTot;oLnL, yellow), the three-particle leading-twist hadronic photon correction at LL (FV3;Pp;hLoLton, red). We further turn to investigate the numerical impact of the perturbative correction at NLO and the QCD resummation of the parametrically large logarithms of m2= 2 for b the leading-twist hadronic photon contribution computed from the LCSR technique. It is evident from gure 6 that the NLO QCD correction can decrease the tree-level prediction of the twist-two hadronic photon contribution by an amount of O (20 for the factorization scale varied in the interval [3:0; 5:0] GeV and the NLL resumma40)% tion e ect can yield O (10 %) enhancement to the NLO QCD results within the same range of . Hence, the dominant radiative correction to the leading-twist hadronic photon contribution originates from the NLO QCD correction to the hard matching coefcient entering the factorization formula (3.38) rather than from resummation of the large logarithms m2= 2 b . However, the renormalization scale dependence of the resummation improved theory predictions in the allowed region indeed becomes weaker compared with the NLO calculation. We further plot the photon-energy dependence of the ratio RF2PVL;Tphoton(n p) FV2;PpLhTo;tNonLL(n p)=FV2;PpLhTo;tLonL(n p) characterizing the perturbative QCD corrections at NLL in gure 6, where the theory uncertainties due to the variations of the renormalization scale are also displayed. form factor FV (mB) at LL (dashed), NLO (dotted), and NLL (solid) accuracy, respectively. Right : the photon-energy dependence of the ratio RF2PVL;Tphoton(n p) p)=FV2;PpLhTo;tLonL(n p) with the uncertainties from the variations of the renormalization scale . FV2;PpLhTo;tNonLL(n B ! Taking into account the fact that the QCD sum rule calculation of the second Gegenbauer moment of the twist-two photon DA a2( 0) su ers from the large theory uncertainties due to the strong sensitivity to the input parameters [16], we plot the leading-twist hadronic photon correction to the vector form factor FV (n p) in a wide range of a2( 0) in gure 7. One can readily observe that the variation of the Gegenbauer moment a2( 0) 2 [ 0:2; 0:2] can only give rise to a minor impact on the theory prediction of the B ! form factor FV (mB) at maximal recoil numerically. However, the \P-wave" conformal spin contribution from the leading-twist photon DA will become signi cant for the evaluation of the form factor FV (n p) with the decrease of the photon energy: approximately O(35%) at n p = 3 GeV. To further understand the systematic uncertainty due to the truncation of the conformal expansion at \P-wave", we also display the theory predictions for the ` form factors including the \D-wave" e ect from the fourth Gegenbauer moment a4( 0) in gure 7. It is apparent that the sensitivity of the leading-twist hadronic photon contribution on a4( 0) is rather weak numerically for n p 2 [3 GeV; mB] in the \reasonable" interval 0:2 a4( 0) 0:2. In the light of such observation, the yet higher Gegenbauer moments of the twist-two photon DA are not expected to bring about notable impact on the prediction of the subleading-power contribution to the B ! ` form factors induced by the photon light-cone DAs. We present our nal predictions for the B ! ` form factors including the newly computed two-particle and three-particle hadronic photon corrections with theory uncertainties in gure 8. The dominant theory uncertainties originate from the rst inverse moment B( 0), the factorization scale entering the leading-power \direct photon" contribution, and the second Gegenbauer moment a2( 0) of the twist-two photon DA. However, the symmetry breaking e ect between the two B ! form factors due to the subleading-power and n p = mB (right panel) on the second Gegenbauer moment of the photon light-cone DA a2( 0) with di erent values of the fourth Gegenbauer moment: a4( 0) = 0:2 (dashed), a4( 0) = 0 (solid) and a4( 0) = 0:2 (dotted). ` form factors as well as their di erence computed from (4.15) with the theory uncertainties from variations of di erent input parameters added in quadrature. local contribution and the higher-twist hadronic photon corrections su ers from much less uncertainty than the individual form factors at 3 GeV n p mB. Having in our hands the theoretical predictions for the B ! constraints on the inverse moment ` form factors, we proceed to discuss the theory B( 0) taking advantage of the future measurements on the (partially) integrated branching fractions with a photon-energy cut to get rid of the soft photon radiation. It is straightforward to derive the di erential decay width for ` ; E inverse moment B( 0) for Ecut = 1:5 GeV (blue band) and Ecut = 2:0 GeV (green band). Ecut) on the rst B ! ` in the rest frame of the B-meson (see also [6, 8]) d (B ! ` ) d E 6 2 em G2F jVubj2 mB E3 1 FV2 (n p) + FA2 (n p) ; (5.12) and the integrated branching fractions with the phase-space cut on the photon energy read BR(B ! ` ; E Ecut) = B Z mB=2 Ecut dE d (B ! d E ` ) (5.13) where B indicates the lifetime of the B-meson. Our predictions for the partial branching fractions of the radiative leptonic decay B ! ` including the hadronic photon corrections to the form factors are displayed in gure 9 with the variation of the inverse moment B( 0) in the interval [0:2; 0:6] GeV. It can be observed that the integrated branching fractions BR(B ! ` ; E Ecut) grow rapidly with the decrease of the inverse moment due to the dependence of the two form factors on 1= B( 0) at leading-power in =mb. Since the photon-energy cut E 1 GeV implemented in the Belle measurements [47] is not su ciently large to perform perturbative QCD calculations of the B ! we will not employ the experimental bound BR(B ! ` ; E Ecut) < 3:5 form factors, 10 6 with the full Belle data sample reported in [47] for the determination of B( 0) at the moment. Instead, we prefer to explore the solid theory constraints on the rst inverse moment by comparing our predictions of the (partially) integrated branching fractions with the improved measurements at the Belle II experiment, with the tighter phase-space cut on the photon energy, thanks to the much higher designed luminosity of the SuperKEKB accelerator. tion to the B ! We computed perturbative QCD corrections to the leading-twist hadronic photon contribu` form factors employing the LCSR method. QCD factorization for the B ! vacuum-to-photon correlation function (3.1) has been demonstrated explicitly at one loop with the OPE technique and the NDR scheme of the Dirac matrix 5 including the evanescent SCET operator. The perturbative matching coe cient entering the NLO factorization formula (3.38) was obtained by applying the method of regions and the factorization-scale independence of the correlation function (3.1) was further veri ed at O( s) with the evolution equations of the twist-two photon DA and of the bottom-quark mass. Resummation of the parametrically large logarithms of O(ln(mb2= 2)) was achieved at NLL accuracy with the two-loop RG equation of the light-ray tensor operator. Implementing the continuum subtraction with the aid of the parton-hadron duality and the Borel transformation, the NLL resummation improved LCSR for the twist-two hadronic photon correction to the form factors was subsequently constructed with the spectral representations of the factorization formula (3.38). The subleading-power correction to the B ! from the leading-twist photon DA was shown to preserve the symmetry relation between the two form factors due to the helicity conservation, in agreement with the observation made in [12]. Along the same vein, we proceed to compute the two-particle and three-particle highertwist hadronic photon corrections to the B ! ` form factors at tree level, up to the twist-four accuracy. The symmetry relation between the two form factors FV (n p) and FA(n p) was found to be violated by both the two-particle and three-particle higher-twist e ects of the photon light-cone DAs. In addition, our calculations explicitly indicate that the correspondence between the heavy-quark expansion and the twist expansion is generally invalid for the soft contributions to the exclusive B-meson decays, in analogy to the similar pattern observed in the context of the pion-photon form factor [31].1 Adding up di erent pieces contributing to the B ! ` amplitude, we further investigated the phenomenological impacts of the subleading-power hadronic photon contributions, employing the conformal expansion of the photon DAs at the \P-wave" accuracy. Numerically, the NLL twist-two hadronic photon correction was estimated to give rise to an approximately O(30%) reduction of the leading-power contribution, computed from QCD factorization, with the default values of theory inputs. By contrast, the higher-twist hadronic photon contributions at LO in O( s) was found to be of minor importance at 3 GeV n p mB, albeit with the rather conservative uncertainty ranges for the nonperturbative parameters collected in table 1. Moreover, we observed that the dominant radiative e ect of the leading-twist hadronic photon contribution comes from the NLO b QCD correction instead of the QCD resummation of the parametrically large logarithms m2= 2. To understand the systematic uncertainty from the truncation of the Gegenbauer expansion at the second order, we explored the numerical impact of the fourth moment of the leading-twist photon DA in a wide interval 4( 0) 2 [ 0:2; 0:2] and observed that 1The large scale Q2 in the pion-photon transition form factor F ! (Q2) plays a similar role of the heavy-quark mass in B-meson decays. HJEP05(218)4 the dependence of the twist-two hadronic photon correction to the B ! on 4( 0) was rather moderate at n p 3 GeV, at least in the framework of the LCSR method. Our main theory predictions for the B ! ` form factors with the uncertainties from variations of di erent input parameters added in quadrature were displayed in gure 8 and the poor constraint on the rst inverse moment of the B-meson DA brought about one of the major uncertainties for the theory calculations. In this respect, the improved measurements of the partial branching fractions BR(B ! ` ; E with the tighter phase-space cut on the photon energy to validate the perturbative QCD calculations from the Belle II experiment will be of value to provide solid constraints on the B( 0) Ecut) inverse moment B( 0), when combined with the theory predictions including the power suppressed contributions of di erent origins. B ! Further improvements of the theory descriptions of the B ! ` form factors in QCD can be pursued in distinct directions. First, it would be of interest to perform the NLO QCD corrections to the twist-three hadronic photon corrections with the LCSR approach for a systematic understanding of the higher-twist contributions. The technical challenge of accomplishing this task lies in the demonstration of QCD factorization for the vacuumto-B-meson correlation function (3.1) in the presence of the non-trivial mixing of the twoparticle and three-particle light-ray operators under the QCD renormalization. Second, exploring the subleading-power contributions to the radiative leptonic B-meson decay in the framework of SCET directly will be indispensable for deepening our understanding of factorization properties for more complicated exclusive B-meson decays, where the rapidity divergences of the convolution integrals entering the corresponding factorization formulae already emerge at leading power in the heavy quark expansion. Earlier attempts to address this ambitious question have been undertaken in di erent contexts (for an incomplete list, see for instance [5, 48{50]). Third, computing the subleading-power corrections to the form factors from the higher-twist B-meson DAs will be of both conceptual and phenomenological value to investigate general properties of the twist expansion in heavy-quark e ective theory (see [8] for a preliminary discussion with an incomplete decomposition of the three-particle vacuum-to-B-meson matrix element on the light-cone). To this end, we will need to employ the RG equations for these higher-twist B-meson DAs at one loop following the discussions presented in [51], where the evolution equations of the twist-four B-meson DAs at one loop were demonstrated to be completely integrable and therefore can be solved exactly. We are therefore anticipating dramatic progress toward better understanding of the strong interaction dynamics of the radiative leptonic decay B ! Acknowledgments Y.M.W acknowledges support from the National Youth Thousand Talents Program, the Youth Hundred Academic Leaders Program of Nankai University, the Fundamental Research Grant of National Universities, and the NSFC with Grant No. 11675082 and 11735010. The work of Y.L.S is supported by Natural Science Foundation of Shandong Province, China under Grant No. ZR2015AQ006. { 29 { Here we collect the dispersion representations of the various convolution integrals entering the NLL factorization formula of the vacuum-to-photon correlation function (3.50) for the sake of constructing the LCSR for the B ! ` form factors. : 1 1 (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) 1 r2 1 r2 0 + 0 1 r3 0 (r3 1) 2 r2 r3 6 1 1 1 1 1 1 0 0 Ims Ims Ims Ims Ims sgn(r2 r3) ln 1 dz 1 z (z; ) 1 z r3 z r2 1 r2 (z r3 +z r2 1) : z r3 +z r2 1+i 0 r1 r3 + (r3 r2) ln2(1 r2) : z r3 +z r2 1 1 r2 (z; ) : 1 r2 r1 r2 1 r1 1 r3 (z; ) : 1 r1 1 r2 1 1 r1 1 r3 1 : 1 1 1 1 1 1 z r3 +z r2 1+i 0 r1 r2 ln j1 z r3 z r2j (z r3 +z r2 1) (z r3 +z r2 1) 0 (z; ) (z; ) + (r3 r2) ln2(1 r2) : dz (z; ) 0 0 0 0 0 Ims Ims 0 1 1 1 1 1 + (r3 r2) 0 (z; ) ln2(1 r2) : z r3 +z r2 1+i 0 r1 r3 z r2 (z r3 +z r2) z r3 +z r2 1+i 0 r1 r2 r2 1 r2 ; r3 r2 2 r2 4 r2 r2 4 (z r3 +z r2 1) : 2 r2 4 +2 +2 (r3 r2) z r3 +z r2 1+i 0 r3(r1 r3) z r2 (r3 r2)+ 4 z (r3 1) (1 z r2)(r3 r2) 2 (1 3 z r2) z r3 +z r2 1+i 0 2 (r1 r2) 1 r1 1 r2 r3 r2 (z; ) 3 (r3 1) 0 1 1 1 1 1 1 1 2 r2 r2(1 r2)zz (r2 r3) ln(1 r2) r2 r3 1 r2 1 r2 (r3 1) dz ln(z r3 +z r2 1) (z r3 +z r2 1) 00(z; ) ln(r3 1) 0 (z = 1; ) dz ln(z r3 +z r2 1) (z r3 +z r2 1) 0 (z; ) 6 1 dz (z; ) Ims Ims Ims 0 Z 1 0 0 r2 r3 dz (z; ) 16 r2 15 (r3 1) 2 (1 r2) r3 r2 z r2 dz 0 Z 1 0 (z; ) z : 1 r2 r3 r2 1 r1 r3 4 z 3 15 2 1 r2 r3 r2 ; (A.7) (A.8) (A.9) (A.10) ln j1 r3j : (A.11) (A.12) In this appendix we will collect the operator-level de nitions of the two-particle and threeparticle photon DAs on the light-cone up to the twist-four accuracy as presented in [16]. ( ) (z; )+ A(z; ) x 2 = i gem Qq hqqi( ) (p + 2 gem Qq hqqi( ) i q x dz eiz p x 0 5 q(0)j0i p x h (p)jq(x) Wc(x; 0) gs G (v x) q(0)j0i dz eiz p x h (z; ) : 0 Z 1 dz eiz p x (a)(z; ) : = i gem Qq hqqi( ) (p ) [D i] ei( q+v g)p x S( i; ) : h (p)jq(x) Wc(x; 0) gs Ge (v x) i 5 q(0)j0i = i gem Qq hqqi( ) (p ) [D i] ei( q+v g)p x Se( i; ) : h (p)jq(x) Wc(x; 0) gs Ge (v x) = gem Qq f3 ( ) p (p h (p)jq(x) Wc(x; 0) gs G (v x) i = gem Qq f3 ( ) p (p ) [D i] ei( q+v g)p x V ( i; ) : h (p)jq(x) Wc(x; 0) gem Qq F (v x) q(0)j0i gs G (v x) q(0)j0i = i gem Qq hqqi( ) (p ) [D i] ei( q+v g)p x S ( i; ) : q(0)j0i = gem Qq f3 ( ) dz eiz p x (v)(z; ) : p ) [D i] ei( q+v g)p x A( i; ) : p p p (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) Z Z Z 5 q(0)j0i Z q(0)j0i Z = gem Qq hqqi( ) p gem Qq hqqi( ) p gem Qq hqqi( ) gem Qq hqqi( ) h h (p x (p x p p x )(p p x )(p p x p x g? 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Yu-Ming Wang, Yue-Long Shen. Subleading-power corrections to the radiative leptonic B → γℓν decay in QCD, Journal of High Energy Physics, 2018, 184, DOI: 10.1007/JHEP05(2018)184