Two-body non-leptonic heavy-to-heavy decays at NNLO in QCD factorization
SSI-HEP-2016-12QFET-2016-06October 16, 2018
Two-body non-leptonic heavy-to-heavydecays at NNLO in QCD factorization
Tobias Huber a , Susanne Kr¨ankl a , Xin-Qiang Li ba Naturwissenschaftlich-Technische Fakult¨at,Universit¨at Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany b Institute of Particle Physics andKey Laboratory of Quark and Lepton Physics (MOE),Central China Normal University, Wuhan, Hubei 430079, P. R. China
Abstract
We evaluate in the framework of QCD factorization the two-loop vertex correc-tions to the decays ¯ B ( s ) → D ( ∗ )+( s ) L − and Λ b → Λ + c L − , where L is a light mesonfrom the set { π, ρ, K ( ∗ ) , a } . These decays are paradigms of the QCD factorizationapproach since only the colour-allowed tree amplitude contributes at leading power.Hence they are sensitive to the size of power corrections once their leading-powerperturbative expansion is under control. Here we compute the two-loop O ( α s ) cor-rection to the leading-power hard scattering kernels, and give the results for theconvoluted kernels almost completely analytically. Our newly computed contribu-tion amounts to a positive shift of the magnitude of the tree amplitude by ∼ a r X i v : . [ h e p - ph ] S e p Introduction
Non-leptonic two-body decays of bottom mesons and baryons are interesting for phe-nomenological studies of the quark flavour sector of the Standard Model (SM) of particlephysics. They yield observables like branching ratios and CP asymmetries that are rele-vant for studying the CKM mechanism of quark flavour mixing and allow access to thequantities of the unitarity triangle (cf. refs. [1–3]).Oscillations and decays of B -mesons received considerable attention for the first timein the 1980s and 90s when the experiments ARGUS at DESY and CLEO at Cornellstarted to collect a lot of statistics. In the last decade, non-leptonic two-body B ( s ) -decays have been extensively measured at the asymmetric e + e − colliders ( B -factories)at SLAC and KEK, but also in hadronic environments such as the Tevatron, and theresults obtained by the Babar, Belle, D0 and CDF collaborations have reached a highlevel of precision (see, e.g. [4]). In recent years the LHCb experiment at the LHC atCERN has become the main player as far as experimental physics of the bottom quarkis concerned. A large data set on bottom mesons and baryons has been accumulated,and results related to non-leptonic decays have been published (cf. [5, 6]) and furtheranalyses are ongoing. In the near future also Belle II will contribute significantly tofurther improve the measurements [7].With the plethora of precise experimental data on non-leptonic decays at hand, the-oretical predictions at the same level of accuracy are very much desired. However, thetheoretical description of non-leptonic two-body B ( s ) decays is notoriously complicated.A straightforward computation of the hadronic matrix elements which describe the weaktransition is not feasible due to the presence of the strong interaction in the purelyhadronic initial and final states. This circumstance entails QCD effects from many dif-ferent scales which are, moreover, largely separated. In a first approach, known as na¨ıvefactorization, the hadronic transition matrix elements were factorized into a product ofa form factor and a decay constant [8]. Subsequent studies built on flavour symme-tries of the light quarks [9] and on factorization frameworks such as perturbative QCD(pQCD) [10, 11] and QCD factorization (QCDF) [12–14], to mention the most prominentones. Certain combinations of these approaches can also be found (see e.g. [15]).In the present work we adopt the QCDF framework and consider non-leptonic heavy-to-heavy transitions, which at the quark-level are mediated by the weak decay b → c ¯ ud ( s ),where we treat the bottom and the charm quark as massive and the light quarks asmassless. Performing an expansion of the amplitude in powers of Λ QCD /m b , where Λ QCD is the typical hadronic scale, a systematic separation of QCD effects from different scalescan be achieved and corrections to na¨ıve factorization be systematically included. Takingthe decay ¯ B → D + L − as an example, the transition amplitude in the heavy-mass limitis then given by [13] (cid:104) D + L − |Q i | ¯ B (cid:105) = (cid:88) j F B → Dj ( m L ) (cid:90) du T ij ( u )Φ L ( u ) , (1)where the local four-fermion operators Q i describe the underlying weak decay. The1 B → Dj form factors and the light-cone distribution amplitude (LCDA) Φ L of the lightmeson contain long-distance effects and can be obtained from non-perturbative methodslike QCD sum rules and lattice QCD. The hard-scattering kernels T ij , on the other hand,only receive contributions from scales of O ( m b ) and are accessible in a perturbativeexpansion in the strong coupling α s . After the convolution over the momentum fraction u of the valence quark inside the light meson, they yield a perturbative contribution tothe topological tree amplitude a ( D + L − ). Taking the decay ¯ B → D + π − as a specificexample, the latter is defined via [13] A ( ¯ B → D + π − ) = i G F √ V ∗ ud V cb a ( D + π − ) f π F B → D ( m π ) ( m B − m D ) . (2)The leading-power hard-scattering kernels have been known to next-to-leading order(NLO) accuracy for more than a decade for both heavy-to-light [12, 14, 16] and heavy-to-heavy [13] decays. In the latter case, expanding the LCDA in Gegenbauer moments upto the first moment α L , the topological tree amplitude a to NLO reads [13] | a ( ¯ B → D + L − ) | = (1 . +0 . − . ) − (0 . +0 . − . ) α L , | a ( ¯ B → D ∗ + L − ) | = (1 . +0 . − . ) − (0 . +0 . − . ) α L . (3)For the light meson being π or ρ we have α π ( ρ )1 = 0 and for the kaon | α K | < | a | (cid:39) .
05 for heavy-to-heavy decays in QCDF to NLO accuracy.A quasi-universality was also found upon extracting a from experimental data [17].However, the favoured central value | a | (cid:39) .
95 for the decays ¯ B → D ( ∗ )+ L − ( L = π , K ),with errors in the individual channels at the 10 – 20% level, is considerably lower.In recent years next-to-next-to-leading order (NNLO) corrections to heavy-to-lightdecays have become available [18–31], and besides the prospects of increasing precisionon the experimental side, there is multiple motivation to go beyond NLO in heavy-to-heavy transitions as well: First, the NLO correction is small since it is proportional toa small Wilson coefficient and, in addition, is colour-suppressed. At NNLO the coloursuppression gets lifted and the large Wilson coefficient re-enters, and therefore the NNLOcorrection could be comparable in size to the NLO term. Moreover, it is interesting tosee whether the quasi-universality of a persists at NNLO. At leading power the de-cays ¯ B ( s ) → D ( ∗ )+( s ) L − receive only vertex corrections to the colour-allowed tree topology.Interactions with the spectator quark as well as the weak annihilation topology are power-suppressed [13] and there are neither contributions from penguin operators nor is therea colour-suppressed tree topology. Therefore, a precise knowledge of the colour-allowedtree amplitude a allows to reliably estimate the size of power corrections to eq. (1) bycomparison to experimental data, and at the same time provides a test of the QCDFframework. This requires that the perturbative expansion of the hard scattering kernel isunder control, and that also the uncertainties of the non-perturbative input parameters(form factors, decay constants, LCDAs) can be minimized. In the present work we there-fore calculate the two-loop vertex correction to the leading-power hard scattering kernels2n the framework of QCDF. Parts of the computational procedure were already presentedin [32,33]. Here, we give the full result of the technically challenging two-loop calculation.Besides, we present an updated phenomenological analysis of ¯ B ( s ) → D ( ∗ )+( s ) L − decays,with a light meson from the set L = { π, ρ, K ( ∗ ) , a } , using the most recent values fornon-perturbative input parameters (for another recent analysis, see [34]).Recently, non-leptonic Λ b decays have received considerable attention as well. Dataon Λ b → Λ + c L − with L being π or K [35] and on baryonic form factors have becomeavailable [36]. Therefore, we extend our study to these decays. Factorization has not yetbeen systematically established for baryonic decays, but was discussed in ref. [37]. As asystematic derivation of the baryonic factorization formula is beyond the scope of thiswork we adopt the factorization formula eq. (4) of ref. [37], with appropriate modificationsto take perturbative corrections into account.This article is organized as follows: In section 2 we present our theoretical frameworkby specifying our operator basis in the effective weak Hamiltonian. Subsequently, wederive the master formulas for the hard scattering kernels by performing a matching ontoSoft-Collinear Effective Theory. In section 3 we discuss the calculation of the two-loopFeynman diagrams and specify the input to the master formulas. The analytical resultsof the hard scattering kernels after the convolution with the LCDAs are presented insection 4. In section 5 we give the formulas for converting from the pole to the MS schemefor the b - and c -quark masses. We present the results of our extensive phenomenologicalanalysis in section 6, and conclude in section 7. We work in the effective five-flavour theory where the top quark, the heavy gauge bosons W ± , Z and the Higgs boson are integrated out and their effects are absorbed into short-distance Wilson coefficients. The decays ¯ B ( s ) → D ( ∗ )+( s ) L − and Λ b → Λ + c L − are mediatedat parton level by a b → c ¯ ud ( s ) transition for L = π, ρ, a ( K, K ∗ ). The correspondingQCD amplitude is computed in the framework of the effective weak Hamiltonian [14, 38],which for the problem at hand simply reads H eff = G F √ V cb V ∗ ud ( C Q + C Q ) + h.c. . (4)We restrict our notation to the case of a b → c ¯ ud transition. The expressions for a strangequark in the final state are obtained by obvious replacements. The local current-currentoperators in the Chetyrkin-Misiak-M¨unz (CMM) basis [39, 40] read Q = ¯ cγ µ (1 − γ ) T A b ¯ dγ µ (1 − γ ) T A u , (5) We use the same symbol a for both, the meson a (1260) and the colour-allowed tree amplitude a ( D + L − ). We think that in each case it is clear from the context which quantity we refer to. q q q Figure 1: The tree-level Feynman diagram for the b → c ¯ ud transition in full (five flavour)QCD: the black square represents the vertex of the effective weak interaction. Themomenta q and q belong to the quark lines with masses m b and m c , respectively, and q + q = q is the momentum of the light meson. All momenta are taken to be incoming. Q = ¯ cγ µ (1 − γ ) b ¯ dγ µ (1 − γ ) u , (6)where Q and Q are referred to as colour-octet and colour-singlet operator, respectively.The use of the CMM basis allows for a consistent treatment of γ in the na¨ıve dimensionalregularization scheme with fully anti-commuting γ .Moreover, as the computation will be performed in dimensional regularization, wehave to augment our physical operators Q , by a set of evanescent operators, for whichwe adopt the convention [41, 42] E (1)1 = (cid:2) ¯ cγ µ γ ν γ ρ (1 − γ ) T A b (cid:3) (cid:2) ¯ dγ µ γ ν γ ρ (1 − γ ) T A u (cid:3) − Q , (7) E (1)2 = [¯ cγ µ γ ν γ ρ (1 − γ ) b ] (cid:2) ¯ dγ µ γ ν γ ρ (1 − γ ) u (cid:3) − Q , (8) E (2)1 = (cid:2) ¯ cγ µ γ ν γ ρ γ σ γ λ (1 − γ ) T A b (cid:3) (cid:2) ¯ dγ µ γ ν γ ρ γ σ γ λ (1 − γ ) T A u (cid:3) − E (1)1 − Q , (9) E (2)2 = (cid:2) ¯ cγ µ γ ν γ ρ γ σ γ λ (1 − γ ) b (cid:3) (cid:2) ¯ dγ µ γ ν γ ρ γ σ γ λ (1 − γ ) u (cid:3) − E (1)2 − Q . (10)These unphysical operators vanish in D = 4 dimensions but contribute if D (cid:54) = 4 sincethey mix under renormalization with the physical operators. At two-loop accuracy theset of operators (5) – (10) closes under renormalization. We construct the master formulas for the hard scattering kernels by performing a match-ing from the effective weak Hamiltonian onto Soft-Collinear Effective Theory (SCET)with three light flavours. The procedure follows similar lines than the derivation of themaster formulas for the hard kernels in heavy-to-light transitions [28].The kinematics of the b → c ¯ ud transition is shown in the tree-level Feynman diagramdepicted in figure 1. The b and the c quark are considered to be massive and carrymomenta q and q , respectively. The massless d and ¯ u quarks share the momentum q with q = uq and q = (1 − u ) q ≡ ¯ uq , where u ∈ [0 ,
1] is the momentum fraction of the In the following we refer to this side of the matching equation as the QCD side. q = m b , q = m c , and q = 0.We consider a reference frame in which the b quark within the B meson moves withmomentum q b = m b v + k , where k is a residual momentum of order of the typical hadronicscale Λ QCD , and v is the velocity of the B meson. The b quark can then be described by aheavy-quark field h v which satisfies the equation of motion /vh v = h v . We further choosea reference frame such that the energetic light meson moves in the light-cone direction n + . The light-like vectors n + and n − = 2 v − n + then fulfill the constraints n ± = 0 and n + n − = 2. As the quark and the anti-quark in the light meson nearly move in the samedirection we can describe them by the same type of collinear SCET field χ , which satisfiesthe equations of motion /n + χ = 0 and ¯ χ/n + = 0. In the derivation of the factorizationformula (1) the power counting m c /m b ∼ O (1) was adopted. Hence, we treat the charmquark as a heavy quark and consequently describe it – in analogy to the b quark – byanother heavy-quark field h v (cid:48) with velocity v (cid:48) and equation of motion /v (cid:48) h v (cid:48) = h v (cid:48) .The amplitudes in full QCD and in SCET are made equal by adjusting the corre-sponding hard coefficients at the matching scale. We express the renormalized matrixelements of the QCD operators (5) and (6) as a linear combination of a basis of SCEToperators, (cid:104)Q i (cid:105) = (cid:88) a =1 [ H ia (cid:104)O a (cid:105) + H (cid:48) ia (cid:104)O (cid:48) a (cid:105) ] , (11)where H ia and H (cid:48) ia are the matching coefficients. The basis of SCET operators is givenby O = ¯ χ /n − − γ ) χ ¯ h v (cid:48) /n + (1 − γ ) h v , (12) O = ¯ χ /n − − γ ) γ α ⊥ γ β ⊥ χ ¯ h v (cid:48) /n + (1 − γ ) γ ⊥ β γ ⊥ α h v , (13) O = ¯ χ /n − − γ ) γ α ⊥ γ β ⊥ γ γ ⊥ γ δ ⊥ χ ¯ h v (cid:48) /n + (1 − γ ) γ ⊥ δ γ ⊥ γ γ ⊥ β γ ⊥ α h v , (14) O (cid:48) = ¯ χ /n − − γ ) χ ¯ h v (cid:48) /n + (1 + γ ) h v , (15) O (cid:48) = ¯ χ /n − − γ ) γ α ⊥ γ β ⊥ χ ¯ h v (cid:48) /n + (1 + γ ) γ ⊥ α γ ⊥ β h v , (16) O (cid:48) = ¯ χ /n − − γ ) γ α ⊥ γ β ⊥ γ γ ⊥ γ δ ⊥ χ ¯ h v (cid:48) /n + (1 + γ ) γ ⊥ α γ ⊥ β γ ⊥ γ γ ⊥ δ h v . (17)Here, the perpendicular component of a Dirac matrix is defined by γ µ = /n + n µ − /n − n µ + γ µ ⊥ . (18)Moreover, we have omitted the Wilson lines which render the non-local light currents¯ χ ( tn − )[ . . . ] χ (0) gauge invariant. One therefore has to keep in mind that the coefficients5 ia are also functions of the variable t , and the products H ( (cid:48) ) ia (cid:104)O ( (cid:48) ) a (cid:105) in eq. (11) are infact convolutions. We also remark that the SCET operator basis is chosen such thatall operators with index a > O and O (cid:48) . The operators (12) – (14) have the same structure as those in [28]for heavy-to-light transitions, but with a heavy-quark field h v (cid:48) instead of an anti-collinearSCET field ξ in direction n − . For heavy-to-heavy transitions this set of operators hasto be extended by those in (15) – (17) which have a different chirality structure, to takeinto account the non-vanishing mass of the charm quark. For technical details on theoperators see [19, 43].We first consider the expansion of the left-hand side of eq. (11) in terms of on-shellQCD amplitudes. The expression for the renormalized matrix elements reads (cid:104)Q i (cid:105) = (cid:26) A (0) ia + α s π (cid:104) A (1) ia + Z (1) ext A (0) ia + Z (1) ij A (0) ja (cid:105) + (cid:16) α s π (cid:17) (cid:104) A (2) ia + Z (1) ij A (1) ja + Z (2) ij A (0) ja + Z (1) ext A (1) ia + Z (2) ext A (0) ia + Z (1) ext Z (1) ij A (0) ja + ( − i ) δm (1) b A ∗ (1) ia + ( − i ) δm (1) c A ∗∗ (1) ia + Z (1) α A (1) ia (cid:105) + O (cid:0) α s (cid:1)(cid:27) (cid:104)O a (cid:105) (0) + ( A ↔ A (cid:48) ) (cid:104)O (cid:48) a (cid:105) (0) . (19)Here, a sum over a = 1 , , α s is the MS strong coupling constant withfive active flavours. The index i = 1 , j includes physical as well as evanescent operators from (5) – (10), hence j = 1 , . . . ,
6. The A ( l ) are the bare l -loop on-shell amplitudes and A ∗ (1) ( A ∗∗ (1) ) is the one-loop bare on-shell amplitude with a b ( c ) quark mass insertion on the heavy b ( c ) line. Theprimed amplitudes are defined analogously. The renormalization factors Z ij , Z ext and Z α stem from operator renormalization, wave-function renormalization of all external legsand coupling renormalization, respectively. They are defined in a perturbative expansion Z = 1 + ∞ (cid:88) k =1 (cid:16) α s π (cid:17) k Z ( k ) . (20)The operator renormalization is performed in the MS scheme, whereas for the mass andthe wave-function renormalization the on-shell scheme is applied. Renormalized matrixelements of evanescent operators vanish also beyond tree level. Nevertheless, these oper-ators cannot be neglected right from the beginning as they yield physical contributionsto the products Z (1) ij A (0) ja , Z (1) ij A (1) ja , and Z (2) ij A (0) ja .Similarly, we can write down the expression for the renormalized matrix elements ofthe SCET operators that enter the right-hand side of eq. (11) and obtain (cid:104)O a (cid:105) = (cid:26) δ ab + ˆ α s π (cid:104) M (1) ab + Y (1) ext δ ab + Y (1) ab (cid:105) + (cid:18) ˆ α s π (cid:19) (cid:104) M (2) ab + Y (1) ext M (1) ab + Y (1) ac M (1) cb + ˆ Z (1) α M (1) ab + Y (2) ext δ ab Y (1) ext Y (1) ab + Y (2) ab (cid:105) + O (cid:0) ˆ α s (cid:1)(cid:27) (cid:104)O b (cid:105) (0) . (21)Here, a = 1 , , b = 1 , , α s has three light flavours and M ( l ) are the bare l -loop SCET amplitudes. The Y ( l ) ext , Y ( l ) ab and ˆ Z ( l ) α are the l -loop wave-function, operator and coupling renormalization con-stants, respectively. They are defined in a perturbative expansion analogous to eq. (20)except that the strong coupling has only three light flavours. The corresponding expres-sion for the primed operators from eqs. (15) – (17) is given by substituting M → M (cid:48) and O → O (cid:48) in eq. (21).Eq. (21) can be simplified to a large extent. In dimensional regularization the on-shellrenormalization constants Y ext are equal to unity. Moreover, the bare on-shell amplitudesonly contain scaleless integrals, which vanish in dimensional regularization. We thusarrive at the following simplified expression of eq. (21) (cid:104)O a (cid:105) = (cid:26) δ ab + ˆ α s π Y (1) ab + (cid:18) ˆ α s π (cid:19) Y (2) ab + O (cid:0) ˆ α s (cid:1) (cid:27) (cid:104)O b (cid:105) (0) , (22)which for the primed operators takes a similar form.For relating the matching coefficients H ia and H (cid:48) ia in eq. (11) to the hard scatteringkernels we introduce two factorized QCD operators Q ( (cid:48) )QCD = [¯ q /n − − γ ) q ][¯ c /n + (1 ∓ γ ) b ] , (23)which are by definition the products of the two currents in brackets. The upper signcorresponds to the un-primed operator. The renormalized operators Q ( (cid:48) )QCD are thenmatched onto the renormalized SCET operators O and O (cid:48) by adjusting the correspond-ing hard coefficients. This can be done separately for the light-to-light and heavy-to-heavycurrents. For the renormalized light-to-light current we make the ansatz (cid:20) ¯ q /n − − γ ) q (cid:21) = C ¯ qq (cid:20) ¯ χ /n − − γ ) χ (cid:21) . (24)The heavy-to-heavy currents with different chiralities mix in the matching. Thus, wemake the ansatz (cid:2) ¯ c /n + (1 − γ ) b (cid:3) = C LL F F (cid:2) ¯ h v (cid:48) /n + (1 − γ ) h v (cid:3) + C LR F F (cid:2) ¯ h v (cid:48) /n + (1 + γ ) h v (cid:3) , (25) (cid:2) ¯ c /n + (1 + γ ) b (cid:3) = C RL F F (cid:2) ¯ h v (cid:48) /n + (1 − γ ) h v (cid:3) + C RR F F (cid:2) ¯ h v (cid:48) /n + (1 + γ ) h v (cid:3) . (26)Since these equations are symmetric under interchanging P L ↔ P R we have C LL F F = C RR F F ≡ C D F F and C LR F F = C RL F F ≡ C ND F F . Finally, we obtain Q QCD = (cid:20) ¯ q /n − − γ ) q (cid:21)(cid:2) ¯ c /n + (1 − γ ) b (cid:3) = C ¯ qq C D F F O + C ¯ qq C ND F F O (cid:48) , (27) Q (cid:48) QCD = (cid:20) ¯ q /n − − γ ) q (cid:21)(cid:2) ¯ c /n + (1 + γ ) b (cid:3) = C ¯ qq C ND F F O + C ¯ qq C D F F O (cid:48) . (28)7ince by construction Q QCD and Q (cid:48) QCD factorize into a light-to-light and a heavy-to-heavy current, the matrix element of these operators is the product of an LCDA and thefull QCD form factor with the corresponding helicity structure.We now consider the two hard scattering kernels ˆ T i and ˆ T (cid:48) i that are defined by thefollowing expression (cid:104)Q i (cid:105) = ˆ T i (cid:104)Q QCD (cid:105) + ˆ T (cid:48) i (cid:104)Q (cid:48) QCD (cid:105) + (cid:88) a> [ H ia (cid:104)O a (cid:105) + H (cid:48) ia (cid:104)O (cid:48) a (cid:105) ] . (29)Comparing eqs. (11) and (29), ˆ T i and ˆ T (cid:48) i can be related to the matching coefficients asfollows (cid:18) ˆ T i ˆ T (cid:48) i (cid:19) = (cid:18) C ¯ qq C D F F C ¯ qq C ND F F C ¯ qq C ND F F C ¯ qq C D F F (cid:19) − (cid:18) H i H (cid:48) i (cid:19) . (30)Plugging in the matching coefficients as expansions in the five-flavour coupling α s , thematrix can be inverted order-by-order in α s . We remark that C ¯ qq = 1 + O ( α s ), i.e. itreceives a correction at two loops only since at one loop only scaleless integrals contribute.The explicit one-loop expressions for the heavy-to-heavy coefficients will be derived insection 3. For the diagonal coefficients we have C D F F = 1 + O ( α s ). In contrast, the non-diagonal matching coefficients C ND F F that induce the chirality mixing only arise beyondtree level, C ND F F = O ( α s ).Putting everything together, the master formulas for the hard scattering kernels readˆ T (0) i = A (0) i ˆ T (1) i = A (1) nfi + Z (1) ij A (0) j ˆ T (2) i = A (2) nfi + Z (1) ij A (1) j + Z (2) ij A (0) j + Z (1) α A (1) nfi − ˆ T (1) i (cid:104) C D(1)
F F + Y (1)11 − Z (1) ext (cid:105) − C ND(1)
F F ˆ T (cid:48) (1) i + ( − i ) δm (1) b A ∗ (1) nfi + ( − i ) δm (1) c A ∗∗ (1) nfi − (cid:88) b (cid:54) =1 H (1) ib Y (1) b . (31)The expression for the primed kernels ˆ T (cid:48) i is given by eq. (31) with the replacement A ↔ A (cid:48) , H ↔ H (cid:48) , ˆ T ↔ ˆ T (cid:48) . Note that the quantities H ( l ) , A ( l ) and the hard kernels ˆ T ( l ) dependon the quark mass ratio z c = m c /m b and the momentum fraction u of the quark insidethe light meson (as do the corresponding primed quantities). Whenever they appearalongside a renormalization factor Y ( l ) such as H (1) ib Y (1) b we must keep in mind that theseexpressions must be interpreted as a convolution product (cid:82) du (cid:48) H (1) ib ( z c , u (cid:48) ) Y (1) b ( u (cid:48) , u ).The amplitudes A ( l ) nfi in eq. (31) are termed “non-factorizable”. At one-loop thecorresponding amplitudes are given by all Feynman diagrams with one gluon connectingthe heavy and the light current. The one-loop Feynman diagrams where the gluon isattached solely to either the light or the heavy current are part of the LCDA and theform factor, respectively. The Feynman diagrams contributing to A (2) nfi can be found infigures 15 and 16 of ref. [13] and in addition include the one-loop self-energy insertions to8igure 2: Sample of Feynman diagrams that contribute to the two-loop hard scatteringkernels.the “non-factorizable” one-loop amplitudes. A sample of two-loop diagrams is shown infigure 2. A (2) nfi is technically the most challenging contribution to the two-loop kernels.Therefore, we briefly describe their evaluation in the next section and, moreover, specifythe remaining input to eq. (31). The final expression of the hard scattering kernels mustbe free of ultraviolet and infrared divergences. We comment on this at the end of thenext section.Finally, we remark that eq. (31) has a structure similar to the corresponding expres-sions for the two-loop hard scattering kernel in the right-insertion contribution to thedecay B → ππ , which is given in eq. (24) in [28]. The main difference is three-fold:First, we find two contributions ˆ T and ˆ T (cid:48) to the hard scattering kernel as a result of theextended operator basis. Second, we encounter the off-diagonal element C ND(1)
F F ˆ T (cid:48) (1)1 dueto the mixing of the heavy-to-heavy currents with different chirality structures. Finally,we have a mass counterterm for the massive charm quark in eq. (31). We work in dimensional regularization with D = 4 − (cid:15) and expand the amplitudes in theparameter (cid:15) . The Feynman diagrams contributing to the bare two-loop amplitude A (2) nf then contain up to 1 /(cid:15) poles stemming from ultraviolet (UV) and infrared (IR) regions.We calculated them by applying commonly-used multi-loop techniques, including a newmethod for evaluating the master integrals. The procedure goes as follows: First, we de-compose all tensor integrals into scalar ones by applying the Passarino-Veltman decompo-sition [44]. We then perform the reduction of the Dirac structures to the SCET operatorbasis given in eqs. (12) – (17) in Mathematica by using simple algebraic transformations.The number of remaining scalar two-loop integrals exceeds several thousands and can besimplified by using the Laporta algorithm [45, 46], which is based on integration-by-partsidentities [47]. Here, we apply the implementations AIR [48] (in
Maple ) and FIRE [49](in
Mathematica ) of this algorithm to reduce the large number of integrals to a smallset of master integrals. Many of the latter are already known from several B → ππ calculations [25, 26, 28]. In addition, we find 23 yet unknown two-loop master integrals.Since most of them depend on two scales (the momentum fraction u and the quark-mass9atio z c = m c /m b ), an analytic solution by common techniques is hardly feasible. Wetherefore evaluate them by applying the approach of differential equations in a canonicalbasis recently advocated in [50]. The solution is given by iterated integrals and falls intothe class of Goncharov polylogarithms [51]. We obtain analytic results for all 23 masterintegrals. Details on their calculation and the result of all master integrals can be foundin [33]. Here we give the explicit expressions for the renormalization factors and matching coeffi-cients that enter the master formula, and in the end comment on the cancellation of thepoles in (cid:15) once all pieces of the master formula are plugged in.The operator renormalization factors Z ij of the effective weak Hamiltonian were cal-culated to two-loop accuracy in the MS scheme in [41,42]. The explicit one- and two-loopexpressions read Z (1) = 1 (cid:15) (cid:18) −
43 512 29 (cid:19) , (32) Z (2) = 1 (cid:15) (cid:32) − n f T f (4 n f T f − ( n f T f − (8 n f T f − n f T f −
39 4 n f T f −
524 19 (cid:33) + 1 (cid:15) (cid:32) n f T f + n f T f − − n f T f − n f T f −
172 1384 − n f T f + − n f T f − − (cid:33) . (33)Here, n f = 5 is the total number of active quark flavours and T f = 1 /
2. The row indexof these matrices corresponds to ( Q , Q , E (1)1 , E (1)2 , E (2)1 , E (2)2 ) and the column index to( Q , Q ). The strong coupling constant is renormalized in the MS scheme as well, whereasthe renormalization of the masses and the wave-functions is performed in the on-shellscheme. The corresponding renormalization factors are well known and shall not berepeated here.In eq. (31) we further encounter the SCET operator renormalization factor Y thatcan be split into the following two parts Y ( u (cid:48) , u ) = Z Jh δ ( u − u (cid:48) ) + Z BL ( u (cid:48) , u ) . (34)Here, Z Jh and Z BL are the renormalization factors for the HQET heavy-to-heavy and theSCET light-to-light current, respectively. Since one collinear sector in SCET is equiva-lent to full QCD, the renormalization constant Z BL coincides with the ERBL kernel inQCD [52,53]. We take Z BL from [54], which for pseudoscalar and longitudinally polarizedvector mesons reads Z BL ( v, w ) = δ ( v − w ) − α s π C F (cid:15) (cid:40) w ¯ w (cid:20) v ¯ w Θ( w − v ) w − v + w ¯ v Θ( v − w ) v − w (cid:21) + − δ ( v − w )10 (cid:104) vw Θ( w − v ) + ¯ v ¯ w Θ( v − w ) (cid:105) (cid:41) + O ( α s ) . (35)The plus-distribution for symmetric kernels f is defined as follows, (cid:90) dw [ f ( v, w )] + g ( w ) = (cid:90) dwf ( v, w ) [ g ( w ) − g ( v )] . (36)The renormalization factor Z Jh can be obtained in a matching of the heavy-to-heavyQCD current ¯ c /n + (1 − γ ) b to the HQET current ¯ h v (cid:48) /n + (1 − γ ) h v . In this process alsothe matching coefficients C F F can be determined. Beyond tree-level the QCD currentalso mixes into the chirality-flipped HQET current ¯ h v (cid:48) /n + (1 + γ ) h v . Hence, we make thefollowing ansatz for the renormalized currents¯ c /n + (1 ∓ γ ) b = C D F F (cid:2) ¯ h v (cid:48) /n + (1 ∓ γ ) h v (cid:3) + C ND F F (cid:2) ¯ h v (cid:48) /n + (1 ± γ ) h v (cid:3) , (37)where we have already made use of the fact that both equations are symmetric underinterchanging P L ↔ P R . The renormalization factor Z Jh is defined via the on-shell one-loop matrix element of the HQET currents (cid:104) ¯ h v (cid:48) /n + (1 ∓ γ ) h v (cid:105) (1) = (cid:16) Y (1) ext + Z (1) Jh (cid:17) (cid:104) ¯ h v (cid:48) /n + (1 ∓ γ ) h v (cid:105) (0) . (38)The one-loop renormalized matrix elements of the QCD currents can be calculatedstraightforwardly. Inserting their explicit expressions in eq. (37) we can identify Z (1) Jh as the pole term in (cid:15) , that is Z (1) Jh = C F (cid:15) (cid:18) ( z c + 1) log( z c ) z c − − (cid:19) , (39)which correctly reproduces the IR behaviour of QCD currents in the effective theory. The C F F , on the other hand, are given by the coefficients that are finite in (cid:15) . Their explicitexpressions read ( L ≡ log( µ /m b )) C D(1)
F F = C F (cid:20) L (cid:18) ( z c + 1) log( z c ) z c − − (cid:19) + ( z c + 1) log ( z c )2 − z c + (5 z c + 1) log( z c )2( z c − − (cid:21) , (40) C ND(1)
F F = C F (cid:18) √ z c log( z c ) z c − (cid:19) . (41)As a last step the contribution of the sum (cid:80) b (cid:54) =1 H (1) ib Y (1) b in eq. (31) needs to be furtherspecified (the primed quantities are obtained by obvious substitutions). We find thatonly H (1) ib with i = 1 and b = 2 yields a non-vanishing contribution. A straightforwardcalculation yields H (1)12 = A (1) nf + Z (1)1 j A (0) j . The operator renormalization factor Y (1)21 has already been used in the NNLO calculation of the vertex corrections to the decay B → ππ and is given in eq. (45) of [25]. Its explicit expression reads Y (1)21 ( u (cid:48) , u ) = 16 C F (cid:18) u (cid:48) Θ( u − u (cid:48) ) u + (1 − u (cid:48) )Θ( u (cid:48) − u )1 − u (cid:19) . (42)11ith this we have specified all input to the master formulas and are now ready to producean expression for the hard scattering kernels.The final expressions for the hard scattering kernels are free of poles in (cid:15) , even thoughmost of the individual terms in eq. (31) contain divergences. At the one-loop level wechecked the cancellation of all poles analytically. We find that our expressions for thefinite pieces of the one-loop kernels agree with the results given in eqs. (89) and (90)in [13] . Some of the one-loop quantities that enter the two-loop master formula (lastequation in (31)) have to be evaluated to higher orders in the (cid:15) -expansion since theymultiply poles in (cid:15) contained in the renormalization factors. We checked that in the limit m c → O ( (cid:15) ) piece of the one-loop hard scattering kernel coincides with the one usedin [28].At two loops we could check the pole cancellation numerically to an accuracy of1 × − or better for 12 different points in the u - z c plane. To this end, we evaluatethe Goncharov polylogarithms and the harmonic polylogarithms [55] that are containedin A (2) nf numerically with the C++ routine GiNaC [56] and the Mathematica program
HPL [57, 58], respectively. The explicit results for the two-loop hard scattering kernels arelengthy, not very illuminating, and enter the physical quantities only after convolutionwith the LCDAs. For these reasons we refrain from presenting them explicitly here, butthey can be obtained from the authors upon request. However, after the convolutionof the hard scattering kernels with an expansion of the LCDAs in terms of Gegenbauerpolynomials up to the second moment, the expressions simplify considerably and we canexpress the result almost entirely in terms of harmonic polylogarithms. At this level weconvolute also the pole terms in (cid:15) and checked that for the convoluted kernels all polescancel analytically. We give the corresponding finite parts in the next section.
The light meson LCDAs are expanded in a basis of Gegenbauer polynomials C / k ( x ) withGegenbauer moments α Lk ,Φ L ( u, µ ) = 6 u (1 − u ) (cid:32) ∞ (cid:88) k =1 α Lk ( µ ) C / k (2 u − (cid:33) . (43)Following [13] we assume that the leading-twist LCDA is close to its asymptotic formΦ L ( u, µ ) = 6 u (1 − u ) and truncate the expansion after the second moment. The first twoGegenbauer polynomials read C / ( x ) = 3 x and C / ( x ) = (5 x − For performing this comparison one has to take into account that the one-loop result given in eq. (90)in [13] was calculated in the “traditional operator basis” given in eq. (V.1) in [38]. (cid:90) du ˆ T i ( u, µ )Φ L ( u, µ ) = V (0) i ( µ ) + (cid:88) l ≥ (cid:16) α s π (cid:17) l (cid:88) k =0 α Lk ( µ ) V ( l ) ik ( µ ) , (44) (cid:90) du ˆ T (cid:48) i ( u, µ )Φ L ( u, µ ) = V (cid:48) (0) i ( µ ) + (cid:88) l ≥ (cid:16) α s π (cid:17) l (cid:88) k =0 α Lk ( µ ) V (cid:48) ( l ) ik ( µ ) √ z c , (45)with α L ( µ ) ≡
1. At tree-level we obtain V (0)1 ( µ ) = 0 , V (cid:48) (0)1 ( µ ) = 0 , (46) V (0)2 ( µ ) = 1 , V (cid:48) (0)2 ( µ ) = 0 . (47)In the following we use the abbreviations L ≡ log( µ /m b ) and H (cid:126)a ( z c ) ≡ H (cid:126)a for theharmonic polylogarithms of argument z c . The one-loop results for the convoluted colour-octet kernels then read V (1)10 ( µ ) = − L (cid:20) − z c ( z c − H + 2 ( z c + 10 z c + 1)3( z c − H + 2 z c ( z c + 1)( z c − H − z c ( z c + 1)( z c − H + π z c ( z c + 1)3( z c − + − z c + 18 z c − z c − (cid:21) + iπ (cid:20) − z c ( z c − H + 2(2 z c + 5 z c − z c − (cid:21) ,V (1)11 ( µ ) = (cid:20) − z c ( z c + 3)( z c − H − z c ( z c − z c − z c − H − z c ( z c + 6 z c + 1)( z c − H − z c − z c − z c + 1)3( z c − H + π z c ( z c + 6 z c + 1)3( z c − + − z c + 155 z c + 155 z c − z c − (cid:21) + iπ (cid:20) − z c ( z c + 3)( z c − H + − z c + 46 z c + 4 z c + 23( z c − (cid:21) ,V (1)12 ( µ ) = (cid:20) − z c ( z c + 3 z c + 1)( z c − H + 2( z c + 1) ( z c + 28 z c + 1)( z c − H + 2 z c ( z c + 29 z c + 29 z c + 1)( z c − H − z c ( z c + 4 z c + 4 z c + 1)( z c − H + π z c ( z c + 4 z c + 4 z c + 1)( z c − + − z c + 1368 z c + 4478 z c + 1368 z c − z c − (cid:21) + iπ (cid:20) − z c ( z c + 3 z c + 1)( z c − H + 2 z c ( z c + 29 z c + 29 z c + 1)( z c − (cid:21) , (cid:48) (1)10 ( µ ) = (cid:20) z c ( z c + 2)3( z c − H + 4 ( z c + 4 z c + 1)3( z c − H − z c + 1)3( z c − H − z c + 1)( z c − H − π z c + 4 z c + 1)9( z c − − z c + 1)( z c − (cid:21) + iπ (cid:20) z c ( z c + 2)3( z c − H − z c + 1)3( z c − (cid:21) ,V (cid:48) (1)11 ( µ ) = (cid:20) − z c ( z c + 5 z c + 2)( z c − H + 2 (19 z c + 28 z c + 1)3( z c − H + 8 (5 z c + 14 z c + 5)3( z c − H − z c + 7 z c + 7 z c + 1)( z c − H + π z c + 7 z c + 7 z c + 1)3( z c − + 2 (41 z c + 206 z c + 41)9( z c − (cid:21) + iπ (cid:20) − z c ( z c + 5 z c + 2)( z c − H + 2 (19 z c + 28 z c + 1)3( z c − (cid:21) ,V (cid:48) (1)12 ( µ ) = (cid:20) z c ( z c + 10 z c + 12 z c + 2)( z c − H − z c + 181 z c + 73 z c + 1)3( z c − H − z c + 127 z c + 127 z c + 23)3( z c − H + 8 ( z c + 12 z c + 24 z c + 12 z c + 1)( z c − H − π z c + 12 z c + 24 z c + 12 z c + 1)3( z c − − z c + 827 z c + 827 z c + 73)9( z c − (cid:21) + iπ (cid:20) z c ( z c + 10 z c + 12 z c + 2)( z c − H − z c + 181 z c + 73 z c + 1)3( z c − (cid:21) . (48)The one-loop colour-singlet kernels vanish as the corresponding colour factors are zero, V (1)2 k ( µ ) = V (cid:48) (1)2 k ( µ ) = 0 , for k = 0 , , . (49)At two loops the result is rather lengthy. Here, we only present the full result for the µ -dependent part which governs the scale dependence. For the µ -independent part weprovide a fitted function in z c that agrees with the original result at the per mill level inthe range of physical values 0 . ≤ z c ≤ .
2. The full result is attached in electronic formto the arXiv submission of the present work. For the convoluted colour-octet kernels weobtain V (2)10 ( µ ) = − L + (cid:20) − z c z c − H + 58 ( z c + 10 z c + 1)9( z c − H + 58 z c ( z c + 1)3( z c − H − z c ( z c + 1)3( z c − H + π z c ( z c + 1)9( z c − − z c − z c + 527)27( z c − (cid:21) L + iπ (cid:20) − z c z c − H + 58 (2 z c + 5 z c − z c − (cid:21) L + (cid:20) . z c − . z c − . − . z c − . z c ) (cid:21) + iπ (cid:20) − . z c + 37 . z c . . z c + 0 . z c ) (cid:21) ,V (2)11 ( µ ) = (cid:20) − z c ( z c + 3)9( z c − H − z c ( z c − z c − z c − H − z c ( z c + 6 z c + 1)9( z c − H −
238 ( z c − z c − z c + 1)27( z c − H + π z c ( z c + 6 z c + 1)27( z c − −
119 (11 z c − z c − z c + 11)81( z c − (cid:21) L + iπ (cid:20) − z c + 3) z c z c − H −
238 (2 z c − z c − z c − z c − (cid:21) L + (cid:20) − . z c + 71 . z c + 228 . . z c + 20 . z c ) (cid:21) + iπ (cid:20) − . z c − . − . z c − . z c ) (cid:21) ,V (2)12 ( µ ) = (cid:20) − z c ( z c + 3 z c + 1)3( z c − H + 274( z c + 1) ( z c + 28 z c + 1)9( z c − H + 274 z c ( z c + 29 z c + 29 z c + 1)9( z c − H − z c ( z c + 4 z c + 4 z c + 1)3( z c − H + π z c ( z c + 4 z c + 4 z c + 1)9( z c − −
137 (7 z c − z c − z c − z c + 7)270( z c − (cid:21) L + iπ (cid:20) − z c ( z c + 3 z c + 1)3( z c − H + 274 z c ( z c + 29 z c + 29 z c + 1)9( z c − (cid:21) L + (cid:20) − . z c + 86 . z c − . . z c + 0 . z c ) (cid:21) + iπ (cid:20) − . z c + 49 . z c − . z c + 32 . − . z c + 0 . z c + 15 . z c ) + 1 . ( z c ) (cid:21) ,V (cid:48) (2)10 ( µ ) = (cid:20) z c ( z c + 2)9( z c − H + 116 ( z c + 4 z c + 1)9( z c − H − z c + 1)9( z c − H − z c + 1)3( z c − H − π
58 ( z c + 4 z c + 1)27( z c − − z c + 1)3( z c − (cid:21) L + iπ (cid:20) z c ( z c + 2)9( z c − H z c + 1)9( z c − (cid:21) L + (cid:20) . z c − . z c + 81 . z c − . − . z c + 0 . z c − . z c ) + 0 . ( z c ) (cid:21) + iπ (cid:20) − . z c + 35 . z c − . . z c + 1 . z c ) (cid:21) ,V (cid:48) (2)11 ( µ ) = (cid:20) − z c ( z c + 5 z c + 2)9( z c − H + 238 (19 z c + 28 z c + 1)27( z c − H + 952 (5 z c + 14 z c + 5)27( z c − H −
476 ( z c + 7 z c + 7 z c + 1)9( z c − H + π
238 ( z c + 7 z c + 7 z c + 1)27( z c − + 238 (41 z c + 206 z c + 41)81( z c − (cid:21) L + iπ (cid:20) − z c ( z c + 5 z c + 2)9( z c − H + 238 (19 z c + 28 z c + 1)27( z c − (cid:21) L + (cid:20) − . z c + 316 . z c − .
270 + 1 . z c − . z c − . z c ) (cid:21) + iπ (cid:20) − . z c − . z c + 37 . . z c − . z c + 22 . z c ) (cid:21) ,V (cid:48) (2)12 ( µ ) = (cid:20) z c ( z c + 10 z c + 12 z c + 2)9( z c − H −
274 (45 z c + 181 z c + 73 z c + 1)27( z c − H −
548 (23 z c + 127 z c + 127 z c + 23)27( z c − H + 1096 ( z c + 12 z c + 24 z c + 12 z c + 1)9( z c − H − π
548 ( z c + 12 z c + 24 z c + 12 z c + 1)27( z c − −
274 (73 z c + 827 z c + 827 z c + 73)81( z c − (cid:21) L + iπ (cid:20) z c ( z c + 10 z c + 12 z c + 2)9( z c − H −
274 (45 z c + 181 z c + 73 z c + 1)27( z c − (cid:21) L + (cid:20) − . z c + 115 . z c + 3 . . z c − . z c + 37 . z c ) (cid:21) + iπ (cid:20) − . z c + 202 . z c − . z c + 84 . . z c − . z c + 34 . z c ) (cid:21) . (50)16he result for the convoluted colour-singlet kernels takes the form V (2)20 ( µ ) = 4 L + (cid:20) z c ( z c − H − z c + 10 z c + 1)( z c − H − z c ( z c + 1)( z c − H + 24 z c ( z c + 1)( z c − H − π z c ( z c + 1)( z c − + 8 (13 z c − z c + 13)3( z c − (cid:21) L + iπ (cid:20) z c ( z c − H − z c + 5 z c − z c − (cid:21) L + (cid:20) . z c + 92 . . z c + 5 . z c ) (cid:21) + iπ (cid:20) − . z c + 23 . − . z c − . z c ) (cid:21) ,V (2)21 ( µ ) = (cid:20) z c ( z c + 3)( z c − H + 4 z c ( z c − z c − z c − H + 24 z c ( z c + 6 z c + 1)( z c − H + 4 ( z c − z c − z c + 1)( z c − H − π z c ( z c + 6 z c + 1)( z c − + 2 (11 z c − z c − z c + 11)3( z c − (cid:21) L + iπ (cid:20) z c + 3) z c ( z c − H + 8 z c − z c − z c − z c − (cid:21) L + (cid:20) . z c − . − . z c − . z c ) (cid:21) + iπ (cid:20) − . z c + 48 . z c + 16 . . z c + 2 . z c ) (cid:21) ,V (2)22 ( µ ) = (cid:20) z c ( z c + 3 z c + 1)( z c − H − z c + 1) ( z c + 28 z c + 1)( z c − H − z c ( z c + 29 z c + 29 z c + 1)( z c − H + 144 z c ( z c + 4 z c + 4 z c + 1)( z c − H − π z c ( z c + 4 z c + 4 z c + 1)( z c − + 7 z c − z c − z c − z c + 75( z c − (cid:21) L + iπ (cid:20) z c ( z c + 3 z c + 1)( z c − H − z c ( z c + 29 z c + 29 z c + 1)( z c − (cid:21) L + (cid:20) − . z c + 246 . z c − . z c + 23 . . z c + 0 . z c + 4 . z c ) (cid:21) + iπ (cid:20) . z c − . − . z c + 0 . z c − . z c ) (cid:21) , (cid:48) (2)20 ( µ ) = (cid:20) − z c ( z c + 2)( z c − H − z c + 4 z c + 1)( z c − H + 4(5 z c + 1)( z c − H + 24( z c + 1)( z c − H + π z c + 4 z c + 1)3( z c − + 24( z c + 1)( z c − (cid:21) L + iπ (cid:20) z c + 1)( z c − − z c ( z c + 2)( z c − H (cid:21) L + (cid:20) . z c + 2 . z c − . . z c − . z c − . z c ) − . ( z c ) (cid:21) + iπ (cid:20) . z c − . − . z c + 0 . z c − . z c ) (cid:21) ,V (cid:48) (2)21 ( µ ) = (cid:20) z c ( z c + 5 z c + 2)( z c − H − z c + 28 z c + 1)( z c − H −
16 (5 z c + 14 z c + 5)( z c − H + 24 ( z c + 7 z c + 7 z c + 1)( z c − H − π z c + 7 z c + 7 z c + 1)( z c − − z c + 206 z c + 41)3( z c − (cid:21) L + iπ (cid:20) z c ( z c + 5 z c + 2)( z c − H − z c + 28 z c + 1)( z c − (cid:21) L + (cid:20) − . z c + 426 . z c − . z c + 0 . − . z c (cid:21) + iπ (cid:20) − . z c + 141 . z c − . z c + 64 . z c − . − . z c − . z c ) (cid:21) ,V (cid:48) (2)22 ( µ ) = (cid:20) − z c ( z c + 10 z c + 12 z c + 2)( z c − H + 4 (45 z c + 181 z c + 73 z c + 1)( z c − H + 8 (23 z c + 127 z c + 127 z c + 23)( z c − H −
48 ( z c + 12 z c + 24 z c + 12 z c + 1)( z c − H + π z c + 12 z c + 24 z c + 12 z c + 1)( z c − + 4 (73 z c + 827 z c + 827 z c + 73)3( z c − (cid:21) L + iπ (cid:20) z c + 181 z c + 73 z c + 1)( z c − − z c ( z c + 10 z c + 12 z c + 2)( z c − H (cid:21) L + (cid:20) − . z c + 118 . z c − . − . z c − . z c − . z c ) (cid:21) + iπ (cid:20) − . z c + 201 . z c − . z c
18 7 . z c + 7 . . z c − . z c + 0 . z c + 4 . z c ) (cid:21) . (51)Finally, we checked with the full result (without interpolation in z c ) that in the limit m c → µ = m b coincides with the result for the vertex corrections to thecolour-allowed tree topology of the decay B → ππ given in eq. (48) of [28] . The convoluted kernels in eqs. (44) and (45) are given in the pole scheme, where m c and m b appearing in L ≡ log( µ /m b ) and z c = m c /m b denote the pole quark masses, andthe renormalization scale µ ∼ m b . In order to discuss the scheme dependence of theconvoluted kernels, we also give the results in the MS scheme for the quark masses. Sincethe LO kernels are constant and the NLO colour-singlet kernels vanish, the conversionfrom the pole to the MS scheme will only affect the NNLO colour-octet kernels V (2)1 k and V (cid:48) (2)1 k . To this end, using the one-loop relation between pole- and MS-quark mass, m q = m q ( µ ) (cid:20) α s π (cid:18)
43 + log (cid:18) µ m q ( µ ) (cid:19)(cid:19)(cid:21) , (52)we find that the corresponding convoluted kernels in the MS scheme are obtained via therelation V ( (cid:48) )MS(2)1 k ( µ ) = V ( (cid:48) )(2)1 k + ∆ V ( (cid:48) )1 k , ∆ V k = − z c ln( z c ) ∂V (1)1 k ∂z c − (cid:20)
323 + 8 L (cid:21) ∂V (1)1 k ∂L , ∆ V (cid:48) k = − √ z c ln( z c ) ∂ √ z c V (cid:48) (1)1 k ∂z c − (cid:20)
323 + 8 L (cid:21) ∂V (cid:48) (1)1 k ∂L , (53)where now L ≡ log( µ /m b ( µ )) and z c = m c ( µ ) /m b ( µ ), with µ ∼ m b ( m b ).The tree-level and one-loop kernels will have the same functional dependence as inthe pole scheme, but now depend on the above new abbreviations in the MS scheme. Attwo loops we explicitly give the terms that have to be added,∆ V = 323 L + (cid:20) − z c ( z c + 4 z c + 1)( z c − H − z c ( z c + 2)( z c − H + 32 z c (5 z c + 1)( z c − H + 96 z c ( z c + 1)( z c − H − z c ( z c − z c − z c − H − z c ( z c + 4 z c + 1)( z c − H + 96 z c ( z c + 1)( z c − H + π z c ( z c + 4 z c + 1)3( z c − H + 1289 (cid:21) + iπ (cid:20) z c (5 z c + 1)( z c − H z c ( z c + 2)( z c − H (cid:21) , ∆ V = (cid:20) z c (23 z c + 68 z c + 5)3( z c − H + 64 z c (7 z c + 34 z c + 7)3( z c − H − z c ( z c + 9 z c + 6)( z c − H − z c ( z c + 15 z c + 15 z c + 1)( z c − H + 16 z c (73 z c + 430 z c + 73)9( z c − H + 64 z c (7 z c + 34 z c + 7)3( z c − H − z c ( z c + 15 z c + 15 z c + 1)( z c − H + π z c ( z c + 15 z c + 15 z c + 1)3( z c − H (cid:21) + iπ (cid:20) H z c (23 z c + 68 z c + 5)3( z c − − z c ( z c + 9 z c + 6) H ( z c − (cid:21) , ∆ V = (cid:20) z c (45 z c + 181 z c + 73 z c + 1)( z c − H + 32 z c (23 z c + 127 z c + 127 z c + 23)( z c − H − z c ( z c + 10 z c + 12 z c + 2)( z c − H + 32 z c (23 z c + 127 z c + 127 z c + 23)( z c − H − z c ( z c + 12 z c + 24 z c + 12 z c + 1)( z c − H − z c ( z c + 12 z c + 24 z c + 12 z c + 1)( z c − H + 16 z c (73 z c + 827 z c + 827 z c + 73)3( z c − H + π z c ( z c + 12 z c + 24 z c + 12 z c + 1)( z c − H (cid:21) + iπ (cid:20) z c (45 z c + 181 z c + 73 z c + 1)( z c − H − z c ( z c + 10 z c + 12 z c + 2)( z c − H (cid:21) , (54)∆ V (cid:48) = (cid:20) −
16 (9 z c + 26 z c + 1)3( z c − H −
16 (5 z c + 26 z c + 5)3( z c − H − H ( z c + 7 z c + 1)3( z c − + 16 z c ( z c + 11 z c + 6)( z c − H + 16 ( z c + 17 z c + 17 z c + 1)3( z c − H −
16 (5 z c + 26 z c + 5)3( z c − H + 32 ( z c + 17 z c + 17 z c + 1)3( z c − H − π z c + 17 z c + 17 z c + 1)9( z c − H (cid:21) + iπ (cid:20) z c ( z c + 11 z c + 6)3( z c − H z c + 26 z c + 1)3( z c − H (cid:21) , ∆ V (cid:48) = (cid:20)
16 (31 z c + 239 z c + 113 z c + 1)3( z c − H + 256 ( z c + 11 z c + 11 z c + 1)3( z c − H − z c ( z c + 22 z c + 35 z c + 6)( z c − H −
16 ( z c + 28 z c + 70 z c + 28 z c + 1)( z c − H + 8 (47 z c + 1105 z c + 1105 z c + 47)9( z c − H + 256 ( z c + 11 z c + 11 z c + 1)3( z c − H −
32 ( z c + 28 z c + 70 z c + 28 z c + 1)( z c − H + π z c + 28 z c + 70 z c + 28 z c + 1)3( z c − H (cid:21) + iπ (cid:20) z c + 239 z c + 113 z c + 1)3( z c − H − z c ( z c + 22 z c + 35 z c + 6)( z c − H (cid:21) , ∆ V (cid:48) = (cid:20) −
16 (69 z c + 1098 z c + 1558 z c + 274 z c + 1)3( z c − H −
16 (35 z c + 686 z c + 1558 z c + 686 z c + 35)3( z c − H + 96 z c ( z c + 39 z c + 130 z c + 74 z c + 6)( z c − H + 32 ( z c + 45 z c + 204 z c + 204 z c + 45 z c + 1)( z c − H − z c + 3700 z c + 10442 z c + 3700 z c + 79)9( z c − H −
16 (35 z c + 686 z c + 1558 z c + 686 z c + 35)3( z c − H + 64 ( z c + 45 z c + 204 z c + 204 z c + 45 z c + 1)( z c − H − π
16 ( z c + 45 z c + 204 z c + 204 z c + 45 z c + 1)3( z c − H (cid:21) + iπ (cid:20) z c ( z c + 39 z c + 130 z c + 74 z c + 6)( z c − H − z c + 1098 z c + 1558 z c + 274 z c + 1)3( z c − H (cid:21) . (55)21 Phenomenological applications
In this section we perform an extensive phenomenological analysis of ¯ B ( s ) → D ( ∗ )+( s ) L − andΛ b → Λ + c L − decays in QCDF. Like before, L is a light meson from the set { π, ρ, K ( ∗ ) , a } .We take into account the expressions through to NNLO for the hard scattering kernels,and the most recent values for non-perturbative input parameters, which we specify be-low. We analyze the impact of the NNLO correction on the topological tree amplitude a ( D ( ∗ )+ L − ), and subsequently predict the branching ratios for the mesonic decays. Af-terwards, we perform tests of QCD factorization by considering suitably chosen ratiosof non-leptonic to either semi-leptonic or non-leptonic channels. Finally, we give thetheoretical predictions for baryonic decays. Here we collect in Table 1 the theoretical input parameters entering our numerical anal-ysis throughout this paper. They include the SM parameters such as the CKM matrixelements, quark masses, and the strong coupling constant, as well as the hadronic pa-rameters such as meson decay constants, transition form factors, and the Gegenbauermoments of light mesons. Three-loop running is used for α s throughout this paper. Fur-thermore, we use a two-loop relation between pole and MS mass to convert the top-quarkpole mass m pole t to the scale-invariant mass m t ( m t ) [75].For the B → D ( ∗ ) transition form factors, we adopt the parameterization proposed byCaprini, Lellouch, and Neubert (CLN) [76], with the relevant parameters extracted fromexclusive semileptonic b → c(cid:96)ν (cid:96) decays [66]. For the B s → D ( ∗ ) s transition form factors,on the other hand, we use the results obtained by QCD sum-rule techniques, assuming apolar dependence on q that is dominated by the nearest resonance [69, 77]. However, todiscuss the SU(3)-breaking effects in the form-factor and decay-constant ratios, we adoptthe most recent lattice QCD results for the ratios [70, 78] F B s → D s ( m π ) F B → D ( m π ) = 1 . ± . stat . ± . syst . ,F B s → D s ( m π ) F B → D ( m K ) = 1 . ± . stat . ± . syst . ,f K f π = 1 . ± . . (56)Neither of the form-factor ratios shows significant deviation from the U-spin symmetry.For the Λ b → Λ c transition form factors, we use the most recent high-precision latticeQCD calculation with 2 + 1 dynamical flavours [36]. Here the q dependence of the formfactors is parameterized in a simplified z expansion [79], modified to account for pion-mass and lattice-spacing dependence. All relevant formulas and input data can be foundin eq. (79) and Tables VII – IX of [36]. Following the procedure recommended in [36],we calculate the central value, statistical uncertainty, and total systematic uncertainty of22able 1: Summary of theoretical input parameters. The Gegenbauer moments of lightmesons are evaluated at µ = 1 GeV. QCD and electroweak parameters G F [10 − GeV − ] α s ( m Z ) m Z [GeV] m W [GeV]1 . . ± . . .
385 [62]
Quark masses [GeV] m pole t m pole b m pole c m t ( m t ) m b ( m b ) m c ( m c )173 . ± .
76 4 . ± .
06 1 . ± .
07 163 . ± .
72 4 . ± .
03 1 . ± .
025 [62, 63]
CKM matrix elements | V ud | | V us | | V cb | exclusive [10 − ]0 . ± . . ± . . ± . Lifetimes and masses of B d,s and Λ b τ B d [ps] τ B s [ps] τ Λ b [ps] m B d [MeV] m B s [MeV] m Λ b [MeV]1 . ± .
004 1 . ± .
004 1 . ± .
010 5279 .
61 5366 .
79 5619 .
51 [62, 66] B → D ( ∗ ) transition form factors F (1) | V cb | [10 − ] ρ R R R B → D . ± .
53 1 . ± .
054 – – – [66, 67] B → D ∗ . ± .
45 1 . ± .
026 1 . ± .
033 0 . ± .
020 0 . ± .
10 [66, 68] B s → D ( ∗ ) s transition form factors F + F A A A F (0) 0 . ± . . ± . . ± .
06 0 . ± .
01 0 . ± .
07 [69]M res [GeV] 6 . . . . . Light-meson decay constants and Gegenbauer moments π K ρ K ∗ a (1260) f L [MeV] 130 . ± . . ± . ± ± ±
10 [70–73] α L – − . ± .
04 – − . ± .
04 – α L . ± .
08 0 . ± .
08 0 . ± .
07 0 . ± . − . ± .
02 [71–74]23ny observable depending on the form-factor parameters according to eqs. (82) – (84) in[36]. Furthermore, we have also taken into account the correlation matrices between theform-factor parameters.The decay constants f π and f K are averaged over the two-flavour lattice QCD re-sults [70], while f ρ and f K ∗ are determined from experiments [71]. The light-mesonGegenbauer moments are determined by the QCD sum rule approach [71, 72] and thelattice QCD calculation [74]. For the hadronic inputs of the axial-vector meson a (1260),we use the results presented in ref. [73]. It is noted that the Gegenbauer moments areevaluated at µ = 1 GeV, and are evolved to the characteristic scale µ ∼ m b [59–61]. Weuse LL running of the Gegenbauer moments for the tree-level and the one-loop amplitude,but NLL running in the two-loop amplitude. Moreover, the running of the Gegenbauermoments is performed in the four-flavour scheme. a ( D ( ∗ )+ L − ) We are now in the position to perform a numerical analysis of the coefficients a ( D ( ∗ )+ L − )according to the expressions a ( D + L − ) = (cid:88) i =1 C i ( µ ) (cid:90) du (cid:104) ˆ T i ( u, µ ) + ˆ T (cid:48) i ( u, µ ) (cid:105) Φ L ( u, µ ) ,a ( D ∗ + L − ) = (cid:88) i =1 C i ( µ ) (cid:90) du (cid:104) ˆ T i ( u, µ ) − ˆ T (cid:48) i ( u, µ ) (cid:105) Φ L ( u, µ ) , (57)into which eqs. (44) and (45) have to be inserted. Using the NNLO Wilson coefficients C i ( µ ) in the CMM basis [42], together with the input parameters collected in Table 1,our final numerical results for a ( D + K − ) are given as a ( D + K − ) = 1 .
025 + [0 .
029 + 0 . i ] NLO + [0 .
016 + 0 . i ] NNLO = (1 . +0 . − . ) + (0 . +0 . − . ) i , (58)where the number without bracket is the LO contribution, which has no imaginary part,and the following two numbers are the NLO and NNLO terms, respectively. The totalerrors comprise the uncertainties, added in quadrature, from the variation of the scales µ ∈ [ m b / , m b ] and µ ∈ [ m W / , m W ], the quark masses, the Gegenbauer moments,and α s ( m Z ). Unless stated otherwise, the numbers given here and below are obtainedwith the b - and c -quark masses renormalized in the pole scheme, which is set as ourdefault scheme. It is observed that both the NLO and NNLO contributions add alwaysconstructively to the LO result. We also observe that the new two-loop correction is quitesmall in the real, but rather large in the imaginary part. It amounts to approximately60% (2%) of the total imaginary (real) part of a ( D + K − ). We emphasize that thesizable NNLO correction to the imaginary part does not indicate a breakdown of theperturbative expansion, but is due to the fact that the imaginary part vanishes at LO,and its NLO term is colour suppressed and proportional to the small Wilson coefficient24 (cid:236) ŁŁ ŁŁ Re @ a H D + K - LD I m @ a H D + K - L D LO NLO NNLO
Figure 3: Graphical representation of a ( D + K − ) in the complex plane at LO, NLO andNNLO. The theoretical error estimates are also indicated. C ( µ ). Moreover, the impact of the imaginary part on | a ( D + K − ) | is only marginal.Graphical representations of a ( D + K − ) are shown in figure 3 at LO, NLO and NNLO.Due to the truncation of the perturbative expansion, the obtained values in eq. (58)depend on the renormalization scale µ , which is usually considered as a measure of theaccuracy of the approximation at a given order in the perturbative expansion. This isshown in figure 4 for a ( D + K − ) up to NNLO, where results both in the pole (blue) andin the MS (red) scheme for b - and c -quark masses are given. We observe a pronouncedstabilization of the scale dependence for the real part, but not for the imaginary part. Thisis again explained by the fact that the imaginary part vanishes at LO. It is also observedthat the dependence on the b - and c -quark mass scheme is quite small, especially for thereal part. We finally remark that also within a given quark-mass scheme the dependenceof a ( D + K − ) on the value of z c is minor. The dependence of a ( D + K − ) on the secondGegenbauer moment is small, too.It is also interesting to mention that, even up to NNLO, the coefficients a ( D ( ∗ )+ L − )are quasi-universal, with very small process-dependent non-factorizable corrections, a factthat was observed already at NLO in ref. [13]. This is clearly seen from the followingnumerical results for different final states: a ( D + K − ) = (1 . +0 . − . ) + (0 . +0 . − . ) i ,a ( D + π − ) = (1 . +0 . − . ) + (0 . +0 . − . ) i ,a ( D ∗ + K − ) = (1 . +0 . − . ) + (0 . +0 . − . ) i ,a ( D ∗ + π − ) = (1 . +0 . − . ) + (0 . +0 . − . ) i . (59)25 m @ GeV D R e @ a H D + K - L D LONLONNLO m @ GeV D I m @ a H D + K - L D LONLONNLO
Figure 4: The dependence of the coefficient a ( D + K − ) on the renormalization scale µ both in the pole (blue) and in the MS (red) scheme for b - and c -quark masses. Dashed,dashed-dotted and solid lines represent the LO, NLO, and NNLO results, respectively. It is generally believed that the factorization theorem is well established in class-I decaysof the form ¯ B ( s ) → D ( ∗ )+( s ) L − , where the spectator anti-quark of the initial ¯ B ( s ) mesons isabsorbed only by the D ( ∗ )+( s ) mesons [13, 80]. We now present in Table 2 our predictionsfor the branching ratios of these decays through to NNLO. The explicit formulas for thebranching ratios can be found in [13] and shall not be repeated here. The experimen-tal data is taken from the Particle Data Group (PDG) [62] and/or the Heavy FlavorAveraging Group (HFAG) [66]. For the vector and axial-vector final states, the resultsrefer to the longitudinal polarization amplitudes only, with the longitudinal polarizationfractions taken from [81] for ¯ B d → D ∗ + ρ − and [82] for ¯ B s → D ∗ + s ρ − , respectively.From Table 2, one can see that our predictions for the branching ratios of these decaysgenerally come out higher than the experimental data, especially for ¯ B d → D ( ∗ )+ π − and¯ B d → D ( ∗ )+ ρ − decays, where the difference in central values is at the 20 – 30% level.Taking into account the uncertainties, the deviation is at the level of 2 – 3 σ . Comparedto ref. [13], which found at NLO rather good agreement between theory and experiment,essentially three things have changed: First, using the latest extraction from [66–68] ournumerical values for the form factors are about 10% larger than the ones used in [13].Second, the NNLO corrections add another positive shift of 2 – 3% on the amplitudelevel. Third, the experimental central values have slightly decreased since the analysis of[13]. All three effects shift theory and experiment further apart.Given the fact that the results show rough agreement within errors for ¯ B d → D ( ∗ )+ K ( ∗ ) − decays, which receive only contributions from colour-allowed tree topologies, this mayindicate a non-negligible impact from the W -exchange topologies appearing only in¯ B d → D ( ∗ )+ π − and ¯ B d → D ( ∗ )+ ρ − decays. For ¯ B s decays, on the other hand, sincethe B s → D ( ∗ ) s transition form factors have so far received only little theoretical at-tention [69, 83–88], especially by the lattice QCD community [78, 89], our theoreticalpredictions are still plagued by larger uncertainties due to these hadronic parameters.26able 2: CP-averaged branching ratios (in units of 10 − for b → c ¯ ud and 10 − for b → c ¯ us transitions) of ¯ B ( s ) → D ( ∗ )+( s ) L − decays. The vector- and axial-vector final states refer tothe longitudinal polarization amplitudes only. The theoretical errors shown correspond tothe uncertainties due to renormalization scales µ and µ , the CKM as well as the hadronicparameters, added in quadrature. The experimental data is taken from refs. [62,66,81,82].Decay mode LO NLO NNLO Exp.¯ B d → D + π − .
58 3 . +0 . − . . +0 . − . . ± . B d → D ∗ + π − .
15 3 . +0 . − . . +0 . − . . ± . B d → D + ρ − .
51 10 . +1 . − . . +1 . − . . ± . B d → D ∗ + ρ − .
45 8 . +0 . − . . +0 . − . . ± . B s → D + s π − .
00 4 . +1 . − . . +1 . − . . ± . B s → D ∗ + s π − .
05 2 . +0 . − . . +0 . − . . ± . B s → D + s ρ − .
31 10 . +3 . − . . +3 . − . . ± . B s → D ∗ + s ρ − .
86 6 . +1 . − . . +1 . − . . ± . B d → D + K − .
74 2 . +0 . − . . +0 . − . . ± . B d → D ∗ + K − .
37 2 . +0 . − . . +0 . − . . ± . B d → D + K ∗− .
79 5 . +0 . − . . +0 . − . . ± . B d → D ∗ + K ∗− .
30 4 . +0 . − . . +0 . − . –¯ B s → D + s K − .
05 3 . +1 . − . . +1 . − . –¯ B s → D ∗ + s K − .
53 1 . +0 . − . . +0 . − . –¯ B s → D + s K ∗− .
15 5 . +1 . − . . +1 . − . –¯ B s → D ∗ + s K ∗− .
02 3 . +0 . − . . +0 . − . –¯ B d → D + a − .
82 11 . +1 . − . . +1 . − . . ± . B d → D ∗ + a − .
12 10 . +1 . − . . +1 . − . –¯ B s → D + s a − .
23 11 . +3 . − . . +3 . − . –¯ B s → D ∗ + s a − .
44 7 . +1 . − . . +1 . − . –27 .4 Test of factorization To further test the factorization hypothesis in class-I decays of B -mesons into heavy-light final states, as well as to probe the non-factorizable corrections to the coeffi-cients a ( D ( ∗ )+ L − ), we now consider either ratios of non-leptonic to semi-leptonic decayrates [13, 17, 90, 91], or ratios of two non-leptonic decay rates [13, 91], both of which areessentially free of CKM and hadronic uncertainties.As suggested firstly by Bjorken [90], a particularly clean and direct method to testthe factorization hypothesis is provided by dividing the non-leptonic ¯ B d → D ( ∗ )+ L − decay rates by the corresponding differential semi-leptonic ¯ B d → D ( ∗ )+ (cid:96) − ¯ ν (cid:96) decay ratesevaluated at q = m L , where (cid:96) refers to either an electron or a muon, and q is thefour-momentum squared transferring to the lepton pair. In this way, the coefficients a ( D ( ∗ )+ L − ) can be extracted directly from experimental data through the relation [13,91] R ( ∗ ) L ≡ Γ( ¯ B d → D ( ∗ )+ L − ) d Γ( ¯ B d → D ( ∗ )+ (cid:96) − ¯ ν (cid:96) ) /dq | q = m L = 6 π | V ij | f L | a ( D ( ∗ )+ L − ) | X ( ∗ ) L , (60)where V ij is, depending on the constituent quark content of the meson L , the appropriateCKM matrix element. With the light lepton mass neglected, X L = X ∗ L = 1 for a vector oraxial-vector meson, whereas for a pseudoscalar X ( ∗ ) L deviates from unity only by calculableterms of order m L /m B , which are numerically below the percent level; explicit expressionsfor X ( ∗ ) L can be found, for example, in ref. [91]. To get the differential semi-leptonic decayrates at q = m L in eq. (60), we use the CLN parameterization for the B → D ( ∗ ) transitionform factors [76], with the relevant parameters summarized in Table 1. Explicitly, we getnumerically (in units of 10 − GeV − ps − ) d Γ( ¯ B d → D ( ∗ )+ (cid:96) − ¯ ν (cid:96) ) dq (cid:12)(cid:12)(cid:12)(cid:12) q = m L = . +0 . − . (2 . ± . , for L = π − . +0 . − . (2 . ± . , for L = ρ − . +0 . − . (2 . ± . , for L = K − . +0 . − . (2 . ± . , for L = K ∗− . +0 . − . (2 . ± . , for L = a − . (61)Together with the data on the branching ratios of non-leptonic decays given in Table 2,we arrive at the experimental values for | a ( D ( ∗ )+ L − ) | collected in Table 3, where, forcomparison, our theoretical predictions at different orders are also shown.From Table 3, one can see clearly that our theoretical predictions based on theQCDF approach result in an essentially universal value of | a ( D ( ∗ )+ L − ) | (cid:39) .
07 (1 .
05) atNNLO (NLO), being consistently higher than the central values favoured by the currentexperimental data. The deviation is again at the level of 2 – 3 σ . Similar results wereobtained in [17], yet without inclusion of the NNLO correction. It would be, therefore,very encouraging to determine directly the ratios of non-leptonic and semi-leptonic de-cay rates at current and future experiments such as LHCb and Belle II. Compared tothe NLO analysis in [13], where theory predictions for | a ( D ( ∗ )+ L − ) | were found to be28able 3: Theoretical predictions for | a ( D ( ∗ )+ L − ) | at different orders in perturbationtheory. For comparison, the coefficients | a ( D ( ∗ )+ L − ) | determined from current dataare shown in the last column. The experimental errors are estimated by adding theuncertainties of the non-leptonic branching ratios and the semi-leptonic decay rates inquadrature, while the uncertainties from the decay constants are not taken into account. | a ( D ( ∗ )+ L − ) | LO NLO NNLO Exp. | a ( D + π − ) | .
025 1 . +0 . − . . +0 . − . . ± . | a ( D ∗ + π − ) | .
025 1 . +0 . − . . +0 . − . . ± . | a ( D + ρ − ) | .
025 1 . +0 . − . . +0 . − . . ± . | a ( D ∗ + ρ − ) | .
025 1 . +0 . − . . +0 . − . . ± . | a ( D + K − ) | .
025 1 . +0 . − . . +0 . − . . ± . | a ( D ∗ + K − ) | .
025 1 . +0 . − . . +0 . − . . ± . | a ( D + K ∗− ) | .
025 1 . +0 . − . . +0 . − . . ± . | a ( D + a − ) | .
025 1 . +0 . − . . +0 . − . . ± . B ( s ) → D ( ∗ )+( s ) L − decay rates, fol-lowing the notations used in refs. [13, 91]. As a quasi-universal | a ( D ( ∗ )+ L − ) | is predictedin the QCDF approach, these ratios could be used to test the factorization hypothe-sis, as well as the SU(3) relations in B -meson decays into heavy-light final states [17].Our results of such an analysis are presented in Table 4, where the experimental data isobtained using the corresponding branching fractions collected in Table 2.From Table 4, one can see that, within the errors, our theoretical predictions aregenerally consistent with the current experimental data, indicating therefore no evidencefor any significant deviation from the factorization hypothesis for these class-I B -mesondecays into heavy-light final states. The last two ratios shown in Table 4 could also beused to determine the ratio of fragmentation functions f d /f s , a key quantity for precisemeasurements of absolute B s -meson decay rates at hadron colliders [17, 92].One possible interpretation of our findings that, on the one hand, the non-leptonic tosemi-leptonic ratios come out larger in theory compared to experiment, and on the otherhand the non-leptonic ratios in general agree with experiment, might be non-negligiblepower corrections which could be negative in sign and 10 – 15% in size on the amplitude29able 4: Predictions for the ratios of non-leptonic ¯ B ( s ) → D ( ∗ )+( s ) L − decay rates atdifferent orders. The experimental data is obtained using the corresponding branchingfractions collected in Table 2.Ratios LO NLO NNLO Exp. Br( ¯ B d → D ∗ + π − )Br( ¯ B d → D + π − ) .
880 0 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D + ρ − )Br( ¯ B d → D + π − ) .
654 2 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D + ρ − )Br( ¯ B d → D ∗ + π − ) .
016 3 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D ∗ + K − )Br( ¯ B d → D + K − ) .
865 0 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D + K ∗− )Br( ¯ B d → D + K − ) .
747 1 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D + K ∗− )Br( ¯ B d → D ∗ + K − ) .
019 2 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D + K − )Br( ¯ B d → D + π − ) .
077 0 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D ∗ + K − )Br( ¯ B d → D ∗ + π − ) .
075 0 . +0 . − . . +0 . − . . ± . Br( ¯ B d → D + K ∗− )Br( ¯ B d → D + ρ − ) .
050 0 . +0 . − . . +0 . − . . ± . Br( ¯ B s → D + s π − )Br( ¯ B d → D + K − ) .
67 14 . +1 . − . . +1 . − . . ± . Br( ¯ B s → D + s π − )Br( ¯ B d → D + π − ) .
120 1 . +0 . − . . +0 . − . . ± . D -meson wave function and on the value of thefirst inverse moment λ B of the B -meson distribution amplitude, these two contributionscould in principle interfere constructively, and in this case their total effect could indeedadd up to −
10% in the amplitude.Another possibility which has essentially the same effect would be to reduce the valuesof | V cb | times the form factors by ∼ | V cb | and the form factors cancelout are in very good agreement with experiment. On the other hand, the semi-leptonicrate is measured very precisely and the current form factors times | V cb | are extracted byHFAG [66] from a global fit to all available data, whose result we quote in Table 1. Hencethey are optimized to describe the shape of the semi-leptonic rate and therefore shouldbe trustworthy. One could even conclude from this that the experimental extraction of30able 5: Predictions for the branching fractions (in units of 10 − for b → c ¯ ud and 10 − for b → c ¯ us transitions) of Λ b → Λ + c L − decays, as well as some ratios between them. Theexperimental data is taken from PDG [62] and HFAG [66].Decay mode LO NLO NNLO Exp.Λ b → Λ + c π − .
60 2 . +0 . − . . +0 . − . . +0 . − . Λ b → Λ + c ρ − .
46 7 . +1 . − . . +1 . − . –Λ b → Λ + c a − .
57 10 . +1 . − . . +1 . − . –Λ b → Λ + c K − .
02 2 . +0 . − . . +0 . − . . ± . b → Λ + c K ∗− .
86 4 . +0 . − . . +0 . − . – Br(Λ b → Λ + c µ − ¯ ν )Br(Λ b → Λ + c π − ) .
88 17 . +2 . − . . +2 . − . . +4 . − . b → Λ + c K − )Br(Λ b → Λ + c π − ) (%) 7 .
77 7 . +0 . − . . +0 . − . . ± . Br(Λ b → Λ + c π − )Br( ¯ B d → D + π − ) .
73 0 . +0 . − . . +0 . − . . ± . | a ( D ( ∗ )+ L − ) | from eq. (60) is independent of the product of | V cb | and the form factor.We emphasize that without a rigorous treatment of power corrections in the QCDFapproach nothing more can be said at the present stage. In any case, the QCDF approachper se is not invalidated. Λ b → Λ + c L − decays While the Λ b baryons are not produced at an e + e − B -factory, they account for about20% of the b -hadrons produced at the LHC [93]. Remarkably, the number of Λ b baryonsproduced is comparable to the number of B u or B d mesons, and is significantly higherthan the number of B s mesons. Due to the half-integer-spin of Λ b , its decays providecomplementary information compared to the corresponding mesonic ones. Therefore,this may open up a new field for flavour physics. For a review, see e.g. refs. [94–97].Here we study the two-body non-leptonic Λ b → Λ + c L − decays, for which the factorizationassumption is believed to be reliable [37, 98–101]. As demonstrated especially in ref. [37],the proof of factorization at leading order in Λ QCD /m b,c for these decays follows closelythat for ¯ B d → D ( ∗ )+ π − [80]. These decays provide, therefore, a testing ground for differentQCD models and factorization assumptions used in B -meson case. It is straightforwardto generalize the expressions in [37] to take radiative corrections through to NNLO intoaccount.Using the most recent lattice QCD results for Λ b → Λ c transition form factors [36], wepresent in Table 5 our predictions for the branching fractions of Λ b → Λ + c L − decays, as31ell as some ratios between them, where the experimental data is taken from HFAG [66].From Table 5, one can see that, contrary to the observation made in mesonic decays, ourpredictions for the branching ratios of these decays now come out lower than the exper-imental data; especially the higher-order corrections always increase the LO predictionsand shift our predictions closer to the experimental data. Our predictions for the tworatios Br(Λ b → Λ + c µ − ¯ ν ) / Br(Λ b → Λ + c π − ) and Br(Λ b → Λ + c K − ) / Br(Λ b → Λ + c π − ) areboth consistent with the current data, indicating that the non-factorizable effects shouldbe small in these decays. Moreover, we emphasize the fact that the non-leptonic to semi-leptonic ratio in the baryonic case is consistent with experiment, but shows a tensionin the mesonic case (see section 6.4). The discrepancy between our prediction and thecurrent experimental data for Br(Λ b → Λ + c π − ) / Br( ¯ B d → D + π − ) makes it interesting toevaluate directly the form-factor ratios of Λ b → Λ c and B → D transitions by the latticecommunity. We have calculated the NNLO vertex corrections to the colour-allowed tree topologyin the framework of QCDF for the mesonic decays ¯ B ( s ) → D ( ∗ )+( s ) L − and the baryonicdecays Λ b → Λ + c L − , with L = { π, ρ, K ( ∗ ) , a } . The calculation of the two-loop correctionto the hard scattering kernels requires the evaluation of several dozens of genuine two-scale Feynman diagrams, which describe these heavy-to-heavy transitions at the quarklevel. We performed this calculation by means of techniques that have become standardin the business of multi-loop computations. It might be worth noting that we evaluatedall master integrals analytically [33] in a so-called canonical basis [50], a result whichcatalyzed the convolution with the LCDA and enabled us to obtain the convoluted kernelsalmost completely analytically.The NNLO contributions yield a positive shift to the colour-allowed tree amplitude a , which is sizable for its imaginary part, but small for its real part and its magnitude.Moreover, the amplitude only mildly depends on the ratio of the heavy-quark masses z c = m c /m b . The dependence on the factorization scale gets reduced for the real partcompared to the NLO result. This reduction does not occur in the imaginary parts,which is expected, as the latter only arise beyond LO. We performed our analysis usingthe pole scheme for the heavy-quark masses. A change to the MS scheme does not showany significant shift of the amplitude within the range of physical values for the heavy-quark masses. Moreover, the results for the different final states only slightly dependon the light meson LCDA and hence, we can confirm the quasi-universality of the treeamplitude to NNLO accuracy.In our phenomenological analysis we evaluated the branching ratios to NNLO accu-racy, and with the latest values for the non-perturbative input parameters. We find thatfor ¯ B d decays the central values of the theoretical predictions are in general higher com-pared to the experimental values. Within the given uncertainties the quantities agree atthe 2 – 3 σ level for π and ρ in the final state, and slightly better for K and K ∗ . Compared32o the analysis at NLO [13], our increased values for the form factors and the amplitude,together with decreased experimental values, have shifted theory and experiment furtherapart. For ¯ B s decays, the theory predictions are still plagued by large uncertainties whichare mainly due to poorly known form factors. For the baryonic decays, on the other hand,the predicted branching fractions turn out to be 20 −
30% smaller than the experimentalones. It would be interesting to understand the reason for this difference in the ¯ B d andthe Λ b decays. We therefore propose a systematic analysis of factorization for Λ b decaysin the future.Moreover, we analyzed ratios of non-leptonic and semi-leptonic decay rates in orderto further probe the factorization theorem to NNLO. The ratios of different non-leptonicrates turn out to be in good agreement, comparing theoretical prediction and experiment.In the case of non-leptonic to semi-leptonic ratios, on the other hand, the values for | a ( D ( ∗ )+ L − ) | that we extract from experiment are lower by 2 – 3 σ compared to theNNLO theory predictions (see also [17]).One possibility to interpret the entity of these results could be non-negligible powercorrections. Given the uncertainties of the branching ratios and a they could be negativein sign and 10 – 15% in size on the amplitude level. They could cure the non-leptonic tosemi-leptonic ratios, without destroying the agreement in the non-leptonic ratios, espe-cially if they were of a certain universality. It will also be very interesting to investigatewhat this would imply for the power corrections in charmless non-leptonic decays. Recentanalyses that address weak annihilation in charmless non-leptonic decays can be foundin [102–104].Another, yet less favourable option would be reduced values of | V cb | times the formfactors. As stated already in section 6, without a rigorous treatment of power correctionsin the QCDF approach nothing more can be said at the present stage. Acknowledgements
We would like to thank Martin Beneke and Oleg Tarasov for collaboration at an initialstage of this project. Moreover, we would like to thank Guido Bell, Martin Beneke,Thorsten Feldmann, and Bj¨orn O. Lange for very helpful discussions, and Guido Belland Martin Beneke for comments on the manuscript. This work was supported in partby the NNSFC of China under contract Nos. 11675061 and 11435003 (XL), and by DFGForschergruppe FOR 1873 “Quark Flavour Physics and Effective Field Theories” (THand SK). XL is also supported in part by the SRF for ROCS, SEM, and by the self-determined research funds of CCNU from the colleges’ basic research and operation ofMOE (CCNU15A02037). XL acknowledges hospitality from Siegen University during thefinal stages of this work. 33 eferences [1] M. Artuso et al. , Eur. Phys. J. C (2008) 309 [arXiv:0801.1833 [hep-ph]].[2] M. Antonelli et al. , Phys. Rept. (2010) 197 [arXiv:0907.5386 [hep-ph]].[3] R. Aaij et al. 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