NNLO vertex corrections in charmless hadronic B decays: Real part
aa r X i v : . [ h e p - ph ] F e b TTP09-02SFB/CPP-09-14
NNLO vertex corrections in charmlesshadronic B decays: Real part Guido Bell Institut f¨ur Theoretische Teilchenphysik,Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany
Abstract
We compute the real part of the 2-loop vertex corrections for charmless hadronic B decays, completing the NNLO calculation of the topological tree amplitudes inQCD factorization. Among the technical aspects we show that the hard-scatteringkernels are free of soft and collinear infrared divergences at the 2-loop level, whichfollows after an intricate subtraction procedure involving evanescent four quarkoperators. The numerical impact of the considered corrections is found to be mod-erate, whereas the factorization scale dependence of the topological tree amplitudesis significantly reduced at NNLO. We in particular do not find an enhancement ofthe phenomenologically important ratio | C/T | from the perturbative calculation. E-mail:[email protected]
Introduction
The study of hadronic B meson decays into a pair of light (charmless) mesons revealsinteresting information about the underlying four quark interactions and the related phe-nomenon of CP violation. While these decay modes are intensively investigated at currentand future B physics experiments, the main challenge for precise theoretical predictionsconsists in the computation of the hadronic matrix elements. QCD factorization [1], orits field theoretical formulation in the language of Soft-Collinear Effective Theory [2],is a systematic framework to compute these matrix elements from first principles. Thestarting point is a factorization formula, which holds in the heavy quark limit m b → ∞ , h M M | Q i | ¯ B i ≃ F BM + (0) f M Z du T Ii ( u ) φ M ( u )+ ˆ f B f M f M Z dωdvdu T IIi ( ω, v, u ) φ B ( ω ) φ M ( v ) φ M ( u ) , (1)where the perturbatively calculable hard-scattering kernels T I,IIi encode the short-distancestrong-interaction effects and the non-perturbative physics is confined to some process-independent hadronic parameters such as decay constants f M , light-cone distributionamplitudes φ M and a transition form factor F BM + at maximum recoil q = 0.In this work we address perturbative corrections to the factorization formula (1).Whereas next-to-leading order (NLO) corrections to the hard-scattering kernels T I,IIi areknown from the pioneering work in [1], partial next-to-next-to-leading order (NNLO)corrections have recently been worked out [3, 4, 5, 6]. The α s corrections to the kernels T IIi ( spectator scattering ) are by now completely determined to NNLO: the corrections forthe topological tree amplitudes have been computed in [3] and the ones for the so-calledpenguin amplitudes in [4].In contrast to this the computation of α s corrections to the kernels T Ii ( vertex cor-rections ) is to date incomplete. Whereas we computed the imaginary parts of the hard-scattering kernels for the topological tree amplitudes in [5, 6], we complete the NNLOcalculation of the tree amplitudes in this work by computing the respective real parts.Partial results of this calculation, in particular the analytical expressions of the required2-loop Master Integrals, have already been given in [6].The organization of this paper is as follows: The technical aspects of the NNLOcalculation are presented in Section 2. We start by briefly recalling our definitions andconventions and make some remarks concerning the computation of the 2-loop diagrams.We then show in some detail how to extract the hard-scattering kernels from the matrixelements which are formally infrared divergent. This subtraction procedure, which be-comes particularly involved for the colour-suppressed tree amplitude, is complicated dueto the presence of evanescent four quark operators which arise in intermediate steps ofthe calculation. Our analytical results for the hard-scattering kernels are summarized inSection 3. We briefly discuss the numerical impact of the considered NNLO correctionsin Section 4, before we conclude in Section 5. Several technical issues of the calculationand the explicit expressions of the hard-scattering kernels are relegated to the Appendix.1 NNLO calculation
The calculation of the real parts of the topological tree amplitudes proceeds along thesame lines as the one of the imaginary parts that we presented in [5]. Still, the currentcalculation turns out to be considerably more complex in several respects. First, itrequires the calculation of a larger amount of 2-loop integrals, which are in additionmore complicated since they involve up to three (instead of one) massive propagators.Second, the renormalization procedure and the infrared subtractions reveal their full 2-loop complexity only in the current calculation as a consequence of the fact that thetree level contribution is real. In the following we summarize the technical aspects ofthe calculation and refer for a more detailed description of the general strategy to [5](cf. also [6]).
The topological tree amplitudes can be derived from the hadronic matrix elements of thecurrent-current operators in the effective weak Hamiltonian H eff = G F √ V ∗ ud V ub ( C Q + C Q ) + h.c. (2)As we apply Dimensional Regularization (DR) to regularize ultraviolet (UV) and infrared(IR) singularities, evanescent four-quark operators appear in intermediate steps of thecalculation. The full operator basis required for the present calculation becomes Q = (cid:2) ¯ uγ µ L T A b (cid:3) (cid:2) ¯ dγ µ L T A u (cid:3) ,Q = [¯ uγ µ L b ] (cid:2) ¯ dγ µ L u (cid:3) ,E = (cid:2) ¯ uγ µ γ ν γ ρ L T A b (cid:3) (cid:2) ¯ dγ µ γ ν γ ρ L T A u (cid:3) − Q ,E = [¯ uγ µ γ ν γ ρ L b ] (cid:2) ¯ dγ µ γ ν γ ρ L u (cid:3) − Q ,E ′ = (cid:2) ¯ uγ µ γ ν γ ρ γ σ γ τ L T A b (cid:3) (cid:2) ¯ dγ µ γ ν γ ρ γ σ γ τ L T A u (cid:3) − E − Q ,E ′ = [¯ uγ µ γ ν γ ρ γ σ γ τ L b ] (cid:2) ¯ dγ µ γ ν γ ρ γ σ γ τ L u (cid:3) − E − Q , (3)with colour matrices T A and L = 1 − γ . We stress that previous studies within QCDfactorization, as e.g. [1, 3, 4], have often been formulated in a different operator basiswith a Fierz-symmetric definition of the physical operators. As has been argued in [5], itis more convenient for the current calculation to use the operator basis (3) since it allowsto work with a naive anticommuting γ beyond NLO [7].There are two different insertions of a four-quark operator which are illustrated inFigure 1 of [5]. The first one gives rise to the colour-allowed tree amplitude α ( M M ),which corresponds to the flavour content [¯ q s b ] of the decaying ¯ B meson, [¯ q s u ] of the recoil We write d = 4 − ε and use an anticommuting γ according to the NDR scheme. This operator basis has been named
CMM basis in [5] (denoted by a hat). M and [¯ ud ] of the emitted meson M . The colour-suppressed tree amplitude α ( M M ) follows from the second insertion and belongs to the flavour contents [¯ q s b ],[¯ q s d ] and [¯ uu ], respectively. In [5] we did not consider the second type of insertions sincewe could derive the imaginary part of the colour-suppressed amplitude from the one ofthe colour-allowed amplitude using Fierz-symmetry arguments .In the current calculation we cannot proceed along the same lines, since a Fierz-symmetric operator basis has not yet been worked out to NNLO . We therefore considerboth types of insertions in this work, which also provides an independent cross-check ofour previous result for the imaginary part of the colour-suppressed tree amplitude. The main task of the calculation consists in the computation of a large number of 2-loopdiagrams (shown in Figure 2 of [5]). We use an automatized reduction algorithm, whichis based on integration-by-parts techniques [11], to express these diagrams in terms of anirreducible set of Master Integrals (MIs). In addition to the MIs that appeared in thecalculation of the imaginary part of the NNLO vertex corrections (cf. Figure 3 of [5]), wefind 22 MIs which are shown in Figure 1. In total the current calculation requires thecomputation of 36 MIs to up to five orders in the ε -expansion.Apart from the MIs that involve the charm quark mass, the analytical results for theMIs from Figure 1 can be found in [6] . The MIs can be expressed in terms of HarmonicPolylogarithms (HPLs) [14] of weight w ≤ H (0; x ) = ln( x ) , H (0 , , x ) = Li ( x ) ,H (1; x ) = − ln(1 − x ) , H (0 , , x ) = S , ( x ) ,H ( − x ) = ln(1 + x ) , H (0 , , , x ) = Li ( x ) ,H (0 , x ) = Li ( x ) , H (0 , , , x ) = S , ( x ) ,H (0 , − x ) = − Li ( − x ) , H (0 , , , x ) = S , ( x ) .H ( − , , x ) ≡ H ( x ) , H (0 , − , , x ) ≡ H ( x ) , (4)where we introduced a shorthand notation for the last two HPLs . Moreover, the massivenon-planar 6-topology MI (last diagram from Figure 1) involves a constant in the finiteterm which, until recently, was only known numerically, C = − . C = − π /
270 [16]. To do so we introduced a second operator basis named traditional basis in [5] (denoted by a tilde). We emphasize that the operator basis from Section 8 in [8] is not Fierz-symmetric and the one fromAppendix A in [9] is presumably not either [10]. Part of these results have recently been confirmed by various groups [12, 13]. The explicit expression of H ( x ) in terms of Nielsen Polylogarithms can be found e.g. in equation(10) of [15]. On the other hand H ( x ) has to be evaluated numerically (in Section 3.2 we find, however,analytical expressions in the convolutions with the light-cone distribution amplitude of the meson M ). Figure 1:
Additional Master Integrals that appear in the calculation of the realparts of the NNLO vertex corrections. Dashed/double/wavy internal lines denotepropagators with mass / m b / m c . Dashed/solid/double external lines correspond tovirtualities / um b / m b . Dotted propagators are taken to be squared. The charm mass dependent MIs can be found in [6, 15]. In this case there existanalytical results apart from the finite terms of two 4-topology MIs. We may, however,evaluate these contributions numerically to implement charm mass effects in the currentanalysis.
The calculation of the renormalized matrix elements requires standard counterterms fromQCD and the effective Hamiltonian. We write the renormalized matrix elements as h Q i i = Z ψ Z ij h Q j i bare , (5)where Z ψ contains the wave-function renormalization factors of the quark fields and Z isthe operator renormalization matrix in the effective theory. Here and below we introducea shorthand notation for the perturbative expansions, h Q i i (bare) = ∞ X k =0 (cid:16) α s π (cid:17) k h Q i i ( k )(bare) , Z ij = δ ij + ∞ X k =1 (cid:16) α s π (cid:17) k Z ( k ) ij . (6)It turns out that the wave-function renormalization factors in Z ψ can be neglected in thecalculation of the hard-scattering kernels since they are absorbed by the form factor andthe light-cone distribution amplitude in the factorization formula, which are defined in4erms of full QCD fields (rather than HQET or SCET fields), for details cf. Section 4.2of [5]. We renormalize the coupling constant in the MS-scheme, Z (1) g = − (cid:18) C A − n f (cid:19) ε , (7)and the b -quark mass in the on-shell scheme, Z (1) m = − C F (cid:18) e γ E µ m b (cid:19) ε Γ( ε ) 3 − ε − ε . (8)The 1-loop and 2-loop MS operator renormalization matrices can be inferred from [8, 17] Z (1) = −
43 512 29 ! ε ,Z (2) = − n f − + n f − + n f − + n f −
39 + 2 n f − + n f
524 19 ! ε + + n f − + n f − n f − − n f − + n f − n f − − ! ε , (9)where the lines refer to the physical operators and the columns to the full operator basisincluding the evanescent operators from (3). In order to extract the hard-scattering kernels T i we rewrite the renormalized matrixelements in the factorized form h Q i i = F · T i ⊗ Φ + . . . (10)where F denotes the form factor, Φ the product of decay constant and distributionamplitude, ⊗ the convolution integral and the ellipsis the spectator scattering term whichwe disregard in the following. As has been discussed in detail in Section 4.2 of [5], onlynaively non-factorizable (nf) 1-loop diagrams contribute to the NLO kernels, h Q i i (1)nf + Z (1) ij h Q j i (0) = F (0) · T (1) i ⊗ Φ (0) . (11)Similarly, the calculation of the NNLO kernels involves only non-factorizable 2-loop dia-grams (but factorizable (f) 1-loop diagrams), h Q i i (2)nf + Z (1) ij h h Q j i (1)nf + h Q j i (1)f i + Z (2) ij h Q j i (0) = F (0) · T (2) i ⊗ Φ (0) + F (1)amp · T (1) i ⊗ Φ (0) + F (0) · T (1) i ⊗ Φ (1)amp , (12)5here the subscript ”amp” (amputated) has been introduced to denote corrections with-out wave-function renormalization. We see that the calculation of the NNLO kernels re-quires the NLO kernels to O ( ε ) as they enter (12) in combination with the IR-divergentform factor correction F (1)amp ∼ /ε . As a consequence the factorization formula hasto be extended in intermediate steps of the calculation to include evanescent operators,which have to be renormalized such that their (IR-finite) matrix elements vanish (fordetails cf. Section 4.3 of [5]).At NNLO the subtraction procedure becomes somewhat involved. It is particularlycomplicated in the calculation of the colour-suppressed tree amplitude, where a Fierz-evanescent operator appears at tree level. In the following we discuss the subtractionprocedure in some detail. Throughout this section we concentrate on the real parts ofthe hard-scattering kernels, since the respective imaginary parts have already been givenin [5]. We refer to Appendix A for the explicit expressions of the auxiliary coefficientfunctions t i ( u ) that we introduce below. Colour-allowed tree amplitude
To NNLO we find three operators that contribute to the right hand side of (10). Inthe position space representation they correspond to products of a local heavy-to-lightcurrent ¯ u ( x )Γ b ( x ) and a non-local light-quark current ¯ d ( y )[ y, x ]Γ u ( x ), where the usualgauge link factor [ y, x ] is understood. We choose the basis of Dirac structures Γ ⊗ Γ as O = [ γ µ L ] ⊗ [ γ µ L ] , O E = [ γ µ γ ν γ ρ L ] ⊗ [ γ µ γ ν γ ρ L ] − O , O E ′ = [ γ µ γ ν γ ρ γ σ γ τ L ] ⊗ [ γ µ γ ν γ ρ γ σ γ τ L ] − O E − O , (13)such that the factorized hadronic matrix element of O gives the standard QCD formfactor and the light-cone distribution amplitude of the emitted meson M . The operators O E and O E ′ are evanescent.We first compute (11) to O ( ε ) to determine the NLO kernels. We find that thecolour-singlet kernels vanish, T (1)2 = T (1)2 ,E = T (1)2 ,E ′ = 0, while the colour-octet kernelsbecomeRe T (1)1 ( u ) = C F N c (cid:18) µ m b (cid:19) ε (cid:26) t ( u ) − L + (cid:16) t ( u ) + 3 L (cid:17) ε + (cid:16) t ( u ) − L (cid:17) ε + O ( ε ) (cid:27) , Re T (1)1 ,E ( u ) = − C F N c (cid:18) µ m b (cid:19) ε (cid:26) t E, ( u ) + 2 L + (cid:16) t E, ( u ) − L (cid:17) ε + O ( ε ) (cid:27) , (14)and T (1)1 ,E ′ = 0 with L = ln µ /m b . The IR subtractions on the right hand side of (12)require in addition form factor and wave function corrections to the operators O and O E (they can be found in Section 4.3 of [5]). We finally perform the convolutions of the NLO We do not consider colour-octet operators since their hadronic matrix elements vanish. F (0) Re T (1)1 Φ (1)amp = C F N c (cid:18) µ m b (cid:19) ε (cid:26) t ( u ) ε + t ( u ) + O ( ε ) (cid:27) F (0) Φ (0) (15)and an additional µ -dependent contribution to the physical kernel from F (0) E Re T (1)1 ,E Φ (1)amp ,E → C F N c (cid:26) L + t E, ( u ) + O ( ε ) (cid:27) F (0) Φ (0) . (16) Colour-suppressed tree amplitude
In this case we find an analogous set of operators,˜ O = [ γ µ L ] ˜ ⊗ [ γ µ L ] , ˜ O E = [ γ µ γ ν γ ρ L ] ˜ ⊗ [ γ µ γ ν γ ρ L ] −
16 ˜ O , ˜ O E ′ = [ γ µ γ ν γ ρ γ σ γ τ L ] ˜ ⊗ [ γ µ γ ν γ ρ γ σ γ τ L ] −
20 ˜ O E −
256 ˜ O , (17)but the fields are now given in the wrong ordering ¯ u ( y )[ y, x ]Γ b ( x ) and ¯ d ( x )Γ u ( x ) (indi-cated by ˜ ⊗ ), which does not yield a form factor and a light-cone distribution amplitude.The latter follow from the factorized hadronic matrix element of the operator O = ¯ d ( x ) γ µ Lb ( x ) ⊗ ¯ u ( y )[ y, x ] γ µ Lu ( x ) , (18)which is the Fierz-symmetric counterpart of ˜ O . We therefore extend the right hand sideof (10) to include four operators in this case: the physical operator O , the evanescentoperators ˜ O E and ˜ O E ′ and the Fierz-evanescent operator ˜ O F ≡ ˜ O − O .The IR subtractions turn out to be particularly complicated in this case, due tothe fact that the evanescent operator ˜ O F already appears in the tree level calculation.As a consequence the naive split-up into non-factorizable diagrams, which contribute tothe hard-scattering kernels, and factorizable diagrams, which give form factor and wavefunction corrections, is spoiled. In NLO we find that equations (30) and (31) of [5] shouldbe replaced by h ˆ Q i i (1)nf + Z (1) ij h ˆ Q j i (0) = ˆ F (0) · ˆ T (1) i ⊗ ˆΦ (0) + ˆ∆ (1) F,i , h ˆ Q i i (1)f + Z (1) ψ h ˆ Q i i (0) = ˆ F (1) · ˆ T (0) i ⊗ ˆΦ (0) + ˆ F (0) · ˆ T (0) i ⊗ ˆΦ (1) − ˆ∆ (1) F,i , (19)where ˆ∆ (1) F,i contains the (non-vanishing) 1-loop counterterms of the form factor and wavefunction corrections for the Fierz-evanescent operator ˜ O F . In other words, the split-upin the above example of the colour-allowed tree amplitude followed from the fact thatthe corresponding counterterms vanish for the physical operator O (i.e. ∆ ( k ) i = 0).From the first equation in (19) we see that we can neglect the factorizable 1-loopdiagrams in the computation of the NLO kernels. In order to account for the counterterm We introduce the ”hat” notation to distinguish these quantities from those of the preceding section. (1)
F,i , we compute the UV-divergences of the 1-loop diagrams from Figure 5and 6 of [5] with an insertion of the Fierz-evanescent operator ˜ O F . We find that thecounterterms (ct) are given byˆ F (1) F ˆΦ (0) F (cid:12)(cid:12) ct ( u ) = C F (cid:26) ˆ F (0) ˆΦ (0) ( u ) + 14 ε ˆ F (0) E ˆΦ (0) E ( u ) (cid:27) , ˆ F (0) F ˆΦ (1) F (cid:12)(cid:12) ct ( u ) = 2 C F Z dw V E ( u, w ) (cid:26) ˆ F (0) ˆΦ (0) ( w ) + 14 ε ˆ F (0) E ˆΦ (0) E ( w ) (cid:27) , (20)with V E ( u, w ) from equation (45) of [5]. Convoluting these expressions with the LOkernels, ˆ T (0)1 ,F = C F /N c and ˆ T (0)2 ,F = 1 /N c , yields the additional counterterm contributionsˆ∆ (1) F, = C F ˆ∆ (1) F, = 2 C F N c (cid:26) ˆ F (0) ˆΦ (0) + 14 ε ˆ F (0) E ˆΦ (0) E (cid:27) . (21)With this prescription the NLO kernels turn out to be free of IR-singularities. Evaluatingthe first equation of (19) to O ( ε ) gives (in terms of T (1)1 from (14)),ˆ T (1)1 ( u ) + C F = − ˆ T (1)2 ( u )2 N c = − T (1)1 ( u ) N c − C F N c (cid:18) µ m b (cid:19) ε (cid:26)(cid:16) ˆ t ( u ) + 2 L (cid:17) ε + (cid:16) ˆ t ( u ) − L (cid:17) ε + O ( ε ) (cid:27) , ˆ T (1)1 ,E ( u ) = − ˆ T (1)2 ,E ( u )2 N c = C F N c (cid:18) µ m b (cid:19) ε (cid:26) L + ˆ t E, ( u ) + (cid:16) ˆ t E, ( u ) − L (cid:17) ε + O ( ε ) (cid:27) , ˆ T (1)1 ,F ( u ) = − ˆ T (1)2 ,F ( u )2 N c = − T (1)1 ( u ) N c − C F N c (cid:18) µ m b (cid:19) ε (cid:26) t ( u ) ε + O ( ε ) (cid:27) . (22)We next compute form factor and wave function corrections to O , ˜ O E and ˜ O F . Proceedingalong the lines outlined in Section 4.3 of [5], we obtain ˆ F (1)amp,E ˆΦ (0) E = C F (cid:20) − (cid:18) e γ E µ m b (cid:19) ε Γ( ε ) 24 ε (1 + ε )(1 − ε ) (cid:21) ˆ F (0) ˆΦ (0) − C F (cid:20) ε + (cid:18) e γ E µ m b (cid:19) ε Γ( ε ) 1 − ε + 16 ε − ε ε (1 − ε )(1 − ε ) (cid:21) ˆ F (0) E ˆΦ (0) E + C F (cid:20) ε − (cid:18) e γ E µ m b (cid:19) ε Γ( ε )4(1 − ε ) (cid:21) ˆ F (0) E ′ ˆΦ (0) E ′ − C F (cid:18) e γ E µ m b (cid:19) ε Γ( ε ) 24 ε (1 + ε )(1 − ε ) ˆ F (0) F ˆΦ (0) F , ˆ F (1)amp,F ˆΦ (0) F = C F (cid:20) − (cid:18) e γ E µ m b (cid:19) ε Γ( ε ) (cid:18) − ε + 6 ε − ε ε (1 − ε )(1 − ε ) − − ε + 2 ε ε (1 − ε ) (cid:19)(cid:21) ˆ F (0) ˆΦ (0) The corrections to the physical operator O can be found in [5]. C F (cid:20) ε − (cid:18) e γ E µ m b (cid:19) ε Γ( ε )4(1 − ε ) (cid:21) ˆ F (0) E ˆΦ (0) E − C F (cid:18) e γ E µ m b (cid:19) ε Γ( ε ) 1 − ε + 6 ε − ε ε (1 − ε )(1 − ε ) ˆ F (0) F ˆΦ (0) F (23)and for the wave function correctionsˆ F (0) E ˆΦ (1)amp,E = 48 C F (cid:20) V E ⊗ ˆ F (0) ˆΦ (0) (cid:21) − C F ε (cid:20)(cid:16) V + 3 V E (cid:17) ⊗ ˆ F (0) E ˆΦ (0) E (cid:21) + C F ε (cid:20) V E ⊗ ˆ F (0) E ′ ˆΦ (0) E ′ (cid:21) , ˆ F (0) F ˆΦ (1)amp,F = 2 C F (cid:20) V E ⊗ ˆ F (0) ˆΦ (0) (cid:21) + C F ε (cid:20) V E ⊗ ˆ F (0) E ˆΦ (0) E (cid:21) − C F ε (cid:20) V ⊗ ˆ F (0) F ˆΦ (0) F (cid:21) , (24)where ⊗ represents a convolution and V is the Efremov-Radyushkin-Brodsky-Lepage(ERBL) kernel [18] (given explicitly in equation (43) of [5]). We finally compute theconvolutions of the NLO kernels with the wave function corrections. For the physicaloperator O we getˆ F (0) ˆ T (1)1 ˆΦ (1)amp = − N c ˆ F (0) ˆ T (1)2 ˆΦ (1)amp = − N c F (0) T (1)1 Φ (1)amp + C F N c (cid:26) ˆ t ( u ) + O ( ε ) (cid:27) ˆ F (0) ˆΦ (0) , (25)while the evanescent operators give again µ -dependent corrections to the physical kernelsˆ F (0) E ˆ T (1)1 ,E ˆΦ (1)amp ,E = − N c ˆ F (0) E ˆ T (1)2 ,E ˆΦ (1)amp ,E → C F N c (cid:26) L + ˆ t E, ( u ) + O ( ε ) (cid:27) ˆ F (0) ˆΦ (0) , ˆ F (0) F ˆ T (1)1 ,F ˆΦ (1)amp ,F = − N c ˆ F (0) F ˆ T (1)2 ,F ˆΦ (1)amp ,F → C F N c (cid:26) L + ˆ t F, ( u ) + O ( ε ) (cid:27) ˆ F (0) ˆΦ (0) . (26)According to (12) we now have assembled all pieces to perform the IR subtractions inNNLO. However, as we have seen above in the calculation of the NLO kernels, the naivesplit-up into factorizable and non-factorizable contributions is spoiled for the colour-suppressed amplitude. In analogy to (19) we therefore have to account for an additionalcontribution ˆ∆ (2) F,i on the right hand side of (12), which represents the 2-loop countertermsof the form factor and wave function corrections for the Fierz-evanescent operator ˜ O F .The calculation of this counterterm contribution requires a rather complicated 2-loopcalculation on its own. We refer to Appendix B for the details of this calculation andquote the contribution to the physical kernel only,ˆ∆ (2) F, = C F ˆ∆ (2) F, → − C F N c (cid:26) C F ε + (cid:20)(cid:16) L (cid:17) C F + 116 C A − n f (cid:21) ε + (cid:18)
28 + π
12 + 7 L + 12 L (cid:19) C F − C A − n f + O ( ε ) (cid:27) ˆ F (0) ˆΦ (0) . (27)9 Vertex corrections in NNLO
As we have seen in the last section, the NNLO calculation of the hard-scattering kernelsrequires a rather complex subtraction procedure of UV- and IR-divergences. The factthat the kernels turn out to be free of any singularities represents both a non-trivialconfirmation of the factorization framework and a stringent cross-check of our calculation.
In terms of the Wilson coefficients C i of the physical operators Q i from the operator basis(3), the topological tree amplitudes take to NNLO the form α ( M M ) = C + α s π C F N c (cid:26) C V (1) + α s π h C V (2)1 + C V (2)2 i + O ( α s ) (cid:27) + . . .α ( M M ) = C F N c C + C N c + α s π C F N c (cid:26) (cid:18) C − C N c (cid:19) V (1) − C A C + α s π (cid:20)(cid:18) C − C N c (cid:19) V (2)1 + (cid:18) C F N c C + C N c (cid:19) V (2)2 + 2 C A C V (1) + (cid:18) C F − C A − n f (cid:19) C A C (cid:21) + O ( α s ) (cid:27) + . . . (28)where the ellipsis refer to the terms from spectator scattering which we disregard inthe following. In this notation the α s corrections have been expressed in terms of theconvolution V (1) = Z du (cid:16) − L + g ( u ) + iπg ( u ) (cid:17) φ M ( u ) , (29)where L = ln µ /m b and (recall that ¯ u = 1 − u ) g ( u ) = − − u + 2 ln ¯ u,g ( u ) = −
22 + 3(1 − u )¯ u ln u + (cid:20) ( u ) − ln u − − u ¯ u ln u − ( u → ¯ u ) (cid:21) . (30)If we transform these expressions into the Fierz-symmetric operator basis that has beenused in many previous QCD factorization analyses, we reproduce the NLO result from [1].In NNLO we find the convolutions V (2)1 = Z du (cid:26)(cid:16) C F − C A + 2 n f (cid:17) L + (cid:26)(cid:16) C A − n f (cid:17) g ( u ) − C A − n f + C F h ( u )+ iπ h(cid:16) C A − n f (cid:17) g ( u ) + C F h ( u ) i(cid:27) L C F h ( u ) + C A h ( u ) + ( n f − h ( u ; 0) + h ( u ; z ) + h ( u ; 1)+ iπ h C F h ( u ) + C A h ( u ) + ( n f − h ( u ; 0) + h ( u ; z ) + h ( u ; 1) i(cid:27) φ M ( u ) ,V (2)2 = Z du (cid:26) L + (cid:16) − g ( u ) − iπg ( u ) (cid:17) L + h ( u ) + iπh ( u ) (cid:27) φ M ( u ) , (31)where n f = 5 represents the number of active quark flavours and z = m c /m b . Theexplicit expressions for the NNLO kernels h − , which specify the imaginary parts of thetopological tree amplitudes, can be found in [5]. As a new result we obtained the realparts of the topological tree amplitudes to NNLO, which have been given in terms of anew set of kernels h − that are listed in Appendix C.Partial structures of our NNLO result can be cross-checked. First, we verified thatthe scale dependence between the Wilson coefficients, the coupling constant, the hard-scattering kernels and the light-cone distribution amplitude cancels in the tree amplitudes α i ( M M ) to O ( α s ) as it should . Second, we compared the terms proportional to n f with the analysis of the large β -limit in [19] and found agreement. Finally, we reproducedthe imaginary part of the colour-suppressed amplitude from our earlier analysis in [5],which was derived on the basis of Fierz-symmetry arguments. We expand the light-cone distribution amplitude of the emitted meson M into the eigen-functions of the 1-loop evolution kernel, φ M ( u ) = 6 u ¯ u " ∞ X n =1 a M n C (3 / n (2 u − , (32)where a M n and C (3 / n are the Gegenbauer moments and polynomials, respectively. It isconvenient to truncate this expansion at n = 2, which allows us to perform the convolutionintegrals in our final expression (31) explicitly. The convolution with the NLO kernelresults in , Z du g ( u ) φ M ( u ) = −
452 + 112 a M − a M , (33)whereas the convolutions with the NNLO kernels become Z du h ( u ) φ M ( u ) = 534760 − ζ + 3744 ζ + (cid:16) − ζ + 72 ln 2 (cid:17) π + 239 π We emphasize that this cancellation would have been incomplete, if µ -dependent contributions fromthe mixing of evanescent operators as e.g. in (16) or (26) had been missed. We refer to [5] for the convolutions with the kernels g and h − . (cid:26) ζ − ζ − (cid:16) − ζ + 24 ln 2 (cid:17) π − π (cid:27) a M + (cid:26) − ζ + 74304 ζ + (cid:16) − ζ − (cid:17) π + 797 π (cid:27) a M , Z du h ( u ) φ M ( u ) = 348 − a M + 32940 a M , Z du h ( u ) φ M ( u ) = 1280960 − ζ + 6564 ζ + (cid:16) − ζ −
48 ln 2 (cid:17) π + 134 π (cid:26) ζ − ζ − (cid:16) − ζ − (cid:17) π − π (cid:27) a M + (cid:26) − ζ + 129204 ζ + (cid:16) − ζ − (cid:17) π + 727 π (cid:27) a M , Z du h ( u ) φ M ( u ) = − ζ − ζ − (cid:16) − ζ −
36 ln 2 (cid:17) π − π (cid:26) ζ − ζ − (cid:16) − ζ + 684 ln 2 (cid:17) π + 151 π (cid:27) a M + (cid:26) − ζ − ζ − (cid:16) − ζ − (cid:17) π − π (cid:27) a M . (34)12e finally perform the convolution with the hard-scattering kernel h ( u ; z f ), which stemsfrom the diagrams with a closed fermion loop. As this contribution depends on the mass m f = z f m b of the internal quark, we parameterize the convolution as Z du h ( u ; z f ) φ M ( u ) = H , ( z f ) + H , ( z f ) a M + H , ( z f ) a M . (35)For massless quarks we may perform the convolution integral analytically, Z du h ( u ; 0) φ M ( u ) = 49318 − π − (cid:18)
403 + 2 π (cid:19) a M + (cid:18) − π (cid:19) a M , (36)whereas we obtain numerical results for massive internal quarks. In Table 1 we summarizethe contributions from closed fermion loops for massless quarks ( z q = 0), for a b -quark( z b = 1) and for a charm quark ( z c ∈ [0 . , . µ = m b and z c = m c /m b = 0 .
3, which yields (with C F = 4 / C A = 3, n f = 5) V (1) = ( − . − . i ) + (5 . − . i ) a M + ( − . a M ,V (2)1 = ( − . − . i ) + (660 . − . i ) a M + ( − . − . i ) a M ,V (2)2 = (322 .
19 + 320 . i ) + ( − .
97 + 154 . i ) a M + (3 . − . i ) a M . (37)We find relatively large coefficients for the NNLO terms and expect only a minor impactof the higher Gegenbauer moments in the symmetric case with a M = 0.We conclude with a remark concerning the large β -limit that has been consideredin [19]. In this approximation we get V (2)1 (cid:12)(cid:12) β ≃ ( − . − . i ) + (380 . − . i ) a M + ( − . − . i ) a M , (38)whereas the contribution from V (2)2 is completely missed. As a consequence the NNLOcontribution to α is substantially underestimated in this approximation, whereas theone to α deviates from the full NNLO result between ∼
15% for the imaginary partand ∼
40% for the real part. This illustrates the importance of performing exact 2-loopcalculations. z f .
25 0 .
275 0 . .
325 0 .
35 1 H , .
81 17 .
12 16 .
43 15 .
72 14 .
99 14 . − . H , − . − . − . − . − . − . − . H , .
56 2 .
08 1 .
94 1 .
81 1 .
68 1 .
57 0 . Fermionic contribution in the notation of (35). The first columnrefers to massless quarks, the last column to the b -quark and the other columnsto the charm quark for different physical values of z c = m c /m b . Numerical analysis
We conclude with a brief analysis of the numerical impact of the considered NNLO cor-rections. As a phenomenological analysis of hadronic B decays is beyond the scope of thepresent paper, we focus on the perturbative structure of the topological tree amplitudesand discuss their remnant uncertainties. In particular, we now combine our results withthe NNLO corrections from 1-loop spectator scattering that have been worked out in [3]. In contrast to the vertex corrections considered in this work, the spectator scatteringterm is sensitive to two perturbative scales: the hard scale µ h ∼ m b and a dynamicallygenerated intermediate (hard-collinear) scale µ hc ∼ (Λ QCD m b ) / . The hard scatteringkernels from spectator scattering therefore factorize further into coefficient functions H IIi ,encoding the hard effects, and a universal hard-collinear jet-function J || . Renormaliza-tion group techniques can be used to resum parametrically large logarithms of the formln m b / Λ QCD in terms of an evolution kernel U || . Following the first paper of [3], weimplement the spectator scattering contribution to the topological tree amplitudes as C i ( µ ) T IIi ( µ ) ⊗ [ ˆ f B φ B ]( µ ) ⊗ φ M ( µ ) ⊗ φ M ( µ ) → C i ( µ h ) H IIi ( µ h ) ⊗ U || ( µ h , µ hc ) ⊗ J || ( µ hc ) ⊗ [ ˆ f B φ B ]( µ hc ) ⊗ φ M ( µ hc ) ⊗ φ M ( µ h ) . (39)Since the spectator scattering starts at O ( α s ), the resummation is required here inthe next-to-leading-logarithmic (NLL) approximation. Unfortunately, a complete NLLresummation is not possible since the evolution kernel U || is known in the leading-logarithmic (LL) approximation only [20].We therefore proceed along the lines of our earlier analysis [5], where we worked inthe LL approximation which is consistent for the imaginary parts that are of O ( α s ).According to this, we implement the LL evolution of the HQET decay constant and theGegenbauer moments to evolve the hadronic parameters from their input scales to theones required in (39). The B meson distribution amplitude is modeled according to [21],which implies λ B (1GeV) = (0 . ± . σ (1GeV) = 1 . ± . σ (1GeV) = 3 . ± .
8. The 1-loop matching corrections tothe hard functions H IIi [3] and the jet function J || [20, 22] are implemented neglectingcrossed terms of O ( α s ). We finally adopt the BBNS model from [1] to estimate the sizeof power corrections to the factorization formula.In the spectator scattering term we compute the Wilson coefficients from the effectiveweak Hamiltonian in the NLL approximation with 2-loop running coupling constant.Quantities referring to the hard scale are evaluated in a theory with n f = 5 flavours andthose referring to the hard-collinear scale with n f = 4. One should keep in mind that the Wilson coefficients in the spectator scattering term refer to adifferent operator basis than the one used in the current work (namely the Fierz-symmetric traditionalbasis that we denoted by a tilde in [5]). .2 Tree amplitudes in NNLO We finally evaluate the topological tree amplitudes for the B → ππ channels using theinput parameters from our earlier analysis [5] and computing the Wilson coefficientsin the vertex corrections in the next-to-next-to-leading-logarithmic (NNLL) approxima-tion [8, 23] with 3-loop running coupling constant [24] and Λ (5) MS = 205 MeV. Under thesespecifications the NNLO prediction of the topological tree amplitudes becomes α ( ππ ) = 1 . (cid:12)(cid:12) V (0) + (cid:2) .
022 + 0 . i (cid:3) V (1) + (cid:2) .
024 + 0 . i (cid:3) V (2) − . (cid:12)(cid:12) S (1) − (cid:2) .
014 + 0 . i (cid:3) S (2) − . (cid:12)(cid:12) P = 1 . +0 . − . + (0 . +0 . − . ) i,α ( ππ ) = 0 . (cid:12)(cid:12) V (0) − (cid:2) .
174 + 0 . i (cid:3) V (1) − (cid:2) .
030 + 0 . i (cid:3) V (2) + 0 . (cid:12)(cid:12) S (1) + (cid:2) .
032 + 0 . i (cid:3) S (2) + 0 . (cid:12)(cid:12) P = 0 . +0 . − . − (0 . +0 . − . ) i. (40)Here we disentangled the contributions of the various terms in the factorization formula,namely the tree level result V (0) (”naive factorization”), NLO (1-loop) vertex corrections V (1) , NNLO (2-loop) vertex corrections V (2) , NLO (tree level) spectator scattering S (1) ,NNLO (1-loop) spectator scattering S (2) and the modelled power corrections P .The new contributions from this work consist in the real parts of the terms denoted by V (2) . For the colour-allowed amplitude α ( ππ ), this correction is slightly larger than the α s terms due to an numerical enhancement from the Wilson coefficients in the effectiveHamiltonian . On the other hand, the colour-suppressed amplitude α ( ππ ) receives a The numbers for the imaginary parts differ slightly from those of [5], since we now evaluate theWilson coefficients throughout in the NNLL approximation. We remark that a similar enhancement is unlikely to exist at even higher order of the perturbativeexpansion, since the NNLO expressions already reveal the full complexity. µ h µ hc f B F Bπ + λ B a π X H Re( α ) +0 . − .
011 +0 . − .
007 +0 . − .
003 +0 . − .
008 +0 . − .
009 +0 . − .
008 +0 . − . Im( α ) +0 . − .
011 +0 . − .
003 +0 . − .
001 +0 . − .
003 +0 . − .
003 +0 . − .
004 +0 . − . Re( α ) +0 . − .
008 +0 . − .
023 +0 . − .
014 +0 . − .
025 +0 . − .
026 +0 . − .
033 +0 . − . Im( α ) +0 . − .
028 +0 . − .
004 +0 . − .
002 +0 . − .
003 +0 . − .
003 +0 . − .
006 +0 . − . Table 2:
Dominant uncertainties of our final predictions for the colour-allowedtree amplitudes α ( ππ ) and the colour-suppressed tree amplitude α ( ππ ) fromscale variations, hadronic input parameters and modelled power corrections. | α /α | from the perturbative calculation.In Table 2 we list the uncertainties of our NNLO predictions stemming from scalevariations, hadronic input parameters and the modelled power corrections. The valuesof the first two columns follow from varying the perturbative scales independently inthe ranges µ h = 4 . +4 . − . GeV and µ hc = 1 . +0 . − . GeV. As the dependence on the hardscale tends to cancel between vertex corrections and spectator scattering, we vary bothcontributions independently and take the larger interval (from the vertex corrections) asour estimate for higher order perturbative corrections. The scale dependence of the vertexcorrections is also illustrated in Figure 2, where we read off that it gets substantiallyreduced for the real parts at NNLO, whereas the reduction is less pronounced for theimaginary parts.For our final error estimate in (40) we added the individual uncertainties from Ta-ble 2 in quadrature. Whereas the colour-allowed amplitude α ( ππ ) can be computedprecisely in the factorization framework, the situation is less fortunate for the colour-suppressed amplitude α ( ππ ). Due to large cancellations between the vertex corrections,the colour-suppressed amplitude becomes particularly sensitive to the spectator scatter-ing contribution and is therefore subject to rather large uncertainties related mainly toour restricted knowledge of the hadronic input parameters. PSfrag replacements µ h Re( α ) V - PSfrag replacements µ h Re( α ) V µ h Im( α ) V - PSfrag replacements µ h Re( α ) V - - - - - PSfrag replacements µ h Re( α ) V µ h Im( α ) V Figure 2:
Dependence of the tree amplitudes α i ( ππ ) as a function of the hard scale µ h (vertexcorrections only). The dotted (black) lines refer to LO, the dashed (orange/light gray) lines toNLO and the solid (blue/dark gray) lines to NNLO. Conclusion
We computed the real parts of the 2-loop vertex corrections for charmless hadronic B meson decays, completing the NNLO calculation of the topological tree amplitudes inthe QCD factorization framework. We in particular showed how to compute the colour-suppressed tree amplitude without making use of Fierz-symmetry arguments and foundthat the hard-scattering kernels are free of IR-singularities and the resulting convolutionswith the light-cone distribution amplitude of the emitted light meson are finite, whichdemonstrates factorization at the 2-loop order.The numerical impact of the considered corrections was found to be moderate, al-though they can be of similar size as the NLO corrections. The scale dependence ofthe real parts of the topological tree amplitudes is significantly reduced at NNLO, whichallows for a precise determination of the colour-allowed amplitude α . In contrast to this,it remains difficult to compute the colour-suppressed amplitude α in the factorizationframework, since it is subject to substantial uncertainties from hadronic input parame-ters and potential 1 /m b corrections. In particular, we do not find an enhancement of thephenomenologically important ratio | α /α | from the perturbative calculation. Acknowledgements
We are grateful to Gerhard Buchalla for interesting discussions and helpful comments onthe manuscript. This work was supported by the DFG Sonderforschungsbereich/Trans-regio 9.
A Auxiliary coefficient functions
In the calculation of the colour-allowed tree amplitude, the NLO kernels have been givenin (14) in terms of the coefficient functions t ( u ) = 4Li ( u ) − ln u + 2 ln u ln ¯ u + ln ¯ u + (2 − u ) (cid:16) ln u ¯ u − ln ¯ uu (cid:17) − π − ,t ( u ) = − ( u ) − , ( u ) − u Li ( u ) + ln u − u ln ¯ u + ln u ln ¯ u − ln ¯ u + 2 − u u ¯ u Li ( u ) − − u ¯ u (cid:16) ln u − ln u ln ¯ u (cid:17) + 6 − u + 2¯ uπ ¯ u ln u + 4 − u u ln ¯ u − − u + 5 uπ u ln ¯ u + (7 − u ) π u + 2 ζ − ,t ( u ) = 10Li ( u ) − , ( u ) + 10S , ( u ) − u Li ( u ) + 10 ln ¯ u S , ( u ) −
712 ln u + 5 ln ¯ u Li ( u ) + 43 ln u ln ¯ u − ln u ln ¯ u + 13 ln u ln ¯ u + 712 ln ¯ u + 2 − u + 6 u u ¯ u Li ( u ) − − u + 3 u u ¯ u (cid:16) S , ( u ) + ln ¯ u Li ( u ) (cid:17) − − u u ln ¯ u
17 2 − u u (cid:16) u − u ln ¯ u + 3 ln u ln ¯ u (cid:17) − − u ) + 17¯ uπ u ln u + 3(6 − u − u ) + u ¯ uπ u ¯ u Li ( u ) + 24 − u + 5¯ uπ u ln u ln ¯ u + (29 − u ) π u + 6(12 − u ) + 7 uπ u ln ¯ u + 24(7 − u ) + (10 − u ) π u ln u − π − u (7 − u ) + (2 + 23 u − u ) π + 24 u ¯ uζ u ¯ u ln ¯ u + 10 − u ¯ u ζ − ,t E, ( u ) = − − u (cid:16) ln u ¯ u − ln ¯ uu (cid:17) + 163 ,t E, ( u ) = − − u u ¯ u Li ( u ) + 1 − u u ln u + u u ln u ln ¯ u − − u u ln ¯ u − − u )3 (cid:16) ln u ¯ u − ln ¯ uu (cid:17) − (6 − u ) π u + 12 (41)and the convolutions of the NLO kernels with the wave function corrections, cf. (15) and(16), involve t ( u ) = 4Li ( u ) + 4S , ( u ) − u Li ( u ) + 23 ln u − u ln ¯ u −
23 ln ¯ u − Li ( u ) u ¯ u − − u u ¯ u (cid:16) u ln u + 2¯ u ln u ln ¯ u − ¯ u ln ¯ u (cid:17) − u ln ¯ u + (4 − u ) π u − − ζ ,t ( u ) = 12Li ( u ) − , ( u ) + 12S , ( u ) − (cid:16) ln u + ln ¯ u (cid:17) Li ( u ) + 12 ln u S , ( u )+ 4 ln ¯ u S , ( u ) + (cid:16) u + 4 ln u ln ¯ u + 2 ln ¯ u (cid:17) Li ( u ) −
34 ln u + 73 ln u ln ¯ u −
12 ln u ln ¯ u −
13 ln u ln ¯ u + 34 ln ¯ u − − u + 3 u u ¯ u Li ( u ) + 5 − u u ln u + 1 + u − u u ¯ u S , ( u ) + 2 − u + 6 u u ¯ u ln u Li ( u ) − − u + 3 u u ¯ u ln ¯ u Li ( u )+ 2 − u + 9 u u ¯ u ln u ln ¯ u − − u u ¯ u ln u ln ¯ u − − u u ln ¯ u − − u + 15 u − u ¯ uπ u ¯ u Li ( u ) − − u + 4¯ uπ u ln u − − u + 27 u − u ¯ uπ u ¯ u ln u ln ¯ u + 3(14 − u ) + 8 uπ u ln ¯ u + 8 − u − π − uζ u ln u + 3(2 − u ) ζ ¯ u − π
60 + (23 − u ) π u − − u + 45 u − (14 − u + 6 u ) π − u ¯ uζ u ¯ u ln ¯ u − , E, ( u ) = − − u ) u ¯ u Li ( u ) − u ln u ln ¯ u − u − u − π ¯ u + 50 . (42)In the calculation of the colour-suppressed tree amplitude, the NLO kernels in (22) containthe coefficient functionsˆ t ( u ) = u ¯ u ln u − ln ¯ u + 8 + iπ, ˆ t ( u ) = u ¯ u (cid:16) Li ( u ) − ln u + ln u ln ¯ u + 4 ln u (cid:17) + 12 ln ¯ u − u − (3 − u ) π u + 20+ iπ (cid:16) − ln ¯ u (cid:17) , ˆ t E, ( u ) = ¯ uu ln ¯ u − ln u + 6 + iπ, ˆ t E, ( u ) = − ¯ uu (cid:16) Li ( u ) + ln ¯ u − u (cid:17) + 12 ln u − u + 14 − π iπ (cid:16) − ln u (cid:17) (43)and the convolutions with the NLO kernels, cf. (25) and (26), give rise toˆ t ( u ) = π − u ¯ u ln u + 2 ln u ln ¯ u − ln ¯ u − ln ¯ uu , ˆ t E, ( u ) = 1¯ u (cid:16) ( u ) + 3 u ln u − π (cid:17) − u ) u ln ¯ u + 27 + 3 iπ, ˆ t F, ( u ) = 1¯ u (cid:16) u )Li ( u ) − u ln u − u ln u ln ¯ u + 3 u ln u − (2 + u ) π (cid:17) + ¯ uu ln ¯ u − u ) u ln ¯ u + 29 + iπ (cid:16) − u ¯ u ln u + 2¯ uu ln ¯ u (cid:17) . (44) B Calculation of 2-loop counterterms ˆ∆ (2) F,i
We present the calculation of the 2-loop counterterms ˆ∆ (2)
F,i , that are required in theNNLO calculation of the colour-suppressed tree amplitude as described in Section 2.4.The counterterms receive three contributionsˆ∆ (2)
F,i = ˆ T (0) i,F ⊗ (cid:26) ˆ F (2) F ˆΦ (0) F + ˆ F (1) F ˆΦ (1) F + ˆ F (0) F ˆΦ (2) F (cid:27) ct (45)where ⊗ represents a convolution and ”ct” refers to the counterterm contributions of theform factor and the wave function corrections. Notice that the wave function correctionsactually correspond to local corrections to the decay constant, as a consequence of thefact that the tree level kernels ˆ T (0) i,F are constant (cf. also (20) and (21)).We first consider the mixed term ˆ F (1) F ˆΦ (1) F , which involves the calculation of the dia-grams from Figure 3. The first diagram vanishes due to a scaleless loop integral and the19igure 3: Diagrams that contribute to the mixed contribution ˆ F (1) F ˆΦ (1) F . Thesymbol ⊗ in the lower (upper) line refers to an insertion of the 1-loop counter-term from the form factor (wave function) correction of the operator ˜ O F . second diagram yields δ = − C F (cid:18) e γ E µ m b (cid:19) ε Γ( ε ) (cid:26)(cid:18) − ε + 2 ε ε (1 − ε ) + 6(1 + ε )(1 − ε ) (cid:19) ˆ F (0) ˆΦ (0) + 6(1 + ε )(1 − ε ) ˆ F (0) F ˆΦ (0) F + 1 − ε + 16 ε − ε ε (1 − ε )(1 − ε ) ˆ F (0) E ˆΦ (0) E + 116 ε (1 − ε ) ˆ F (0) E ′ ˆΦ (0) E ′ (cid:27) . (46)We are left with the 2-loop counterterm from the last diagram of Figure 3, which requiresthe calculation of the UV-divergences of the 2-loop diagram from Figure 4. For this it isconvenient to apply the method proposed in [25] (sometimes called IR-rearrangement ),which allows to set all masses and external momenta to zero. The calculation then reducesto the evaluation of 2-loop tadpole integrals, which depend on a single mass scale (anartificial scale that has been introduced to separate UV- and IR-divergences). Computingthe 1-loop counterterms with the same prescription and accounting for the wave-functionrenormalization, we get δ = C F (cid:26)(cid:18) ε + 5 (cid:19) ˆ F (0) ˆΦ (0) − (cid:18) ε − ε (cid:19) ˆ F (0) E ˆΦ (0) E + 116 ε ˆ F (0) E ′ ˆΦ (0) E ′ + 6 ε ˆ F (0) F ˆΦ (0) F (cid:27) . (47)Next we compute the form factor correction ˆ F (2) F ˆΦ (0) F (the corresponding diagrams areshown in Figure 5a). The first diagram gives again the contribution δ from (46). Onthe other hand the computation of the 2-loop counterterm from the second diagramof Figure 5a is rather involved. It requires the calculation of the UV-divergences of acouple of 2-loop diagrams (shown e.g. in Figure 1 of [15]) and the corresponding 1-loopcounterterms. Proceeding as before with the method of IR-rearrangement and accountingfor the 2-loop wave-function renormalization in the MS-scheme [26], Z (2)2 ,b = Z (2)2 ,q = C F (cid:26)(cid:18) C F + C A (cid:19) ε + (cid:18) C F − C A + 12 n f (cid:19) ε (cid:27) , (48)Figure 4: The UV-divergences of this 2-loop diagram contribute to δ . a) (b) Figure 5:
Diagrams that contribute to ˆ F (2) F ˆΦ (0) F (a) and ˆ F (0) F ˆΦ (2) F (b). yields the 2-loop form factor counterterm for the Fierz-evanescent operator ˜ O F δ = C F (cid:26) (cid:20)(cid:18) C F − C A + 13 n f (cid:19) ε + (cid:18) − C F + 14936 C A + 518 n f (cid:19)(cid:21) ˆ F (0) ˆΦ (0) + (cid:20)(cid:18) − C F − C A + 112 n f (cid:19) ε + (cid:18) C F + 53144 C A − n f (cid:19) ε (cid:21) ˆ F (0) E ˆΦ (0) E + (cid:20) C F ε + (cid:18) − C F + 132 C A (cid:19) ε (cid:21) ˆ F (0) E ′ ˆΦ (0) E ′ + (cid:18) C F − C A + 13 n f (cid:19) ε ˆ F (0) F ˆΦ (0) F (cid:27) . (49)We finally account for the wave function correction ˆ F (0) F ˆΦ (2) F from Figure 5b. The firstdiagram again vanishes due to a scaleless integral and the second diagram yields, in aconvolution with a constant kernel, again the contribution δ from (49).To summarize, in terms of the individual contributions δ i from (46), (47) and (49),the 2-loop counterterms required in the calculation of the colour-suppressed amplitudebecome ˆ∆ (2) F, = C F ˆ∆ (2) F, = C F N C (cid:26) δ + δ + 2 δ (cid:27) . (50) C NNLO hard-scattering kernels
Our final expressions for the real parts of the NNLO vertex corrections from (31) involvethe following set of hard-scattering kernels, h ( u ) = (cid:20) − u + 8 u − u )¯ u Li ( u ) − − u + 36 u − u + 4 u ) u ¯ u S , ( u ) −
24 ln u Li ( u ) − − u + 71 u − u + 24 u ) u ¯ u (cid:18) ln ¯ u Li ( u ) − ζ ln u (cid:19) + 17 − u + 40 u − u u ¯ u Li ( u ) + 4 ln u Li ( u ) + 23 ln u −
43 ln u ln ¯ u − − u + 20 u u ¯ u ln u ln ¯ u Li ( u ) −
51 + 16 u + 268 u + 4 u u ¯ u ln u ln ¯ u We performed this calculation for arbitrary bilinear quark currents, which allows us to performseveral cross-checks. We in particular verified that the anomalous dimension of the vector currentvanishes at the 2-loop level and reproduced the one of the scalar and the tensor current from [27].
21 3 − u + 351 u − u + 266 u − u + 17 u − u + 4 u u ¯ u Li ( u ) − − u + 109 u − u − u u ¯ u ln u Li ( u ) − u (3 − u + 2 u )6¯ u ln u − − u + 69 u − u − u + 3 u − u u ¯ u ln u ln ¯ u + 3 − u − u − u u (cid:18) Li ( − u ) − ln u Li ( − u ) − ln u + π u ) (cid:19) − (¯ u − u )(6(5 − u + 60 u − u ) + 4(19 − u − u ) π )24 u ¯ u Li ( u )+ (cid:18) − u − u + 49 u − u u ¯ u − π (cid:19) ln u + (cid:18) − u + 302 u − u u ¯ u + (701 − u + 2794 u − u ) π u ¯ u (cid:19) ln u ln ¯ u − (cid:18) − u u − (96 − u + 59 u + 9 u + 3 u − u ) π u ¯ u (cid:19) ln u + (849 − u + 4496 u − u + 2024 u − u + 328 u ) π u ¯ u + (5 − u + 938 u − u + 1366 u − u + 230 u ) π u ¯ u + 15078 − (3 − u + 358 u − u + 32 u + 216 u − u ) ζ u ¯ u + ( u ↔ ¯ u ) (cid:21) + (cid:20) − u )¯ u Li ( u ) − − u + 36 u − u + 4 u ) u ¯ u S , ( u ) − (¯ u − u )(12 − u + 25 u − u + 2 u ) u ¯ u (cid:18) u Li ( u ) + ln u ln ¯ u − ζ ln u (cid:19) − u − u )(3 − u + 5 u ) u ¯ u Li ( u ) − − u + 85 u − u + 18 u u ¯ u ln u Li ( u ) − − u + 15 u − u + 4 u ) u ¯ u (cid:18) ln u ln ¯ u − π (cid:19) Li ( u )+ 3 − u + 293 u − u + 36 u + 74 u − u + 14 u u ¯ u Li ( u ) − − u + 7 u u ln u − − u + 57 u − u + 16 u − u u ¯ u ln u ln ¯ u + 3 + 3 u + 5 u + 3 u u (cid:18) Li ( − u ) − ln u Li ( − u ) − ln u + π u ) (cid:19)
22 21 − u − u + 182 u − u u ¯ u Li ( u ) + 42 + u − u + 91 u u ¯ u ln u + (¯ u − u )(3 u ¯ u (21 − u + 13 u ) + 4(33 − u + 65 u − u + 4 u ) π )12 u ¯ u ln u ln ¯ u − u − − u − u + 37 u − u ) π u ¯ u ln u + (¯ u − u )(48 − u + 85 u − u + 2 u ) π u ¯ u − (¯ u − u )(147 − u + 74 u ) π u ¯ u − (¯ u − u )(3 − u + 96 u + 28 u − u ) ζ u ¯ u − ( u ↔ ¯ u ) (cid:21) ,h ( u ) = (cid:20) − − u )2¯ u ln u + 3(1 − u )2 u ¯ u ln u ln ¯ u − − u )2¯ u ln u + (1 − u ) π u ¯ u + ( u ↔ ¯ u ) (cid:21) + (cid:20) ( u ) − u Li ( u ) + 43 ln u − u ln ¯ u − − u u ¯ u Li ( u )+ 25 − u u ln u + 13¯ u ln u ln ¯ u − u ln u − π u − ( u ↔ ¯ u ) (cid:21) ,h ( u ) = (cid:20) (1 + u )(3 − u + 3 u )3 u ¯ u (cid:18) H ( u ) + π ln(1 + u ) (cid:19) −
48 ln u Li ( u )+ 2 u u (cid:18) H ( u ) − π Li ( − u ) (cid:19) + 4(6 − u + 16 u − u )¯ u Li ( u ) + 8 ln u Li ( u ) − − u + 156 u − u + 18 u − u ) u ¯ u S , ( u ) + 43 ln u −
83 ln u ln ¯ u − − u + 308 u − u + 107 u − u + u ) u ¯ u (cid:18) ln ¯ u Li ( u ) − ζ ln u (cid:19) − − u + 88 u − u + 82 u − u + 16 u u ¯ u Li ( u ) + 3 − u + 6 u − u u ln u − (¯ u − u )(1 − u + 2 u )(13 − u + 2 u ) u ¯ u ln u ln ¯ u Li ( u ) − − u + 235 u − u − u u ¯ u ln u Li ( u ) − − u + 308 u − u + 71 u + 33 u − u u ¯ u ln u ln ¯ u + 2(3 − u + 374 u − u + 280 u − u + 17 u − u + 4 u ) u ¯ u Li ( u )23 − u + 149 u − u − u + 6 u − u u ¯ u ln u ln ¯ u + 2(3 − u − u − u ) u (cid:18) Li ( − u ) − ln u Li ( − u ) − ln u + π u ) (cid:19) + (¯ u − u ) × (cid:18) − u + 109 u + 16 u − u u ¯ u + (1 − u + 2 u )(83 − u + 14 u ) π u ¯ u (cid:19) Li ( u )+ (cid:18) − u + 26 u + 13 u − u u ¯ u − π (cid:19) ln u + (cid:18) − u + 421 u − u + 132 u u ¯ u + (139 − u + 856 u − u + 164 u + 138 u − u ) π u ¯ u (cid:19) ln u ln ¯ u − (cid:18) − u u − (96 − u + 55 u + 15 u + 3 u − u ) π u ¯ u (cid:19) ln u + (191 − u + 1262 u − u + 973 u − u + 165 u ) π u ¯ u − (580 − u + 2239 u − u + 476 u ) π u ¯ u − − u + 119 u − u + 4 u + 78 u − u ) ζ u ¯ u + 16594 + ( u ↔ ¯ u ) (cid:21) + (cid:20) (1 + u )(1 + 4 u − u )3 u ¯ u (cid:18) H ( u ) + π ln(1 + u ) (cid:19) −
36 ln u Li ( u ) + 14 ln u Li ( u )+ 2(1 − u + 3 u − u )3¯ u (cid:18) H ( u ) − π Li ( − u ) (cid:19) + 4(2 − u + 18 u − u )¯ u Li ( u ) − − u + 156 u − u + 18 u − u ) u ¯ u S , ( u ) − (¯ u − u )(13 − u ¯ u ) u ¯ u Li ( u ) − − u + 77 u − u + 13 u − u + u ) u ¯ u (cid:18) u Li ( u ) + ln u ln ¯ u − ζ ln u (cid:19) − − u + 66 u − u + 18 u − u u ¯ u (cid:18) ln u ln ¯ u − π (cid:19) Li ( u ) − ln u + 83 ln u ln ¯ u + 2(3 − u + 339 u − u + 100 u + 59 u − u + 14 u ) u ¯ u Li ( u ) − − u + 227 u − u + 38 u u ¯ u ln u Li ( u ) − − u + 14 u u ln u − − u + 165 u − u + 34 u − u u ¯ u ln u ln ¯ u + 2(3 + 3 u + 5 u + 3 u ) u (cid:18) Li ( − u ) − ln u Li ( − u ) − ln u + π u ) (cid:19) (cid:18) − u + 161 u + 440 u − u u ¯ u − − u + 3 u )(¯ u + 2 u − u ) π u ¯ u (cid:19) Li ( u )+ (cid:18) − u − u + 55 u u ¯ u − π (cid:19) ln u + (¯ u − u )(13 + 169 u − u + 4 u ) π u ¯ u + (cid:18) (1 + u )(246 − u )4 u ¯ u + (87 − u − u ) π u ¯ u (cid:19) ln u ln ¯ u − (cid:18) u − (96 − u − u + 34 u − u ) π u ¯ u + 4 ζ (cid:19) ln u − (¯ u − u )(580 − u + 187 u ) π u ¯ u + 8(¯ u − u ) π u ¯ u ln 2 − u − u )(1 − u + 42 u − u + 3 u ) ζ u ¯ u − ( u ↔ ¯ u ) (cid:21) ,h ( u ) = (cid:20) − (1 + u )(3 − u + 3 u )6 u ¯ u (cid:18) H ( u ) + π ln(1 + u ) (cid:19) + 18 ln u Li ( u ) − u u (cid:18) H ( u ) − π Li ( − u ) (cid:19) − − u + 20 u − u ¯ u Li ( u ) − u Li ( u )+ 4(5 − u + 5 u + 5 u + 4 u − u ) u ¯ u S , ( u ) −
12 ln u + ln u ln ¯ u + 4(5 − u + 16 u − u + 3 u − u + u ) u ¯ u (cid:18) ln ¯ u Li ( u ) − ζ ln u (cid:19) + (¯ u − u ) (5 − u + 9 u − u + 6 u )4 u ¯ u Li ( u ) + 3 + u − u + 2 u u ln u + (¯ u − u )(5 − u + 15 u − u + 2 u )4 u ¯ u ln u ln ¯ u Li ( u )+ 10 − u + 32 u + 7 u − u + 21 u − u u ¯ u ln u ln ¯ u − − u + 157 u − u + 134 u − u + 32 u − u + 8 u u ¯ u Li ( u )+ 6 − u + 47 u − u − u u ¯ u ln u Li ( u ) − (4 + 35 u − u + 438 u − u ) π u ¯ u + (2 − u + u )(3 − u + 5 u + 2 u − u )4 u ¯ u ln u ln ¯ u − − u − u − u u (cid:18) Li ( − u ) − ln u Li ( − u ) − ln u + π u ) (cid:19) − (¯ u − u ) × (cid:18) (4 − u + 2 u )(1 − u − u )4 u ¯ u + (19 − u + 73 u − u + 14 u ) π u ¯ u (cid:19) Li ( u )25 (cid:18) u − u + 115 u − u u ¯ u − π (cid:19) ln u − (cid:18) − u + 24 u − u + 21 u u ¯ u + (127 − u + 652 u − u ) π u ¯ u (cid:19) ln u ln ¯ u + (cid:18) − u u − (12 − u + 8 u + 11 u + 3 u − u ) π u ¯ u (cid:19) ln u − (23 − u + 176 u − u + 449 u − u + 125 u ) π u ¯ u + (6 − u + 134 u + 26 u − u + 540 u − u ) ζ u ¯ u − u ↔ ¯ u ) (cid:21) + (cid:20) − (1 + u )(1 + 4 u − u )6 u ¯ u (cid:18) H ( u ) + π ln(1 + u ) (cid:19) + 2 ln u Li ( u ) − u Li ( u ) − − u + 3 u − u u (cid:18) H ( u ) − π Li ( − u ) (cid:19) + 19 − u + 36 u − u ¯ u Li ( u )+ 4(5 − u + 5 u + 5 u + 4 u − u ) u ¯ u S , ( u ) + (¯ u − u )(¯ u + u )(5 + 6 u − u )4 u ¯ u Li ( u ) + 5 − u + 16 u − u − u + u + u u ¯ u (cid:18) u Li ( u ) + ln u ln ¯ u − ζ ln u (cid:19) + 3 − u + 36 u − u u ¯ u (cid:18) ln u ln ¯ u − π (cid:19) Li ( u ) −
112 ln u −
23 ln u ln ¯ u − (¯ u − u )(7 − u − u + 48 u − u )16 u ¯ u ln u ln ¯ u − − u + 237 u + 242 u − u + 546 u − u + 84 u u ¯ u Li ( u )+ 6 − u + 19 u + 7 u + 48 u u ¯ u ln u Li ( u ) + 30 − u + 7 u u ln u + 18 − u + 78 u − u + 217 u − u u ¯ u ln u ln ¯ u − u + 4 u + 3 u u (cid:18) Li ( − u ) − ln u Li ( − u ) − ln u + π u ) (cid:19) − (¯ u − u )(6 − u + 283 u ) π u ¯ u − (cid:18) − u − u + 3610 u − u u ¯ u − (2 − u + 12 u − u − u + 9 u − u ) π u ¯ u (cid:19) Li ( u ) − u − u ) π u ¯ u ln 2 − (cid:18)
72 + 1703 u − u + 1805 u u ¯ u + π (cid:19) ln u (cid:18) (¯ u − u )(3 − u + 10 u )6 u ¯ u − (33 − u + 35 u − u + 14 u ) π u ¯ u (cid:19) ln u ln ¯ u + (cid:18) u − (12 + 26 u − u + 80 u − u ) π u ¯ u + 4 ζ (cid:19) ln u − (¯ u − u )(429 + 84 u − u + 2320 u − u ) π u ¯ u + 3(¯ u − u )(2 − u − u + 50 u − u ) ζ u ¯ u − ( u ↔ ¯ u ) (cid:21) . (51)The definition of the functions H , ( x ) can be found in Section 2.2. The diagrams witha closed fermion loop give for massless internal quarks h ( u ; 0) = (cid:20) ( u )¯ u + 1 − u u ln u + 1 + u u ln u ln ¯ u − − u )6¯ u ln u − (1 + u ) π u + ( u ↔ ¯ u ) (cid:21) + (cid:20)
43 Li ( u ) −
23 ln u + 43 ln u ln ¯ u − − u u Li ( u ) + 35 − u u ln u − u ln u ln ¯ u −
13 + 24¯ uπ u ln u + π u − ( u ↔ ¯ u ) (cid:21) , (52)and for an internal b -quark h ( u ; 1) = (cid:20) u (cid:18) Li ( − x b ) − S , ( − x b ) − ln(1 + x b )Li ( − x b ) −
112 ln x b x b −
16 ln (1 + x b ) + π x b x b (cid:19) − − u + 60 u + 19 u u ¯ u ln u ln ¯ u − y b (6 + u ) u (cid:18) Li ( − x b ) −
14 ln x b x b + 12 ln (1 + x b ) + π (cid:19) + 4 u ¯ u Li ( u ) − − u + 504 u − u + 405 u − u + 29 u u ¯ u Li ( u ) − u − u )6¯ u ln u − (40 − u + 120 u − u ) π u ¯ u − − u + 6 u ) ζ u ¯ u + 5(61 − u )24 u ¯ u + ( u ↔ ¯ u ) (cid:21) + (cid:20) − − u )3 u (cid:18) Li ( − x b ) − S , ( − x b ) − ln(1 + x b )Li ( − x b ) −
112 ln x b x b −
16 ln (1 + x b ) + π x b x b (cid:19) + 54 − u + 81 u u ¯ u ln u ln ¯ u
27 2 y b (38 + 29 u )9 u (cid:18) Li ( − x b ) −
14 ln x b x b + 12 ln (1 + x b ) + π (cid:19) − u − u )3¯ u Li ( u ) − − u + 389 u u ¯ u ln u − − u + 504 u − u + 405 u − u + 29 u u ¯ u Li ( u )+ (42 − u + 39 u ) π u ¯ u + 4 ζ u ¯ u + 261 − u u ¯ u − ( u ↔ ¯ u ) (cid:21) , (53)where we introduced the shorthand notation x b = 12 ( y b − , y b = r uu . (54)We finally refrain from presenting the charm quark contribution, which is rather com-plicated and depends on two parameterizations for the 4-topology Master Integrals thatwe could not solve in a closed analytical form (cf. the discussion in [15]). We may stillevaluate these Master Integrals numerically in Section 3.2 to perform the convolutionwith the light-cone distribution amplitude of the emitted meson M . References [1] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. (1999) 1914 [arXiv:hep-ph/9905312];M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B (2000)313 [arXiv:hep-ph/0006124];M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B (2001)245 [arXiv:hep-ph/0104110].[2] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D (2001)114020 [arXiv:hep-ph/0011336];C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D (2002) 054022[arXiv:hep-ph/0109045];M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Nucl. Phys. B (2002)431 [arXiv:hep-ph/0206152].[3] M. Beneke and S. Jager, Nucl. Phys. B (2006) 160 [arXiv:hep-ph/0512351];N. Kivel, JHEP (2007) 019 [arXiv:hep-ph/0608291];V. Pilipp, PhD thesis, LMU M¨unchen, 2007, arXiv:0709.0497 [hep-ph];V. Pilipp, Nucl. Phys. B (2008) 154 [arXiv:0709.3214 [hep-ph]].[4] M. Beneke and S. Jager, Nucl. Phys. B (2007) 51 [arXiv:hep-ph/0610322];A. Jain, I. Z. Rothstein and I. W. Stewart, arXiv:0706.3399 [hep-ph].[5] G. Bell, Nucl. Phys. B (2008) 1 [arXiv:0705.3127 [hep-ph]].286] G. Bell, PhD thesis, LMU M¨unchen, 2006, arXiv:0705.3133 [hep-ph].[7] K. G. Chetyrkin, M. Misiak and M. Munz, Nucl. Phys. B (1998) 279[arXiv:hep-ph/9711280].[8] M. Gorbahn and U. Haisch, Nucl. Phys. B (2005) 291 [arXiv:hep-ph/0411071].[9] A. J. Buras, M. Gorbahn, U. Haisch and U. Nierste, JHEP (2006) 002[arXiv:hep-ph/0603079].[10] M. Gorbahn, private communication.[11] F. V. Tkachov, Phys. Lett. B (1981) 65;K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B (1981) 159.[12] R. Bonciani and A. Ferroglia, JHEP (2008) 065 [arXiv:0809.4687 [hep-ph]];H. M. Asatrian, C. Greub and B. D. Pecjak, Phys. Rev. D (2008) 114028[arXiv:0810.0987 [hep-ph]].[13] M. Beneke, T. Huber and X. Q. Li, Nucl. Phys. B (2009) 77 [arXiv:0810.1230[hep-ph]].[14] E. Remiddi and J. A. M. Vermaseren, Int. J. Mod. Phys. A (2000) 725[arXiv:hep-ph/9905237].[15] G. Bell, Nucl. Phys. B (2009) 264 [arXiv:0810.5695 [hep-ph]].[16] T. Huber, arXiv:0901.2133 [hep-ph].[17] P. Gambino, M. Gorbahn and U. Haisch, Nucl. Phys. B (2003) 238[arXiv:hep-ph/0306079].[18] A. V. Efremov and A. V. Radyushkin, Phys. Lett. B (1980) 245;G. P. Lepage and S. J. Brodsky, Phys. Rev. D (1980) 2157.[19] T. Becher, M. Neubert and B. D. Pecjak, Nucl. Phys. B (2001) 538[arXiv:hep-ph/0102219];C. N. Burrell and A. R. Williamson, Phys. Rev. D (2006) 114004[arXiv:hep-ph/0504024].[20] R. J. Hill, T. Becher, S. J. Lee and M. Neubert, JHEP (2004) 081[arXiv:hep-ph/0404217];M. Beneke and D. Yang, Nucl. Phys. B (2006) 34 [arXiv:hep-ph/0508250].[21] S. J. Lee and M. Neubert, Phys. Rev. D (2005) 094028 [arXiv:hep-ph/0509350].[22] T. Becher and R. J. Hill, JHEP (2004) 055 [arXiv:hep-ph/0408344];G. G. Kirilin, arXiv:hep-ph/0508235. 2923] C. Bobeth, M. Misiak and J. Urban, Nucl. Phys. B (2000) 291[arXiv:hep-ph/9910220].[24] O. V. Tarasov, A. A. Vladimirov and A. Y. Zharkov, Phys. Lett. B (1980) 429;S. A. Larin and J. A. M. Vermaseren, Phys. Lett. B (1993) 334[arXiv:hep-ph/9302208].[25] M. Misiak and M. Munz, Phys. Lett. B (1995) 308 [arXiv:hep-ph/9409454];K. G. Chetyrkin, M. Misiak and M. Munz, Nucl. Phys. B (1998) 473[arXiv:hep-ph/9711266].[26] E. Egorian and O. V. Tarasov, Teor. Mat. Fiz. (1979) 26 [Theor. Math. Phys. (1979) 863].[27] R. Tarrach, Nucl. Phys. B (1981) 384;D. J. Broadhurst and A. G. Grozin, Phys. Rev. D52