The perturbative QCD factorization of ρ γ ⋆ →π
aa r X i v : . [ h e p - ph ] S e p The perturbative QCD factorization of ργ ⋆ → π Shan Cheng ∗ and Zhen-Jun Xiao , †
1. Department of Physics and Institute of Theoretical Physics,Nanjing Normal University, Nanjing, Jiangsu 210023, People’s Republic of China, and2. Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems,Nanjing Normal University, Nanjing 210023, People’s Republic of China (Dated: May 7, 2018)In this paper, we firstly varify that the factorization hypothesis is valid for the exclusiveprocess ργ ⋆ → π at the next-to-leading order (NLO) with the collinear factorization ap-proach, and then extend this proof to the case of the k T factorization approach. We particu-larly show that at the NLO level, the soft divergences in the full quark level calculation couldbe canceled completely as for the πγ ⋆ → π process where only the pseudoscalar π mesoninvolved, and the remaining collinear divergences can be absorbed into the NLO hadronwave functions. The full amplitudes can be factorized as the convolution of the NLO wavefunctions and the infrared-finite hard kernels with these factorization approaches. We alsowrite out the NLO meson distribution amplitudes in the form of nonlocal matrix elements. PACS numbers: 11.80.Fv, 12.38.Bx, 12.38.Cy, 12.39.St
I. INTRODUCTION
As the fundamental tool of the perturbative Quantum Chromodynamics(QCD)[1] with a largemomentum translation, the factorization theorem [2] assume that the hard part of the relevantprocesses is infrared-finite and can be calculated, while the non-perturbative dynamics of thesehigh-energy QCD processes can be canceled at the quark level or absorbed into the input universalhadron wave functions. The physical quantities can be written as the convolutions of the hard partkernels and the universal processes-independent wave functions, and then the perturbative QCDhas the prediction power. The collinear factorization [3, 4] and the k T factorization [5–7], withthe distinction whether to keep the transversal momenta in the propergators, are the two popularfactorization approaches applied on the kard QCD processes.We know that the theoretical study for the exclusive processes are in general more difficult thanthat for the inclusive processes [8]. Because in the exclusive processes, the pQCD factorizationin it’s standard form may be valid only for the large momentum transfer processes; while in theinclusive processes, like the deep-inelastic scattering, the leading twist factorization approximationis adequate already at Q ∼ Gev. So the intensively investigation for the factorization theoremsor the factorization approaches for the exclusive processes is unavoidable.In recent years, based on the factorization hypothesis, the collinear factorization and k T fac-torization for the exclusive processes πγ ⋆ → γ ( π ) and B → γ ( π ) lν have been testified both atthe leading order (LO) and the next-to-leading order (NLO) level, and then these factorization ∗ [email protected] † [email protected] proofs were developed into all-orders with the induction approach[9–11]. The NLO hard ker-nels for these exclusive processes have also been calculated for example in Refs. [12–16]. TheseNLO evaluations showed that the positive corrections from the leading twist would be cancelledpartly by the negative corrections from the NLO twist, resulting in a small net NLO correction tothe leading order hard kernels, which further verified the feasibility of the perturbative QCD tothose considered exclusive processes. But all these proofs and calculations are only relevant forthe pseudo-scalar mesons, the exclusive processes with vector mesons have not been included atpresent. The study of the electromagnetic form factor processes between the vector meson and thepseudo-scalar meson is an important way to understand the internal structure of hadrons. Thereare many works on this subject: (a) ρ meson transition and electromagnetic form factors are pre-dicted at the NLO level in the QCD sum rule analysis[17]; (b) space-like and time-like pion-rhotransition form factors were investigated in Ref. [18] in the light-cone formalism; (c) the mesontransition form factors were studied within a model of QCD based on the Dyson-Schwinger equa-tions in[19]; and (d) the transition form factor of ργ ⋆ → π was also extracted from the otherprocesses in the extended hard-wall AdS/QCD model[20] recently.In this paper, we also consider the rho-pion transition process. By inserting the Fierz iden-tity into the relevant expressions and employing the eikonal approximation, we can factorize thefermion flow and the momentum flow effectively. By summing over all the color factors, we canexpress these irreducible convolutions into three parts: with the additional gluon momentum flow,not flow and partly flow into the leading order hard kernel. We will do the factorization prooffor the exclusive process ργ ⋆ → π at the NLO level, from the collinear factorization to the k T factorization approach. With the light-cone kinetics, we will obtain the gauge invariant nonlocalmatrix element for the pion meson and rho meson wave functions along the light-cone direction inthe collinear factorization, and lightly deviate from the light-cone direction in the k T factorization.At the NLO level, we clearly verified that the soft divergences will be canceled in the quark leveldiagrams, and the collinear divergences can be absorbed into the NLO wave functions, then wecan obtain an infrared-finite next-to-leading order hard kernel in principle.The paper is organized as following. The leading order dynamical analysis is presented in thesecond section. In section-III we prove that the collinear factorization approach is valid for the ρ → π transition process at the next-to-leading order. The collinear factorization approach isextended to the k T factorization approach for this ρ → π transition process in section-VI. Thesummary and some discussions will appear at the final section. II. COLLINEAR FACTORIZATION OF ργ ⋆ → π In this section we will prove the collinear factorization of the transition ργ ⋆ → π . We firstlyconsider the two sets of leading order transition amplitudes, and then use the Fierz identity and theeikonal approximation to factorize the fermion currents and the momentum currents at the NLOlevel, in order to obtain the NLO transition amplitudes for each sub-diagram in the convolutedforms of the LO hard transition amplitudes and the gauge invariant nonlocal NLO distributionamplitudes(DAs) along the light-core(LC) direction. We finally sum up all the sub-diagrams foreach set to collect all the color factors. The key point of the factorization is to find and absorb theinfrared divergences, so we will not consider the self-energy corrections to the internal quark linesbecause they don’t generate infrared divergences. ρ π ( a ) k k p − k p − k ( b )( d )( c ) FIG. 1. The four leading-order quark diagrams for the ργ ⋆ → π form factor with the symbol • representingthe virtual photon vertex. A. Leading Order Hard Kernel
The LO quark diagrams for the ργ ⋆ → π transition are shown in the Fig. 1, where the vir-tual photon vertex represented by the dark spot have been placed at the four different positionsrespectively. In the light-cone coordinator system, the incoming ρ meson carry the momenta p = Q √ (1 , , T ) , and the outgoing π carry the momenta p = Q √ (0 , , T ) . Besides the mo-menta, the initial ρ would carry the longitudinal polarization vector ǫ µ ( L ) = √ γ ρ (1 , − γ ρ , T ) and the transversal polarization vector ǫ µ ( T ) = (0 , , T ) . The momenta carried by the anti-quark of the initial and final state meson are defined as k = Q √ ( x , , T ) and k = Q √ (0 , x , T ) with x and x being the momentum fraction carried by the anti-partons inside ρ and π .As the spin-1 particle, the wave functions for ρ meson should contain both longitudinal andtransverse components[21]. Φ ρ ( p , ǫ T ) = i √ N c h M ρ ǫ/ T φ vρ ( x ) + ǫ/ T p/ φ Tρ ( x ) + M ρ iǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ( x ) i , Φ ρ ( p , ǫ L ) = i √ N c (cid:2) M ρ ǫ/ L φ ρ ( x ) + ǫ/ L p/ φ tρ ( x ) + M ρ φ sρ ( x ) (cid:3) , (1)in which φ ρ and φ Tρ are twist-2 (T2) DAs, φ t/sρ , φ t/sρ are twist-3 (T3) DAs, and the unit vector n/v isdefined as (1 , , ) / (0 , , ) . The pseudoscalar π meson wave function up to twist-3 is also givenas in Refs. [22–24] Φ π ( p ) = − i √ N c (cid:8) γ p/ φ aπ ( x ) + m π γ (cid:2) φ pπ ( x ) + ( v/n/ − φ tπ ( x ) (cid:3)(cid:9) , (2)with the twist-2 DA φ aπ and twist-3 DAs φ pπ and φ tπ . The operator product expansion (OPE)[25]states that amplitudes from the twist-3 DAs are suppressed by the hierarchy M ρ /Q and m π /Q atthe large momenta transition region, when compared with the twist-2 DAs of the ρ and π mesonwave functions respectively. We can classify the LO transition amplitudes into four sets by thetwists’ analysis of the initial and final meson wave functions: T2&T2; T2&T3; T3&T2 and finallyT3&T3. Fortunately, we just need to consider the first two sub-diagrams Fig. 1(a) and Fig. 1(b)directly, because the amplitudes of sub-diagram Fig. 1(c) (Fig. 1(d) ) can be obtained by simplereplacement x i → − x i ( i = 1 , from the amplitudes of Fig. 1(a) ( Fig. 1(b)). The standardcalculations show that only the T3&T2 set ( the twist-3 DAs of the rho meson and the twist-2 DAsof the pion meson) contribute to the LO transition amplitude of Fig. 1(a), which can be written asthe following form, G (0) a, ( x , x ) = ieg s C F ǫ/ T M ρ φ vρ + M ρ iǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ] γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α ( p − k ) ( k − k ) , (3)where γ α should be chosen as γ − . Similarly, only the crossed sets of T2&T3 (Set-I) and T3&T2(Set-II) contribute to the LO transition amplitudes of Fig. 1(b), which can be written as the formof G (0) b, ( x , x ) = ieg s C F ǫ/ T p/ φ Tρ ] γ α [ γ m π φ Pπ ] γ α ( p/ − k/ ) γ µ ( p − k ) ( k − k ) , (4)where the γ α can be γ − or γ α ⊥ ; G (0) b, ( x , x ) = ieg s C F ǫ/ T M ρ φ vρ + M ρ iǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ] γ α [ γ p/ φ Aπ ] γ α ( p/ − k/ ) γ µ ( p − k ) ( k − k ) , (5)where the γ α = γ α ⊥ . The LO transition amplitudes as given in Eqs. (3,4,5) are all transversal dueto the γ from the final pion meson wave function, the γ µ from the virtual photon vertex and thepolarization vector ǫ of the initial ρ meson.From the expressions of the LO transition amplitudes G (0) ( x , x ) as given in Eqs. (3,4,5), onecan see that there are clear qualitative differences between the ργ ⋆ → π studied in this paper andthe πγ ⋆ → π investigated previously in Refs. [9, 11, 13, 15]:(i) In the πγ ⋆ → π transition, the initial and final state meson are the same pion. Consequently,only the contribution from Fig. 1(a) should be calculated explicitly, while the contributionsfrom Figs. 1(b,c,d) can be obtained from those of Fig. 1(a) by direct kinetic transformations[13, 15]. Furthermore, only the T2&T2 and T3&T3 terms contribute to the LO transitionamplitudes because of the presence of matrix γ in both the initial and final state pion meson.(ii) For ργ ⋆ → π transition, however, the initial and final state meson are the vector ρ andpseudo-scalar pion. The possible contributions from Figs. 1(a) and 1(b) are rather dif-ferent and should be calculated explicitly. For ργ ⋆ → π transition, in fact, only thetransversal component Φ ρ ( p , ǫ T ) of initial rho meson in Eq. (1) contribute to the LO rho-pion transition amplitude, and this LO transition amplitude receive the contributions from G (0) a, ( x , x ) in Eq. (3 )(i.e. the crossed-set T3&T2 ) from Fig. 1(a), and from G (0) b, ( x , x ) and G (0) b, ( x , x ) in Eqs. (4,5)(i.e. the crossed sets T2&T3 and T3&T2 ) from Fig. 1(b). B. O ( α s ) corrections to Fig.1(a) A complete amplitude for a physical process in QCD is usually defined in three spaces: the spinspace, the momenta space and the color space. So the factorization theorems need to deal with all ( a ) ( b )( d ) ( g )( c )( e ) ( f )( h ) ( i ) ( j ) ( k ) FIG. 2. O ( α s ) corrections to Fig. 1(a) with an additional gluon (blue curves) emitted from the initial ρ meson. these three spaces in the QCD processes. We can factorize the fermion currents in the spin spaceby using the Fierz identity, I ij I lk = 14 I ik I lj + 14 ( γ ) ik ( γ ) lj + 14 ( γ α ) ik ( γ α ) lj + 14 ( γ γ α ) ik ( γ α γ ) lj + 18 ( σ αβ γ ) ik ( σ αβ γ ) lj , (6)where I is the identity matrix and σ αβ is defined by σ αβ = i [ γ α , γ β ] / , the different terms inEq. (6) stand for different twists’ contributions. The eikonal approximation is used to factorize themomenta currents in the momentum space. And at last we need to sum over all the color factors toobtain the gauge-independent high order DAs. In this section we will show the NLO factorizationof the ρ → π transition process, according to the LO transition amplitudes expressed in Eqs. (3,4,5)for the sub-diagrams Figs. 1(a,b). We try to factorize these NLO transition amplitudes into theconvolutions of the LO hard amplitudes and the NLO meson DAs.We here firstly testify that the collinear factorization is valid at the NLO level for the Fig. 1(a),where the LO transition amplitude as given in Eq. (3) contains the T3&T2 contribution only. Sowe just need to consider the twist-3 DAs for the initial ρ meson and the twist-2 DA for the finalstate π meson in this NLO factorization proofs.There are two types infrared divergences from O ( α s ) corrections to Fig. 1(a) induced by anadditional gluon as illustrated in Fig. 2 and Fig. 4, which are distinguished by the direction ofthe additional gluon momentum. We firstly identify these infrared divergences for the O ( α s ) correction with the additional ”blue” gluon emitted from the initial ρ meson as shown in Fig. 2,where the gluon momenta may be parallel to the rho meson momenta p .It’s easy to find that the amplitudes in Eqs. (7,8,9) are reducible for sub-diagrams Fig. 2(a,b,c),because we can factorize this amplitudes by simply inserting the Fierz identity. The symmetryfactor / in the self-energy diagrams Eqs. (7,9) represent the freedom to chose the most outsidevertex of the additional gluon. The soft divergences from the l ∼ ( λ, λ, λ ) region are canceledin these reducible amplitudes G (1)2 a, ( x ; x ) , G (1)2 b, ( x ; x ) , G (1)2 c, ( x ; x ) , which is determined bythe QCD dynamics that the soft gluon don’t resolve the color structure of the rho meson. G (1)2 a, = 12 eg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k ) ( p − k ) ( p − k + l ) l · γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α ( p/ − k/ ) γ ρ ′ ( p/ − k/ + l/ ) γ ρ ′ = 12 φ (1) ,vρ,a ⊗ G (0) ,va, ( x ; x ) + 12 φ (1) ,aρ,a ⊗ G (0) ,aa, ( x ; x ) , (7) G (1)2 b, = − eg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k − l ) ( p − k + l ) ( k − l ) l · γ ρ ′ ( k/ − l/ ) γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α ( p/ − k/ + l/ ) γ ρ ′ = φ (1) ,vρ,b ⊗ G (0) ,va, ( ξ , x ) + φ (1) ,aρ,b ⊗ G (0) ,aa, ( ξ , x ) , (8) G (1)2 c, = 12 eg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k ) ( p − k ) ( p − k + l ) l · γ ρ ′ ( k/ − l/ ) γ ρ ′ k/ γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α = 12 φ (1) ,vρ,c ⊗ G (0) ,va, ( x , x ) + 12 φ (1) ,aρ,c ⊗ G (0) ,aa, ( x , x ) , (9)where the LO hard amplitudes G (0) ,va, ( ξ , x ) and G (0) ,aa, ( ξ , x ) in Eq. (8) with the gluon momentaflowing into the LO hard kernel are of the following form G (0) ,va, ( ξ ; x ) = ieg s C F ǫ/ T M ρ φ vρ ] γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α ( p − k ) ( k − k − l ) , (10) G (0) ,aa, ( ξ ; x ) = ieg s C F M ρ iǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ ] γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α ( p − k ) ( k − k − l ) . (11)The NLO DAs φ (1) ρ in Eqs. (7,8,9), which absorbed all the infrared singularities from thosereducible sub-diagrams Figs. 2(a,b,c), can be written as the following form φ (1) ,vρ,a = − ig s C F γ b ⊥ γ ⊥ b ( p/ − k/ ) γ ρ ′ ( p/ − k/ + l/ ) γ ρ ′ ( p − k ) ( p − k + l ) l ,φ (1) ,aρ,a = − ig s C F γ γ µ ′ ⊥ γ ⊥ µ ′ γ ( p/ − k/ ) γ ρ ′ ( p/ − k/ + l/ ) γ ρ ′ ( p − k ) ( p − k + l ) l ; φ (1) ,vρ,b = ig s C F γ b ⊥ γ ρ ′ ( k/ − l/ ) γ ⊥ b ( p/ − k/ + l/ ) γ ρ ′ ( k − l ) ( p − k + l ) l ,φ (1) ,aρ,b = ig s C F γ γ µ ′ ⊥ γ ρ ′ ( k/ − l/ ) γ ⊥ µ ′ γ ( p/ − k/ + l/ ) γ ρ ′ ( k − l ) ( p − k + l ) l ; φ (1) ,vρ,c = − ig s C F γ b ⊥ γ ρ ′ ( k/ − l/ ) γ ρ ′ k/ γ ⊥ b ( k − l ) ( k ) l ,φ (1) ,aρ,c = − ig s C F γ γ µ ′ ⊥ γ ρ ′ ( k/ − l/ ) γ ρ ′ k/ γ ⊥ µ ′ γ ( k − l ) ( k ) l . (12)The additional gluons in sub-diagrams Figs. 2(d,e,f,g) generate the collinear divergences only,because one vertex of the gluon is attached to the LO hard part and then the soft region is stronglysuppressed by /Q . For these amplitudes, we choose the radiative gluon momenta being parallelto the initial rho meson momenta p to evaluate the collinear divergences. All the amplitudes forthose sub-diagrams in Fig. 2(d,e,f,g) are listed in Eqs. (13,15,16,17). For Fig. 2(d) we find G (1)2 d, = − ieg s T r [ T a T c T b ] f abc N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k ) ( p − k + l ) ( k − k − l ) l · γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ β ( p/ − k/ + l/ ) γ γ F αβγ ∼ φ (1) ,vρ,d ⊗ [ G (0) ,va, ( x ; x ) − G (0) ,va, ( ξ ; x )]+ 916 φ (1) ,aρ,d ⊗ [ G (0) ,aa, ( x ; x ) − G (0) ,aa, ( ξ ; x )] , (13)with φ (1) ,vρ,d = − ig s C F γ b ⊥ γ ⊥ b ( p/ − k/ + l/ ) γ ρ v ρ ( p − k + l ) l ( v · l ) ,φ (1) ,aρ,d = − ig s C F γ γ µ ′ ⊥ )( γ ⊥ µ ′ γ )( p/ − k/ + l/ ) γ ρ v ρ ( p − k + l ) l ( v · l ) . (14)In Eq. (13), we have F αβρ ′ = g αβ (2 k − k − l ) ρ ′ + g βρ ′ ( k − k + 2 l ) α + g ρ ′ α ( k − k − l ) β , and wefind that only the terms proportional to g αβ and g ρ ′ α contribute to the LO hard kernel with γ α = γ − .Then we can factorize the amplitude G (1)2 d, into the NLO twist-3 transversal rho DAs φ (1) ,vρ,d and φ (1) ,aρ,d in Eq. 14, convoluted with the LO hard amplitudes G (0) ,va, ( x ; x ) and G (0) ,aa, ( x ; x ) , to whichthe gluon momenta flow or not flow in.For Fig. 2(e) we have G (1)2 e, = ieg s T r [ T a T c T b ] f abc N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k ) ( k − l ) ( k − k − l ) l · γ γ ( k/ − l/ ) γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ β F αβγ ∼ , (15)where F αβγ = g αβ (2 k − k − l ) γ + g βγ ( k − k − l ) α + g γα ( k − k +2 l ) β . The possible contributionsfrom the three terms in the tensor F αβγ is either suppressed by the kinetics or excluded by therequirement that the Gamma matrix in the NLO amplitudes should hold the LO content γ α = γ − .Then we can assume that the infrared contribution from the sub-diagram Fig. 2(e) can be neglectedsafely. The kinetic suppression is also happened for the amplitudes of Figs. 2(f,g), theses two sub-diagrams also do not provide infrared correction to the LO hand kernel G (0) ,v/aa, , i.e., G (1)2 f, = eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k ) ( p − k + l ) ( p − k + l ) l · γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ ρ ′ ( p/ − k/ + l/ ) γ α ( p/ − k/ + l/ ) γ ρ ′ ∼ , (16) G (1)2 g, = − eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k − l ) ( p − k − l ) ( k − l ) l · γ ρ ′ ( k/ − l/ ) γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ ρ ′ ( p/ − k/ − l/ ) γ α ∼ . (17)For sub-diagrams Figs. 2(h,i,j,k), however, the additional gluon generates the collinear diver-gences as well as the soft divergences, because both ends of the gluon are attached to the externalquark lines. As the partner with the soft divergences, the collinear divergences are also evaluatedby setting the radiative gluon momenta being parallel to the initial rho meson momenta p . Theamplitudes for all these four sub-diagrams are given in Eqs. (18,19,20,21).For Figs. 2(h,i) we have G (1)2 h, = eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( k − k ) ( p − k + l ) ( p − k + l ) ( p − k + l ) l · γ α [ γ p/ φ Aπ ] γ ρ ′ ( p/ − k/ + l/ ) γ µ ( p/ − k/ + l/ ) γ α ( p/ − k/ + l/ ) γ ρ ′ ∼ ( −
18 ) φ (1) ,vρ,d ⊗ G (0) ,va, ( x , x ) + ( −
18 ) φ (1) ,aρ,d ⊗ G (0) ,aa, ( x , x ) , (18) G (1)2 i, = − eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( k − k − l ) ( p − k ) ( p − k + l ) ( k + l ) l · γ α ( k/ + l/ ) γ ρ ′ [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α ( p/ − k/ + l/ ) γ ρ ′ ∼ ( 18 ) φ (1) ,vρ,d ⊗ G (0) ,va, ( ξ ; x ) + ( 18 ) φ (1) ,aρ,d ⊗ G (0) ,aa, ( ξ ; x ) . (19)For Figs. 2(j,k), we find that G (1)2 j, and G (1)2 k, don’t provide the NLO correction to the LOamplitude G (0) a, , because of the confine of the Gamma matrixes to extract the LO amplitude G (0) a, ,then the infrared contribution of these two amplitudes can also be neglected safely. G (1)2 j, = eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( k − k ) ( p − k ) ( k − l ) ( k − l ) l · γ ρ ′ ( k/ − l/ ) γ α ⊥ ( k/ − l/ ) γ ρ ′ [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ ⊥ α ∼ , (20) G (1)2 k, = − eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( k − k − l ) ( p − k − l ) ( k − l ) ( p − k − l ) l · γ ρ ′ ( k/ − l/ ) γ α ⊥ [ γ p/ φ Aπ ] γ ρ ′ ( p/ − k/ − l/ ) γ µ ( p/ − k/ − l/ ) γ ⊥ α ∼ , (21)For the irreducible infrared amplitudes as shown in Eqs. (13,15-21), we have the followingobservations:(i) We sum up the amplitudes for the irreducible sub-diagrams Figs. 2(d,f,h,i) together, in whichthe additional gluon is radiated from the initial up-line quark. G (1)2 up, ( x ; x ) = G (1)2 d, ( x ; x ) + G (1)2 f, ( x ; x ) + G (1)2 h, ( x ; x ) + G (1)2 i, ( x ; x )= φ (1) ,vρ,d ⊗ (cid:18) (cid:19) h G (0) ,va, ( x ; x ) − G (0) ,va, ( ξ ; x ) i + φ (1) ,aρ,d ⊗ (cid:18) (cid:19) h G (0) ,aa, ( x ; x ) − G (0) ,aa, ( ξ ; x ) i . (22) ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) FIG. 3. O ( α s ) effective diagrams for the initial transversal rho meson wave function, which collect allthe collinear divergences from the initial rho meson in the irreducible NLO quark diagrams. The verticaldouble line denotes the Wilson line along the light cone, whose Feynman rule is v ρ / ( v · l ) as described inEq. (25,26). The summation of the amplitudes for the sub-diagrams Figs. 2(e,g,j,k), in which the ad-ditional gluon is radiated from the initial down-line quark, would give the zero infraredcontribution. The infrared divergences only come from the gluon radiated from the up-linequark of rho meson as shown in Fig. 2, while the infrared contributions from the down-linequark are excluded either by the dynamics or the kinetics.(ii) By comparing the amplitudes G (1)2 h, with G (1)2 i, , We find that the soft divergences from theirreducible sub-diagrams Fig. 2(h) and Fig. 2(i) will be canceled completely by the simplereplacement ξ → x . Combining with the cancellation of the soft divergences in the sub-diagrams Figs. 2(a,b,c), there is no soft divergence in the quark level for the Fig. 2.(iii) The NLO corrections to the LO sub-diagram Fig. 1(a) with the collinear gluon emitted fromthe initial state do have the collinear divergences, but they can be absorbed into the NLOrho meson DAs φ (1) ,vρ,d and φ (1) ,aρ,d . From Eqs. (13,16,18,19), one can write out the Feynmanrules for the perturbative calculation of the NLO twist-3 transversal ρ meson wave functions φ (1) ,vρ,d and φ (1) ,aρ,d as a nonlocal hadronic matrix element with the structure γ ⊥ / and ( γ γ ⊥ ) / sandwiched respectively: φ (1) ,vρ,d = 12 N c P +1 Z dy − π e − ixp +1 y − < | q ( y − ) γ ⊥ − ig s ) Z y − dzv · A ( zv ) q (0) | ρ ( p ) >, (23) φ (1) ,aρ,d = 12 N c P +1 Z dy − π e − ixp +1 y − < | q ( y − ) γ γ ⊥ − ig s ) Z y − dzv · A ( zv ) q (0) | ρ ( p ) > . (24) The integral variable z runs from to ∞ for the upper eikonal line as showed in Fig. 3(a),and runs from ∞ back to y − for the lower eikonal line as showed in Fig. 3(b). The choiceof the light-cone coordinate y − = 0 represents the fact that the collinear divergences fromthe sub-diagrams of Fig. 2 don’t cancel exactly.(iv) The factor v ρ / ( v · l ) in Eq. (14) is the Feynman rule associated with the Wilson line, whichis required to remain gauge invariance of the nonlocal matrix element in the NLO rho wavefunctions and has been included in Eqs. (23,24) of the NLO wave functions. We can retrievethis factor by Fourier transformation of the gauge field from A ( zv ) to A ( l ) in these NLO0wave functions: Z ∞ dzv · A ( zv ) → Z dl e iz ( v · l + iǫ ) Z ∞ dzv · e A ( l ) = i Z dl v ρ v · l e A ρ ( l ) , (25) Z y − dzv · A ( zv ) → Z dl e iz ( v · l + iǫ ) Z y − dzv · e A ( l ) = − i Z dl v ρ v · l e il + y − e A ρ ( l ) . (26)The Fourier factor e il + y − in Eq. (26) will lead to the function δ ( ζ − x + i + /p +1 ) , which meansthat the gluon momentum l has flowed into the LO hard kernel as described in Eqs. (10,11).(v) The NLO irreducible amplitudes for Fig. 2 in the collinear region can be written as theconvolutions of the NLO DAs and the LO hard amplitudes. The collinear factorization isvalid for the NLO corrections for the Fig. 1(a) with the additional gluon emitted from theinitial rho meson.(vi) The sub-diagrams Figs. 3(a,b,e) are the effective-diagrams for the additional gluon radiatedfrom the left-up quark line, the sub-diagrams Figs. 3(c,d,f) represent the effective-diagramsfor the additional gluon radiated from the left-down anti-quark line. We can also sort thesesix effective-diagrams in Fig. 3 into three sets by the flowing of the gluon momenta: (a)the first set contains the effective diagram 3(a) and 3(c) with no gluon momenta flow intothe LO hard amplitudes; (b) the second set is made of the effective diagram 3(b) and 3(d)with the gluon momenta flow into the LO hard amplitudes; and (c) the third set includesthe effective diagram 3(e) and 3(f) with the gluon momenta flow partly into the LO hardamplitudes.Now we consider the infrared divergences from O ( α s ) radiative corrections to Fig. 1(a) withthe additional collinear gluon emitted from the final π meson as shown in Fig. 4, where the gluonmomenta may be collinear with the pion meson momenta p .Since the sub-diagrams Figs. 4(a,b,c) are reducible diagrams, we can factorize them directlyby inserting the Firez identity into proper places as being done for Figs. 2(a,b,c) previously. Thesymmetry factor / are also exist in G (1)4 a, and G (1)4 c, . And the soft divergences in these re-ducible amplitudes G (1)4 a, , G (1)4 b, , G (1)4 c, as given in Eqs. (27.28,29) will also be cancelled eachother exactly. G (1)4 a, = 12 eg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ] γ α [ γ p/ φ Aπ ] γ ρ ′ ( p − k ) ( k − k ) ( p − k ) ( p − k + l ) l · ( p/ − k/ + l/ ) γ ρ ′ ( p/ − k/ ) γ µ ( p/ − k/ ) γ α = 12 G (0) a, ( x ; x ) ⊗ φ (1) ,Aπ,a , (27) G (1)4 b, = − eg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k + l ) ( k − k + l ) ( p − k + l ) ( k − l ) l · γ ρ ′ [ γ p/ φ Aπ ] γ ρ ′ ( p/ − k/ + l/ ) γ µ ( p/ − k/ + l/ ) γ α = G (0) a, ( x ; ξ ) ⊗ φ (1) ,Aπ,b , (28)1 ( b ) ( c )( d ) ( e ) ( f ) ( g )( h ) ( i ) ( j ) ( k )( a ) FIG. 4. O ( α s ) corrections to Fig. 1(a) with an additional gluon (blue curves) emitted from the final π meson. G (1)4 c, = 12 eg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ] γ α k/ γ ρ ′ ( k/ − l/ )( p − k ) ( k − k ) ( k − l ) ( k ) l · γ ρ ′ [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α = 12 G (0) a, ( x , x ) ⊗ φ (1) ,Aπ,c , (29)where φ (1) ,Aπ,i with i = ( a, b, c ) are the NLO DAs, which absorbed all the infrared singularities fromthese reducible sub-diagrams Figs. 2(a,b,c) and can be written in the following forms: φ (1) ,Aπ,a = − ig s C F γ γ + ] γ ρ ′ ( p/ − k/ + l/ ) γ ρ ′ ( p/ − k/ )[ γ − γ ]( p − k ) ( p − k + l ) l ; φ (1) ,Aπ,b = ig s C F k/ − l/ ) γ ρ ′ [ γ γ + ] γ ρ ′ ( p/ − k/ + l/ )[ γ − γ ]( p − k + l ) ( k − l ) l ; φ (1) ,Aπ,c = − ig s C F γ − γ ] k/ γ ρ ′ ( k/ − l/ ) γ ρ ′ [ γ γ + ]( k − l ) ( k ) l . (30)The infrared singularity analysis for Fig. 2 are also valid for Fig. 4. The sub-diagrams in thesecond row of Fig. 4 also contain the collinear singularity only, while the third row sub-diagramsmay contain both collinear and soft divergences. Before discussing the infrared behaviour of theseirreducible sub-diagrams in Figs. 4(d-k), we here firstly define those LO hard amplitudes which2either appeared in Eq. 28 or will appeare in the NLO irreducible amplitudes, G (0) a, ( x ; ξ ) = ieg s C F ǫ/ T M ρ φ vρ + M ρ iǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ ]( p − k + l ) ( k − k + l ) · γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ + l/ ) γ α , (31) G (0) a, ( x ; ξ , x ) = ieg s ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k + l ) ( k − k ) · γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α , (32) G ′ (0) a, ( x ; ξ , x ) = ieg s ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k + l ) · γ α [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α . (33)In the collinear region l k p , we can find the equal relation G (0) a, ( x ; ξ , x ) = G ′ (0) a, ( x ; ξ , x ) for the newly defined LO hard amplitudes as shown in Eqs. (32,33).The transition amplitude for Fig. 4(d) can be written as the form of G (1)4 d, = − ieg s T r [ T c T b T a ] f abc N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k + l ) ( k − k ) ( k − k + l ) ( p − k + l ) l · γ α [ γ p/ φ Aπ ] γ β ( p/ − k/ + l/ ) γ µ ( p/ − k/ + l/ ) γ γ F αβγ = h G (0) a, ( x ; ξ , x ) − G (0) a, ( x ; ξ ) i ⊗ φ (1) ,Aπ,d , (34)with the tensor F αβγ = g αβ ( k − k − l ) γ + g βγ ( k − k + 2 l ) α + g γα (2 k − k − l ) β , in whichonly terms proportional to g βγ and g γα contribute to the LO hard kernel G (0) a, . The NLO twist-2pion DA φ (1) ,Aπ,d is defined in the following form φ (1) ,Aπ,d = − ig s C F γ γ + ] γ ρ ( p/ − k/ + l/ )[ γ − γ ] n ρ ′ ( p − k + l ) l ( n · l ) . (35)Here the eikonal approximation has been employed to obtain the convolution forms for theseirreducible amplitudes.For Fig. 4(e), similarly, we have G (1)4 e, = ieg s T r [ T c T b T a ] f abc N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k ) ( k − k + l ) ( k − l ) l · γ α ( k/ − l/ ) γ β [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ γ F αβγ = h G (0) a, ( x ; x ) − G ′ (0) a, ( x ; ξ , x ) i ⊗ φ (1) ,Aπ,e , (36)where F αβγ = g αβ ( k − k + 2 l ) γ + g βγ ( k − k − l ) α + g γα (2 k − k − l ) β , and only the termsproportional to g βγ and g γα contribute to the LO hard kernel G (0) a, . The NLO twist-2 pion DA φ (1) ,Aπ,e is defined in the form of φ (1) ,Aπ,e = ig s C F γ γ + ] γ ρ ( k/ − l/ )[ γ − γ ] n ρ ′ ( k − l ) l ( n · l ) , (37)3where the additional gluon is emitted from the right-down anti-parton line. Then the ampli-tudes for the remaining irreducible sub-diagrams in Fig. 4 can be written with the definitionsin Eqs. (31,32,33,35,37): G (1)4 f, = eg s C F N c N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k ) ( p − k + l ) l ( p − k + l ) · γ α [ γ p/ φ Aπ ] γ ρ ′ ( p/ − k/ + l/ ) γ µ ( p/ − k/ + l/ ) γ ρ ′ ( p/ − k/ ) γ α = h G (0) a, ( x ; x ) − G (0) a, ( x ; ξ , x ) i ⊗ φ (1) ,Aπ,d , (38) G (1)4 g, = − eg s C F N c N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k + l ) ( k − l ) l ( p − k + l ) · γ α ( k/ − l/ ) γ ρ ′ [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ ρ ′ ( p/ − k/ + l/ ) γ α = h G ′ (0) a, ( x ; ξ , x ) − G (0) a, ( x ; ξ ) i ⊗ φ (1) ,Aπ,e , (39) G (1)4 h, = eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k + l ) ( k − k ) ( p − k + l ) l ( p − k + l ) · γ α [ γ p/ φ Aπ ] γ ρ ′ ( p/ − k/ + l/ ) γ µ ( p/ − k/ + l/ ) γ α ( p/ − k/ + l/ ) γ ρ ′ = G (0) a, ( x ; ξ , x ) ⊗ ( −
18 ) φ (1) ,Aπ,d , (40) G (1)4 i, = − eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k + l ) ( k − k + l ) ( p − k + l ) l ( k + l ) · γ ρ ′ ( k/ + l/ ) γ α [ γ p/ φ Aπ ] γ ρ ′ ( p/ − k/ + l/ ) γ µ ( p/ − k/ + l/ ) γ α =0 , (41) G (1)4 j, = eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ] γ ρ ′ ( k/ − l/ ) γ α ( p − k ) ( k − k ) ( k − l ) l ( k − l ) · ( k/ − l/ ) γ ρ ′ [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α =0 , (42) G (1)4 k, = − eg s T r [ T c T a T c T a ]2 N c [ ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k + l ) ( k − l ) l ( p − k − l ) · γ α ( k/ − l/ ) γ ρ ′ [ γ p/ φ Aπ ] γ µ ( p/ − k/ ) γ α ( p/ − k/ − l/ ) γ ρ ′ = G ′ (0) a, ( x ; ξ , x ) ⊗ ( 18 ) φ (1) ,Aπ,e . (43)The infrared contributions from the NLO amplitudes G (1)4 j, and G (1)4 j, are zero, since the Gammamatrixes in these two amplitudes are γ α = γ α ⊥ instead of the γ α = γ − for the LO amplitudes.In order to investigate the NLO collinear factorization of the Fig. 4 and to extracte the NLOtwist-2 pion meson DA, we make the summation over all the irreducible amplitudes in Fig. 4 intotwo sets: the first set includes the sub-diagrams with the gluon radiated from the right-up quark line4 ( d )( b ) ( c ) ( e ) ( f )( a ) FIG. 5. O ( α s ) effective diagrams for the final pion meson wave function, with vertical double line denotingthe Wilson line along the light cone, whose Feynman rule is n ρ ′ / ( n · l ) . of the final pion meson, while the second set containes the sub-diagrams with the gluon radiatedfrom the right-down quark line.We firstly sum up the infrared amplitudes for the irreducible sub-diagrams in Figs. 4(d,f,h,i)with the gluon radiated from the right-up quark line: G (1)4up , ( x ; x ) = G (1)4 d, ( x ; x ) + G (1)4 f, ( x ; x ) + G (1)4 h, ( x ; x ) + G (1)4 i, ( x ; x )= (cid:20) G (0) a, ( x ; x ) − G (0) a, ( x ; ξ ) − G (0) a, ( x ; x , ξ ) (cid:21) ⊗ φ (1) ,Aπ,d ; (44)For the second set of the irreducible sub-diagrams in Figs. 4(e,g,j,k)( where the gluon radiatedfrom the right-down anti-quark line), similarly, we make the summation and then find the infraredamplitude: G (1)4down , ( x ; x ) = G (1)4 e, ( x ; x ) + G (1)4 h, ( x ; x ) + G (1)4 j, ( x ; x ) + G (1)4 k, ( x ; x )= (cid:20) − G (0) a, ( x ; ξ ) + 916 G (0) a, ( x ; x ) + 916 G (0) a, ( x ; x , ξ ) (cid:21) ⊗ φ (1) ,Aπ,e . (45)Because the IR singularities in Eqs. (41,42) are suppressed, then the soft divergences in Eq. (40)and Eq. (43) from the collinear region can’t be cancelled by their counterparts described in Eq. (41)and Eq. (42) respectively. But these remained soft divergences in Eqs. (40,43) could be canceledeach other exactly, because the NLO DA φ (1) ,Aπ,d in Eq. (35) is equivalent to the DA φ (1) ,Aπ,e in Eq. (37).At the quark level, finally, no soft divergences are left after summation of the NLO contributionsfrom all the sub-diagrams as shown in Fig. 4.All the remaining collinear divergences can be absorbed into the NLO twist-2 pion mesonDA φ (1) ,Aπ . From the expressions as given in Eqs. (34,36,38,39,40,43), we can define the Feynmanrules for the perturbative calculation of the twist-2 pion wave function φ (1) ,Aπ as a nonlocal hadronicmatrix element with the structure ( γ − γ ) / sandwiched: φ (1) ,Aπ = 12 N c P − Z dy + π e − ixp − y + < π ( p ) | q ( y + )( − ig s ) Z y + dzn · A ( zn ) γ − γ q (0) | >, (46)which has the same form as the in the πγ ⋆ → π [9]. The relevant effective diagrams for the pionmeson wave function are also showed in Fig. 5, and here only the first four diagrams in Fig. 5are useful to this sort NLO corrections described in Fig. 4 because the corrections with the gluonmomentum partly flowing into LO hard kernel are cancelled in Eqs. (44,45). We can also derivedthe Feynman rule ( n ρ ′ / ( n · l ) ) for the Wilson line in Fig. 5 from the O ( α s ) component of the pionwave function by the similar Fourier transformation as for Fig. 3. Then collinear factorization istherefore valid for the NLO corrections for the Fig. 1(a) when the additional gluon emitted fromthe final pion meson.5 ( a ) ( b ) ( c )( d ) ( f ) ( g )( h ) ( i ) ( k )( j )( e ) FIG. 6. O ( α s ) correction to Fig. 1(b) with the additional gluon (blue curves) emitted from the initial ρ meson. ( a ) ( b ) ( c )( d ) ( e ) ( f ) ( g )( k )( j )( i )( h ) FIG. 7. O ( α s ) corrections to Fig. 1(b) with the additional gluon (blue curves) emitted from the final π meson. C. O ( α s ) correction to Fig. 1(b) In this subsection, we study the feasibility of the collinear factorization for the NLO correctionsto Fig. 1(b) with the same approach as we did for Fig. 1(a). With the requirement to hold the LOcontents as shown in Eqs. (4,5) in the NLO factorization proof, we will consider both the T2&T3and T3&T2 sets for the DAs of the initial and final state meson in the NLO transition processas illustrated in Fig. 6 and Fig. 7. We try to use the collinear factorization approach to separatethe infrared divergences of the amplitudes for Fig. 6 and Fig. 7 with the additional blue gluonsradiated from the initial rho meson and final pion meson respectively.Firstly, the reducible sub-diagrams Figs. 6(a,b,c) and Figs. 7(a,b,c) are factorized easily by6simple inserting of the Fierz identity defied in Eq. 6. And the soft divergences will be canceledexactly in these reducible amplitudes similarly as we verified for Figs. 2(a,b,c)and Figs. 4(a,b,c).We can then extract out the NLO twist-2 transversal rho meson DA’s and the NLO twist-3 pionmeson DA’s in the following forms: φ (1) ,Tρ,a = − ig s C F γ a ⊥ γ − ][ γ ⊥ a γ + ]( p/ − k/ ) γ ρ ′ ( p/ − k/ + l/ ) γ ρ ′ ( p − k ) ( p − k + l ) l ,φ (1) ,Tρ,b = ig s C F γ a ⊥ γ − ] γ ρ ′ ( k/ − l/ )[ γ ⊥ a γ + ]( p/ − k/ + l/ ) γ ρ ′ ( k − l ) ( p − k + l ) l ,φ (1) ,Tρ,c = − ig s C F γ a ⊥ γ − ] γ ρ ′ ( k/ − l/ ) γ ρ ′ k/ [ γ ⊥ a γ + ]( k ) ( k − l ) l ; (47) φ (1) ,Pπ,a = − ig s C F γ γ ρ ′ ( p/ − k/ + l/ ) γ ρ ′ ( p/ − k/ ) γ ( p − k ) ( p − k + l ) l ,φ (1) ,Pπ,b = ig s C F k/ − l/ ) γ ρ ′ γ γ ρ ′ ( p/ − k/ + l/ ) γ ( k − l ) ( p − k + l ) l ,φ (1) ,Pπ,c = − ig s C F γ k/ γ ρ ′ ( k/ − l/ ) γ ρ ′ γ ( k ) ( k − l ) l . (48)Secondly, the NLO transversal NLO twist-2 rho meson DA φ (1) ,Tρ,d and the NLO twist-3pion meson DAs φ (1) ,Pπ,d ) can be extracted from the irreducible sub-diagrams Figs. 6(d,e,f,g) andFigs. 7(d,e,f,g) respectively and are of the following form: φ (1) ,Tρ,d = − ig s C F γ a ⊥ γ − ][ γ ⊥ a γ + ]( p/ − k/ + l/ ) γ ρ v ρ ( p − k + l ) l ( v · l ) . (49) φ (1) ,Pπ,d = − ig s C F γ γ ρ ( p/ − k/ + l/ ) γ v ρ ( p − k + l ) l ( v · l ) , (50)Thirdly, the NLO transversal NLO rho meson DA φ (1) ,T/v/aρ,e and the NLO twist-3 pion me-son DAs φ (1) ,Pπ,e can also be extracted from the irreducible sub-diagrams Figs. 6(h,i,j,k) andFigs. 7(h,i,j,k) respectively, and can be written in the following form: φ (1) ,Tρ,e = ig s C F γ a ⊥ γ − ] γ ρ ( k/ − l/ )[ γ ⊥ a γ + ] v ρ ( k − l ) l ( v · l ) (cid:20) − ( k − k ) ( k − k − l ) (cid:21) ,φ (1) ,vρ,e = ig s C F γ b ⊥ γ ρ ( k/ − l/ ) γ a v ρ ( k − l ) l ( v · l ) (cid:20) − ( k − k ) ( k − k − l ) (cid:21) ,φ (1) ,aρ,e = ig s C F γ γ µ ′ ⊥ ] γ ρ ( k/ − l/ )[ γ ⊥ µ ′ γ ] v ρ ( k − l ) l ( v · l ) (cid:20) − ( k − k ) ( k − k − l ) (cid:21) ; (51) φ (1) ,Pπ,e = ig s C F γ ( k/ − l/ ) γ ρ γ n ρ ( k − l ) l ( n · l ) . (52)The hard LO amplitude G (0) b, ( ξ , x ) , G (0) ,v/ab, ( ξ , x ) and G ( ′ )(0) b, ( x , ξ , x ) with the gluon mo-menta flowing or partly flowing into the original LO hard amplitudes, are defined in the collinear7region l k p for Fig. 6 in the following form: G (0) b, ( ξ ; x ) = ieg s C F ǫ/ T p/ φ Tρ ] γ α [ γ m π φ Pπ ] γ α ( p/ − k/ + l/ ) γ µ ( p − k + l ) ( k − k − l ) , (53) G (0) ,vb, ( ξ ; x ) = ieg s C F ǫ/ T M ρ φ vρ ] γ α [ γ p/ φ Aπ ] γ α ( p/ − k/ + l/ ) γ µ ( p − k + l ) ( k − k − l ) , (54) G (0) ,ab, ( ξ ; x ) = ieg s C F M ρ iǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ ] γ α [ γ p/ φ Aπ ] γ α ( p/ − k/ + l/ ) γ µ ( p − k + l ) ( k − k − l ) , (55) G (0) b, ( x , ξ ; x ) = ieg s C F ǫ/ T p/ φ Tρ ] γ α [ γ m π φ Pπ ] γ α ( p/ − k/ + l/ ) γ µ ( p − k + l ) ( k − k ) , (56) G ′ (0) b, ( x , ξ ; x ) = ieg s C F ǫ/ T p/ φ Tρ ] γ α [ γ m π φ Pπ ] γ α ( p/ − k/ ) γ µ ( p − k ) ( k − k − l ) , (57)where the γ α in Eqs. (56,57) could be γ + or γ α ⊥ . When we set γ α = γ + , the amplitude G (0) b, ( x , ξ , x ) becomes G (0) ,Lb, ( x , ξ , x ) , while G ′ (0) b, ( x , ξ , x ) becomes G ′ (0) ,Lb, ( x , ξ , x ) .When we choose γ α = γ α ⊥ , the amplitude G (0) b, ( x , ξ , x ) becomes G (0) ,Tb, ( x , ξ , x ) , while G ′ (0) b, ( x , ξ , x ) becomes G ′ (0) ,Tb, ( x , ξ , x ) . And we can find that in the collinear region l k p ,these two newly defined LO amplitudes in Eqs. (56,56) should be equal.From the irreducible sub-diagrams of Fig. 7 in the collinear region l k p , the LO hard ampli-tudes G (0) b, / ( x ; ξ ) and G ( ′′ )(0) b, ( x ; ξ , x ) with the gluon momentum flowing or partly flowinginto the original LO hard amplitudes can be defined in the following form: G (0) b, ( x ; ξ ) = ieg s C F ǫ/ T p/ φ Tρ ] γ α [ γ m π φ Pπ ] γ α ( p/ − k/ ) γ µ ( p − k ) ( k − k − l ) , (58) G (0) b, ( x ; ξ ) = ieg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k ) ( k − k − l ) · γ α [ γ p/ φ Aπ ] γ α ( p/ − k/ ) γ µ , (59) G (0) b, ( x ; ξ , x ) = ieg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k + l ) ( k − k ) · γ α [ γ p/ φ Aπ ] γ α ( p/ − k/ ) γ µ , (60) G ′′ (0) b, ( x ; ξ , x ) = ieg s C F ǫ/ T M ρ φ vρ + iM ρ ǫ µ ′ νρσ γ γ µ ′ ǫ ν T n ρ v σ φ aρ ]( p − k + l ) ( k − k + l ) · γ α [ γ p/ φ Aπ ] γ α ( p/ − k/ ) γ µ . (61)By summing up the amplitudes from those irreducible sub-diagrams Figs. 6(d,e,f,g,h,i,j,k), thetotal NLO amplitudes with the crossed-twist DAs T2&T3 (i.e. the NLO set-I amplitude) can bewritten in a convolution of the NLO rho wave function and the LO hard amplitudes, with the gluon8momentum not flowing, flowing or partly flowing into the LO hard kernels: G (1)6 , ( x ; x ) = φ (1) ,Tρ,d ⊗ n G (0) ,Lb, ( x ; x ) − G (0) ,Lb, ( ξ ; x ) − G (0) ,Lb, ( x , ξ ; x )+ G (0) ,Tb, ( x ; x ) − G (0) ,Tb, ( ξ ; x ) + 18 G (0) ,Tb, ( x , ξ ; x ) o + φ (1) ,Tρ,e ⊗ n − G (0) ,Lb, ( ξ ; x ) + 916 G (0) ,Lb, ( x ; x ) + 916 G (0) ,Lb, ( x , ξ ; x )+ G (0) ,Tb, ( x ; x ) − G (0) ,Tb, ( ξ ; x ) − G (0) ,Tb, ( x , ξ ; x ) o . (62)It’s easy to find that the soft divergences from the collinear region for these irreducible amplitudesare canceled each other. At the quark level, consequently, there is no soft divergence left afterthe summation for the contributions from all sub-diagrams in Fig. 6 with the case of the T2&T3DAs. The collinear divergences, generated from the gluon radiated from the up-line quark and thedown-line anti-quark of the initial rho meson in Fig. 6, can be absorbed into the NLO twist-2 rhomeson DA φ (1) ,Tρ,d/e , which is written in the following nonlocal hadronic matrix element in b spacewith the structure ( γ b ⊥ γ + ) / sandwiched: φ (1) ,Tρ = 12 N c P +1 Z dy − π e − ixp +1 y − < | q ( y − ) γ b ⊥ γ + − ig s ) Z y − dzv · A ( zv ) q (0) | ρ ( p ) > . (63)We then give a short summary for the NLO Set-II amplitudes with the crossed-twist DAsT3&T2 for the sub-diagrams Figs. 6(d,e,f,g,h,i,j,k). After the summation, the total infrared di-vergences from the NLO corrections to the LO hard amplitude G (0) b, ( x ; x ) in the l k p regioncan be written in the following form: G (1)6 , ( x ; x ) = φ (1) ,vρ,e ⊗ n G (0) ,vb, ( x ; x ) + 2 G (0) ,vb, ( ξ ; x ) o . (64)We find that the infrared divergences from the Set-II amplitudes for sub-diagrams Figs. 6(d,g,h,i),with the additional gluon radiated from the right-up quark line, are suppressed by the kineticconstraints, then only the sub-diagrams Figs. 6(e,g,j,k) generate infrared divergent corrections tothe Set-II LO amplitudes G (0) b, ( x ; x ) with T3&T2 DAs. The soft divergences from the sub-diagrams with the gluon radiated from the left-down anti-quark line were canceled exactly. Onlythe collinear divergences, generated from the gluon radiated from the left-down anti-quark of theinitial rho meson in Fig. 6, will be absorbed into the NLO twist-3 rho meson DA φ (1) ,vρ . Then wecan factorize the Set-II irreducible amplitudes for Fig. 6 in the collinear region as the convolutionsof the NLO twist-3 DA and LO hard amplitudes. The collinear factorization is therefore valid forthe NLO Set-II corrections for the Fig. 1(b) with the additional gluon emitted from the initial rhomeson.Now, we elaborate the factorization for the infrared divergences in the irreducible sub-diagramsFig. 7(d,e,f,g,h,i,j,k), in which the additional blue gluons are radiated from the final pion meson.The total NLO corrections for the Set-I amplitudes of the sub-diagrams Figs. 7(d,e,f,g,h,i,j,k) withthe T2&T3 DAs from the l k p region are also summed over and can be written in the followingconvolution form: G (1)7 , ( x ; x ) = φ (1) ,Pπ,d ⊗ n h G (0) ,Lb, ( x ; x ) − G (0) ,Lb, ( x ; ξ ) i − h G (0) ,Tb, ( x ; x ) − G (0) ,Tb, ( x ; ξ ) io + φ (1) ,Pπ,e ⊗ h G (0) ,Tb, ( x ; x ) − G (0) ,Tb, ( x ; ξ ) i . (65)9For the NLO Set-I amplitudes in Eq. (65), the soft divergences from sub-diagrams Figs. 7(h,i) canbe canceled by their counterparts from Figs. 7(j,k), then only the collinear divergences are left forthe infrared absorbtion. The collinear divergences can all be absorbed into the NLO pion mesonDAs of φ (1) ,Pπ,d/e , which can be written as the nonlocal hadronic matrix element with the structure asthat in Ref. [11]: φ (1) ,Pπ = 12 N c P − Z dy + π e − ixp − y + < π ( p ) | q ( y + )( − ig s ) Z y + dzn · A ( zn ) γ q (0) | > . (66)As shown in Eq. (65), all Set-I infrared-relevant NLO amplitudes can be written as the convolutionof the LO hard kernel and the NLO π meson DAs ( G (0) b, ⊗ φ (1) ,Pπ,d and G (0) b, ⊗ φ (1) ,Pπ,e ), with the integralmomenta flowing or not flowing into the LO hard amplitudes.By making the summation for the Set-II amplitudes for the sub-diagrams Figs. 7(d,e,f,g,h,i,j,k)with the T3&T2 DAs, we find: G (1)7 , ( x ; x ) = φ (1) ,Aπ,d ⊗ n G (0) b, ( x ; x ) − G (0) b, ( x ; ξ ) + 18 G (0) b, ( x ; ξ , x ) o + φ (1) ,Aπ,e ⊗ n G (0) b, ( x ; x ) − G (0) b, ( x ; ξ ) − G ′′ (0) b, ( x ; ξ , x ) o . (67)For the infrared singularities in the Set-II amplitudes in Eq. (67), the soft divergences fromFigs. 7(i,j) can’t be canceled by their counterparts in Figs. 7(h,k), but these soft divergences arealso diminished because φ (1) ,Aπ,e = φ (1) ,Aπ,d . The remaining collinear singularities in Eq. (67) can beabsorbed into the NLO DAs φ (1) ,Aπ,d/e . All these irreducible NLO amplitudes can be written as theconvolution of the LO hard kernel and the NLO π meson DAs( G (0) b, ⊗ φ (1) ,Aπ,e and G (0) b, ⊗ φ (1) ,Aπ,e ),and the collinear factorization approach is valid for the Fig. 7. III. k T FACTORIZATION OF ργ ⋆ → π In this section, the NLO proof of the factorization theorem is demonstrated with the inclusionof the transversal momentum k T . The k T factorization approach is qualified to deal with the small-x physics[2, 6, 9], because of it’s advantage to avoid the end-point singularity without introducingother non-physics methods.The hierarchy k iT ≪ k · k is holding in the bound wave functions, so the transversal contri-butions on the numerators can be dropped safely and the transversal momentum k T in the LO hardkernels can also be dropped, then factorization proofs made in the above section with the collinearfactorization approach is valid here with the inclusion of the transversal momentum [10, 13].When we extend the proofs for the NLO ρ → π transition from collinear factorization approach to k T factorization approach, the only modification required is to include the transversal integral l T to the NLO wave functions in Eqs. (23,24, 46,63,66), besides the longitudinal integral along thelight cone. This modification can also be understood as the integral deviated from the light conedirection by b in the coordinate space, as illustrated by Fig. 8.The O ( α s ) wave functions at twist-2 and twist-3 as deifined in Eqs. (23,24,46,63,66) can bereproduced by the following nonlocal matrix element in the b space. φ (1) ,Tρ ( x , ξ ; b ) = 12 N c P +1 Z dy − π d b (2 π ) e − ixp +1 y + + i k · b · < | q ( y − ) γ b ⊥ γ + − ig s ) Z y dzv · A ( zv ) q (0) | ρ ( p ) >, (68)0 b n − ( n + ) y − ( y + ) ∞ ( a ) 0 0 b n − ( n + ) y − + b ( y + + b ) ∞ ( b ) ∞ + b FIG. 8. The deviation of the integral (Wilson link) from the light corn by b in the coordinate space for thetwo-parton meson wave function. φ (1) ,vρ ( x , ξ ; b ) = 12 N c P +1 Z dy − π d b (2 π ) e − ixp +1 y + + i k · b · < | q ( y − ) γ ⊥ − ig s ) Z y dzv · A ( zv ) q (0) | ρ ( p ) >, (69) φ (1) ,aρ ( x , ξ ; b ) = 12 N c P +1 Z dy − π d b (2 π ) e − ixp +1 y + + i k · b · < | q ( y − ) γ γ ⊥ − ig s ) Z y dzv · A ( zv ) q (0) | ρ ( p ) > ; (70) φ (1) ,Aπ ( ξ , x ; b ) = 12 N c P − Z dy + π d b (2 π ) e − ixp − y + + i k · b · < π ( p ) | q ( y + )( − ig s ) Z y dzn · A ( zn ) γ − γ q (0) | >, (71) φ (1) ,Pπ ( ξ , x ; b ) = 12 N c P − Z dy + π d b (2 π ) e − ixp − y + + i k · b · < π ( p ) | q ( y + )( − ig s ) Z y dzn · A ( zn ) γ q (0) | > . (72)All these NLO wave functions would reproduce the Feynman rules of Wilson lines. IV. SUMMAREY
In this paper we firstly verified that the factorization hypothesis is valid for the ρ → π transitionprocess at NLO level in the collinear factorization approach, and then we extended this proof tothe case of the k T factorization approach. Because of the difference of the initial vector meson ρ and the final pseudo-scalar meson π , we considered both the two LO sub-diagrams Figs. 1(a) and11(b), with the virtual photon vertex positioned on the initial state quark line and on the final statequark line, respectively.For each LO sub-diagram Fig. 1(a) or Fig. 1(b), we first evaluated the NLO corrections fromthe additional gluon radiated from the initial rho meson as well as from the final pion meson,and then we verified that all the infrared singularities in those four NLO quark level diagrams (Fig.1(a) - Fig.1(d) ) could be absorbed into the NLO meson wave functions. Certainly, we madethis proof both in the collinear factorization approach and in the k T factorization approach. Andwe showed explicitly that every NLO quark level amplitude can be expressed as the convolution ofthe NLO wave functions and the LO hard kernel, with the gluon momenta, which would generatethe infrared singularities, flowing, not flowing or partly flowing into the LO hard amplitudes.Particularly, we find that: (a) only the T3&T2 set with the twist-3 ρ meson DAs and twist-2 pionDAs contribute to the LO amplitude of Fig. 1(a), as defined in Eq. (3); (b) only the collinear sin-gularities would appeare in the NLO diagrams for the LO Fig. 1(a), because the soft singularitiesin these NLO diagrams are either suppressed by the kinetics or canceled each other.For the NLO corrections to the LO Fig. 1(b), however, there exist two kinds of the LO ampli-tudes as described in Eqs. (4,5) with the T2&T3 and T3&T2 combinations of the initial and finalstate meson wave functions and we called them Set-I and Set-II respectively. We further find thatthe NLO corrections to the Set-I and Set-II LO amplitude generate the collinear singularities only,since the soft singularities in these two cases are either suppressed by the kinetics or canceled eachother. The underlying reason is the fact that the soft gluon will not change the color structure of therho and pion mesons. All the remaining infrared singularities from the collinear regions, shouldbe absorbed into the NLO wave functions, and we have also defined the NLO wave functions withdifferent twists in the nonlocal matrix elements, which would help us to understand the fundamen-tal meson wave functions and push us to calculate the NLO hard kernels for this ρ → π transitionprocess. V. ACKNOWLEDEMENT
The authors would like to thank H.N. Li and C.D. Lu for long term collaborations and valu-able discussions. This work is supported by the National Natural Science Foundation of Chinaunder Grant No. 11235005, and by the Project on Graduate Students Education and Innovation ofJiangsu Province, under Grant No. CXZZ13-0391. [1] John. Collins, Foundations of Perturbative QCD (Cambridge University Press, Cambridge, England,1993).[2] G.P. Lepage, and S.J. Brodsky,
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