J/ψ meson production in SIDIS: matching high and low transverse momentum
Daniël Boer, Umberto D'Alesio, Francesco Murgia, Cristian Pisano, Pieter Taels
JJ/ψ meson production in SIDIS: matching high and low transverse momentum
Dani¨el Boer, ∗ Umberto D’Alesio,
2, 3, † Francesco Murgia, ‡ Cristian Pisano,
2, 3, § and Pieter Taels
3, 4, ¶ Van Swinderen Institute for Particle Physics and Gravity,University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Dipartimento di Fisica, Universit`a di Cagliari, Cittadella Universitaria, I-09042 Monserrato, Cagliari, Italy INFN, Sezione di Cagliari, Cittadella Universitaria, I-09042 Monserrato, Cagliari, Italy Centre de Physique Th´eorique, ´Ecole polytechnique, CNRS, I.P. Paris, F-91128 Palaiseau, France (Dated: April 16, 2020)We consider the transverse momentum spectrum and the cos 2 φ azimuthal distribution of J/ψ mesons produced in semi-inclusive, deep-inelastic electron-proton scattering, where the electron andthe proton are unpolarized. At low transverse momentum, we propose factorized expressions interms of transverse momentum dependent gluon distributions and shape functions. We show thatour formulae correctly match with the collinear factorization results at high transverse momentum,obtained in the framework of collinear, nonrelativistic QCD (NRQCD). We conclude that the shapefunctions at high transverse momentum are independent of the quantum numbers of the producedquarkonium state, except for the overall magnitude given by the appropriate NRQCD long distancematrix element.
I. INTRODUCTION
The production of a light hadron h with a specific transverse momentum in semi-inclusive, deep-inelastic electron-proton scattering (SIDIS), e p → e (cid:48) h X , is in general characterized by three different scales: the hard scale of theprocess Q , given by the virtuality of the gauge boson exchanged in the reaction, the nonperturbative QCD scaleΛ QCD , and the magnitude of the hadron transverse momentum q T in a suitable reference frame. Depending on thevalue of q T , two different factorization frameworks can be adopted for the description of this process. Both of themenable to separate the short-distance from the long-distance contributions to the cross section. While the former canbe perturbatively calculated through a systematic expansion in the strong coupling constant, the latter has to beparametrized in terms of parton distributions (PDFs) and fragmentation functions (FFs), which need to be extractedfrom data.More explicitly, collinear factorization is applicable in the so-called high- q T region, namely for q T (cid:29) Λ QCD , wherethe transverse momentum in the final state is generated by perturbative radiation and the cross section is expressedin terms of collinear ( i.e. , integrated over transverse momentum) PDFs and FFs. The other framework, based ontransverse momentum dependent (TMD) factorization [1–3] is valid at low q T , q T (cid:28) Q , and involves TMD PDFs andFFs (or TMDs for short). The high- and low- q T regions overlap for Λ QCD (cid:28) q T (cid:28) Q , where both descriptions cantherefore be applied. If the two results describe the same dynamics, characterized by the same power behavior, theyhave to match in this intermediate region. Conversely, if the two results describe competing mechanisms, they shouldbe considered independently and added together [4].The SIDIS cross section differential in q T and integrated over the azimuthal angle of the final hadron can be expressedin terms of unpolarized, twist-two TMDs in the small- q T region, and its matching with the collinear description hasbeen demonstrated in Ref. [4]. The matching for the analogous observable in Drell-Yan (DY) dilepton production, p p → (cid:96) (cid:96) (cid:48) X , has been proven as well in Refs. [5, 6]. Azimuthal asymmetries in both SIDIS [7] and DY [8–10] processeshave also been widely investigated within the TMD framework. In particular, the matching of the cos φ modulations,which are suppressed by a factor q T /Q with respect to the φ -integrated cross sections and involve twist-three TMDs,has been shown very recently in Ref. [11].In this paper, we analyze J/ψ production in SIDIS, e p → e (cid:48) J/ψ X , along the same lines of Refs. [4, 11]. There areseveral reasons that motivate our study. First of all, since its discovery in 1974, the
J/ψ meson, a charm-anticharmquarkonium bound state with odd charge parity, has always attracted a lot of attention as a probe of the perturbativeand nonperturbative aspects of quantum chromodynamics (QCD) and their interplay. Moreover,
J/ψ production in ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] a r X i v : . [ h e p - ph ] A p r both ep [12–15] and pp collisions [12, 16–19] has been proposed lately as a tool to access gluon TMDs. Similar studieshave also been carried out within the so-called generalized parton model approach [20–27]. From the experimentalpoint of view, these reactions should have a very clean signature due to the large branching ratio of the J/ψ leptonicdecay modes. Hence our findings could be in principle verified at the future Electron-Ion Collider (EIC) planned inthe United States [28, 29].As compared to e p → e (cid:48) h X , the study of e p → e (cid:48) J/ψ X presents an additional complication, namely a secondhard scale given by the
J/ψ mass, M ψ . Since we want to avoid contributions from photoproduction processes, wefocus on the kinematic region where Q (cid:38) M ψ . In principle, each of the two scales, or any combination of them, canbe chosen as the factorization scale µ in the calculation of the cross section. From our analysis it will turn out thatthe choice µ = (cid:113) Q + M ψ allows for a smooth transition of the cross section from the high- to low- q T region.When the J/ψ meson is produced with a large transverse momentum, q T (cid:29) M ψ , nonrelativistic QCD (NRQCD)can be applied. This rigorous theoretical framework implies a separation of short-distance coefficients, which canbe calculated perturbatively as expansions in the strong-coupling constant α s , from long-distance matrix elements(LDMEs), which must be extracted from experiment [30]. The relative importance of the latter can be estimatedby means of velocity scaling rules, i.e. the LDMEs are predicted to scale with a definite power of the heavy-quarkvelocity v in the limit v (cid:28)
1. In this way, the theoretical predictions are organized as double expansions in α s and v . The main feature of this formalism is that the charm-anticharm quark pair forming the bound state can beproduced both in a color-singlet (CS) configuration, with the same quantum numbers as the J/ψ meson, and as anintermediate color-octet (CO) state with different quantum numbers. In the latter case, the pair subsequently evolvesinto a colorless state through the emission of soft gluons.In the small- q T region, q T (cid:28) M ψ , TMD factorization has not yet been proven in a rigorous way for the process e p → e (cid:48) J/ψ X . There are however strong arguments in favor of its validity, if we consider the analogy with e p → e (cid:48) h X , for which TMD factorization holds at all orders [1]. The only difference from the color point of view is that thedominant partonic subprocess is now γ ∗ g → c ¯ c instead of γ ∗ q → q (cid:48) . Hence final state interactions will be resummedin the gauge link of the gluon correlator, which will be in the adjoint representation, rather than in a quark correlatorin the fundamental representation. Since the J/ψ mass does not affect the gauge link structure, we do not expect anyTMD factorization breaking effects due to color entanglement [13].Motivated by these arguments, in the present analysis we put forward factorization formulae, valid at the twist-two level, for the transverse momentum spectrum and the cos 2 φ azimuthal distribution of J/ψ mesons produced inSIDIS. In addition to the usual TMD PDFs, we consider the recently proposed shape functions [31, 32], which arethe generalization of the collinear LDMEs in NRQCD. Alternatively, they can be seen as the analog of the TMD FFsfor light hadron production in SIDIS. By requiring a proper matching with the collinear results, in complete analogywith the TMD cross sections for e p → e (cid:48) h X and p p → (cid:96) (cid:96) (cid:48) X , we are able to assess the role of the shape functionsin the TMD formalism for quarkonium production. This will have important implications for a recent suggestion toextract poorly known CO LDMEs from a comparison between quarkonium production and open heavy quark pairproduction in SIDIS at the EIC [13].The paper is organized as follows. In Section II we define the variables and the reference frames that are adoptedin our calculation. Parametrizing the cross section in terms of different structure functions, we compute the crosssection in the collinear framework after which we take the small- q T limit. Section III is devoted to the computationof the cross section in the TMD regime, under the approximation that the J/ψ meson is collinear with the outgoingheavy-quark pair. The large- q T limit of the result is then taken and compared with the small- q T limit of the collinearcalculation. In Section IV, both results are shown to match after including the smearing of the transverse momentumof the quarkonium in its hadronization, which is encoded in the appropriate shape functions. Conclusions are givenin Section V. II. FROM HIGH TO INTERMEDIATE TRANSVERSE MOMENTUM
In this section the framework of collinear NRQCD is adopted for the description of the process e ( (cid:96) ) + p ( P ) → e ( (cid:96) (cid:48) ) + J/ψ ( P ψ ) + X , (1)where all the particles are unpolarized and their four-momenta are given within brackets. This reaction is describedby the conventional variables x B = Q P · q , y = P · qP · (cid:96) , z = P · P ψ P · q , (2)with q ≡ (cid:96) − (cid:96) (cid:48) and q = − Q . We denote by M p and M ψ the masses of the proton and the J/ψ meson, respectively.In the one-photon exchange approximation, at leading order (LO) in NRQCD, the transverse momentum of the
J/ψ is due to parton emission in the hard scattering process γ ∗ ( q ) + a ( p a ) → cc [ n ]( P ψ ) + a ( p (cid:48) a ) , (3)where parton a can be either a gluon, a quark or an antiquark. The charm-anticharm quark pair is produced in a Fockstate n = S +1 L [ c ] J with spin S , orbital angular momentum L , total angular momentum J and color configuration c = 1 , x = Q p a · q , ˆ z = p a · P ψ p a · q . (4)If we neglect the proton mass and any smearing effects both in the initial and in the final state, we can take p a = ξP , (5)and therefore ˆ x = x B ξ , ˆ z = z , (6)which implies ˆ xp a = x B P and ˆ x ≥ x B . (7) A. Reference frames
A convenient reference frame for the calculation of the structure functions for this process is defined by adoptinglight-cone coordinates with respect to the directions of the relevant hadron four-momenta, P and P ψ . We introducethe light-like vectors n + and n − such that n + · n − = 1. Neglecting the proton mass, n µ + = P µ , n µ − = 1 P · P ψ (cid:32) P µψ − M ψ P · P ψ P µ (cid:33) . (8)Hence the four-momentum of the virtual photon can be written as q µ = − x B (cid:18) − q T Q (cid:19) n µ + + Q x B n µ − + q µ T , (9)with the definition [4] q T = ( − q µ T q T µ ) / , (10)so that q T is positive and q T ≡ q T . From Eq. (9), the off-collinearity of the process is determined as q µ T = q µ + (cid:32) − q T Q + M ψ ˆ z Q (cid:33) x B P µ − z P µψ = q µ + (cid:32) − q T Q − M ψ ˆ z Q (cid:33) ˆ x p µa − z P µψ . (11)At the partonic level, the Mandelstam variables can be expressed asˆ s = ( q + p a ) = Q (cid:18) − ˆ x ˆ x (cid:19) , ˆ t = ( q − P ψ ) = − (1 − ˆ z ) Q − ˆ zq T − − ˆ z ˆ z M ψ , ˆ u = ( p a − P ψ ) = − ˆ z ˆ x Q + M ψ . (12)In the calculation of the cross section, we can perform the following replacement δ (( p (cid:48) a ) ) = δ (( q + p a − P ψ ) ) = 1ˆ zQ δ (cid:32) q T Q + 1 − ˆ z ˆ z M ψ Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z (cid:33) . (13)Moreover, from the kinematical constraintsˆ s ≥ M ψ , − (ˆ s + Q )(ˆ s − M ψ )ˆ s ≤ ˆ t ≤ , − (ˆ s − M ψ ) − Q ≤ ˆ u ≤ − Q M ψ ˆ s , (14)we obtain x B ≤ ˆ x ≤ Q Q + M ψ ≡ ˆ x max , ˆ z ≤ , (15)and, from momentum conservation, ˆ t = M ψ − Q − ˆ s − ˆ u = − − ˆ z ˆ x Q . (16)By comparing the above relation with the second of Eqs. (12), we getˆ t = − − ˆ z ˆ x Q = − − ˆ x (cid:20) ˆ z q T + 1 − ˆ z ˆ z M ψ (cid:21) . (17)Alternatively, this process can be studied in a frame where the three-momenta P and q are collinear and lie on the z -axis. In this frame, the virtual photon has obviously no transverse momentum. The four-momenta of the particlescan be decomposed using two new vectors κ µ + and κ µ − , such that κ = κ − = 0 and κ − · κ + = 1, P µ = κ µ + ,p µa = x B ˆ x κ µ + ,q µ = − x B κ µ + + Q x B κ µ − ,P µψ = M ψ + P ψ ⊥ ˆ z Q x B κ µ + + ˆ z Q x B κ µ − + P µψ ⊥ ,(cid:96) µ = 1 − yy x B κ µ + + Q x B y κ µ − + Qy (cid:112) − y ˆ (cid:96) µ ⊥ , (18)where the following relations between the light-like vectors of the two frames hold κ µ + = n µ + , κ µ − = 2 x B q T Q n µ + + n µ − + 2 x B Q q µ T . (19)The partonic Mandelstam variables in this frame readˆ s = ( q + p a ) = Q (cid:18) − ˆ x ˆ x (cid:19) , ˆ t = ( q − P ψ ) = − (1 − ˆ z ) Q − − ˆ z ˆ z M ψ − z P ψ ⊥ , ˆ u = ( p a − P ψ ) = − ˆ z ˆ x Q + M ψ . (20)By comparing the above expression for ˆ t with the one in Eq. (12), we obtain the relation between the transversemomentum of the photon q T w.r.t. the hadrons, and the transverse momentum of the hadron | P ψ ⊥ | w.r.t. the photonand the proton, q T = 1ˆ z | P ψ ⊥ | . (21) qp a P p a (a) qp a p a P (b) qp a P p a (c) qp a P p a (d) FIG. 1: Representative leading order diagrams for the partonic process γ ∗ ( q ) + a ( p a ) → J/ψ ( P ψ ) + a ( p (cid:48) a ), with a = g, q, ¯ q .The only diagrams contributing to the CS production mechanism are of the type (a), and there are six of them. There are twodiagrams for each type (b), (c), (d). The dominant diagrams in the small- q T limit are those of type (c) and (d). Moreover, using the above expression together with the Sudakov decomposition of P µψ in the two frames, namely P µψ = M ψ ˆ zQ x B n µ + + ˆ z Q x B n µ − = M ψ + P ψ ⊥ ˆ z Q x B κ µ + + ˆ z Q x B κ µ − + P µψ ⊥ , (22)and using Eq. (19), we obtain P µψ ⊥ = − ˆ zq µ T − q T Q ˆ z x B n µ + . (23) B. Calculation of the cross section
Within the collinear NRQCD approach and in a frame where the longitudinal directions are fixed by the protonand the photon, the cross section for the process under study can be written as follows,d σ d y d x B d z d P ψ ⊥ d φ ψ = 164 1(2 π ) y (cid:88) n (cid:90) ˆ x max x B dˆ x ˆ x (cid:90) z dˆ z ˆ z δ (cid:32) q T Q + 1 − ˆ z ˆ z M ψ Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z (cid:33) × (cid:88) a (cid:20) Q f a (cid:16) x B ˆ x , µ (cid:17) L µν H a [ n ] µ H a [ n ] ∗ ν (cid:104) |O ( n ) | (cid:105) (cid:21) z δ (ˆ z − z ) , (24)where φ ψ is the azimuthal angle of the final J/ψ meson, defined with respect to the lepton plane according to theconventions of Ref. [33], and where H a [ n ] µ is the amplitude for the hard scattering subprocess γ ∗ a → c ¯ c [ n ] a , with a = g, q, ¯ q . The corresponding Feynman diagrams, at the perturbative order αα s , are depicted in Fig. 1. The dominantFock states included in the calculation are n = S [1]1 , S [8]0 , S [8]1 , P [8] J , with J = 0 , , f a is the unpolarized PDF, which depends on the light-cone momentum fraction ξ = x B / ˆ x of parton a and on the apriori arbitrary hard factorization scale µ , while (cid:104) |O ( n ) | (cid:105) is the NRQCD LDME.The leptonic tensor L µν can be written as [7] L µν = e (cid:2) − g µν Q + 2( (cid:96) µ (cid:96) (cid:48) ν + (cid:96) ν (cid:96) (cid:48) µ ) (cid:3) = e Q y (cid:26) − [1 + (1 − y ) ] g µν ⊥ + 4(1 − y ) (cid:15) µL (cid:15) νL + 4(1 − y ) (cid:18) ˆ (cid:96) µ ⊥ ˆ (cid:96) ν ⊥ + 12 g µν ⊥ (cid:19) + 2(2 − y ) (cid:112) − y ( (cid:15) µL ˆ (cid:96) ν ⊥ + (cid:15) νL ˆ (cid:96) µ ⊥ ) (cid:27) , (25)where the second equality can be obtained from the first one by replacing the expression for (cid:96) µ in Eq. (18), and wherethe transverse projector g µν ⊥ is given by g µν ⊥ ≡ g µν − κ µ + κ ν − − κ µ − κ ν + = g µν − P · q ( P µ q ν + P ν q µ ) − Q ( P · q ) P µ P ν . (26)Furthermore, we have introduced the longitudinal polarization vector of the exchanged virtual photon, (cid:15) µL ( q ) = 1 Q (cid:18) q µ + Q P · q P µ (cid:19) , (27)which fulfills the relations (cid:15) L ( q ) = 1 and (cid:15) µL ( q ) q µ = 0. From Eq. (21) it follows that the cross section differential in q T can be obtained by simply multiplying Eq. (24) by a factor z ,d σ d y d x B d z d q T d φ ψ = 164 1(2 π ) yz (cid:88) n (cid:90) ˆ x max x B dˆ x ˆ x (cid:90) z dˆ z ˆ z δ (cid:32) q T Q + 1 − ˆ z ˆ z M ψ Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z (cid:33) × (cid:88) a (cid:20) Q f a (cid:16) x B ˆ x , µ (cid:17) L µν H a [ n ] µ H a [ n ] ∗ ν (cid:104) |O ( n ) | (cid:105) (cid:21) δ (ˆ z − z ) . (28)Along the same lines of Ref. [7], the final result can be expressed in terms of four independent structure functions:d σ d y d x B d z d q T d φ ψ = α yQ (cid:26) [1 + (1 − y ) ] F UU,T + 4(1 − y ) F UU,L + (2 − y ) (cid:112) − y cos φ ψ F cos φ ψ UU + (1 − y ) cos 2 φ ψ F cos 2 φ ψ UU (cid:27) , (29)where the first and second subscripts of the structure functions F denote the polarization of the initial electron andproton, respectively, while the third one, when present, specifies the polarization of the exchanged virtual photon.The expressions of the structure functions in the limit q T (cid:28) Q can be calculated from Eq. (28), replacing theDirac delta with its expansion in the small- q T limit derived in Appendix A, namely δ (cid:32) q T Q + 1 − ˆ z ˆ z M ψ Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z (cid:33) = ˆ x max (cid:40) ˆ x (cid:48) (1 − ˆ x (cid:48) ) + δ (1 − ˆ z ) + Q + M ψ Q + M ψ / ˆ z ˆ z (1 − ˆ z ) + δ (1 − ˆ x (cid:48) )+ δ (1 − ˆ x (cid:48) ) δ (1 − ˆ z ) ln (cid:18) Q + M ψ q T (cid:19)(cid:41) , (30)where ˆ x max = Q Q + M ψ , ˆ x (cid:48) = ˆ x ˆ x max . (31)By using the relations 11 − ˆ z = 1 q T Q (1 − ˆ x (cid:48) )ˆ z + M ψ (ˆ z − ˆ x (cid:48) )ˆ x (cid:48) ˆ z , (32)and 11 − ˆ x (cid:48) = ˆ z (1 − ˆ z )ˆ x (cid:48) Q + M ψ ˆ z q T + (1 − ˆ z ) M ψ , (33)we obtain the leading power behavior of the structure functions F UU,T = σ UU,T (cid:34) L (cid:32) Q + M ψ q T (cid:33) f g ( x, µ ) + (cid:0) P gg ⊗ f g + P gi ⊗ f i (cid:1) ( x, µ ) (cid:35) ,F UU,L = σ UU,L (cid:34) L (cid:32) Q + M ψ q T (cid:33) f g ( x, µ ) + ( P gg ⊗ f g + P gi ⊗ f i )( x, µ ) (cid:35) ,F cos 2 φ ψ UU = σ cos 2 φ ψ UU (cid:2) ( δP gg ⊗ f g )( x, µ ) + ( δP gi ⊗ f i )( x, µ ) (cid:3) , (34)where a sum over i = q, ¯ q is understood. These results are valid up to corrections of the order of O (Λ QCD /q T ) and O ( q T /Q ). The structure function F cos φ ψ UU is suppressed by a factor q T /Q with respect to the other ones and will notbe considered in the following. In Eq. (34), we have defined x ≡ x B ˆ x max = x B (cid:32) M ψ Q (cid:33) , (35)and L (cid:32) Q + M ψ q T (cid:33) ≡ C A ln (cid:32) Q + M ψ q T (cid:33) − C A − n f T R , (36)where n f refers to the number of active flavors, T R = 1 / C A = N c , with N c being the number of colors. The symbol ⊗ denotes a convolution in the longitudinal momentum fractions:( P ⊗ f )( x, µ ) = (cid:90) x dˆ x ˆ x P (cid:0) ˆ x, µ (cid:1) f (cid:16) x ˆ x , µ (cid:17) . (37)The well-known LO unpolarized splitting functions read P gg (ˆ x ) = 2 C A (cid:20) ˆ x (1 − ˆ x ) + + 1 − ˆ x ˆ x + ˆ x (1 − ˆ x ) (cid:21) + δ (1 − ˆ x ) 11 C A − n f T R ,P gq (ˆ x ) = P g ¯ q (ˆ x ) = C F − ˆ x ) ˆ x , (38)while the splitting functions of an unpolarized parton into a linearly polarized gluon are [35, 36] δP gg (ˆ x ) = C A − ˆ x ˆ x ,δP gq (ˆ x ) = δP g ¯ q (ˆ x ) = C F − ˆ x ˆ x , (39)with C F = ( N c − / N c . The plus-prescription on the singular parts of the splitting functions is defined, as usual,such that the integral of a sufficiently smooth distribution G is given by (cid:90) z d y G ( y )(1 − y ) + = (cid:90) z d y G ( y ) − G (1)1 − y − G (1) ln (cid:18) − z (cid:19) (40)and 1(1 − y ) + = 11 − y for 0 ≤ y < . (41)Assuming the validity of the common heavy-quark spin symmetry relations [30] (cid:104) |O ( P [8] J ) | (cid:105) = (2 J + 1) (cid:104) |O ( P [8]0 ) | (cid:105) , (42)the cross sections for the partonic processes γ ∗ g → cc [ n ] in Eq. (34) read σ UU,T = α s e c M ψ ( M ψ + Q ) q T (cid:34) (cid:104) |O ( S [8]0 ) | (cid:105) + 4 (7 M ψ + 2 M ψ Q + 3 Q ) M ψ ( M ψ + Q ) (cid:104) |O ( P [8]0 ) | (cid:105) (cid:35) , σ UU,L = α s e c M ψ ( M ψ + Q ) q T (cid:34) Q ( M ψ + Q ) (cid:104) |O ( P [8]0 ) | (cid:105) (cid:35) ,σ cos 2 φ ψ UU = 4 α s e c M ψ ( M ψ + Q ) q T (cid:34) −(cid:104) |O ( S [8]0 ) | (cid:105) + 4 3 M ψ − Q M ψ ( M ψ + Q ) (cid:104) |O ( P [8]0 ) | (cid:105) (cid:35) , (43)where e c is the electric charge of the charm quark in units of the proton charge. We note that the relevant partonicsubprocesses, contributing to the above cross sections in the small- q T region, are only the n = S [8]0 , P [8] J ones thatcorrespond to ˆ t -channel Feynman diagrams of the type (c) and (d) in Fig. 1.The structure functions F UU,T and F UU,L in Eq. (34) exhibit logarithmic (collinear) singularities as q T →
0. Theirbehavior is similar to the analogous structure functions for light hadron production in SIDIS discussed in Ref. [4],where the dominant underlying partonic process is γ ∗ q → q . There are some differences though. In the latter case,the logarithmic term L is given by C F (2 ln Q /q T −
3) instead of Eq. (36). The color factor C F clearly corresponds toa quark initiated process, while C A corresponds to a gluon initiated one. The two different finite terms originate fromthe virtual corrections to the splitting functions P qq and P gg , respectively. Moreover, in light hadron production extraterms appear, containing convolutions of FFs with the P qq and P gq splitting functions, which cannot be present in ourcalculation for quarkonium production within the NRQCD framework. We also point out that the structure function F cos 2 φ ψ UU does not contain any large logarithm in the region q T (cid:28) Q , whereas the corresponding observable for lighthadron production diverges logarithmically and is suppressed by an overall factor q T /Q . Finally, the appearance ofa logarithm ln( Q + M ψ ) /q T , instead of ln Q /q T , suggests Q + M ψ as the natural choice for the hard scale in ourprocess. III. FROM SMALL TO INTERMEDIATE TRANSVERSE MOMENTUM
The process γ ∗ g → cc [ n ] has been calculated in Ref. [13] within the TMD framework, taking into account theintrinsic transverse momentum effects of the gluons inside the proton. If we neglect smearing effects in the final state, i.e. if we assume that the final J/ψ meson is collinear to the c ¯ c pair originally produced in the hard scattering process,the cross section can be cast in the following formd σ d y d x B d z d q T d φ ψ = α yQ (cid:110) [1 + (1 − y ) ] F UU,T + 4(1 − y ) F UU,L + (1 − y ) cos 2 φ ψ F cos 2 φ ψ UU (cid:111) δ (1 − z ) (44)with F UU,T = 2 π α s e c M ψ ( M ψ + Q ) (cid:34) (cid:104) |O ( S [8]0 ) | (cid:105) + 4 (7 M ψ + 2 M ψ Q + 3 Q ) M ψ ( M ψ + Q ) (cid:104) |O ( P [8]0 ) | (cid:105) (cid:35) f g ( x, p T ) (cid:12)(cid:12)(cid:12)(cid:12) p T = q T , F UU,L = 2 π α s e c M ψ ( M ψ + Q ) 16 Q ( M ψ + Q ) (cid:104) |O ( P [8]0 ) | (cid:105) f g ( x, p T ) (cid:12)(cid:12)(cid:12)(cid:12) p T = q T , F cos 2 φ ψ UU = 2 π α s e c M ψ ( M ψ + Q ) (cid:34) −(cid:104) |O ( S [8]0 ) | (cid:105) + 4 3 M ψ − Q M ψ ( M ψ + Q ) (cid:104) |O ( P [8]0 ) | (cid:105) (cid:35) p T M p h ⊥ g ( x, p T ) (cid:12)(cid:12)(cid:12)(cid:12) p T = q T , (45)where f g and h ⊥ g are, respectively, the unpolarized and linearly polarized gluon TMDs inside an unpolarized pro-ton [37–41].We note that, beyond the parton model approximation, for those processes where TMD factorization is valid, softgluon radiation to all orders is included into an exponential Sudakov factor, which can be split and its parts absorbedinto the TMD PDFs and FFs involved in the reaction, whereas the remaining perturbative corrections are collectedinto a hard factor H . As a consequence of the regularization of their ultraviolet and rapidity divergences, TMDsdepend on two different scales, not explicitly shown in the above equations. In the following we will take these twoscales to be equal to each other and denote them by µ , to be identified with a typical hard scale of the process.TMDs can be calculated perturbatively in the limit p T ≡ | p T | (cid:29) Λ QCD . This can be better achieved in the impactparameter space. To this aim, we focus first on f g ( x, p T ) and its Fourier transform, which is defined as (cid:98) f g ( x, b T ; µ ) ≡ π (cid:90) d p T e i b T · p T f g ( x, p T ; µ ) = (cid:90) ∞ d p T p T J ( b T p T ) f g ( x, p T ; µ ) , (46)with J n being the Bessel function of the first kind of order n . The perturbative part of the gluon TMD, valid in thelimit b T (cid:28) / Λ QCD , reads (cid:98) f g ( x, b T ; µ ) = 12 π (cid:88) a = q, ¯ q,g ( C g/a ⊗ f a )( x, µ b ) e − S A ( b T ,µ ) , (47)where f a ( x, µ ) are the collinear parton distribution functions for a specific (anti)quark flavor or a gluon a , and µ b = b /b T with b = 2 e − γ E ≈ . C g/a and the perturbative Sudakov exponent S A ,which resums large logarithms of the type ln( b T µ ), are calculable in perturbative QCD. The coefficient functions canbe expanded in powers of α s as follows C g/a ( x, µ b ) = δ ga δ (1 − x ) + ∞ (cid:88) k =1 C ( k ) g/a ( x ) (cid:18) α s ( µ b ) π (cid:19) k , (48)while the perturbative Sudakov factor at LO reads S A ( b T , µ ) = C A π (cid:90) µ µ b d ζ ζ α s ( ζ ) (cid:18) ln µ ζ − − n f /C A (cid:19) = C A π α s (cid:18)
12 ln µ µ b − − n f /C A µ µ b (cid:19) . (49)Therefore, in the small- b T limit, namely b T (cid:28) / Λ QCD , and at LO in α s we find (cid:98) f g ( x, b T ; µ ) = 12 π (cid:40) f g ( x, µ b ) − α s π (cid:34)(cid:18) C A µ µ b − C A − n f µ µ b (cid:19) f g ( x, µ b ) − (cid:88) a (cid:16) C (1) g/a ⊗ f a (cid:17) ( x, µ b ) (cid:35)(cid:41) . (50)Using the DGLAP equations we can evolve f g from a scale µ down to another scale µ b < µ and obtain f g ( x, µ b ) = f g ( x, µ ) − α s π ( P gg ⊗ f g + P gi ⊗ f i )( x, µ ) ln µ µ b + O ( α s ) , (51)where a sum over i = q, ¯ q is understood. By substituting the above expression into Eq. (50) we get (cid:98) f g ( x, b T ; µ ) = 12 π (cid:26) f g ( x, µ ) − α s π (cid:20)(cid:18) C A µ µ b − C A − n f µ µ b (cid:19) f g ( x, µ )+ ( P gg ⊗ f g + P gi ⊗ f i )( x, µ ) ln µ µ b − (cid:88) a (cid:16) C (1) g/a ⊗ f a (cid:17) ( x, µ ) (cid:21)(cid:27) . (52)Transforming back to momentum space, we find the transverse momentum distribution in the region p T (cid:29) Λ QCD , f g ( x, p T ; µ ) = 12 π (cid:90) d b T e − i b T · p T (cid:98) f g ( x, b T ; µ )= α s π p T (cid:20)(cid:18) C A ln µ p T − C A − n f (cid:19) f g ( x, µ ) + ( P gg ⊗ f g + P gi ⊗ f i )( x, µ ) (cid:21) , (53)where we have used the following integrals (cid:90) d b T e − i b T · q T ln µ µ b = − πq T ln µ q T , (cid:90) d b T e − i b T · q T ln µ µ b = − πq T . (54)By substituting the expression for f g ( x, p T ) given in Eq. (53), evolved to the scale µ = Q + M ψ , into Eq. (45), wefind that the TMD structure functions F UU,T and F UU,L do not exactly match the corresponding collinear ones inthe small- q T limit given in Eq. (34). In the intermediate region Λ QCD (cid:28) q T (cid:28) Q , we get F UU,T = F UU,T − σ UU,T C A ln (cid:32) Q + M ψ q T (cid:33) , F UU,L = F UU,L − σ UU,L C A ln (cid:32) Q + M ψ q T (cid:33) . (55)0This suggests that one needs to include smearing effects in the final state as well, through the inclusion of a suitableshape function [31, 32], to be convoluted with the TMD in momentum space. Imposing the validity of the matchingwill give us the LO expression of the shape function, as we shall see in the next section.We now turn to the polarized gluon distribution h ⊥ g ( x, p T ) and the structure function F cos 2 φ ψ UU . From Ref. [35], weknow the perturbative tail of h ⊥ g ( x, p T ) at LO in terms of the unpolarized collinear PDFs f a ( x, µ ), p T M p h ⊥ g ( x, p T ; µ ) = α s π p T C A (cid:90) x dˆ x ˆ x (cid:18) ˆ xx − (cid:19) f g (ˆ x, µ ) + C F (cid:88) i = q, ¯ q (cid:90) x dˆ x ˆ x (cid:18) ˆ xx − (cid:19) f i (ˆ x, µ ) . (56)The above expression, together with Eq. (45), leads to F cos 2 φ ψ UU = 4 α s e c M ψ ( M ψ + Q ) q T (cid:34) −(cid:104) |O ( S [8]0 ) | (cid:105) + 4 3 M ψ − Q M ψ ( M ψ + Q ) (cid:104) |O ( P [8]0 ) | (cid:105) (cid:35) × C A (cid:90) x dˆ x ˆ x (cid:18) ˆ xx − (cid:19) f g (ˆ x, µ ) + C F (cid:88) i = q, ¯ q (cid:90) x dˆ x ˆ x (cid:18) ˆ xx − (cid:19) f i (ˆ x, µ ) = σ cos 2 φ ψ UU [( δP gg ⊗ f g )( x, µ ) + ( δP gi ⊗ f i )( x, µ )] = F cos 2 φ ψ UU , (57)which shows the exact matching of the TMD and collinear results in the intermediate region for the structure function F cos 2 φ ψ UU . This is achieved without the need of any shape function because of the absence of a logarithmic term at theperturbative order we are considering. IV. TMD FACTORIZATION AND MATCHING WITH THE COLLINEAR FRAMEWORK
On the basis of the above considerations, TMD factorized expressions for the structure functions F UU,T and F UU,L have to take into account smearing effects [13], encoded in the shape function ∆ [ n ] [31, 32], which can be thought asa generalization of the long distance matrix elements of NRQCD in collinear factorization. We start by assuming thevalidity of the following formulae, F UU,T = (cid:88) n H [ n ] UU,T C (cid:2) f g ∆ [ n ] (cid:3) ( x, q T ; µ ) , F UU,L = (cid:88) n H [ n ] UU,L C (cid:2) f g ∆ [ n ] (cid:3) ( x, q T ; µ ) , (58)where the H represent the hard parts, which can be calculated pertubatively. Moreover, we have introduced thetransverse momentum convolution C (cid:2) f g ∆ [ n ] (cid:3) ( x, q T ; µ ) = (cid:90) d p T (cid:90) d k T δ ( q T − p T − k T ) f g ( x, p T ; µ ) ∆ [ n ] ( k T , µ ) . (59)As expected, in absence of smearing, ∆ [ n ] ( k T ; µ ) = (cid:104) |O ( n ) | (cid:105) δ ( k T ), and the convolution in Eq. (59) reduces to theproduct of the LDME (cid:104) |O ( n ) | (cid:105) with the gluon TMD f g ( x, q T ). Furthermore, this convolution can be expressed asfollows C (cid:2) f g ∆ [ n ] (cid:3) ( x, q T ; µ ) = (cid:90) d b T e − i b T · q T (cid:98) f g ( x, b T ; µ ) (cid:98) ∆ [ n ] ( b T , µ ) , (60)where we have introduced the Fourier transform of the shape function, (cid:98) ∆ [ n ] ( b T , µ ) = 12 π (cid:90) d k T e i b T · k T ∆ [ n ] ( k T , µ ) . (61)We are now able to show that the following expression of (cid:98) ∆ [ n ] , valid at LO in α s , (cid:98) ∆ [ n ] ( b T , µ ) = 12 π (cid:104) |O ( n ) | (cid:105) (cid:18) − α s π C A µ µ b (cid:19) , (62)1which leads to ∆ [ n ] ( k T , µ ) = α s π k T C A (cid:104) |O ( n ) | (cid:105) ln µ k T (63)in the limit | k T | (cid:29) Λ QCD , will solve the matching issue of the TMD and collinear results in the region Λ
QCD (cid:28) q T (cid:28) Q .In fact, by plugging it together with Eq. (52) into Eq. (60), with the choice µ = Q + M ψ , we get C (cid:2) f g ∆ [ n ] (cid:3) ( x, q T ) = (cid:104) |O ( n ) | (cid:105) π (cid:90) d b T e − i b T · q T (cid:26) f g ( x, µ ) − α s π (cid:20)(cid:18) C A µ µ b − C A − n f µ µ b (cid:19) f g ( x, µ )+ ( P gg ⊗ f g + P gi ⊗ f i )( x, µ ) ln µ µ b − (cid:88) a (cid:16) C (1) g/a ⊗ f a (cid:17) ( x, µ ) (cid:35)(cid:41) (cid:18) − α s π C A µ µ b (cid:19) = (cid:104) |O ( n ) | (cid:105) π (cid:90) d b T e − i b T · q T (cid:26) f g ( x, µ ) − α s π (cid:20)(cid:18) C A ln µ µ b − C A − n f µ µ b (cid:19) f g ( x, µ )+ ( P gg ⊗ f g + P gi ⊗ f i )( x, µ ) ln µ µ b − (cid:88) a (cid:16) C (1) g/a ⊗ f a (cid:17) ( x, µ b ) (cid:35)(cid:41) = α s π q T (cid:104) |O ( n ) | (cid:105) (cid:20)(cid:18) C A ln µ q T − C A − n f (cid:19) f g ( x, µ ) + ( P gg ⊗ f g + P gi ⊗ f i )( x, µ ) (cid:21) . (64)Substituting the last formula in the LO expressions for F UU,T and F UU,L in Eqs. (58), we recover the correct resultsfor F UU,T and F UU,L in Eqs. (34).Along the same lines of Ref. [11], in which a TMD factorized formula has been proposed at the twist-three level forthe cos φ asymmetry for light-hadron production in SIDIS, we can conjecture that the formulae in Eq. (58) are validto all orders in α s , provided one includes also the nonperturbative contributions of the TMD gluon distribution andshape function.At this point, it may be good to stress the difference between the shape function and the wave function of the J/ψ meson. In the lowest-order picture a quarkonium state consists of a heavy quark and an antiquark. In thecenter-of-mass frame of the quarkonium the momenta k and k of the heavy quark and antiquark add up to zero.Their difference defines the relative velocity v = | v | : k ≡ k − k = m Q v , and NRQCD involves an expansion in v (cid:28)
1. The wave function Ψ( k ) of the quarkonium in momentum space is expected to be positronium-like, with atail that depends on L : the orbital angular momentum of the quark-antiquark pair. What enters in the expressionsfor L = 0 states is (cid:82) d k Ψ( k ) = R (0), the wave function at the origin, and for L = 1 states its derivative R (cid:48) (0). Onthe other hand, the shape function ∆ [ n ] of the QQ system is a function of k + k and, in LO NRQCD, is equal toa delta function in k + k . Upon radiating additional gluons and quarks, this becomes smeared out. That is thereason why in Ref. [13] the shape function was referred to as smearing function. For lack of better input, the modeladopted there for this function was based on the expectations for the wave function Ψ( k ). However, what the resultspresented in this paper show is that the shape function (or at least its perturbative tail) is actually independent of L .An L -independent shape or smearing function would imply that it would not be an obstacle to the extraction of theCO matrix elements from a comparison between quarkonium production and open heavy quark pair production inSIDIS as proposed in Ref. [13], implying a much more robust result. Note that the transverse momentum dependenceof the shape function in Eq. (63) not only implies independence from the L quantum number, but actually from anyquantum number of the produced quarkonium. Only the overall magnitude of the shape function is a function ofthese quantum numbers determined by the relevant LDME.For the angular dependent structure function we expect the following result to hold, even if due to the absence ofa collinear divergence a shape function is not strictly needed: F cos 2 φ ψ UU = (cid:88) n H cos 2 φ ψ [ n ] UU C (cid:2) w h ⊥ g ∆ [ n ] h (cid:3) ( x, q T ; µ ) , (65)where C (cid:2) w h ⊥ g ∆ [ n ] h (cid:3) ( x, q T ; µ ) = (cid:90) d p T (cid:90) d k T δ ( q T − p T − k T ) w ( p T , k T ) h ⊥ g ( x, p T ; µ ) ∆ [ n ] h ( k T , µ ) , (66)with w ( p T , k T ) being a transverse momentum dependent weight function. The shape function ∆ [ n ] h could be in generaldifferent from ∆ [ n ] : the determination of its perturbative tail would require a similar study at higher order in α s . We2note however that a full calculation of the cross section for J/ψ production in SIDIS at the order α α s , within thecollinear NRQCD framework, is still missing [42].Based on the fact that the p T dependence for h ⊥ g in the gluon correlator has a rank-two tensor structure in thenoncontracted transverse momentum, and unpolarized vector-meson production generally has a rank-zero structure,we consider a shape function ∆ [ n ] h of rank zero (ignoring a possible contribution from a linearly polarized quark-pairstate) and a weight function expression: w ( p T , k T ) = 1 M p q T (cid:2) p T · q T ) − p T q T (cid:3) . (67)Furthermore, the convolution in Eq. (66) can be rewritten as C (cid:2) w h ⊥ g ∆ [ n ] h (cid:3) ( x, q T ; µ ) = πM p (cid:90) ∞ d b T b T J ( b T q T ) (cid:98) h ⊥ g (2)1 ( x, b T ; µ ) (cid:98) ∆ [ n ] h ( b T , µ ) , (68)where we have introduced the second derivative w.r.t. b T of the Fourier transform of the linearly polarized gluon TMDdistribution, (cid:98) h ⊥ g (2)1 ( x, b T ; µ ) = 2 (cid:18) − M p ∂∂b T (cid:19) (cid:98) h ⊥ g ( x, b T ; µ ) = 2 M p (cid:90) ∞ dp T p T p T b T J (cid:0) b T | p T | (cid:1) h ⊥ g (cid:0) x, p T ; µ (cid:1) . (69)Concerning the L -independence of ∆ [ n ] h at high transverse momentum, we cannot draw any conclusion, due tothe absence of a logarithmic dependence. Moreover, we cannot conclude any L -independence for small transversemomentum for either ∆ [ n ] or ∆ [ n ] h , but the suggestions of Ref. [13] and the proposed cross-checks, do allow anexperimental investigation of this. Shape functions need to be experimentally extracted, just like the LDMEs andFFs have to. The EIC can play an important role in this regard. V. CONCLUSIONS
Let us recapitulate the main points of this work. Our starting point is the assumption that transverse momentumdependent factorization is valid for
J/ψ production in SIDIS at small q T . This Ansatz is a very reasonable one, sinceSIDIS for light hadrons is one of the few processes for which TMD factorization is proven at all orders in α s . Buildingfurther on this premise, we calculate the cross section for e p → e (cid:48) J/ψ X in two different regimes. At low q T , thecross section is factorized in terms of TMD PDFs and generic shape functions (first introduced in Refs. [31, 32] asthe generalization of the NRQCD LDMEs to TMD factorization), while at high q T the factorization involves collinearPDFs and NRQCD matrix elements. Our Ansatz of TMD factorization then requires the consistency condition thatboth descriptions match in the intermediate region Λ (cid:28) q T (cid:28) Q . The perturbative calculations for the leading-power SIDIS structure functions F UU,T and F UU,L allow, at least at LO accuracy in the strong coupling constant, todeduce from the matching the specific form of the color-octet shape function at large transverse momentum. Moreover,the angular structure function F cos 2 φ ψ UU matches without any dependence on a shape function whatsoever, due to theabsence of a logarithmic divergence.We therefore conclude that the assumption of TMD factorization for J/ψ production in SIDIS, and the necessary(but not sufficient) condition of matching at intermediate q T , imposes certain properties to the perturbative structureof the shape functions. In particular, we find that their perturbative tails are independent of the quantum numbersof the produced quarkonium state, except for their overall magnitude given by the collinear NRQCD LDMEs. Oneconsequence of this is that the feasibility of extracting the CO LDMEs by comparing quarkonium and open heavy-quark production at an EIC, proposed in Ref. [13], is not hampered.We note that our perturbative result agrees with the findings in recent work based on the soft-collinear effectivetheory (SCET) approach [32], although we are unable to draw conclusions on the nonperturbative structure of theshape functions with our method. Corroborating the results in Ref. [32] from a different and arguably simplerapproach, we thus believe that our study could contribute towards a full proof of TMD factorization for quarkoniumproduction in SIDIS. VI. ACKNOWLEDGEMENTS
We thank Werner Vogelsang for useful discussions on the derivation of Eq. (A25). The work of U.D., F.M. and C.P.is supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No.3824093 (STRONG 2020). U.D. acknowledges financial support by Fondazione Sardegna under the project Quarkoniumat LHC energies, CUP F71I17000160002 (University of Cagliari).
Appendix A: Expansion of the momentum conserving delta-function for q T (cid:28) Q We consider the integral I = (cid:90) dˆ z g (ˆ z ) (cid:90) ˆ x max dˆ x f (ˆ x ) δ ( F (ˆ x, ˆ z )) , (A1)where f and g are two generic functions, ˆ x max = Q Q + M ψ , (A2)and F (ˆ x, ˆ z ) = q T Q + 1 − ˆ z ˆ z M ψ Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z . (A3)By introducing the variable ˆ x (cid:48) = ˆ x ˆ x max , with 0 ≤ ˆ x (cid:48) ≤ , (A4)the integral becomes I = ˆ x max (cid:90) dˆ z ˆ z g (ˆ z ) (cid:90) dˆ x (cid:48) ˆ x (cid:48) f (ˆ x (cid:48) ) δ ( G (ˆ x (cid:48) , ˆ z )) , (A5)with G (ˆ x (cid:48) , ˆ z ) = q T Q ˆ x (cid:48) ˆ z + M ψ Q (1 − ˆ z )(ˆ x (cid:48) − ˆ z ) − ˆ z (1 − ˆ z )(1 − ˆ x (cid:48) ) . (A6)After performing the first integration over ˆ x (cid:48) , the integral in Eq. (A5) can be written as I = ˆ x max (cid:90) dˆ z ˜ g (ˆ z )(1 − ˆ z ) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) (cid:32) M ψ ˆ zQ (cid:33) ˜ f (ˆ x (cid:48) ) , (A7)where ˜ g (ˆ z ) = ˆ z (cid:32) M ψ ˆ zQ (cid:33) − g (ˆ z ) , ˜ f (ˆ x (cid:48) ) = ˆ x (cid:48) f (ˆ x (cid:48) ) , (A8)and ˆ x (cid:48) (ˆ z ) = (cid:32) M ψ Q (cid:33) (cid:34) M ψ ˆ zQ + q T Q ˆ z − ˆ z (cid:35) − . (A9)By using the identity ˜ g (ˆ z ) ˜ f (ˆ x (cid:48) ) = (˜ g (ˆ z ) − ˜ g (1)) ˜ f (1) + ˜ g (1) ˜ f (1) + ˜ g (ˆ z )( ˜ f (ˆ x (cid:48) ) − ˜ f (1)) (A10)the integral in Eq. (A5) can be split in three parts, I = ˆ x max ( I + I + I ) , (A11)4with I = (cid:90) dˆ z ˜ g (ˆ z ) − ˜ g (1)(1 − ˆ z ) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) (cid:32) M ψ ˆ zQ (cid:33) ˜ f (1) (A12) I = ˜ g (1) (cid:90) dˆ z − ˆ z ) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) (cid:32) M ψ ˆ zQ (cid:33) ˜ f (1) , (A13) I = (cid:90) dˆ z ˜ g (ˆ z ) ˜ f (ˆ x (cid:48) ) − ˜ f (1)(1 − ˆ z ) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) (cid:32) M ψ ˆ zQ (cid:33) , (A14)
1. Integral I In the calculation of the integral I , we can directly take the limit q T → I = (cid:90) dˆ z ˜ g (ˆ z ) − ˜ g (1)1 − ˆ z ˜ f (1)= (cid:90) dˆ z ˜ g (ˆ z ) ˜ f (1) 1(1 − ˆ z ) + = (cid:90) dˆ z (cid:90) dˆ x (cid:48) ˜ g (ˆ z ) ˜ f (ˆ x (cid:48) ) 1(1 − ˆ z ) + δ (1 − ˆ x (cid:48) )= (cid:90) dˆ z (cid:90) dˆ x (cid:48) g (ˆ z ) f (ˆ x (cid:48) ) (cid:32) M ψ ˆ zQ (cid:33) − ˆ z (1 − ˆ z ) + δ (1 − ˆ x (cid:48) )= ˆ x − (cid:90) dˆ z (cid:90) ˆ x max dˆ x g (ˆ z ) f (ˆ x ) (cid:32) M ψ ˆ zQ (cid:33) − ˆ z (1 − ˆ z ) + δ (1 − ˆ x/ ˆ x max ) , (A15)where the plus-distribution is defined in Eq. (40).
2. Integral I We can perform the integral I exactly and then keep the leading term in the expansion in powers of q T /Q , I = ˜ g (1) (cid:90) dˆ z − ˆ z ) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) (cid:32) M ψ ˆ zQ (cid:33) ˜ f (1)= ˜ g (1) ˜ f (1) (cid:34) ln Q q T + ln (cid:32) M ψ Q (cid:33)(cid:35) + ˜ g (1) ˜ f (1) q T Q (cid:32) M ψ Q (cid:33) − (cid:34)(cid:32) M ψ Q (cid:33) ln Q q T + (cid:32) M ψ Q − M ψ Q (cid:33) ln (cid:32) M ψ Q (cid:33) + M ψ Q ln M ψ Q − M ψ Q (cid:35) + O (cid:18) q T Q (cid:19) ≈ ˜ g (1) ˜ f (1) ln (cid:18) Q + M ψ q T (cid:19) = (cid:90) dˆ z (cid:90) dˆ x (cid:48) ˜ g (ˆ z ) ˜ f (ˆ x (cid:48) ) ln (cid:18) Q + M ψ q T (cid:19) δ (1 − ˆ x (cid:48) ) δ (1 − ˆ z )5= (cid:90) dˆ z (cid:90) dˆ x (cid:48) g (ˆ z ) f (ˆ x (cid:48) ) (cid:32) M ψ Q (cid:33) − ln (cid:18) Q + M ψ q T (cid:19) δ (1 − ˆ x (cid:48) ) δ (1 − ˆ z )= (cid:90) dˆ z (cid:90) ˆ x max dˆ x g (ˆ z ) f (ˆ x ) ln (cid:18) Q + M ψ q T (cid:19) δ (1 − ˆ x/ ˆ x max ) δ (1 − ˆ z ) . (A16)
3. Integral I The integral I is given by I = (cid:90) dˆ z ˜ g (ˆ z ) ˜ f (ˆ x (cid:48) (ˆ z )) − ˜ f (1)(1 − ˆ z ) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) (cid:32) M ψ ˆ zQ (cid:33) = (cid:90) dˆ z g (ˆ z ) ˆ z − ˆ z ˜ f (ˆ x (cid:48) (ˆ z )) − ˜ f (1) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) . (A17)We can trade the integration variable ˆ z with ˆ x (cid:48) by inverting the relation ˆ x (cid:48) = x (cid:48) (ˆ z ). This amounts to solve a secondorder equation in ˆ z with solutionsˆ z − = M ψ Q ˆ x (cid:48) M ψ /Q + 1 − ˆ x (cid:48) q T Q M ψ Q (cid:32) M ψ Q (cid:33) − ˆ x (cid:48) ( M ψ /Q + 1 − ˆ x (cid:48) )(1 − ˆ x (cid:48) ) + O (cid:18) q T Q (cid:19) , ˆ z + = 1 − q T Q (cid:32) M ψ Q (cid:33) − ˆ x (cid:48) − ˆ x (cid:48) + O (cid:18) q T Q (cid:19) . (A18)The first solution is not physically acceptable, since momentum conservation implies that q T = 0 when ˆ z = 1 and0 < x (cid:48) <
1. Therefore we take ˆ z = ˆ z + , which is the only solution surviving in the massless limit. We note that, if weneglect terms of the order q T /Q , we obtain the requirement ˆ x (cid:48) ≤ − q T / ( Q + M ψ ) since ˆ z ≥
0. Moreover we finddˆ z = q T Q (cid:32) M ψ Q (cid:33) − − ˆ x (cid:48) ) dˆ x (cid:48) , ˆ z (1 − ˆ z ) (cid:104) M ψ / (ˆ zQ ) + q T ˆ z/ ( Q (1 − ˆ z )) (cid:105) = Q q T (1 − ˆ x (cid:48) ) . (A19)By substituting in the expression for I , taking into account that g (ˆ z ) → g (1) as q T →
0, we obtain I = (cid:32) M ψ Q (cid:33) − g (1) (cid:90) dˆ x (cid:48) ˜ f (ˆ x (cid:48) ) − ˜ f (1)1 − ˆ x (cid:48) = (cid:32) M ψ Q (cid:33) − g (1) (cid:90) dˆ x (cid:48) ˜ f (ˆ x (cid:48) )(1 − ˆ x (cid:48) ) + = (cid:90) dˆ z g (ˆ z ) (cid:90) dˆ x (cid:48) f (ˆ x (cid:48) ) (cid:32) M ψ Q (cid:33) − ˆ x (cid:48) (1 − ˆ x (cid:48) ) + δ (1 − ˆ z )= (cid:90) dˆ z g (ˆ z ) (cid:90) ˆ x max dˆ x f (ˆ x ) ˆ x/ ˆ x max (1 − ˆ x/ ˆ x max ) + δ (1 − ˆ z ) . (A20)
4. The sum I + I + I Summing up the different contributions we find that the integral I = (cid:90) dˆ z g (ˆ z ) (cid:90) ˆ x max dˆ x f (ˆ x ) δ (cid:32) q T Q + 1 − ˆ z ˆ z M ψ Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z (cid:33) , (A21)6when q T (cid:28) Q , is equal to I = ˆ x max (cid:90) dˆ z g (ˆ z ) (cid:90) ˆ x max dˆ x f (ˆ x ) (cid:40) ˆ x/ ˆ x max (1 − ˆ x/ ˆ x max ) + δ (1 − ˆ z ) + Q + M ψ Q + M ψ / ˆ z ˆ z (1 − ˆ z ) + δ (1 − ˆ x/ ˆ x max )+ δ (1 − ˆ x/ ˆ x max ) δ (1 − ˆ z ) (cid:34) ln Q q T + ln (cid:32) M ψ Q (cid:33)(cid:35)(cid:41) , (A22)and therefore our final result reads δ (cid:32) q T Q + 1 − ˆ z ˆ z M ψ Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z (cid:33) = ˆ x max (cid:40) ˆ x (cid:48) (1 − ˆ x (cid:48) ) + δ (1 − ˆ z ) + Q + M ψ Q + M ψ / ˆ z ˆ z (1 − ˆ z ) + δ (1 − ˆ x (cid:48) )+ δ (1 − ˆ x (cid:48) ) δ (1 − ˆ z ) ln (cid:18) Q + M ψ q T (cid:19)(cid:41) , (A23)where ˆ x max = Q Q + M ψ , ˆ x (cid:48) = ˆ x ˆ x max . (A24)In the limit M ψ →
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1, ˆ x (cid:48) → ˆ x and we recover the known relation [43] δ (cid:18) q T Q − (1 − ˆ x )(1 − ˆ z )ˆ x ˆ z (cid:19) = ˆ x (1 − ˆ x ) + δ (1 − ˆ z ) + ˆ z (1 − ˆ z ) + δ (1 − ˆ x ) + δ (1 − ˆ x ) δ (1 − ˆ z ) ln Q q T , (A25)which is valid in the region q T (cid:28) Q . As is clear from comparing Eq. (A23) with (A25), the inclusion of a heavymass gives rise to a logarithm ln( Q + M ψ ) /q T instead of ln Q /q T , which can be traced back to the extra termln(1 + M ψ /Q ) in Eq. (A22). 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