Abstract
We consider the transverse momentum spectrum and the cos 2ϕ azimuthal distribution of J/ψ mesons produced in semi-inclusive, deep-inelastic electron-proton scattering, where the electron and the proton are unpolarized. At low transverse momentum, we propose factorized expressions in terms of transverse momentum dependent gluon distributions and shape functions. We show that our formulae, at the order αs, correctly match with the collinear factorization results at high transverse momentum. The latter are computed at the order {alpha}_s^2 in the framework of nonrelativistic QCD (NRQCD), with the inclusion of the intermediate {}^3{S}_1^{left[1right]} color-singlet Fock state, as well as the subleading color-octet ones that are relatively suppressed by a factor v4 in the NRQCD velocity parameter v. We show that the {}^1{S}_0^{left[8right]} and {}^3{P}_J^{left[8right]} (J = 0, 1, 2) contributions diverge in the small transverse momentum region and allow us to determine the perturbative tails of the shape functions, which carry the same quantum numbers. These turn out to be identical, except for the overall magnitude given by the appropriate NRQCD long distance matrix element.
Highlights
When the J/ψ meson is produced with a large transverse momentum, qT ΛQCD, a collinear factorization approach based on fixed-order perturbative quantum chromodynamics (QCD) can be applied
On the basis of the above considerations, TMD factorized expressions for the structure functions FUU,T and FUU,L have to take into account smearing effects [13], encoded in the shape function ∆[n] [32, 33], which can be thought as a generalization of the long distance matrix elements of nonrelativistic QCD (NRQCD) in collinear factorization
Our starting point is the assumption that transverse momentum dependent factorization is valid for J/ψ production in SIDIS at small qT
Summary
Collinear factorization and NRQCD are adopted for the description of the process e( ) + p(P ) → e( ) + J/ψ(Pψ) + X ,. Where the first and second subscripts of the structure functions F denote the polarization of the initial electron and proton, respectively, while the third one, when present, specifies the polarization of the exchanged virtual photon. We note that the partonic subprocesses contributing to the cross sections in the small-qT limit given in eq (2.27) are only the n = 1S0[8], 3PJ[8] ones, which correspond to t-channel Feynman diagrams of the types (c) and (d) in figure 1. The other partonic subprocesses, namely the gluon induced 3S1[1,8] channels and the quark-induced 1S0[8] channel depicted in figures 1 (a)-(b), are suppressed and vanish as qT2 → 0 They are not relevant for our study of the matching of the collinear and TMD results and will not be considered in the following. The appearance of a logarithm ln(Q2 + Mψ2)/qT2 , instead of ln Q2/qT2 , suggests Q2 + Mψ2 as the natural choice for the hard scale in the process under study
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