Abstract
The quasi parton distribution is a spatial correlation of quarks or gluons along the $z$ direction in a moving nucleon which enables direct lattice calculations of parton distribution functions. It can be defined with a nonperturbative renormalization in a regularization independent momentum subtraction scheme (RI/MOM), which can then be perturbatively related to the collinear parton distribution in the $\overline{\text{MS}}$ scheme. Here we carry out a direct matching from the RI/MOM scheme for the quasi-PDF to the $\overline{\text{MS}}$ PDF, determining the non-singlet quark matching coefficient at next-to-leading order in perturbation theory. We find that the RI/MOM matching coefficient is insensitive to the ultraviolet region of convolution integral, exhibits improved perturbative convergence when converting between the quasi-PDF and PDF, and is consistent with a quasi-PDF that vanishes in the unphysical region as the proton momentum $P^z\to \infty$, unlike other schemes. This direct approach therefore has the potential to improve the accuracy for converting quasi-distribution lattice calculations to collinear distributions.
Highlights
One of the great successes of QCD are factorization theorems, such as those that enable us to make predictions for deep inelastic scattering (DIS) and Drell-Yan processes at hadron colliders [1]
We have described the procedure of nonperturbative renormalization of quasi-parton distribution function (PDF) in the regularization independent momentum subtraction scheme (RI/MOM) scheme
The z-dependent renormalization constant is obtained by imposing Eq (9) on the off-shell quark matrix element of the spatial correlation operator in lattice QCD
Summary
One of the great successes of QCD are factorization theorems, such as those that enable us to make predictions for deep inelastic scattering (DIS) and Drell-Yan processes at hadron colliders [1]. The exponential factor eδmjzj introduces counterterms that cancel the linear divergences δm ∼ Λ, whereas the rest of the renormalization factor Zz depends on the end points of the Wilson line, including the coordinates 0 and z, and includes only logarithmic divergences We can generalize this renormalization relation to gauge-invariant nonlocal quark bilinear operators, as was proven in [14,15], so that the quasi-PDF renormalization can be split into two parts. In order to specify a well defined renormalization scheme with the split in Eq (16), a distinct definition must be given for δm and Zψ It would be useful if we can redefine the quasi-PDF to make it free of linear divergence, such as the treatment for transverse momentum distributions [43,44], or the gradient flow method [45], but a practical solution on the lattice has not yet been proposed or carried out. We exploit the independence of the RI/MOM scheme to the choice of UV regulator to carry out this matching calculation using dimensional regularization
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