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We investigate the non-perturbative contribution of instantons to current matching kernels used in the context of the large momentum effective theory (LaMET). We derive explicitly these contributions using first principle semi-classical calculus for the unpolarized and polarized quark parton distributions and the matching kernel, and show that they are part of a trans-series expansion. These contributions are substantial at current lattice matching momenta.
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Gluon dressing of the light quarks within hadrons is very strong and extremely important in that it dynamically generates most of the observable mass through the breaking of chiral symmetry. The quark and gluon parton densities, $q(x)$ and $g(x)$, are necessarily interrelated since any gluon emission and absorption process, especially dressing of a quark, contributes to $g(x)$ and modifies $q(x)$. Guided by long-established results for the parton-in-parton distributions from a strict 1-loop perturbative analysis of a quark target, we extend the non-perturbative QCD approach based on the Rainbow-Ladder truncation of the Dyson-Schwinger equations to describe the interrelated valence $q_{rm v}(x)$ and the dressing-gluon $g(x)$ for a hadron at its intrinsic model scale. We employ the pion description from previous DSE work that accounted for the gluon-in-quark effect and introduce a simple model of the nucleon for exploratory purposes. We find typically mbox{$langle x rangle_g sim 0.20$} for both pion and nucleon at the model scale, and the valence quark helicity contributes 52% of nucleon spin. We deduce both $q_{rm v}(x)$ and $g(x)$ from 30 calculated Mellin moments, and after adopting existing data analysis results for $q_{rm sea}(x)$, we find that NLO scale evolution produces $g(x)$ in good agreement with existing data analysis results for the pion at 1.3 GeV and the nucleon at 5 GeV$^2$. At the scale 2 GeV typical of lattice-QCD calculations, we obtain mbox{$langle x rangle_g^{rm N} = 0.42$} in good agreement with 0.38 from the average of recent lattice-QCD calculations.
The Collins-Soper kernel, which governs the energy evolution of transverse-momentum dependent parton distribution functions (TMDPDFs), is required to accurately predict Drell-Yan like processes at small transverse momentum, and is a key ingredient for extracting TMDPDFs from experiment. Earlier we proposed a method to calculate this kernel from ratios of the so-called quasi-TMDPDFs determined with lattice QCD, which are defined as hadronic matrix elements of staple-shaped Euclidean Wilson line operators. Here we provide the one-loop renormalization of these operators in a regularization-independent momentum subtraction (RI$^prime$/MOM) scheme, as well as the conversion factor from the RI$^prime$/MOM-renormalized quasi-TMDPDF to the $overline{rm MS}$ scheme. We also propose a procedure for calculating the Collins-Soper kernel directly from position space correlators, which simplifies the lattice determination.
We investigate the impact of displaced heavy quark matching scales in a global fit. The heavy quark matching scale $mu_{m}$ determines at which energy scale $mu$ the QCD theory transitions from $N_{F}$ to $N_{F}+1$ in the Variable Flavor Number Scheme (VFNS) for the evolution of the Parton Distribution Functions (PDFs) and strong coupling $alpha_S(mu)$. We study the variation of the matching scales, and their impact on a global PDF fit of the combined HERA data. As the choice of the matching scale $mu_{m}$ effectively is a choice of scheme, this represents a theoretical uncertainty; ideally, we would like to see minimal dependence on this parameter. For the transition across the charm quark (from $N_{F}=3$ to $4$), we find a large $mu_m=mu_{c}$ dependence of the global fit $chi^2$ at NLO, but this is significantly reduced at NNLO. For the transition across the bottom quark (from $N_{F}=4$ to $5$), we have a reduced $mu_{m}=mu_b$ dependence of the $chi^2$ at both NLO and NNLO as compared to the charm. This feature is now implemented in xFitter 2.0.0, an open source QCD fit framework.
We investigate the dressed quark-gluon vertex combining two established non-perturbative approaches to QCD: the Dyson-Schwinger equation (DSE) for the quark propagator and lattice-regularized simulations for the quark, gluon and ghost propagators. The vertex is modeled using a generalized Ball-Chiu ansatz parameterized by a single form factor $tilde X_0$ which effectively represents the quark-ghost scattering kernel. The solution space of the DSE inversion for $tilde X_0$ is highly degenerate, which can be dealt with by a numerical regularization scheme. We consider two possibilities: (i) linear regularization and (ii) the Maximum Entropy Method. These two numerical approaches yield compatible $tilde X_0$ functions for the range of momenta where lattice data is available and feature a strong enhancement of the generalized Ball-Chiu vertex for momenta below 1 GeV. Our ansatz for the quark-gluon vertex is then used to solve the quark DSE which yields a mass function in good agreement with lattice simulations and thus provides adequate dynamical chiral symmetry breaking.
Transverse momentum dependent parton distribution functions (TMDPDFs) provide a unique probe of the three-dimensional spin structure of hadrons. We construct spin-dependent quasi-TMDPDFs that are amenable to lattice QCD calculations and that can be used to determine spin-dependent TMDPDFs. We calculate the short-distance coefficients connecting spin-dependent TMDPDFs and quasi-TMDPDFs at one-loop order. We find that the helicity and transversity distributions have the same coefficient as the unpolarized TMDPDF. We also argue that the same is true for pretzelosity and that this spin universality of the matching will hold to all orders in $alpha_s$. Thus, it is possible to calculate ratios of these distributions as a function of longitudinal momentum and transverse position utilizing simpler Wilson line paths than have previously been considered.
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