Abstract
We discuss strategies for comparisons of nonperturbative QCD predictions for parton distribution functions (PDFs) with high-energy experiments in the region of large partonic momentum fractions $x$. Analytic functional forms for PDFs cannot be uniquely determined solely based on discrete experimental measurements because of a mathematical property of mimicry of PDF parametrizations that we prove using a representation based on B\'ezier curves. Predictions of nonperturbative QCD approaches for the $x$ dependence of PDFs instead should be cast in a form that enables decisive comparisons against experimental measurements. Predictions for effective power laws of $(1-x)$ dependence of PDFs may play this role. Expectations for PDFs in a proton based on quark counting rules are compared against the effective power laws of $(1-x)$ dependence satisfied by CT18 next-to-next-to-leading order parton distributions. We comment on implications for studies of PDFs in a pion, in particular on the comparison of nonperturbative approaches with phenomenological PDFs.
Highlights
Quantum chromodynamics governs interactions of strongly interacting particles and predicts the existence of bound states of hadronic matter
Analytic functional forms for parton distribution functions (PDFs) cannot be uniquely determined solely based on discrete experimental measurements because of a mathematical property of mimicry of PDF parametrizations that we prove using a representation based on Bezier curves
Can the power laws predicted by the Quark counting rules (QCRs) be tested by experimental hadron measurements? We address this question by presenting a mathematical argument in Sec
Summary
Quantum chromodynamics governs interactions of strongly interacting particles and predicts the existence of bound states of hadronic matter. The QCD factorization formulas approximate the hadronic cross sections in simple inclusive processes in a way that accounts for soft and collinear contributions, and neglects numerically small mass terms The relation of these formulas to the nonperturbative PDFs includes power-suppressed terms that are not controlled to the necessary extent. The power law predicted by the QCRs as well as the asymptotic behavior of the dðxÞ=uðxÞ at x → 1—a consequence of the spin-flavor extension of the original QCRs [22]—are consistent with the behavior of the actual phenomenological PDFs, alternative behaviors are not ruled out We will examine this (1 − x) falloff in the recent CT18 next-to-next-to-leading-order global analysis [23]. We address the dependence of the empirical power-law exponent on the functional form of the PDFs, factorization scale, and the type of the scattering process in the remainder of Sec. III.
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