[Paper Review] On the positivity of MSbar parton distributions

Alessandro Candido, Stefano Forte+2 more|arXiv (Cornell University)|Jul 31, 2023

Particle physics theoretical and experimental studiesPhysics and Astronomy3 citations

TL;DR

This paper establishes that $¯{\mathrm{MS}}$ parton distribution functions (PDFs) are non-negative in the perturbative regime by analyzing their derivation from positive-definite physical observables and bare PDFs in dimensional regularization. It confirms that positivity holds above a scale of approximately $Q^2 \gtrsim 2\,\text{GeV}^2$, reconciling earlier arguments with recent findings that bare PDFs can become negative at low scales.

ABSTRACT

We revisit our argument that shows that parton distribution Functions (PDFs) in the MSbar{ scheme are non-negative in the perturbative region, with the main goals of elucidating its domain of validity and clarifying its theoretical underpinnings. We specifically discuss recent results proving that PDFs can turn negative at sufficiently low scale, we clarify quantitatively various aspects of our derivation of positivity in the perturbative region, and we provide an estimate for the scale above which PDF positivity holds.

Motivation & Objective

• To clarify the domain of validity for the positivity of $¯{\mathrm{MS}}$ parton distribution functions (PDFs) in perturbative QCD.
• To reconcile conflicting results: while recent work shows bare PDFs can be negative at low scales, this paper argues $¯{\mathrm{MS}}$ PDFs remain positive in the perturbative region.
• To quantify the scale above which $¯{\mathrm{MS}}$ PDF positivity holds, based on NLO corrections being smaller than LO terms.
• To assess the equivalence and consistency of two routes to positivity: via positive physical observables and via positive bare PDFs before collinear subtraction.
• To provide a conservative estimate of the perturbative domain where $¯{\mathrm{MS}}$ PDFs are guaranteed non-negative.

Proposed method

• Derives the factorization of deep-inelastic scattering (DIS) structure functions in $d=4-2\epsilon$ dimensions, distinguishing bare PDFs (UV-renormalized but not collinear-subtracted) from renormalized PDFs.
• Analyzes the positivity of bare PDFs in a model with an off-shell quark target, computing the matrix element of a Wilson line operator perturbatively in dimensional regularization.
• Uses the physical factorization scheme (e.g., DIS scheme) to define PDFs via measurable cross-sections, ensuring positivity by construction.
• Applies perturbative QCD factorization to relate bare PDFs to $¯{\mathrm{MS}}$ PDFs through a perturbative scheme change, preserving positivity if NLO corrections are smaller than LO.
• Performs explicit one-loop computations of the bare PDF for a massless off-shell quark, identifying the $\delta(1-x)$ term, splitting function contributions, and finite corrections.
• Compares results with recent findings on low-scale negativity of bare PDFs, showing that while bare PDFs can become negative, the $¯{\mathrm{MS}}$-transformed PDFs remain positive above $Q^2 \gtrsim 2$ GeV².

Experimental results

Research questions

• RQ1 At what scale does the positivity of $¯{\mathrm{MS}}$ PDFs become reliably valid in perturbative QCD?
• RQ2 How do recent results showing negativity of bare PDFs at low scales affect the validity of the $¯{\mathrm{MS}}$ PDF positivity argument?
• RQ3 Are the two routes to positivity—via physical observables and via positive bare PDFs—equivalent, and under what conditions?
• RQ4 What is the quantitative estimate for the lower bound of the perturbative domain where $¯{\mathrm{MS}}$ PDFs are non-negative?
• RQ5 How does the scale dependence of the bare PDF's negativity relate to the physical scale of the target in hadronic processes?

Key findings

• The $¯{\mathrm{MS}}$ PDFs are non-negative in the perturbative regime, provided that next-to-leading order (NLO) corrections are smaller than leading-order (LO) terms.
• The domain of validity for $¯{\mathrm{MS}}$ PDF positivity is conservatively estimated to begin at $Q^2 \gtrsim 2\,\text{GeV}^2$, consistent with standard QCD expectations.
• For an off-shell quark target, the scale at which bare PDFs lose positivity is approximately twice the target virtuality, indicating a model-dependent but physically motivated scale.
• The positivity of $¯{\mathrm{MS}}$ PDFs is preserved when transforming from a physical factorization scheme (where PDFs are defined via measurable cross-sections) to $¯{\mathrm{MS}}$, due to the perturbative nature of the scheme change.
• Explicit one-loop computations confirm that the finite part of the bare PDF contains a contribution $f_f(x) = \frac{1}{2}\delta(1-x) + x - 2$, which is negative for $x \in (0,1)$, but this does not affect the positivity of the final $¯{\mathrm{MS}}$ PDFs in the perturbative region.
• The results are consistent with recent findings that bare PDFs can be negative at low scales, but this does not invalidate the positivity of $¯{\mathrm{MS}}$ PDFs in the perturbative domain.