[Paper Review] Evolution of transverse-momentum-dependent densities

І. О. Чередніков, N. G. Stefanis|arXiv (Cornell University)|Apr 1, 2011

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper investigates the renormalization-group evolution of transverse-momentum-dependent (TMD) parton densities, focusing on the challenges posed by light-cone singularities in gauge links. It compares different operator definitions—off-light-cone, light-cone with subtraction, and light-cone with Mandelstam-Leibbrandt prescription—and demonstrates that only consistent regularization and soft-factor subtraction restore the correct anomalous dimension $γ_{\text{LC}}$, ensuring compatibility with DGLAP evolution upon integration over transverse momentum.

ABSTRACT

We discuss different operator definitions of the transverse-momentum dependent (TMD) parton densities from the point of view of their renormalization-group (RG) properties and UV evolution. We also consider the structure of the gauge links (Wilson lines) in these operator definitions and examine the role of the soft factor in the factorization formula within the TMD approach to semi-inclusive processes.

Motivation & Objective

• To resolve inconsistencies in the renormalization-group evolution of TMD parton densities due to light-cone singularities.
• To compare different operator definitions of TMDs—specifically off-light-cone and light-cone with soft-factor subtraction or ML prescription—regarding their UV and rapidity divergences.
• To establish a consistent framework for TMD evolution that preserves the correct anomalous dimension $γ_{\text{LC}}$ and ensures compatibility with collinear PDF evolution.
• To clarify the role of the soft factor $S$ and its interplay with overlapping UV-soft divergences in TMD factorization.
• To address the unresolved problem of all-order factorization with lightlike gauge links in covariant gauges.

Proposed method

• Analyzes the operator definition of unpolarized quark TMD $f_1(x, \mathbf{k}_\perp|n)$ using path-ordered Wilson lines with lightlike and off-lightlike contours.
• Classifies divergences into three types: UV, rapidity, and overlapping (UV+soft), with the latter requiring special treatment.
• Evaluates four regularization approaches: (1) off-light-cone gauge links, (2) soft-factor subtraction $R^{-1}$, (3) dimensional regularization of $1/q^+$, and (4) Mandelstam-Leibbrandt pole prescription.
• Derives the renormalization-group evolution equations for each definition, showing that only with proper soft-factor subtraction or ML prescription does the anomalous dimension reduce to $\gamma_{\text{LC}}$.
• Uses the generalized renormalization procedure to restore the correct anomalous dimension $\gamma_{\text{LC}} = \frac{3}{4} \frac{\alpha_s C_F}{\pi} + O(\alpha_s^2)$ in the light-cone case.
• Compares the evolution of TMDs in different gauges, showing that off-light-cone definitions yield different evolution kernels $\mathcal{K}_v \neq \mathcal{K}_{\text{DGLAP}}$ than lightlike ones.

Experimental results

Research questions

• RQ1 How do overlapping UV-soft divergences in light-cone TMD definitions affect the anomalous dimension and RG evolution?
• RQ2 Can the anomalous dimension of the TMD parton density be consistently restored to $\gamma_{\text{LC}}$ when using light-cone gauge links?
• RQ3 What is the role of the soft factor $S$ in factorizing semi-inclusive processes, and how does it affect the evolution of TMDs?
• RQ4 Why do off-light-cone gauge links lead to different evolution kernels compared to lightlike ones, and what are the phenomenological consequences?
• RQ5 Is a complete all-order factorization theorem possible with lightlike longitudinal gauge links in covariant gauges?

Key findings

• The standard light-cone TMD definition (2) suffers from overlapping divergences that mix UV and soft singularities, leading to incorrect anomalous dimensions unless corrected.
• Only after soft-factor subtraction $R^{-1}$ or use of the Mandelstam-Leibbrandt prescription do the TMDs evolve with the correct anomalous dimension $\gamma_{\text{LC}} = \frac{3}{4} \frac{\alpha_s C_F}{\pi}$.
• The off-light-cone definition (3) avoids overlapping divergences and yields the correct $\gamma_{\text{LC}}$ directly, but its factorization properties are less transparent.
• The soft factor $R$ is essential for restoring consistency: without it, the anomalous dimension deviates by the cusp anomalous dimension $\gamma_{\text{cusp}}$.
• After proper renormalization, the integrated TMD $\int d^2k_\perp f_1(x, \mathbf{k}_\perp|n)$ reproduces the DGLAP-evolving collinear PDF $f_1(x)$, confirming consistency with collinear QCD.
• In contrast, off-light-cone TMDs with $v$-dependent gauge links evolve with a different kernel $\mathcal{K}_v \neq \mathcal{K}_{\text{DGLAP}}$, implying distinct phenomenological behavior.