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Quark TMDPDFs from Large to Small $x$ : A Unified Description of CSS, DGLAP, and Small- $x$ Double Logarithms

This paper extends a full-kinematic background-field construction to quark TMDPDFs, introducing the MSTT scheme and a rapidity-only scheme that unify the description of large- and small- $x$ dynamics by simultaneously recovering CSS, DGLAP, and small- $x$ double logarithmic evolutions within a single framework.

Original authors: Swagato Mukherjee, Vladimir V. Skokov, Andrey Tarasov, Shaswat Tiwari

Published 2026-10-22

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CC BY 4.0

Original authors: Swagato Mukherjee, Vladimir V. Skokov, Andrey Tarasov, Shaswat Tiwari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside every proton, a chaotic storm of particles called quarks and gluons buzzes with energy, held together by the strongest force in nature. To understand how these particles behave, physicists often look at how they are distributed inside the proton, specifically tracking how much of the proton's forward momentum a single particle carries and how much it moves sideways. For decades, scientists have relied on two different sets of rules to describe this behavior, depending on whether they are looking at particles carrying a large share of the momentum or a tiny fraction. The first set of rules works well for particles with a large share, while the second set, designed for the high-speed, low-momentum particles that dominate the interior of a proton at extreme energies, uses a completely different mathematical approach. This split has left a gap in our understanding, particularly for the upcoming Electron-Ion Collider, a massive machine designed to take high-resolution pictures of the proton's interior. In the middle ground between these two extremes, neither set of rules has been fully reliable, leaving physicists without a single, unified way to describe the full range of particle behavior.

A team of researchers has now constructed a new framework that bridges this gap, offering a single description that works for both large and small shares of momentum. They achieved this by developing a method that treats the background environment of the proton without forcing the particles into a specific order of movement. In previous approaches, scientists often had to assume that particles moved in a strictly ordered line or followed a specific path to make the math work. This new method, however, allows the particles to retain their full complexity, keeping both their forward speed and their sideways motion intact without imposing artificial restrictions. By doing this, the researchers were able to calculate the behavior of quarks at a high level of precision, showing that their new approach naturally produces the correct results for both the large-momentum and small-momentum limits.

The team demonstrated that their unified description smoothly transitions between the two previously separate regimes. When they applied their method to the large-momentum limit, the results matched the established rules used for decades. When they applied it to the small-momentum limit, the results correctly reproduced the double logarithmic patterns that characterize high-energy particle interactions. Crucially, their work also revealed that this single framework retains subtle details about the particles' behavior that were previously lost when scientists tried to force the data into one of the two older, separate categories. This means the new model captures a richer, more complete picture of the proton's inner workings, including effects that are usually ignored or treated as minor corrections.

The researchers also showed how this new framework connects to the standard tools used by experimentalists. They proved that their unified description can be translated into the familiar language of current experiments, ensuring that the new insights can be directly tested against real-world data. By removing the need to choose between two conflicting sets of rules, this work provides a solid foundation for interpreting the complex signals that will be generated by future particle colliders. The study confirms that a single, coherent picture of quark behavior is possible, one that respects the full complexity of the proton's interior without sacrificing mathematical rigor. This achievement does not just fill a gap in theory; it offers a more reliable map for navigating the uncharted territory of high-energy physics, ensuring that the next generation of experiments can see the proton clearly, from its most energetic particles to its most elusive ones.

Technical Summary: Quark TMDPDFs from Large to Small x: A Unified Description of CSS, DGLAP, and Small-x Double Logarithms

Problem Statement
Quantum Chromodynamics (QCD) factorization typically separates high-energy scattering into short-distance perturbative coefficients and long-distance hadronic matrix elements. However, a unified description of Transverse-Momentum Dependent Parton Distribution Functions (TMDPDFs) across the full kinematic range of the momentum fraction $x$ remains challenging.

At large $x$ , dynamics are governed by transverse-momentum ordering, leading to Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution.
At small $x$ , dynamics are governed by longitudinal (rapidity) ordering, leading to Balitsky-Fadin-Kuraev-Lipatov (BFKL) and related non-linear evolutions.
Standard TMD factorization relies on the Collins–Soper–Sterman (CSS) formalism, which assumes a specific separation of scales but does not inherently interpolate between the distinct ordering requirements of the large- $x$ and small- $x$ limits. Furthermore, existing expansions often assume either a strict collinear limit (leading twist) or a strict eikonal limit (small $x$ ), failing to capture the interplay of higher-twist and sub-eikonal corrections in the intermediate kinematic regions relevant for future facilities like the Electron-Ion Collider (EIC).

Methodology
The authors extend the full-kinematic background-field construction, previously developed for gluon TMDPDFs, to the quark sector. The core methodology involves a two-stage factorization and renormalization procedure:

Two-Stage Scale Separation:
- Upper Scales ( $\sigma$ ): Define the boundary between hard scattering modes and quantum modes entering the renormalized TMD operator. This stage handles ultraviolet (UV) divergences.
- Lower Scales ( $\sigma'$ ): Separate perturbative quantum modes from non-perturbative background modes. This stage handles infrared (IR) divergences and defines the quantum-background separation.
- The framework allows the upper and lower schemes to be chosen independently, enabling the extraction of different perturbative limits from the same initial operator.
Background-Field Construction:
- The authors utilize the background-field method where QCD fields are split into perturbative ( $A$ ) and non-perturbative background ( $B$ ) components.
- Crucially, the background fields retain both longitudinal and transverse momenta without imposing collinear or eikonal ordering. This preserves the full kinematic dependence of the TMD operator.
- The calculation is performed at Next-to-Leading Order (NLO) for the leading-power quark TMD beam function.
Scheme Formulation:
- MSTT Scheme: A "full-kinematic" scheme where the lower-scale operators are built from background fields with unrestricted momentum ordering.
- Rapidity-Only Scheme: A scheme utilizing a single longitudinal cutoff to regulate both UV and IR divergences, distinct from the standard two-scale CSS approach.
- Upper-Scale Renormalization: The authors compute the NLO matching for both the standard CSS scheme (using $\mu$ and $\zeta$ ) and a rapidity-only scheme (using a single longitudinal cutoff $\sigma$ ).

Key Contributions and Results

NLO Calculation of Quark TMD Beam Function:
The authors compute the bare quark TMD beam function at NLO in the background-Feynman gauge. Unlike the gluon case, the quark TMD operator at NLO contains no genuine infrared rapidity divergence; potential IR divergences are regulated by the non-zero transverse momentum flow from the background fields.
Upper-Scale Renormalization and Soft Function:
- CSS Scheme: The standard two-scale evolution is recovered. The authors explicitly compute the soft function in the rapidity-only scheme and demonstrate that, at NLO, its contribution is already contained within the renormalized beam function. Thus, adding a separate soft factor would result in double-counting.
- Rapidity-Only Scheme: The authors derive a single evolution equation for the rapidity-only TMDPDFs. This equation acts as a one-scale counterpart to the CSS evolution, governing the UV renormalization dynamics.
Lower-Scale Matching and Evolution:
- MSTT Scheme: In this scheme, the finite perturbative remainder of the beam function is assigned to coefficient functions, while the background-field operators are renormalized at the lower scale $\sigma'$ . For quarks, this lower-scale renormalization is trivial at NLO (no IR divergence), but the scheme retains the full kinematic structure.
- Collinear Limit ( $b_\perp \to 0$ ): The leading term of the MSTT operator expansion reproduces the standard DGLAP logarithm and the DGLAP evolution equation.
- High-Energy Limit ( $x \to 0$ ): The leading term reproduces the double-logarithmic evolution characteristic of small- $x$ quark TMDPDFs.
- Intermediate Kinematics: At finite $b_\perp$ and arbitrary $x$ , the leading-power MSTT operator retains portions of higher-twist collinear and sub-eikonal corrections. It does not require the resummation of independent higher-twist operators to connect the large- $x$ and small- $x$ regimes; the leading-power operator itself contains the necessary dynamics.
Rapidity-Only Lower-Scale Evolution:
The authors formulate a rapidity-only scheme for the lower scales $\sigma'$ . The resulting evolution equation reduces to DGLAP evolution in the collinear limit and to the characteristic double-logarithmic quark evolution at small $x$ . This provides a smooth transition between the two regimes using a single longitudinal cutoff.
Matching Relations:
Explicit matching relations are derived between:
- The MSTT scheme and standard collinear factorization (recovering DGLAP).
- The MSTT scheme and high-energy factorization (recovering small- $x$ double logarithms).
- The MSTT scheme and the lower rapidity-only scheme.
  These relations demonstrate how the same perturbative information is distributed differently between coefficient functions and background-field operators depending on the chosen scheme.

Significance
The paper claims to provide a unified leading-power description of quark TMDPDFs that bridges the gap between large- $x$ (DGLAP) and small- $x$ (BFKL/double-logarithmic) dynamics. By utilizing a background-field construction that does not impose collinear or eikonal ordering, the authors show that the leading-power TMD operator naturally incorporates the logarithmic structures of both limits.

The significance lies in:

Unified Framework: Offering a common operator framework where the interplay between large and small $x$ is handled without assuming a specific ordering a priori.
Handling of Higher-Twist/Sub-Eikonal Effects: The construction retains parts of higher-twist and sub-eikonal corrections within the leading-power operator, offering a more complete description of the TMD structure in intermediate kinematic regions than standard expansions.
Scheme Independence: Demonstrating that different factorization schemes (MSTT vs. Rapidity-Only) describe the same underlying operator dynamics, differing only in how perturbative information is partitioned between coefficients and operators.

The work establishes a theoretical foundation for analyzing TMDPDFs in kinematic regimes where neither strict collinear nor strict eikonal approximations are parametrically guaranteed to dominate, which is particularly relevant for the phenomenology of the future Electron-Ion Collider.

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