⚛️ phenomenology

Threshold resummation for computing large- $x$ parton distribution through large-momentum effective theory

This paper establishes a threshold resummation framework for computing large- $x$ parton distribution functions via large-momentum effective theory by factorizing the matching coefficient into a space-like jet function and heavy-light Sudakov form factors, thereby enabling direct lattice QCD calculations of these critical distributions with NNLO consistency.

Original authors: Xiangdong Ji, Yizhuang Liu, Yushan Su

Published 2026-09-22

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CC BY 4.0

Original authors: Xiangdong Ji, Yizhuang Liu, Yushan Su

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, the building blocks of the atomic nucleus, a chaotic dance of quarks and gluons unfolds. These particles do not sit still; they zip around at nearly the speed of light, bound together by the strong force. To understand how protons behave in high-energy collisions, such as those at the Large Hadron Collider, physicists must know exactly how the momentum of the proton is shared among its constituents. This distribution is described by mathematical functions called parton distribution functions. While scientists have mapped these functions quite well for particles carrying a small fraction of the proton's momentum, the picture becomes hazy for particles carrying a large share. This gap in knowledge is critical because the most energetic collisions, which might reveal new physics beyond our current understanding, are driven by these high-momentum particles.

The difficulty in measuring these high-momentum particles stems from a specific mathematical problem that arises when trying to calculate their behavior. As the momentum fraction approaches its maximum limit, the equations used to describe the interactions generate enormous, unstable numbers that grow uncontrollably. These numbers, known as threshold logarithms, are not just a nuisance; they represent a fundamental breakdown in the standard calculation methods unless they are carefully tamed and reorganized. For decades, physicists have developed techniques to manage these numbers for particles moving through a vacuum, but applying these techniques to the complex environment of a proton inside a computer simulation has remained a significant hurdle.

A team of researchers has now developed a new method to solve this specific problem, paving the way for precise calculations of these elusive high-momentum particles directly from the laws of quantum mechanics. By treating the proton as a system where the internal particles are moving with a very large momentum, the team discovered that the chaotic mathematical terms could be separated into distinct, manageable pieces. They found that the complex interactions could be split into a "hard" core, representing the most violent exchanges of energy, and a "soft" cloud, representing the gentler, lingering interactions that surround the particles. This separation allowed them to isolate the troublesome numbers and reorganize them into a stable, predictable pattern.

The researchers demonstrated that this new approach works by comparing their theoretical predictions against the most advanced calculations available, which involve two layers of complexity in the underlying equations. Their method matched these existing results perfectly, confirming that their way of breaking down the problem was correct. More importantly, they identified a universal rule that governs how these high-momentum particles behave, a rule that applies not just to the proton's internal structure but also to other related measurements in particle physics. This universality means that once this rule is understood, it can be applied to a wide range of problems, simplifying the path forward for many different types of calculations.

The ultimate goal of this work is to enable computers to calculate the distribution of these high-momentum particles with the same precision as other parts of the proton. Currently, experiments struggle to pin down these values because the data is scarce and the theory is difficult to apply. With this new framework, scientists can now perform these calculations directly, without relying on guesswork or fitting unknown shapes to the data. This capability will allow for a much sharper view of the proton's interior, potentially revealing subtle signals of new physics that have been hidden in the noise of previous approximations. The work does not claim to have solved every mystery of the strong force, but it has removed a major barrier that was preventing a clear view of the most energetic parts of the proton.

By establishing a reliable way to handle the mathematical singularities that appear at the edge of the momentum spectrum, the researchers have provided a robust tool for the next generation of particle physics. The method relies on a deep understanding of how space and time interact at the quantum level, specifically how the flow of particles changes as they approach the speed of light. The team's success in matching their predictions with the most complex existing calculations gives high confidence that the method is sound. This achievement marks a significant step toward a complete, first-principles understanding of the proton, turning a region of uncertainty into a domain where precise predictions can be made and tested against future experiments.

Technical Summary: Threshold Resummation for Large- $x$ Parton Distribution Functions via Large-Momentum Effective Theory

Problem Statement
Parton distribution functions (PDFs) at large momentum fraction $x$ are critical for probing physics beyond the Standard Model at colliders and understanding non-perturbative QCD phenomena such as color confinement and spin structure. However, these regions are poorly constrained by experimental data due to large uncertainties. Theoretically, the extraction of large- $x$ PDFs is complicated by "threshold logarithms" of the form $\ln^n(1-x)$ , which arise from additional scales $(1-x)Q^2$ near the threshold. These logarithms spoil the convergence of fixed-order perturbation theory and require resummation. While threshold resummation has been established for Deep Inelastic Scattering (DIS) and Drell-Yan processes using Soft-Collinear Effective Theory (SCET), applying these techniques to lattice QCD calculations of PDFs via Large-Momentum Effective Theory (LaMET) remains a challenge. Specifically, the matching kernel between lattice quasi-PDFs and physical PDFs contains threshold singularities that must be systematically resummed to enable precise lattice determinations.

Methodology
The authors investigate the perturbative matching kernel $C(\xi, xP^z/\mu)$ connecting the quasi-PDF $\tilde{f}$ to the physical PDF $f$ in the threshold limit $\xi \to 1$ (where $\xi = y/x$ ). The core methodology involves:

Factorization in the Threshold Limit: The authors propose a factorization formula for the matching kernel in the limit $\xi \to 1$ . They demonstrate that the kernel factorizes into a hard function $H$ and a jet function $J_f$ :
$C(\xi, xP^z/\mu) \big|_{\xi \to 1} = H\left(\frac{4x^2 P_z^2}{\mu^2}\right) |xP^z| J_f\left(\frac{(1-\xi)xP^z}{\mu}, \frac{4x^2 P_z^2}{\mu^2}\right)$
This structure mirrors DIS factorization but involves a "space-like heavy-quark jet function" due to the static gauge link in the quasi-PDF definition.
Space-Time Picture and Jet Function Definition: By analyzing the space-time picture of the threshold limit, the authors identify a separation of scales. The hard exchanges occur at scale $\zeta_z = 4x^2 P_z^2$ , while soft/semi-hard exchanges between the collinear quark and the static gauge link occur at the scale $\mu_i = |1-\xi|xP^z$ . They define a space-like jet function $\tilde{J}$ using light-like and space-like gauge links.
Analytic Continuation: A key technical step involves relating the newly defined space-like jet function to a previously calculated time-like jet function (from Ref. [70]). The authors prove that the space-like jet function can be obtained from the time-like version via analytic continuation ( $t \to -i|z|$ ) to all orders in perturbation theory. This allows the utilization of existing two-loop results for the time-like case.
Renormalization Group (RGE) Resummation: The authors derive the Renormalization Group Equations (RGEs) for the hard kernel and the jet function. They utilize the cusp anomalous dimension $\Gamma_{\text{cusp}}$ and other process-dependent anomalous dimensions to resum large logarithms. The evolution involves running from the hard scale $\zeta_z$ and the jet scale $\mu_i$ to a common renormalization scale $\mu$ .
NNLO Verification: The formalism is tested against the known Next-to-Next-to-Leading Order (NNLO) calculation of the quasi-PDF matching kernel (Refs. [68, 69]). By comparing the factorized prediction with the exact two-loop result, the authors extract unknown constant terms in the heavy-light Sudakov form factor.

Key Contributions and Results

Factorization Formula: The paper establishes a rigorous factorization formula for the quasi-PDF matching kernel in the threshold limit, separating the hard physics (Sudakov form factors) from the semi-hard jet physics.
Space-Like Jet Function: The authors define and calculate the space-like heavy-quark jet function. They demonstrate its relationship to the time-like jet function via analytic continuation, providing a proof based on parametric-space representation and Wightman functions.
Universal Heavy-Light Sudakov Form Factor: The hard kernel is identified as a pair of universal heavy-light Sudakov form factors. The authors calculate this object up to NNLO, extracting the previously unknown constant terms ( $c_a$ and $c_H$ ). These constants are shown to be universal, appearing also in the large-momentum expansion of quasi-transverse-momentum-dependent PDFs (quasi-TMDPDFs) and quasi-light-cone wave functions (quasi-LFWFs).
RGE Resummation: The authors derive the fully resummed form of the matching kernel in momentum space. They show that the RGE of the matching kernel in the threshold limit precisely matches the threshold limit of the DGLAP evolution kernel, ensuring consistency with standard QCD evolution.
NNLO Agreement: The factorization-based prediction for the threshold limit of the matching kernel is shown to agree completely with the exact NNLO calculation. This confirms the correctness of the derived anomalous dimensions and the factorization structure.

Significance and Claims
The paper claims to pave the way for direct, high-precision calculations of large- $x$ PDFs in lattice QCD. By providing a systematic resummation of threshold logarithms, the work addresses a major theoretical bottleneck that has limited the accuracy of lattice PDF extractions in the large- $x$ region. The authors emphasize that their results reveal a high degree of universality across different large-momentum expansion applications (quasi-PDFs, quasi-TMDPDFs, and quasi-LFWFs).

The authors state that the derived resummed matching kernel will be applied in future work to the numerical determination of pion and proton PDFs. They note that during the writing of the paper, a separate calculation of the two-loop heavy-light Sudakov form factor in the context of quasi-TMDPDFs (Ref. [101]) was published, which agrees with their findings, further validating the universality of the heavy-light Sudakov form factor. The work does not propose new experimental setups but provides the necessary theoretical infrastructure to interpret future lattice data in the large- $x$ regime.

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