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Effects of threshold resummation for large- $x$ PDF in large momentum effective theory

This paper investigates large- $x$ parton distribution functions within large momentum effective theory by factorizing the matching coefficient into hard and soft components to resum threshold double logarithms and leading renormalons, demonstrating that perturbative matching remains reliable when both active and spectator quark momenta significantly exceed the QCD scale.

Original authors: Xiangdong Ji, Yizhuang Liu, Yushan Su, Rui Zhang

Published 2026-09-22

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CC BY 4.0

Original authors: Xiangdong Ji, Yizhuang Liu, Yushan Su, Rui Zhang

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

To understand the universe at its most fundamental level, physicists study the protons and neutrons that make up the visible matter around us. These particles are not solid spheres but complex, swirling clouds of even smaller constituents called quarks and gluons. To map how these pieces move and interact inside a proton, scientists rely on mathematical descriptions known as parton distribution functions. These functions act like a census, telling researchers the probability of finding a quark carrying a specific fraction of the proton's total momentum. While this census is well-understood for quarks carrying a small or moderate share of the momentum, the picture becomes blurry and uncertain for those carrying nearly all of it. This high-momentum region is critical for understanding the strong force that binds matter together and for searching for new physics beyond our current theories, yet it remains one of the most difficult areas to measure or calculate with precision.

For decades, the primary way to learn about these particles has been to smash them together in massive accelerators and analyze the debris. However, this experimental approach hits a wall when trying to study the high-momentum quarks because the data becomes scarce and noisy. In recent years, a different approach has emerged using supercomputers to simulate the laws of quantum physics from the ground up. This method, known as lattice quantum chromodynamics, allows scientists to calculate the properties of protons directly. A specific technique within this field, called large momentum effective theory, has made it possible to extract the momentum distribution of quarks from these computer simulations. The challenge, however, is that the mathematical tools used to translate the computer data into physical reality break down when looking at the highest momentum quarks. The calculations become unstable, filled with large mathematical terms that grow uncontrollably and obscure the true physical signal.

A team of researchers has now developed a refined mathematical framework to fix this instability, specifically targeting the difficult high-momentum region. They focused on the "spectator" quarks—those that are not the primary focus of a collision but are left behind, carrying the remaining momentum. In the standard calculations, the mathematical description of these leftover quarks creates large errors as their momentum approaches the total limit. The researchers realized that these errors stem from a specific type of mathematical behavior that can be separated and handled differently. By splitting the problem into a "hard" part, dealing with the main quark, and a "soft" part, dealing with the spectator, they created a new way to organize the calculation. This separation allows them to systematically sum up the troublesome terms that were previously causing the breakdown, effectively taming the mathematical chaos.

The team tested this new method using the pion, a particle similar to a proton but lighter, which serves as a clean laboratory for these calculations. They applied their improved framework to data generated from lattice simulations at specific momentum values, around 1.9 and 2.4 billion electron volts. The results showed that when the momentum of the spectator quark is sufficiently high, the new method produces stable and reliable predictions that match the expected physical behavior. However, they also found a clear limit: when the spectator momentum drops too low, approaching the scale where the strong force becomes dominant and non-perturbative, the mathematical expansion fails, and the method can no longer provide a trustworthy answer. This finding is crucial because it defines the precise boundaries of where these computer simulations can be trusted to reveal the structure of matter.

Furthermore, the researchers addressed another subtle source of error known as a renormalon, a type of mathematical divergence that can plague high-precision calculations. By incorporating a technique to resum these divergences, they demonstrated that the convergence of their calculations improves significantly. This means that the results become more consistent as more terms are added to the equation, rather than spiraling out of control. The study confirms that with these advanced resummation techniques, it is possible to achieve a high level of accuracy in describing the internal structure of hadrons, provided the momentum scales involved remain within the perturbative regime. This work does not just offer a better calculation; it provides a rigorous map of where the current theoretical tools work and where they stop, guiding future efforts to understand the deepest layers of the atomic nucleus.

Technical Summary: Effects of Threshold Resummation for Large-x PDF in Large Momentum Effective Theory

Problem Statement
Parton distribution functions (PDFs) at large momentum fraction $x$ are critical for understanding hadron structure and searching for new physics, yet they remain challenging to extract due to large experimental uncertainties and theoretical limitations. In the framework of Large Momentum Effective Theory (LaMET), where light-cone PDFs are accessed via the large momentum expansion of lattice QCD quasi-PDFs, the perturbative matching kernels contain large threshold logarithms of the form $\alpha_s^k \ln^{2k}(1-x)$ . These logarithms arise from incomplete infrared cancellation between real and virtual diagrams near the endpoint $x \to 1$ . Without resummation, these terms spoil the convergence of the perturbative series, rendering fixed-order matching unreliable in the large- $x$ region. Furthermore, the presence of linear power corrections (renormalons) in the perturbative expansion can lead to factorial divergence, limiting the precision of the extraction.

Methodology
The authors develop a systematic framework for threshold resummation within LaMET, extending previous work by incorporating a rigorous factorization of the matching kernel and addressing renormalon ambiguities.

Threshold Factorization: The matching kernel is factorized into a hard kernel, associated with the active quark momentum scale $xP^z$ , and a threshold soft function, associated with the spectator momentum scale $(1-x)P^z$ . This separation is derived by analyzing the infrared modes (collinear and soft-collinear) in both light-cone and quasi-PDFs using an on-shell quark state with a finite mass $m$ as an infrared regulator. The factorization allows the threshold logarithms to be separated into scale-dependent terms that can be resummed via Renormalization Group (RG) equations.
RG Resummation: The authors solve the RG equations for the hard kernel and the soft function. The hard kernel evolves from a hard scale $\mu_h \sim 2|x|P^z$ , while the soft function evolves from a semi-hard scale $\mu_i \sim 2|1-x|P^z$ . This evolution resums the double logarithms $\alpha_s^k \ln^{2k}(1-x)$ . The formalism is adapted to the hybrid renormalization scheme, which is standard for lattice QCD calculations, by incorporating specific counterterms to handle ultraviolet divergences.
Leading Renormalon Resummation (LRR): To address the factorial divergence of the perturbative series (specifically the $n! \beta_0^n$ growth), the authors implement leading renormalon resummation. They identify that the threshold expansion introduces a linear power ambiguity of order $\mathcal{O}(\Lambda_{\text{QCD}}/|x|P^z)$ in the hard kernel's phase angle and a mass renormalon ambiguity of order $\mathcal{O}(\Lambda_{\text{QCD}}/|1-x|P^z)$ in the soft function. By resumming these series using Borel summation with a principal value prescription, and ensuring the same prescription is applied to the quasi-PDF, the authors aim to cancel these ambiguities and achieve linear power accuracy.

Key Contributions

IR-Free Definitions: The paper proposes IR-free definitions for the matching coefficients by including light-cone matrix elements that are scaleless and equal to 1 in dimensional regularization. This clarifies the physical separation of collinear and soft parts in the matrix elements.
Scale Choice and Validity: The authors explicitly demonstrate that perturbative matching breaks down when the spectator momentum $(1-x)P^z$ approaches $\Lambda_{\text{QCD}}$ , as the effective coupling becomes non-perturbative. Conversely, the framework remains valid when both the active ( $xP^z$ ) and spectator ( $(1-x)P^z$ ) momenta are significantly larger than $\Lambda_{\text{QCD}}$ .
Renormalon Cancellation: The work details the cancellation pattern of threshold renormalons between the leading and next-to-leading threshold expansions. It shows that the linear power corrections arising from the hard kernel phase angle and the soft function mass renormalon can be systematically removed, leading to a more convergent perturbative series.
Numerical Implementation: The formalism is applied to pion valence PDFs using lattice data from the BNL/ANL collaboration. The authors compare results with and without threshold resummation and with/without renormalon resummation.

Results

Threshold Resummation Effects: Numerical tests on pion valence PDFs show that threshold resummation significantly improves the stability of the large- $x$ region compared to fixed-order matching. However, the results indicate that for $P^z \sim 2$ GeV, the perturbative expansion breaks down when $2(1-x)P^z \lesssim 0.8$ GeV, highlighting the necessity of the spectator momentum being a perturbative scale.
Renormalon Improvement: The inclusion of leading renormalon resummation (LRR) improves the perturbative convergence. Comparisons between NLO+RGR+TR and NNLO+RGR+TR results show that without renormalon control, there are significant deviations in the moderate $x$ region ( $0.3 < x < 0.7$ ). With LRR, the perturbative convergence is enhanced, suggesting that the large coefficients in the fixed-order expansion are largely driven by renormalon effects.
Scale Dependence: The analysis of scale variations confirms that uncertainties are well-controlled in the region where both $xP^z$ and $(1-x)P^z$ are perturbative, but grow rapidly as the spectator momentum enters the non-perturbative regime.

Significance
The paper claims that this formalism provides a reliable and controllable method for calculating large- $x$ PDFs from lattice QCD data. By combining threshold resummation with leading renormalon resummation, the authors achieve leading (linear) power accuracy, effectively separating perturbative and non-perturbative physics in the threshold region. This approach addresses the breakdown of perturbative matching at large $x$ and offers a pathway to reduce theoretical uncertainties in the extraction of PDFs, which are essential for precision tests of the Standard Model and searches for new physics. The work complements existing factorization schemes by explicitly handling the interplay between the hard and spectator scales and the associated renormalon ambiguities.

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