Understanding the structure of hadrons, such as protons and neutrons, is currently one of the most important goals for researchers studying nuclear and particle physics. Hadrons consist of quarks and gluons, together called partons. The behavior of partons inside hadrons is described by mathematical tools called parton distribution functions (PDFs), which tell us the probability of finding partons carrying a fraction x of the hadron's total momentum. Until recently, PDFs were mainly determined by fitting data obtained from decades of high-energy experiments, a process known as phenomenological modeling. Recently, the interest in calculating PDFs from first principal calculations has gained considerable momentum.
Partons interact via the strong force, one of the four fundamental forces of nature, which is governed by quantum chromodynamics (QCD), known to be notoriously difficult to solve. Lattice QCD is a computational approach that makes QCD tractable, by simulating spacetime into a finite grid or lattice. Even with lattice QCD, calculating PDFs from first principles remains challenging. PDFs are naturally defined using light-cone coordinates, which represent all possible paths where light can travel in space-time. In these coordinates, time always moves forward. In contrast, lattice QCD works in Euclidean spacetime, where time behaves like a fourth spatial dimension. This makes it difficult to directly calculate PDFs.
To circumvent this problem, researchers have developed alternative methods. One such method is short-distance expansion (SDE), in which PDFs are calculated by analyzing parton correlations over very short Euclidean distances using standard QCD. A more recent and powerful method called Large-Momentum Effective Theory (LaMET), allows calculation of PDFs over a broad x-range. While these two methods are equivalent at infinite momentum, they have major differences at finite momentum, where actual calculations are performed.
Now, in a recent study, Distinguished University Professor Xiangdong Ji from the University of Maryland, College Park, USA, who first pioneered LaMET, presented a detailed comparison of LaMET and SDE. "Both LaMET and SDE are widely studied approaches for calculating PDFs and have their strengths in different aspects," explains Prof. Ji. "In this study, I discuss how these approaches contrast and how they complement each other to give us a more complete understanding of partons." The study was published in Volume 8 of the journal Research on May 28, 2025.
In LaMET, approximations of PDFs are calculated using lattice QCD at a large but finite hadron momentum. These approximations can be largely corrected through a process called matching, which produces increasingly accurate representations of true light cone PDFs as momentum increases. This method allows the precise study of local x-dependence of PDFs over a broad x-range. However, LaMET becomes less reliable at very small and very large x values, requiring extremely large momentum, which makes computations inaccessible at present.
This is where SDE can help. It provides global constraints on PDFs. Prof. Ji suggests that SDE can be used to bridge the gaps in LaMET analyses. Particularly, LaMET can be used to calculate PDFs for the middle x-region and the global constraints obtained from SDE, together with theoretical and phenomenological modeling, can be used for end limits of very small and very large x.
He demonstrated this approach by calculating the valence PDFs of pions. The results were successfully validated using high-precision data from the Argonne National Laboratory (ANL)/Brookhaven National Laboratory (BNL) collaboration. The LaMET calculations generated reasonable predictions in the x-range of 0.1 to 0.7. Beyond this range, LaMET's reliability decreased, while phenomenological modeling using global SDE constraints improved predictions, especially for large-x.
"This approach will prove beneficial for generating state-of-the-art lattice QCD PDFs," says Prof. Ji. "These can be used to make predictions for high-energy particle collisions at facilities like the Large Hadron Collider, potentially leading to the discovery of new particles and deepening our understanding of partons, and by extension, hadrons."
Overall, this study marks a significant step forward in using lattice QCD for calculating PDFs and uncovering new insights into parton physics in hadrons.
About the University of Maryland, College Park, USA
Founded in 1856, University of Maryland, College Park is the flagship institution of the state of Maryland. It is the largest university in Maryland and the Washington metropolitan area. brings together world-class scientists and scholars in an unbeatable location near the nation's capital to discover and innovate. Its proximity to Washington, D.C. has resulted in many partnerships with the federal government of the USA. It fosters a culture of environmental sustainability, social innovation and leadership, through teaching, research, service and campus operations. The university strives to provide exceptional opportunities to learn, innovate, and serve the public good.
About Research by Science Partner Journal
Launched in 2018, Research is the first journal in the Science Partner Journal (SPJ) program. Research is published by the American Association for the Advancement of Science (AAAS) in association with Science and Technology Review Publishing House. Research publishes fundamental research in the life and physical sciences as well as important findings or issues in engineering and applied science. The journal publishes original research articles, reviews, perspectives, and editorials. IF=10.7, Citescore=13.3.
Sources: https://doi.org/10.34133/research.0695
