QCD, while elegantly describing how quarks and gluons interact via the strong force, poses extremely difficult computational problems due to its inherent nonlinear and nonperturbative nature, particularly in the low-energy regime relevant for hadron structure. A powerful computational approach to this problem is lattice QCD, which discretizes spacetime into a finite four-dimensional grid—a lattice—allowing the calculation of QCD observables from the bottom up. However, lattice QCD conventionally operates in Euclidean spacetime where time is treated as a spatial dimension, in contrast to the Minkowski spacetime needed for light-cone physics where PDFs are naturally defined. This fundamental mismatch renders direct calculation of PDFs using lattice methods highly nontrivial.

To surmount this obstacle, theorists have innovated alternative techniques that translate lattice computations into meaningful information about PDFs. One prominent method is short-distance expansion (SDE), where correlations between partons at very short Euclidean distances are examined. SDE exploits the operator product expansion and the known behaviors of QCD at short distances to infer PDFs via moment calculations and global constraints. Although SDE has been a staple method offering insights into moments of PDFs, it has limitations in resolving the full x-dependence, especially outside the low moment region.

Another groundbreaking approach that has emerged is Large-Momentum Effective Theory (LaMET), which Xiangdong Ji of the University of Maryland first pioneered. LaMET enables lattice QCD calculations at large but finite hadron momenta, bridging the gap between Euclidean lattice computations and Minkowski light-cone physics. The key innovation lies in using boosted hadron states on the lattice, allowing quasi-distributions—lattice calculable objects in Euclidean space—to be matched perturbatively to true PDFs defined in light-cone coordinates. In the infinite momentum limit, LaMET quasi-PDFs converge to standard PDFs, while at finite momenta, sophisticated matching procedures correct approximations to produce explicit x-dependent distributions.

In a landmark study published in the journal Research on May 28, 2025, Distinguished University Professor Xiangdong Ji presented a thorough analysis comparing LaMET and SDE methodologies. His work highlights their complementary strengths and how a synergy between these approaches can significantly enhance the precision and reliability of lattice QCD-derived PDFs. “Both LaMET and SDE are widely studied approaches for calculating PDFs and have their strengths in different aspects,” Ji explains. By integrating global constraints from SDE and x-dependent precision from LaMET, researchers can develop a more holistic and accurate picture of parton dynamics within hadrons.

One of the major advantages of LaMET is its direct access to the x-dependence of PDFs over a wide intermediate momentum fraction range, typically spanning roughly from 0.1 to 0.7. This intermediate region is particularly relevant for many high-energy processes studied at particle colliders. However, at very small x (corresponding to partons carrying tiny fractions of momentum) and very large x (carrying near-total momentum), LaMET becomes less effective because the requisite hadron boost becomes unrealistically large, posing severe computational difficulties. Here, the SDE method complements by providing global moment constraints that effectively guide and stabilize the extrapolation of LaMET-calculated PDFs in these difficult-to-reach regions.

In practical terms, Professor Ji applied this combined framework to calculate the valence quark PDFs of pions, a system of fundamental interest given their role in the strong interaction and as probes in various experiments. The calculations utilized high-precision lattice QCD computations under the LaMET formalism, supplemented by phenomenological modeling aided with SDE global constraints. Crucially, these theoretical predictions matched remarkably well with experimental data from collaborations at Argonne and Brookhaven National Laboratories, validating the hybrid approach’s effectiveness.

The success of this research paves the way for generating state-of-the-art lattice QCD PDFs that can be used to make powerful predictions for high-energy particle collisions, such as those at the Large Hadron Collider. Enhanced precision in PDFs reduces uncertainties in theoretical models and may help reveal subtle signatures of new physics or novel hadronic phenomena previously obscured by theoretical limitations. This marks a critical stride toward first-principle, nonperturbative understanding of hadron structure—a longstanding quest in nuclear and particle physics.

Beyond immediate practical impacts, Ji’s study underscores a broader scientific narrative: the importance of methodological innovation and cross-validation in theoretical physics. The convergence of LaMET and SDE exemplifies how diverse frameworks can complement and reinforce each other, overcoming intrinsic limitations and deepening the insights into one of nature’s most fundamental forces, the strong interaction.

The implications also extend to other subfields. PDFs are indispensable in interpreting experimental results not only for protons and pions but also for more exotic hadrons and nuclei, thereby influencing research areas spanning from astrophysics to cosmology, where strong interaction physics plays a role in stellar evolution and the early universe.

Moreover, the refinement of lattice QCD techniques empowered by large-scale computational resources, combined with the new theoretical frameworks, heralds a new era where ab initio calculations of hadronic properties move from aspiration to reality. This progress will steadily reduce reliance on phenomenological fits, enabling truly predictive theoretical physics grounded in the fundamental axioms of QCD.

Professor Ji’s work exemplifies the power of theoretical ingenuity coupled with computational advancements, driving particle physics forward. His pioneering contributions to LaMET and his synthesis of complementary techniques represent a milestone in decoding the quark-gluon world—an achievement that will resonate throughout the physics community for years to come.

Subject of Research:
Ab initio calculations in lattice Quantum Chromodynamics focused on parton distribution functions within hadrons.

Article Title:
Ab Initio Lattice Quantum Chromodynamics Calculations of Parton Physics in the Proton: Large-Momentum Effective Theory versus Short-Distance Expansion

News Publication Date:
28-May-2025

Web References:
DOI: 10.34133/research.0695

Image Credits:
Professor Xiangdong Ji, University of Maryland, College Park, USA

Keywords

Lattice QCD, Parton Distribution Functions, Large-Momentum Effective Theory, Short-Distance Expansion, Quantum Chromodynamics, Hadron Structure, Pion Valence PDFs, Ab Initio Calculations, High-Energy Physics, Theoretical Particle Physics

The research team employed a state-of-the-art methodology known as lattice quantum chromodynamics. This advanced computational technique involves dividing space and time into an extremely fine grid, allowing for an intricate simulation of how the strong force, the fundamental interaction that binds quarks into protons and neutrons, operates within the confines of a proton. This unprecedented approach enabled the scientists to capture the dynamics of these powerful forces in a manner that was previously unattainable.

In their study, the researchers reveal the potential implications of their findings, which might represent the most detailed force field map of nature generated to date. The published results in the renowned journal Physical Review Letters detail how quarks respond under the influence of high-energy photons, ultimately unveiling the forces that dictate proton behavior during high-energy collisions. This has significant implications for various domains of particle physics and enhances our understanding of subatomic interactions.

The calculations were led by University of Adelaide’s PhD student Joshua Crawford, who, alongside the collaborative team, sought to analyze and interpret the data collected. Crawford emphasized the sheer magnitude of forces at play within protons, stating that these forces can reach levels up to half a million Newtons. To put this power into perspective, such force is comparable to the weight of ten elephants confined within a space that is significantly smaller than an atomic nucleus.

As the researchers continue to refine their understanding of these forces, they anticipate that their findings may yield valuable insights into high-energy physics experiments, particularly those conducted at the Large Hadron Collider. The LHC, as the largest and most powerful particle accelerator in the world, serves as a key facility for physicists to test the predictions made by various theoretical frameworks concerning particle physics. The connection between the findings regarding proton forces and experiments at the LHC underscores the potential for this knowledge to influence future research and technology.

This revelation that protons exhibit such immense internal forces is noteworthy, as it aids physicists in understanding why protons behave as they do under extreme conditions, such as those experienced during high-energy collisions. By mapping these forces, scientists are not only elucidating the nature of protons but also bridging gaps that existed between theoretical predictions and experimental validation. This relationship between theory and experiment is essential for advancing the field of particle physics and refining our comprehension of subatomic matter.

Moreover, the implications of this work extend beyond just theoretical physics. The new understanding of proton structure has potential applications in medicine, particularly in proton therapy, an innovative treatment that utilizes high-energy protons to precisely target tumors while minimizing damage to surrounding healthy tissue. The advancements in mapping internal forces could pave the way for more effective and targeted therapies in combating cancer, showcasing how fundamental research can have profound real-world implications.

Crawford likened their scientific endeavor to historical advancements in our understanding of light, drawing parallels between the present research and the foundational work done by previous generations of scientists. Just as Edison and others laid the groundwork for transformative technologies related to light, the current exploration of proton dynamics may similarly lead to breakthroughs that revolutionize applications across science and medicine. The emphasis on revealing these invisible forces within protons brings a fresh perspective to proton therapy and other technologies that could benefit from enhanced knowledge of particle interactions.

The research team’s ongoing efforts to unravel the complexities of proton structure signify a commitment to enhancing our collective knowledge of particle physics. Their work not only showcases the capabilities of advanced computational techniques but also highlights the collaborative nature of modern scientific inquiry. As researchers continue to investigate the forces and interactions that govern the behavior of protons, it becomes increasingly clear that the insights gleaned from these studies will contribute to a deeper understanding of the universe’s fundamental building blocks.

In conclusion, the revelations regarding the forces acting within protons mark a significant stride in the exploration of fundamental physics. The intricate dynamics unveiled by this research enhance our understanding of the strong force and its implications for subatomic particles, while also opening doors for potential applications in various scientific fields. With such remarkable insights into the inner workings of protons, the future of particle physics and its applications appears considerably brighter.

Subject of Research: Not applicable
Article Title: Transverse Force Distributions in the Proton from Lattice QCD
News Publication Date: 19-Feb-2025
Web References: Physical Review Letters
References: Not applicable
Image Credits: Joshua Crawford / University of Adelaide

Keywords

Proton structure, quarks, strong force, lattice quantum chromodynamics, particle physics, Large Hadron Collider, proton therapy, subatomic interactions, computational techniques, fundamental forces.