At the Large Hadron Collider and other high-energy experiments, protons travel at nearly the speed of light. The constituents that take part in hard collisions are quarks and gluons, collectively known as partons. Each carries a fraction of its parent proton’s momentum and can interact with a parton from the other proton. Predicting how often a particular reaction occurs requires knowing how likely it is to find each type of parton with a given momentum fraction. Parton distribution functions encode this information, describing the proton’s one-dimensional momentum structure along its direction of motion.

These distributions are mainly extracted from experimental data. Calculating them directly from quantum chromodynamics (QCD), the theory of the strong interaction, has remained difficult. Lattice QCD divides space and time into a grid so that the strong interactions inside a proton can be simulated on a computer. Parton distributions, however, involve light-cone correlations: the spatial separation between two points equals the distance light travels during their time separation. Such correlations involve both space and real time and are not directly accessible to conventional lattice calculations.

Large-momentum effective theory (LaMET), proposed by Xiangdong Ji in 2013, offers a way to make the calculation possible. Researchers first study a proton with a large but finite momentum on the lattice, calculating correlations between quarks or gluons at different spatial positions at the same time. In the large-momentum limit, these equal-time spatial correlations contain the same long-distance structural information as light-cone correlations. Their short-distance differences are accounted for by a calculable theoretical relation known as matching. This connection allows parton distributions to be extracted from spatial correlations that can be computed on the lattice.

Within LaMET, different types of spatial correlations can be used to study the same parton distribution. In this work, the team examined current-current correlators between local quark currents at two positions inside the proton. These can be understood as the proton’s correlated response to two local probes, containing information about quark momentum distributions. The researchers derived a one-loop formula for extracting parton distributions from these correlators and used lattice QCD data to calculate the proton’s isovector parton distribution.

An advantage of this approach is that it does not require a Wilson line between the two probe positions. It therefore avoids the additional divergences associated with this connection, simplifying the treatment of the lattice results.

This exploratory study demonstrates the feasibility of extracting parton distributions from current-current correlators. Calculating the same physical quantity using different correlation functions helps cross-check results and better control computational uncertainties, providing a basis for further understanding the proton’s internal structure.

Jia-Lu Zhang, a postdoctoral researcher at the Tsung-Dao Lee Institute, Shanghai Jiao Tong University, is the first author; Rui Zhang of the Massachusetts Institute of Technology and Christian Zimmermann of the University of Kentucky and Lawrence Berkeley National Laboratory are the corresponding authors; Professor Xiangdong Ji of Shanghai Jiao Tong University and Andreas Schafer of the University of Regensburg are coauthors.

The authors gratefully acknowledge support from the National Natural Science Foundation of China, the State Key Laboratory of Dark Matter Physics, the Thomas and Linda Lau Family Foundation, and the T. D. Lee Scholarship.