Calculations of charge distribution in mesons establish benchmarks for experimental tests and support the commonly used ‘factorization’ method for visualizing the fundamental components of matter.
Theoretical physicists studying nuclear physics at the U.S. Department of Energy’s (DOE) Brookhaven National Laboratory have shown that advanced calculations performed on supercomputers can reliably predict how electric charges are distributed in mesons, which are particles consisting of a quark and an antiquark. Researchers are eager to deepen their understanding of mesons and other quark-based particles—collectively referred to as hadrons—through high-energy experiments at the upcoming Electron-Ion Collider (EIC), currently under construction at Brookhaven Lab. Insights gained from these predictions and the experiments at the EIC are expected to illuminate how quarks and the corresponding gluons—particles that bind them—create the mass and structure of almost all visible matter.
“The main scientific goal of the EIC is to uncover how the characteristics of hadrons, which include mesons as well as the more familiar protons and neutrons, stem from the distributions of their constituent quarks and gluons,” explained Swagato Mukherjee, a theorist at Brookhaven Lab who led the study. The pion, being the lightest meson, is pivotal in the nuclear strong force that binds protons and neutrons within atomic nuclei. By exploring the enigmatic nature of pions, protons, and other hadrons, the EIC aims to help scientists understand the interactions that hold atomic structures together.
The new predictions, recently reported in Physical Review Letters, agree closely with experimental data from lower-energy experiments conducted at DOE’s Thomas Jefferson National Accelerator Facility (Jefferson Lab), which is a collaborative partner in the EIC project, and they also extend into the high-energy territory anticipated for the new facility’s experiments. These projections hold significant value as they will serve as a reference point when EIC experiments commence in the early 2030s.
However, the findings extend beyond merely setting expectations for a specific EIC measurement. As explained in the publication, the researchers utilized their predictions along with additional independent supercomputer calculations to substantiate a widely accepted methodological framework for analyzing particle properties. This framework, known as factorization, simplifies complex physical processes into two separate parts or factors. Confirming the validity of factorization will allow for a broader range of EIC predictions and enhance the accuracy of interpretations regarding experimental observations.
Diving into hadrons
To explore the inner composition of hadrons, the EIC will conduct high-energy collisions using electrons and protons or atomic nuclei. During these interactions, virtual photons—akin to light particles—are emitted from electrons, aiding in uncovering the characteristics of the hadrons—functioning much like a microscope for observing the fundamental components of matter.
Experiments at the EIC will yield precise measurements of different physical scattering events. To convert these exact measurements into detailed images of the matter’s foundational constituents within hadrons, researchers will rely on factorization. This theoretical framework allows for the experimental measurements—like the distribution of electric charges in mesons—to be divided into two factors. This division enables scientists to apply knowledge from two processes to deduce information regarding a third process.
Consider a mathematical expression where X = Y × Z. Here, X represents the experimental measurement, which is constituted of two factors, Y and Z. One factor, Y, represents the arrangement of quarks and gluons inside the hadron, while the other factor, Z, denotes how those quarks and gluons interact with the high-energy virtual photon emitted by the colliding electron.
Calculating the quark/gluon distributions poses significant challenges due to the strong interactions among quarks and gluons within a hadron. These calculations involve billions of variables and are governed by the theory of strong interactions known as quantum chromodynamics (QCD). Typically, solving QCD equations requires simulations on a hypothetical space-time lattice using powerful supercomputers.
In contrast, the interactions between quarks and gluons with the virtual photon are relatively weaker, allowing theorists to derive those values using simpler “pen-on-paper” methods. They can then merge these straightforward calculations with the experimental measurements—both observed and anticipated—and utilize the mathematical relationships between the factors to solve the equation and discern the distributions of quarks and gluons within hadrons.
“But is this method really effective—breaking down one phenomenon into two distinct factors?” queried Qi Shi, a visiting graduate student in the Nuclear Theory Group at Brookhaven Lab. “We had to confirm its validity.”
To validate the method, the researchers approached factorization in reverse. “We flipped the process,” Shi noted.
Shi and Xiang Gao, a postdoctoral researcher in the group, utilized supercomputers and space-time lattice simulations to compute the distributions of quark-antiquark pairs in the mesons (Y, in the previously mentioned equation). They then applied simpler “pen-on-paper” calculations for the quark/gluon interactions with photons (Z) and calculated the predicted value for the experimental measurement (X)—which is the charge distribution within mesons.
Ultimately, the scientists compared these new predictions with those obtained from an independent supercomputer calculation that paralleled the Jefferson Lab measurements at lower energies. By evaluating the two sets of predictions—one derived through factorization and the other computed separately through the lattice simulation—they were able to test the validity of the factorization approach in addressing such issues.
The reverse factorization calculations coincided perfectly with their supercomputer-derived predictions.
“In this instance, we were able to compute everything entirely through lattice calculations,” Shi stated. “We selected this specific scenario because we could compute both sides of the equation independently, demonstrating that factorization is effective.”
Going forward, scientists can employ factorization to predict and analyze additional EIC observables, even if one component cannot be calculated directly.
“This work demonstrates that the factorization method is sound,” commented Peter Petreczky, the group leader and co-author of the study. “Researchers can now leverage forthcoming EIC data in combination with factorization to infer the more intricate quark and gluon distributions in hadrons that might not be directly computable, even with the most advanced computational resources and techniques.”
This research received funding from the DOE Office of Science (NP) and utilized computational resources at the Argonne Leadership Computing Facility, the Oak Ridge Leadership Computing Facility, and the National Energy Research Scientific Computing Center—all DOE Office of Science user facilities at DOE’s Argonne National Laboratory, Oak Ridge National Laboratory, and Lawrence Berkeley National Laboratory, respectively. Portions of the computations were also performed on facilities belonging to the US Lattice Quantum Chromodynamics (USQCD) Collaboration.