Computability of GPDs near $x=\pm\xi$ in Lattice QCD

Inside every proton, the building block of the visible universe, quarks and gluons are not scattered randomly. They are organized in a complex, three-dimensional dance of momentum and position that defines the proton's mass, spin, and the forces holding it together. To map this internal landscape, physicists rely on mathematical objects called generalized parton distributions. Think of these as a detailed blueprint that tells us how likely it is to find a specific quark carrying a certain amount of forward momentum while sitting at a specific spot inside the proton. While scientists have long been able to measure these distributions in high-energy collisions, a complete picture has remained elusive because the most critical regions of the map are incredibly difficult to calculate from first principles. For years, a specific zone near the edges of the momentum range was considered a blind spot, a place where the standard tools of calculation were believed to break down, leaving a gap in our understanding of how the proton's interior transitions from one state to another.

A new study by researchers at the University of Maryland, Shanghai Jiao Tong University, Jagiellonian University, and the Massachusetts Institute of Technology has removed this blind spot. The team demonstrated that the most difficult parts of the proton's internal map are actually accessible to calculation, overturning a long-held assumption that these regions were too messy to compute. By revisiting existing data from supercomputer simulations with a more careful mathematical approach, they showed that the generalized parton distributions remain smooth and continuous at the critical transition points, even though their rate of change shifts abruptly. This discovery opens the door to the first-ever direct predictions for the proton's structure in a region that connects the behavior of quarks moving forward with those moving backward, filling a gap that has persisted for decades.

The challenge the researchers tackled stems from how we try to see inside the proton. In the real world, we probe these structures by smashing particles together at near light speed, a process that reveals the proton's inner workings through the debris of the collision. However, to understand the fundamental laws governing these interactions, physicists must also be able to calculate these distributions directly from the theory of the strong force, known as quantum chromodynamics. Because the proton is a quantum object, these calculations are often performed on a grid of space and time, a method called lattice quantum chromodynamics. The problem arises when the momentum of a quark inside the proton approaches zero relative to the overall motion of the proton. In this specific scenario, the mathematical tools used to translate the grid-based calculations into real-world predictions were thought to fail. The prevailing wisdom suggested that as the momentum difference shrank, the calculations would become unreliable, effectively cutting off a significant portion of the proton's internal map from theoretical study.

The researchers realized that this limitation was an artifact of how the calculations were being handled, not a fundamental barrier in nature. They drew a parallel to a different process called deeply virtual Compton scattering, where a photon bounces off a proton. In that process, physicists have long known that the interaction remains calculable even when the momentum transfer is small, because the soft, low-energy physics that usually causes trouble is naturally suppressed. The team applied this same logic to the lattice calculations. They showed that the problematic soft contributions near the critical momentum points are indeed suppressed, meaning the standard tools can work there if used correctly. This insight meant that the "forbidden" zone was actually reachable, provided the calculations were performed with a specific, more rigorous mathematical technique that respects the continuity of the physical quantities involved.

To prove this, the team went back to data generated by previous experiments on supercomputers. These earlier studies had attempted to calculate the proton's structure at non-zero momentum transfer but had reported strange, jagged discontinuities at the critical points, which seemed to confirm the idea that the calculations were failing. The new team reanalyzed this data, applying a refined matching procedure that carefully handled the mathematical singularities. Instead of accepting the jagged results, they smoothed out the calculation by treating the critical points as limits approached from the surrounding regions. When they did this, the jagged edges vanished. The resulting maps showed that the generalized parton distributions are indeed continuous at the transition points, just as physical intuition and other theoretical models had suggested they should be. The only feature that remained discontinuous was the slope of the curve, a sharp change in direction that is a known signature of the underlying quantum mechanics.

This finding is significant because it validates the use of these powerful calculation methods across the entire range of the proton's internal structure. The researchers confirmed that the calculations are reliable as long as the momentum of the proton is high enough, a condition that is met in current and future experiments. The results provide the first theoretical predictions for the "distribution amplitude-like" region, a zone that acts as a bridge between the behavior of quarks moving with the proton and those moving against it. Before this work, this region was a theoretical void, with little information available from experiments or calculations. Now, scientists have a clear, first-principles prediction for how the proton's interior behaves in this transition zone, offering a new benchmark for future experiments at facilities like the Electron-Ion Collider.

The study also clarified the boundaries of where these calculations can be trusted. While the critical transition points are now accessible, the researchers noted that the calculations still require the proton's momentum to be sufficiently large to avoid infrared sensitivity, a technical term for the influence of very low-energy fluctuations. They identified that the method works well as long as at least one of the relevant momentum scales remains large compared to the energy scale of the strong force. This defines a safe zone for the calculations, ensuring that future predictions will be robust. By resolving the puzzle of the discontinuities and expanding the computable region, the team has provided a more complete and accurate picture of the proton, turning a theoretical blind spot into a well-charted territory.