Leading-Neutron Electroproduction at HERA and the EIC: Sullivan Process, Target Fragmentation, and Pion PDFs

Deep within the heart of matter, protons and neutrons are not solid, indivisible spheres but rather bustling cities of even smaller particles called quarks and gluons. These constituents are held together by the strong force, the most powerful interaction in nature. While we understand how protons are built, the neutron's close relative, the pion, remains a more elusive puzzle. The pion is the lightest particle made of quarks, acting as the messenger that carries the strong force between protons and neutrons, much like a ball thrown back and forth keeps two people connected. Because pions are so light, they play a dominant role in how atomic nuclei stick together over long distances. However, scientists cannot simply place a pion on a table and shoot particles at it; pions are too unstable to exist as stationary targets. To understand the internal structure of the pion, researchers must instead look for ways to create them briefly during high-energy collisions and study how they behave before they vanish.

At the HERA particle collider in Germany, which operated for many years, scientists collided electrons with protons at incredibly high speeds. In these collisions, the protons sometimes broke apart, sending a neutron flying forward in the same direction as the original proton beam. This process, known as leading-neutron production, offered a unique window into the pion's secrets. The theory suggests that before the collision, the proton can briefly fluctuate into a neutron and a virtual pion. If the electron hits this fleeting pion instead of the neutron, the pion acts as a temporary target, revealing its own internal map of quarks and gluons. This method, called the Sullivan process, has been used to map the pion's structure, but previous studies had to ignore a large portion of the data to avoid confusion from other messy collision effects.

A new study by researchers at Academia Sinica in Taiwan revisits these old data with a fresh approach, asking whether the messy parts of the collision could be understood and included rather than discarded. The team used a sophisticated computer program called Pythia to simulate the "target fragmentation" aspect of the collisions. This is the part where the proton breaks apart in a chaotic way that is not related to the pion exchange. By combining this simulation of the chaotic breakup with the theoretical model of the pion exchange, the researchers found they could reproduce the full range of experimental data from HERA without needing to artificially adjust the numbers. This is a significant step forward because it means scientists can now use a much wider variety of data points, including those where the neutron carries less of the original proton's energy. This expanded view allows for a more precise map of the pion, particularly in the region where the quarks carry very small fractions of the pion's momentum, a territory that was previously difficult to explore.

The researchers also tested how sensitive their results were to the specific mathematical rules used to describe how the pion and neutron interact at the moment of their creation. They found that the data from the HERA experiments are indeed sensitive to these details, meaning that future studies must be very careful in how they model these interactions. The study highlights that while the pion-exchange model works very well when the neutron keeps most of the proton's forward momentum, the chaotic breakup of the proton becomes the dominant factor when the neutron carries less energy. By successfully modeling both effects together, the team demonstrated that the entire dataset can be used to refine our understanding of the pion.

Looking ahead, the researchers projected how these measurements could improve with the upcoming Electron-Ion Collider in the United States. This future machine will be far more powerful than HERA, offering a much higher rate of collisions and better detectors to catch the forward-moving particles. The study suggests that by carefully choosing the energy settings of the new collider, scientists can isolate the pion-exchange signal even more effectively, suppressing the chaotic background noise. This will allow for a multidimensional, high-precision measurement of the pion's structure across a vast range of conditions. The ultimate goal is to finally pin down the distribution of sea quarks and gluons inside the pion, completing a crucial piece of the puzzle of how the strong force builds the visible universe.