Parton distribution functions of the proton in a hot isospin-asymmetric nuclear medium and finite nuclei

Inside every atom, the nucleus is a crowded room of protons and neutrons, bound together by forces so strong they defy our everyday intuition. But these protons and neutrons are not solid, indivisible marbles; they are complex systems made of even smaller particles called quarks. In a free proton, floating alone in space, these quarks move with a certain rhythm and distribution, carrying specific amounts of energy and spin. However, when a proton is trapped inside a nucleus, squeezed by its neighbors, the environment changes. The question that has puzzled physicists for decades is whether this crowded environment alters the internal structure of the proton itself. Does the proton's "personality" change when it is bound, or does it remain exactly the same as it is in isolation? This is the core of the mystery known as the EMC effect, named after the European Muon Collaboration that first noticed a strange discrepancy in how electrons scatter off iron compared to lighter elements.

To answer this, a team of researchers from India set out to map the internal landscape of a proton under extreme conditions. They focused on three specific ways the quarks inside a proton can be arranged: how they are distributed by their speed, how they spin in the direction of the proton's motion, and how they spin sideways. These are the fundamental properties that define the proton's identity. The researchers wanted to see how these properties shift when the proton is placed in a "hot" and "asymmetric" nuclear medium. In this context, "hot" means the environment has a high temperature, similar to the conditions found just after the Big Bang or inside colliding heavy ions, while "asymmetric" refers to a mixture where there are significantly more neutrons than protons, a common state in the dense cores of heavy atomic nuclei.

The team used a sophisticated theoretical framework to simulate this environment. Instead of trying to solve the impossible math of every single interaction, they modeled the proton as a central quark watching over a pair of other quarks, known as a diquark, which acts as a spectator. They then calculated how the masses of these particles change when they are immersed in a dense nuclear soup. In the vacuum of empty space, these particles have fixed masses, but inside a nucleus, the intense fields around them effectively make them lighter. The researchers fed these new, lighter masses into their model to see how the quarks would rearrange themselves. They ran these simulations for a range of densities, from empty space all the way up to the saturation density found in the center of an atomic nucleus, and at temperatures of zero and a very high 0.1 GeV.

The results revealed a clear and consistent pattern. As the density of the surrounding nuclear matter increased, the quarks inside the proton began to behave differently. The most noticeable change was in the distribution of their momentum. In a free proton, the quarks tend to cluster around a specific, moderate speed. But as the proton gets squeezed, this cluster flattens out and shifts. The peak of the distribution moves toward higher speeds, meaning the quarks are, on average, carrying a larger fraction of the proton's total momentum. This shift happened for all three types of distributions the team studied, but the effect was not uniform. The changes were most dramatic for the quarks' sideways spin, known as transversity, and their forward spin, known as helicity. The distribution of quarks based purely on their speed changed the least. This suggests that the internal spin structure of the proton is far more sensitive to its environment than its basic speed structure.

The researchers also looked at how temperature and the imbalance between protons and neutrons affected the results. They found that while heat does have an impact, it is the density of the nuclear matter that drives the most significant changes. Increasing the temperature slightly counteracts the effects of density, pushing the quark distributions back toward their original, vacuum-like state, but the density remains the dominant force. When they looked at the difference between a balanced nuclear environment and one packed with extra neutrons, the changes were present but much smaller than those caused by density or heat. The quarks responded most strongly to the sheer pressure of the crowd, rather than the specific mix of neighbors.

To make these findings relevant to the real world, the team applied their calculations to four specific, heavy atomic nuclei: calcium, iron, gold, and lead. They used a model that accounts for how density varies from the center of a nucleus to its edge. In the center of these heavy atoms, the density is highest, and the proton's internal structure is most modified. Interestingly, they found that the modification depends on the local density at the center of the nucleus, not simply on how heavy the nucleus is. For example, the proton in the center of a lead nucleus, which has a lower central density than a calcium nucleus, showed a different degree of modification than one might expect if only the total number of particles mattered. This suggests that the "squeezing" effect is a local phenomenon, determined by the immediate neighborhood of the proton.

The study also calculated specific numbers that describe the proton's spin and structure, known as charges. In a free proton, the axial charge, which measures the total spin contribution of the quarks, is a value of 1.28. In the dense nuclear medium at saturation density, this value drops significantly to 0.844. This confirms that the quarks inside a bound proton contribute less to the overall spin of the proton than they do when the proton is free. The tensor charge, which relates to the sideways spin, also changed, with the up quark's contribution decreasing and the down quark's contribution increasing in magnitude. These shifts provide a quantitative picture of how the proton's internal machinery is rewired by the nuclear environment.

Ultimately, this work provides a detailed map of how the fundamental building blocks of matter adapt to their surroundings. It shows that a proton is not a static object with a fixed internal structure; rather, it is a dynamic system that reshapes itself in response to the pressure and temperature of its environment. The findings suggest that the changes observed in heavy nuclei are driven primarily by the density of the nuclear matter, which alters the effective mass of the quarks and, consequently, their distribution of momentum and spin. While the study relies on theoretical simulations rather than direct measurement, it offers a coherent explanation for why the internal structure of bound nucleons differs from free ones, pointing toward a future where the interplay of density, temperature, and spin can be fully understood in the complex world of nuclear physics.