Global Analysis of Nuclear Parton Distribution Functions using Markov Chain Monte Carlo

Deep inside every atom, protons and neutrons are not solid, indivisible spheres but rather bustling cities of smaller particles called quarks and gluons. These constituents, collectively known as partons, zip around at nearly the speed of light, held together by the strong nuclear force. To understand how matter behaves in the most extreme environments, from the heart of a star to the collisions inside the Large Hadron Collider, physicists must map out exactly how these partons share the momentum of their parent particle. This map is called a parton distribution function. When protons and neutrons are locked together inside a heavy atomic nucleus, the rules change. The crowded nuclear environment distorts the parton maps, creating a complex, modified landscape that scientists call nuclear parton distribution functions. Determining these maps is crucial for predicting the outcomes of high-energy experiments, but it is notoriously difficult because the data is sparse and the mathematical landscape used to find the best fit is often rugged and full of hidden valleys.

For decades, researchers have relied on a standard mathematical shortcut to navigate this terrain. This method, known as the Hessian approach, assumes the landscape around the best solution is a smooth, symmetrical bowl, like a perfect parabola. By measuring the steepness of the bowl's sides, scientists can estimate how uncertain their map is. However, this assumption works best only when the data is abundant and the relationships between variables are simple. In the case of heavy nuclei, where experimental data is limited and the connections between different types of particles are tangled, this smooth-bowl assumption often breaks down. The true landscape might be lumpy, tilted, or even contain multiple separate valleys where the solution could hide. If scientists assume the landscape is smooth when it is actually jagged, their estimates of uncertainty can be misleading, potentially hiding the true complexity of the atomic nucleus.

To address this, a researcher led by Nasim Derakhshanian developed a new way to explore these nuclear maps using a technique called Markov Chain Monte Carlo. Instead of assuming the landscape is a smooth bowl, this method treats the search like a guided walk through a vast, unknown territory. The researcher built a computer program that takes millions of random steps, constantly checking how well each step fits the available experimental data. Over time, the program learns the true shape of the landscape, spending more time in the deep valleys where the solutions are best and less time on the high peaks where the data does not fit. This allows them to sample the entire probability distribution directly, capturing asymmetries, multiple solutions, and complex correlations that the old method would miss.

The researcher applied this new approach to the nucleus of lead, one of the heaviest and most complex atoms used in experiments. They performed two distinct types of analysis to test the robustness of their findings. First, they looked exclusively at data from lead nuclei, creating a "lead-only" map to see the intrinsic structure of the fit without outside influence. This analysis revealed a highly complex and non-trivial structure. The computer sampling uncovered that the parameters describing the valence quarks—the core particles that give the proton its identity—were not sitting in a single, smooth valley. Instead, the landscape contained multiple local minima, or secondary valleys, where the fit could also be quite good. The distribution of these solutions was skewed and asymmetric, meaning the uncertainty was not the same in all directions. The standard Hessian method, which assumes a single, symmetrical valley, failed to capture this multi-peaked reality, effectively smoothing over the complex structure and providing an incomplete picture of the uncertainty.

To see how this complex picture changed when more information was added, the researcher then performed a second analysis that included data from a wide range of lighter nuclei, not just lead. By connecting the data from different atomic masses through a standard mathematical relationship, they introduced new constraints that linked the behavior of lead to that of lighter elements. This "multi-nuclei" analysis significantly altered the landscape. The secondary valleys that were prominent in the lead-only analysis were largely suppressed, and the parameter space became more regular and closer to a single, dominant valley. The inclusion of lighter nuclei acted as a powerful anchor, tightening the constraints on the quark distributions and reducing the uncertainty in the lead map. However, even with these extra constraints, the researcher found that the landscape was not perfectly symmetrical. Subtle non-Gaussian features and asymmetric tails persisted, particularly in the valence quark distributions, which the standard Hessian method would still have missed.

A direct comparison between the two approaches highlighted a critical difference in how they handled the data. In the lead-only analysis, the new method showed that the uncertainty bands for the quark distributions were much wider and more asymmetric than the old method suggested. The old method, relying on the smooth-bowl assumption, underestimated the true range of possibilities. When the multi-nuclei data was added, the uncertainties shrank, but the new method still revealed a more realistic, constrained uncertainty that accounted for the lingering asymmetries. Interestingly, the distribution of gluons—the particles that carry the strong force—remained relatively stable across both analyses. Because the gluon distribution is already well-constrained by direct data from lead collisions at the Large Hadron Collider, adding data from lighter nuclei did not significantly change the picture for gluons, unlike the quark distributions which were more sensitive to the new constraints.

The study also demonstrated a practical way to make these complex results usable for other scientists. The new method generates thousands of individual samples to represent the full probability distribution, which is too large to share directly. The researcher tested different ways to compress this massive dataset into a smaller, manageable set of replicas without losing the essential statistical information. They found that while simply picking a smaller number of samples at regular intervals worked well for the central values, it could distort the uncertainty bands if the sample size became too small. A more sophisticated compression algorithm, which carefully selected samples to preserve the shape and correlations of the full dataset, performed better at retaining the complex features of the distribution, provided the final set was not reduced too drastically.

Ultimately, this work establishes that the statistical landscape of nuclear parton distribution functions is far more intricate than previously assumed. The standard method of approximating the solution space as a smooth, symmetrical bowl is insufficient for capturing the full complexity of nuclear data, especially when dealing with heavy nuclei like lead. By using a method that samples the entire landscape directly, the researcher revealed hidden structures, such as multiple competing solutions and asymmetric uncertainties, that were previously invisible. This approach offers a more reliable and transparent way to quantify the unknowns in our understanding of the atomic nucleus, ensuring that future predictions for high-energy physics experiments are built on a foundation that truly reflects the complexity of the data.