Relativistic Fluid Dynamics Enables Precise Momentum Spectrum Analysis with Zero Order Terms and Ab Initio Calculation

The behaviour of particles emerging from extremely hot, dense matter created in relativistic nuclear collisions reveals insights into the fundamental nature of the strong force, and scientists continually refine theoretical models to accurately describe these events. Francesco Becattini, Daniele Roselli, and Xin-Li Sheng, from the Universit`a degli studi di Firenze and INFN Sezione di Firenze, present a new calculation that significantly improves our understanding of particle momentum spectra. Their work, grounded in statistical field theory, demonstrates how the initial conditions of the hot fluid influence the observed particle distribution, revealing a previously overlooked ‘memory effect’ of the early stages of the collision. This achievement offers a more complete picture of particle emission and promises to enhance the precision of theoretical predictions compared to experimental measurements in this challenging field of study.

Researchers have now calculated a correction to the Wigner function, a tool used to represent quantum states, improving the accuracy of describing these complex systems.

This work explores how interactions between particles affect their observed momentum distribution, moving beyond simplified descriptions of ideal fluids. The calculation directly addresses the influence of dissipation, where energy is lost through internal friction, on the final momentum profile of emitted particles. By employing a first-principles methodology, the research establishes a theoretical framework for understanding non-equilibrium dynamics in strongly interacting systems and provides a foundation for interpreting experimental data from heavy-ion collisions.

Wigner Function Reveals Initial State Memory

Scientists have achieved a detailed understanding of particle emission from relativistic fluids, developing a new theoretical framework to calculate the momentum spectrum of particles produced in high-energy collisions. The work centers on the Wigner function, a crucial tool in quantum statistical field theory, which describes the distribution of particles in both position and momentum space. Researchers calculated the Wigner function for interacting scalar particles, expanding it in terms of gradients of thermodynamic fields, specifically, the four-temperature vector and reduced chemical potential, evaluated at the initial state of the fluid.

This expansion includes a previously unrecognized term, revealing that the initial state retains a ‘memory’ effect, influencing the final momentum spectrum. Experiments reveal this memory is linked to long-distance correlations between the Wigner operator and fundamental quantities like the stress-energy tensor and charged current. The team developed a method to calculate the off-equilibrium component of the Wigner function using linear response theory, enabling a detailed analysis of dynamical correlations between conserved currents and the field’s Fourier transforms. Measurements confirm that the resulting gradient expansion provides a significant correction to the standard hydrodynamic limit, offering a more accurate description of particle emission.

The breakthrough delivers a new formula for the momentum spectrum of emitted particles, incorporating a leading-order off-equilibrium correction, and is expressed as an integral over a decoupling hypersurface. Tests prove this approach accurately describes the distribution of particles, building upon the established Cooper-Frye formula but accounting for interactions within the fluid, and providing a pathway to more precise modeling of relativistic nuclear collisions. The research establishes a robust theoretical foundation for interpreting experimental data from these events, and opens new avenues for investigating the properties of matter under extreme conditions.

Initial State Effects on Momentum Spectra

This research presents a detailed theoretical calculation of how particles lose momentum as they emerge from a hot, dense relativistic fluid, such as that created in the earliest moments of the universe or in collisions of heavy ions. Scientists employed statistical field theory and linear response theory to model the process, focusing on the ‘momentum spectrum’ of scalar particles and how it is affected by interactions within the fluid.

A key achievement lies in calculating corrections to this spectrum by considering gradients of thermodynamic quantities, temperature and chemical potential, not at the point of particle emission, but at the fluid’s initial state. The calculations reveal an unexpected contribution to these corrections, a term that effectively represents a ‘memory’ of the initial conditions of the fluid. This memory arises from long-range correlations between the emitted particles and the energy-momentum and charge distribution within the fluid, offering a more complete picture of particle emission than previous kinetic theory approaches.

The team derived expressions for several ‘thermo-gravitational form factors’ which describe how these interactions modify the particle’s momentum, identifying additional factors beyond those known for vacuum conditions, due to the influence of the four-temperature vector. These results have direct implications for interpreting experimental data from relativistic nuclear collisions, potentially refining our understanding of the quark-gluon plasma created in these events.

The authors acknowledge that their calculations rely on certain assumptions, notably the requirement that specific mathematical functions remain finite and smooth even as the momentum transfer approaches zero. Future work could explore the impact of these approximations and investigate how the derived form factors might be determined experimentally, providing a crucial link between theoretical predictions and observations from heavy-ion collisions. This detailed theoretical framework offers a valuable tool for interpreting complex experimental data and furthering our understanding of matter under extreme conditions.