Dissipative corrections to the particle momentum spectrum of a decoupling fluid

TL;DR

This work derives an ab initio quantum-statistical computation of dissipative corrections to the momentum spectrum of scalar particles emitted from a relativistic fluid at decoupling. By formulating the Wigner function via a gradient expansion in the initial thermo-hydrodynamic fields evaluated on the local-equilibrium hypersurface $\Sigma_0$, the authors uncover a surprising zeroth-order memory term that encodes differences between the initial and decoupling states, in addition to leading gradient corrections that vanish at linear order in the hydrodynamic fields. The corrections are expressed through thermo-gravitational and thermo-charged correlators and reduce to the standard Cooper-Frye result in the quasi-free limit, while introducing non-local, history-dependent effects that can enhance low-$p_T$ yields in heavy-ion collisions. The framework accommodates finite chemical potential and arbitrary decoupling geometries, bridging quantum-statistical field theory with kinetic descriptions of particlization and offering a pathway for first-principles estimates of dissipative corrections in relativistic fluids.

Abstract

We present an \emph{ab initio} calculation within quantum statistical field theory and linear response theory, of the dissipative correction to the momentum spectrum of scalar particles emitted at decoupling (freeze-out) from a relativistic fluid assuming the initial state to be in local thermodynamic equilibrium. We obtain an expansion of the Wigner function of the interacting quantum field in terms of the gradients of the classical thermo-hydrodynamic fields - four-temperature vector and reduced chemical potential - evaluated on the initial local-equilibrium hypersurface, rather than on the decoupling (freeze-out) hypersurface as usual in kinetic theory. The gradient expansion includes an unexpected zeroth order term depending on the differences between thermo-hydrodynamic fields at the decoupling and the initial hypersurface. This term encodes a memory of the initial state which is related to the long-distance persistence of the correlation function between Wigner operator and stress-energy tensor and charged current that is discussed in detail. We address the phenomenological implications of these corrections for the momentum spectra measured in relativistic nuclear collisions.

Figures (5)

• Figure 1: A typical shape of a decoupling hypersurface $\Sigma_D$ in a relativistic nuclear collision in a space-time diagram. The fluid decouples at $\Sigma_{\rm D}$ and the produced particles interact through collisions in the region $\Upsilon$ until all interaction cease and the spectra freeze out. The particles are eventually observed in the asymptotic future $t\to\infty$.
• Figure 2: Schematic illustration of a nuclear collision at high energy. The initial local equilibrium hypersurface is $\Sigma_0$ (solid line) and the decoupling hypersurface $\Sigma_{\rm D}$ (finely dotted line); the Quark Gluon Plasma as a fluid lives in the encompassed region $\Omega$. A point $x$ on $\Sigma_{\rm D}$ is the most suitable place where an approximate expression of the spectrum can be obtained from the Wigner operator.
• Figure 3: The correlation function between the Wigner operator $\widehat{W}^+(x,k)$ and stress-energy tensor (or vector current) operator in a point $y$ features terms which are constant over the worldlines with tangent vector $k$ (solid lines) and terms which are constant over worldlines with tangent vector $v(k,\beta)$ (dashed lines). The integration over $k$ eventually yields correlation functions which are strongly peaked around $y = x$.
• Figure 4: For a system with fixed volume (e.g. a gas within a vessel) in the limit $t \to +\infty$ all worldlines with a slope proportional to $k$ eventually have no intersection with the hypersurface $\Sigma_0$ at $t=0$. Conversely, all worldlines whose tangent four-vector is $\lim_{t \to \infty} \beta(x) = \frac{1}{T_0} (1,{\bf 0})$ worldlines intersect $\Sigma_0$.
• Figure 5: In a relativistic nuclear collision (see also figure \ref{['fig:Gauss Theorem Region']}) particles with (off-shell) four-momentum $k_2$ do not receive zero-order correction from the Wigner function $W^+(x,k)$ at $x$ over the decoupling hypersurface because the world-line does not intersect the initial hypersurface $\Sigma_0$; the converse is true for the momentum $k_1$.