Probing vorticity through femtoscopic correlations

TL;DR

The paper addresses how vorticity generated in heavy-ion collisions affects femtoscopic observables. It combines non-central Au+Au simulations at $E_{\rm kin}=1.23~A\rm{GeV}$ using the UrQMD transport model with a CMF EOS and a coarse-grained flow analysis to connect rotational motion to proton–pion emission patterns via the Koonin–Pratt femtoscopy framework $C_{\vec{v}}(\vec{q})=\int d^3r\,K(\vec{q},\vec{r})\,S_{\vec{v}}(\vec{r})$, thereby identifying a displacement between proton and pion emission centers as a signal of vorticity. The study finds that nonradial flow produces a positive $x$-direction shift and a negative $z$-direction shift in the emission centers, with the angle $\alpha$ between the pair velocity $\vec{v}_{p\pi}$ and the emission-centroid separation $\langle \vec{r}_p-\vec{r}_{\pi} \rangle$ being negative, signaling clockwise rotation around the $y$-axis. Proper modeling of the source function is crucial, as a Gaussian approximation can bias the extracted signals, though agreement improves at higher $v^x_{p\pi}$. The authors propose proton–pion femtoscopy as a spin-independent, experimentally accessible probe of vorticity that complements polarization measurements and could be extended to lower energies or asymmetric systems using spherical-harmonics source decompositions.

Abstract

In heavy-ion collisions, as the two nuclei pass through one another and create hot and dense matter, part of their initial angular momentum is transferred to the fireball, generating a nonzero average vorticity. Understanding heavy-ion collision dynamics and its influence on key observables, including those used to probe the initial state or assess thermodynamics of nuclear matter, requires understanding the magnitude of effects tied to vorticity. In this work, we use simulations of non-central Au+Au collisions at $E_{\rm{kin}}=1.23~A\rm{GeV}$ to show that the rotation of the system impacts the space-time picture of particle emission and, in particular, leaves imprints on proton-pion femtoscopic correlations. Next, we use coarse-graining of the simulation outputs to extract the collective velocity as a function of position and time, shedding light on the dynamical origin of this effect. Moreover, we demonstrate that the displacement between the proton and pion emission centers quantifies the strength of the rotation and propose it as a new signal of vorticity in heavy-ion collisions.

Figures (4)

• Figure 1: Correlation functions of $p\pi^+$ pairs with $\vec{v}_{p \pi} \approx v_{p \pi}^x \vec{e}_x$, for two values of $v^x_{p \pi}$ (top and bottom panel), obtained via Eq. \ref{['eq:Koonin-Pratt']} using source functions from UrQMD simulations and the Coral kernel. The solid red line shows correlations along the $q_x$ relative momentum axis, $C_{\vec{v}}(q,0,0)$, while the dashed blue and dash-dotted green lines correspond to correlations along the $q_y$ and $q_z$ axes, respectively. For $C_{\vec{v}}(q,0,0)$, the asymmetry around $q = 0$ stems from a shift of the $p\pi^+$ source toward positive values of $x$, while the asymmetry in $C_{\vec{v}}(0,0,q)$ reflects a smaller negative shift in the $z$-direction.
• Figure 2: Flow in the $zx$-plane at $t = 20$ fm/c, with arrows depicting the magnitude and direction of the collective flow velocity $\vec{v}_f$; the magnitude of $\vec{v}_f$ is also indicated by the color of the arrows (see legend). In our coordinate setup, the nucleus with a center of mass located at $x > 0$ carries a positive $z$-momentum, while the one at $x<0$ carries a negative $z$-momentum. The expansion accelerates with distance from the center of the collision. The presence of angular momentum in the flow, resulting in a non-zero average vorticity, can be seen in components of $\vec{v}_f$ orthogonal to lines connecting a given point to the center of the collision. In particular, the region where $\vec{v}_f$ is largely aligned with the positive $x$-axis, marked with a violet-shaded area, is located away from $z = 0$.
• Figure 3: Average proton-pion source characteristics plotted against $v^x_{p \pi}$, obtained by directly averaging proton-pion pair displacements from UrQMD simulation outputs (black crosses with red areas denoting errors) and by extracting the source size from $p \pi^+$ femtoscopic correlations $C_{\vec{v}}(\vec{q})$, Fig. \ref{['Cq']} (green dots). Panels (a) and (b) show the average proton-pion source displacement in the $x$- and $z$-directions, respectively. Panel (c) shows the angle $\alpha = \angle( \vec{v}, \langle \vec{r}_p - \vec{r}_{\pi} \rangle)$ which can be used as a measure of vorticity. The differences between source characteristics extracted directly and from a fit to $C_{\vec{v}}(\vec{q})$ can be linked to assuming a Gaussian source shape in the fit.
• Figure 4: Proton (red) and pion (blue) emission points, extracted from UrQMD simulations, for $v_{p \pi}^x = 0.55c$ (top) and $0.73c$ (bottom). The proton source is localized and approximates a Gaussian, while the pion source differs from a simple Gaussian, with more particles emitted from a tail at a negative $z$. The pion source tails are seen to be reduced for larger $v_{p \pi}^x$.