Transits of the QCD critical point

Yukinao Akamatsu, Derek Teaney, Fanglida Yan, and Yi Yin

Phys. Rev. C 100, 044901 – Published 1 October 2019

Abstract

We analyze the evolution of hydrodynamic fluctuations in a heavy-ion collision as the system passes close to the QCD critical point. We introduce two small dimensionless parameters $λ$ and $Δ_{s}$ to characterize the evolution. $λ$ compares the microscopic relaxation time (away from the critical point) to the expansion rate $λ \equiv τ_{0} / τ_{Q}$ , and $Δ_{s}$ compares the baryon to entropy ratio, $n / s$ , to its critical value, $Δ_{s} \equiv (n / s - n_{c} / s_{c}) / (n_{c} / s_{c})$ . We determine how the evolution of critical hydrodynamic fluctuations depends parametrically on $λ$ and $Δ_{s}$ . Finally, we use this parametric reasoning to estimate the critical fluctuations and correlation length for a heavy-ion collision and to give guidance to the experimental search for the QCD critical point.

Received 29 March 2019

DOI:https://doi.org/10.1103/PhysRevC.100.044901

Physics Subject Headings (PhySH)

QCD phase transitions Quark-gluon plasma Relativistic heavy-ion collisions

Nuclear Physics

Authors & Affiliations

Yukinao Akamatsu^1,*, Derek Teaney^2,†, Fanglida Yan^2,‡, and Yi Yin^3,§

¹Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
²Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
³Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*akamatsu@kern.phys.sci.osaka-u.ac.jp
^†derek.teaney@stonybrook.edu
^‡yan.fanglida@stonybrook.edu
^§yiyin3@mit.edu

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Vol. 100, Iss. 4 — October 2019

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Images

Figure 1
(a) A schematic trajectory of a heavy-ion collision passing close to the critical point. The duration of panel (a) is of order $Δ t \sim τ_{Q}$ . (b) A magnification of the critical region in panel (a) by $Δ_{s}$ . The duration of panel (b) is of order $Δ t \sim τ_{Q} Δ_{s}$ . In this regime, the Ising magnetic field $h$ is negligibly small, and the susceptibilities scale as a power $Δ n / n_{c}$ . (c) In this panel, we have rescaled the $Δ n / n_{c}$ and $Δ s / s_{c}$ axes of panel (b) by $Δ_{s}^{b}$ and $Δ_{s}$ , respectively, with $b \equiv (1 - α) / β ≃ 2.7$ . The duration of panel (c) is of order $Δ t \sim t_{cr} \sim τ_{Q} Δ_{s}^{b}$ . At the time $t_{cr}$ , the system leaves the coexistence region and the equilibrium correlation length reaches its maximal value [see Eq. (84)]. Only in panel (c) is the equation of state a nontrivial function (i.e., beyond simple powers) of the scaling variable $z \propto r / h^{1 / β δ}$ .
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Figure 2
The Ising susceptibility and correlation length as a function of time during a transit of the QCD critical point along an adiabatic trajectory characterized by ${\bar{Δ}}_{s}$ . The time axis has been rescaled by $t_{cr} \sim τ_{Q} Δ_{s}^{b}$ ; see Eq. (84). The $y$ axes have been rescaled by an appropriate power of ${\bar{Δ}}_{s}$ so that the curve is independent of ${\bar{Δ}}_{s}$ .
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Figure 3
The time evolution of the correlation function of entropy per baryon fluctuations $δ \hat{s} \equiv δ s - (s / n) δ n$ , when the system passes directly through the critical point, $t_{cr} / t_{kz} = 0$ . The wave number is measured in units of $ℓ_{kz}^{- 1}$ ( $\bar{k} \equiv k ℓ_{kz}$ ), and $N^{\hat{s} \hat{s}}$ has been rescaled by the specific heat $C_{p}$ at the Kibble-Zurek time $C_{p, kz}$ ( ${\bar{N}}^{\hat{s} \hat{s}} \equiv N^{\hat{s} \hat{s}} / C_{p, kz}$ ). (a) The time evolution in the coexistence region $t < 0$ (before the critical point). (b) The time evolution after the system has left the coexistence region $t > 0$ (after the critical point).
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Figure 4
(The upper row) The same as Fig. 3, but the system misses the critical point with $t_{cr} / t_{kz} = 1$ . $t_{cr}$ is the time when the system leaves the coexistence region and $Δ t \equiv t - t_{cr}$ . The time evolution of $N^{\hat{s} \hat{s}}$ (a) when the system is in the coexistence region $t < t_{cr}$ and (b) when the system leaves the coexistence region $t > t_{cr}$ . (The lower row) The same as the upper row but with $t_{cr} / t_{kz} = 3$ .
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Figure 5
A schematic plot showing the dependence of the maximal fluctuations, $N_{\max}^{\hat{n} \hat{n}} / n$ , and the maximal correlation length, $ξ_{\max}$ , on the parameters $Δ_{s}$ and $λ$ during a transit of the critical point. Also shown is the nonequilibrium length $ℓ_{neq}$ ; modes with wavelength longer than $ℓ_{neq}$ fall out of equilibrium during the transit. For $Δ_{s} ≳ 1$ , the adiabatic trajectory misses the critical point completely. In this regime, $N_{\max}^{\hat{n} \hat{n}} / n$ is of order unity, and $ℓ_{neq}$ is of order $ℓ_{\max} \sim ℓ_{0} λ^{- 1 / 2}$ . For $Δ_{s} < 1$ , but larger than $Δ_{kz} = λ^{0.096}$ , the trajectory approaches the critical point. In this (narrow) regime, the dependence of $N_{\max}^{\hat{n} \hat{n}} / n$ and $ξ_{\max}$ on $Δ_{s}$ follows from equilibrium scaling. The nonequilibrium length $ℓ_{neq}$ remains longer than the correlation length $ξ_{\max}$ . For $Δ_{s} ≲ Δ_{kz}$ , equilibrium scaling is irrelevant, and the Kibble-Zurek scaling sets in. In Kibble-Zurek region, $N_{\max}^{\hat{n} \hat{n}} / n$ and $ℓ_{neq}$ scale as $λ^{- 0.37}$ and $ℓ_{0} λ^{- 0.19}$ respectively. In this region, both quantities are independent $Δ_{s}$ , i.e., of how close the adiabatic trajectory is to the critical point.
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