Abstract
We study the behavior of the bulk viscosity $\zeta$ in QCD near a possible critical endpoint in the phase diagram. We verify the expectation that $(\zeta/s)\sim a(\xi/\xi_0)^{x_\zeta}$, where $s$ is the entropy density, $\xi$ is the correlation length, $\xi_0$ is the non-critical correlation length, $a$ is a constant and $x_\zeta\simeq 3$. Using a recently developed equation of state that includes a critical point in the universality class of the Ising model we estimate the constant of proportionality $a$. We find that $a$ is typically quite small, $a\sim O(10^{-4})$. We observe, however, that the result is sensitive to the commonly made assumption that the Ising temperature axis is approximately aligned with the QCD baryon chemical potential axis. If this is not the case, then the critical $\zeta/s$ can approach the non-critical value of $\eta/s$, where $\eta$ is the shear viscosity, even if the enhancement of the correlation length is modest, $\xi/\xi_0\sim 2$.
Highlights
There are several programs dedicated to exploring the phase diagram of quantum chromodynamics (QCD) at heavy ion accelerator laboratories around the world [1]
In this work we have studied the critical bulk viscosity in QCD
We find that the result is very sensitive to the orientation of the Ising axes in the QCD phase diagram
Summary
There are several programs dedicated to exploring the phase diagram of quantum chromodynamics (QCD) at heavy ion accelerator laboratories around the world [1]. There is a general expectation that the large critical exponent in Eq (1), combined with the strong deviation of the QCD equation of state from scale invariance, will lead to a substantial enhancement of the bulk viscosity and to large effects on the evolution of a heavy ion collision near a critical end point [26,27]. Onuki observed that the Ising entropy functional contains a trilinear coupling between ε and ψ2, and as a result fluctuations of the pressure couple to ψ2 [16,17] This means that correlation functions of δP are controlled by the order parameter relaxation rate, and the slow relaxation of ψ leads to a large enhancement in the bulk viscosity. Combining Eqs. (7) and (9) we obtain the slow mode contribution to fluctuations of the pressure δP 1⁄4 nTAanεγψ; ð11Þ where we have defined anε 1⁄4 ð∂εÞ=ð∂ðδnÞÞ
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