Critical behavior of the bulk viscosity in QCD

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, $\xi/\xi_0\sim 2$.