Transport and hydrodynamics in the chiral limit

We analyze the evolution of hydrodynamic fluctuations for QCD matter below $T_c$ in the chiral limit, where the pions (the Goldstone modes) must be treated as additional non-Abelian superfluid degrees of freedom, reflecting the broken $S{U}_L(2)\times S{U}_R(2)$ symmetry of the theory. In the presence of a finite pion mass $m_{\pi }$, the hydrodynamic theory is ordinary hydrodynamics at long distances, and superfluidlike at short distances. The presence of the superfluid degrees of freedom then gives specific contributions to the bulk viscosity, the shear viscosity, and diffusion coefficients of the ordinary theory at long distances which we compute. This determines, in some cases, the leading dependence of the transport parameters of QCD on the pion mass. We analyze the predictions of this computation, as the system approaches the $O(4)$ critical point.

Figure 1

Long wavelength modes (black lines) with $\ell \sim (\nabla ·u{)}^{-1}\sim L$ are described with ordinary hydrodynamics. To model wavelengths of order $\ell \sim (m_{\pi }{)}^{-1}$, the effective theory must treat the soft pion modes explicitly (green lines). These modes can be described with a non-Abelian superfluid theory, due to the fact that the pions are Goldstone bosons. Finally, the microscopic degrees of freedom (purple arrows), which include typical pions with $p\sim {\lambda }^{-1}\sim \pi T$ and other hadronic states, determine the thermodynamic and dissipative parameters of the superfluid. The superfluid modes leave calculable imprints on the transport parameters of the ordinary fluid.

Figure 2

An example of a hydrodynamic pion loop in Kubo furmulas.