Anomalies and time reversal invariance in relativistic hydrodynamics: the second order and higher dimensional formulations

We present two new results on relativistic hydrodynamics with anomalies and external electromagnetic fields, "Chiral MagnetoHydroDynamics" (CMHD). First, we study CMHD in four dimensions at second order in the derivative expansion assuming the conformal/Weyl invariance. We classify all possible independent conformal second order viscous corrections to the energy-momentum tensor and to the U(1) current in the presence of external electric and/or magnetic fields, and identify eighteen terms that originate from the triangle anomaly. We then propose and motivate the following guiding principle to constrain the CMHD: the anomaly--induced terms that are even under the time reversal invariance should not contribute to the local entropy production rate. This allows us to fix thirteen out of the eighteen transport coefficients that enter the second order formulation of CMHD. We also relate one of our second order transport coefficients to the chiral shear waves. Our second subject is hydrodynamics with (N+1)-gon anomaly in an arbitrary 2N dimensions. The effects from the (N+1)-gon anomaly appear in hydrodynamics at (N-1)'th order in the derivative expansion, and we identify precisely N such corrections to the U(1) current. The time reversal constraint is powerful enough to allow us to find the analytic expressions for all transport coefficients. We confirm the validity of our results (and of the proposed guiding principle) by an explicit fluid/gravity computation within the AdS/CFT correspondence.