Vacuum condition and the relation between response parameter and anomaly coefficient in (1+3) dimensions

The role of Israel-Hartle-Hawking vacuum is discussed for anomalous fluid in presence of both the gauge and gravitational anomalies in ($1+3$) dimensions. I show that imposition of this vacuum condition leads to the relation ${\tilde{c}}_{4d}=-8{\pi }^2{c}_m$ between the response parameter (${\tilde{c}}_{4d}$) and the anomaly coefficient ($c_m$). This establishes a connection between the coefficients appearing in a first order and a third order derivative terms in the constitutive relation.