Non-local conservation in the coupling field: effect on critical dynamics

We consider the critical dynamics of a system with an $n$-component non-conserved order parameter coupled to a conserved field with long range diffusion. An exponent $\sigma$ characterizes the long range transport, $\sigma=2$ being the known locally conserved case. With renormalisation group calculations done upto one loop order, several regions are found with different values of the dynamic exponent $z$ in the $\sigma -n$ plane. For $n<4$, there are three regimes, I: nonuniversal, $\sigma$ dependent $z$, II: universal with $z$ depending on $n$ and III': conservation law irrelevant, $z$ being equal to that in the nonconserved case. The known locally conserved case belongs to regions I and II.