  When the branch character has root number -1, the corresponding
anticyclotomic Katz p-adic L-function identically vanishes. In this case, we
study the $\mu$-invariant of the cyclotomic derivative of Katz p-adic
L-function. As an application, this proves the non-vanishing of the
anticyclotomic regulator of a self-dual CM modular form with the root number
-1. The result also plays a crucial role in the recent work of Hsieh on the
Eisenstein ideal approach to a one-sided divisibility of the CM main
conjecture.
