Quantum consistency in supersymmetric theories with R-symmetry in curved space

We discuss consistency at the quantum level in the rigid N = 1 supersymmetric field theories with a U(1)(R) symmetry in four-dimensional curved space which are formulated via coupling to the new-minimal supergravity background fields. By analyzing correlation functions of the current operators in the R-multiplet, we show that the quantum consistency with the (unbroken) supersymmetry requires the U(1)(R) anomaly coefficient, which depends only on the field content of the theory, to vanish. This consistency condition is obtained under the assumption that the supercurrent Ward identity is non-anomalous and that the vacuum is supersymmetric.