Moduli Dependent Non-Holomorphic Contributions of Massive States to   Gravitational Couplings and $C^2$-Terms in $Z_N$-Orbifold Compactifications
  It is pointed out that massive states in D=4, N=1 supergravity-matter theories can, in general, at the 1-loop level contribute non-holomorphic terms to quadratic gravitational couplings. It is then shown in the context of $(2,2)$-symmetric $Z_N$-orbifold theories that, for constant moduli backgrounds, the inclusion of such contributions can result in the cancellation of naked $C^2$-terms. ${\cal R}^2$-terms can also arise but, being ghost free, need not cancel. 
Gabriel Lopes Cardoso, Dieter L\"ust and Burt A. Ovrut