Explicit results for the anomalous three point function and non-renormalization theorems

Two-loop corrections for the correlator of the singlet axial and vector currents in QCD are calculated in the chiral limit for arbitrary momenta. Explicit calculations confirm the non-renormalization theorems derived recently by Vainshtein [A. Vainshtein, Phys. Lett. B 569 (2003) 187] and Knecht et al. [M. Knecht, S. Peris, M. Perrottet, E. de Rafael, JHEP 0403 (2004) 035]. We find that as in the one-loop case also at two loops the correlator has only three independent form-factors instead of four. From the explicit results we observe that the two-loop correction to the correlator is equal to the one-loop result times the constant factor C2(R)αs/π in the MS-bar scheme. This holds for the full correlator, for the anomalous longitudinal as well as for the non-anomalous transversal amplitudes. The finite overall αs dependent constant has to be normalized away by renormalizing the axial current according to Witten's algebraic/geometrical constraint on the anomalous Ward identity [ correlator]. Our observations, together with known facts, suggest that in perturbation theory the correlator is proportional to the one-loop term to all orders and that the non-renormalization theorem of the Adler-Bell-Jackiw anomaly carries over to the full correlator