The 4-loop /β-function in the 2D non-Abelian Thirring model, and comparison with its conjectured ``exact'' form

Recently, B. Gerganov et al. [Phys. Rev. Lett. 86 (2001) 4753] have proposed an "exact" (all-orders) β-function for 2-dimensional conformal field theories with Kac-Moody current-algebra symmetry at any level k, based on a Lie group G, which are perturbed by a current-current interaction. This theory is also known as the non-Abelian Thirring model. We check this conjecture with an explicit calculation of the β-function to 4-loop order, for the classical groups G=SU(N) , SO( N) and SP( N) at level k=0. We find a contribution at 4-loop order, proportional to a higher-order group-theoretical invariant, which is incompatible with the proposed β-function in all possible regularization schemes.