Bubble-resummation and critical-point methods for β -functions at large N

We investigate the connection between the bubble-resummation and critical-point methods for computing the β-functions in the limit of large number of flavours, N, and show that these can provide complementary information. While the methods are equivalent for single-coupling theories, for multi-coupling case the standard critical exponents are only sensitive to a combination of the independent pieces entering the β-functions, so that additional input or direct computation are needed to decipher this missing information. In particular, we evaluate the β-function for the quartic coupling in the Gross–Neveu–Yukawa model, thereby completing the full system at O(1 / N). The corresponding critical exponents would imply a shrinking radius of convergence when O(1 / N²) terms are included, but our present result shows that the new singularity is actually present already at O(1 / N) , when the full system of β-functions is known.