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

Abstract

Abstract We investigate the connection between the bubble-resummation and critical-point methods for computing the $$\beta $$ 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 $$\beta $$ β -functions, so that additional input or direct computation are needed to decipher this missing information. In particular, we evaluate the $$\beta $$ β -function for the quartic coupling in the Gross–Neveu–Yukawa model, thereby completing the full system at $$\mathcal {O}(1/N)$$ O(1/N) . The corresponding critical exponents would imply a shrinking radius of convergence when $$\mathcal {O}(1/N^2)$$ O(1/N2) terms are included, but our present result shows that the new singularity is actually present already at $$\mathcal {O}(1/N)$$ O(1/N) , when the full system of $$\beta $$ β -functions is known.