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

## TL;DR

This paper compares bubble-resummation and critical-point methods for calculating $eta$-functions at large N, revealing their complementarity and providing new insights into the convergence properties of the theory.

## Contribution

It demonstrates the equivalence of the methods for single-coupling theories and extends the analysis to multi-coupling cases, including explicit $eta$-function calculations in the Gross-Neveu-Yukawa model.

## Key findings

- The methods are equivalent for single-coupling theories.
- Additional input is needed for multi-coupling theories to extract full information.
- The $eta$-function for the quartic coupling was computed at $ ext{O}(1/N)$.

## Abstract

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