# A Hermite-Pad\'e perspective on Gell-Mann--Low renormalization group: an application to the correlation function of Lieb-Liniger gas

## TL;DR

This paper demonstrates that Gell-Mann--Low renormalization group can be viewed as an integral Hermite-Padé approximation in the single-variable case, enabling effective interpolation between different regimes without requiring symmetries, exemplified by analyzing the Lieb-Liniger gas.

## Contribution

It reveals a new perspective connecting renormalization group methods with Hermite-Padé approximation, expanding their applicability beyond symmetric cases.

## Key findings

- Accurate extraction of the scaling-law prefactor for the Lieb-Liniger gas.
- Remarkable agreement with ab initio numerical results.
- Introduces a novel approach for interpolating between series expansions.

## Abstract

While Pad\'e approximation is a general method for improving convergence of series expansions, Gell-Mann--Low renormalization group normally relies on the presence of special symmetries. We show that in the single-variable case, the latter becomes an integral Hermite-Pad\'e approximation, needing no special symmetries. It is especially useful for interpolating between expansions for small values of a variable and a scaling law of known exponent for large values. As an example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb-Liniger gas. Using a new result for the 4th-order term in the short-distance expansion, we find a remarkable agreement with known ab initio numerical results.