# $1/N$-expansion for the critical temperature of the Bose gas

## TL;DR

This paper applies the large-N expansion to analyze the critical temperature of a three-dimensional Bose gas with short-range repulsion, providing numerical and analytical insights into its behavior across different interaction strengths.

## Contribution

It introduces a full numerical calculation of the critical temperature using the 1/N-approximation for the Bose gas and estimates its strong-coupling behavior.

## Key findings

- Critical temperature shift grows at small gas parameter
- It reaches a maximum and then decreases, becoming negative
- Provides analytical estimates for strong-coupling limit

## Abstract

We revised the large-$N$ expansion for a three-dimensional Bose system with short-range repulsion in normal phase. Particularly, for the model potential that is characterised only by the $s$-wave scattering length $a$ the full numerical calculations of the critical temperature in the $1/N$-approximation as a function of the gas parameter $an^{1/3}$ are performed. Additionally to the well-known result in the dilute limit we estimated analytically the leading-order strong-coupling behavior of the Bose-Einstein condensation transition temperature. It is shown that the critical temperature shift of the non-ideal Bose gas grows at small $an^{1/3}$, reaches some maximal value and then falls down becoming negative.