The chiral condensate from renormalization group optimized perturbation

TL;DR

This paper applies a novel renormalization group optimized perturbation method to calculate the QCD chiral condensate, achieving stable multi-loop results and comparing favorably with other nonperturbative approaches.

Contribution

It introduces a modified OPT method incorporating RG properties to compute the QCD chiral condensate with high stability across multiple loop orders.

Findings

Stable results at two-, three-, and four-loop orders.

Quantitative estimates of the chiral condensate for n_f=2 and 3.

Moderate suppression of condensate from n_f=2 to 3.

Abstract

Our recently developed variant of variationnally optimized perturbation (OPT), in particular consistently incorporating renormalization group properties (RGOPT), is adapted to the calculation of the QCD spectral density of the Dirac operator and the related chiral quark condensate $\langle \bar q q \rangle$ in the chiral limit, for $n_f=2$ and $n_f=3$ massless quarks. The results of successive sequences of approximations at two-, three-, and four-loop orders of this modified perturbation, exhibit a remarkable stability. We obtain $\langle \bar q q\rangle^{1/3}_{n_f=2}(2\, {\rm GeV}) = -(0.833-0.845) \bar\Lambda_2 $, and $ \langle\bar q q\rangle^{1/3}_{n_f=3}(2\, {\rm GeV}) = -(0.814-0.838) \bar\Lambda_3 $ where the range spanned by the first and second numbers (respectively four- and three-loop order results) defines our theoretical error, and $\bar\Lambda_{n_f}$ is the basic QCD scale in the $\overline{MS}$-scheme. We obtain a moderate suppression of the chiral condensate when going from $n_f=2$ to $n_f=3$. We compare these results with some other recent determinations from other nonperturbative methods (mainly lattice and spectral sum rules).