Chiral Condensate and Spectral Density at full five-loop and partial six-loop orders of Renormalization Group Optimized Perturbation

Abstract

We reconsider our former determination of the chiral quark condensate $⟨\bar{q}q⟩$ from the related QCD spectral density of the Euclidean Dirac operator, using our Renormalization Group Optimized Perturbation (RGOPT) approach. Thanks to the recently available {\em complete} five-loop QCD RG coefficients, and some other related four-loop results, we can extend our calculations exactly to $N^{4}LO$ (five-loops) RGOPT, and partially to $N^{5}LO$ (six-loops), the latter within a well-defined approximation accounting for all six-loop contents exactly predictable from five-loops RG properties. The RGOPT results overall show a very good stability and convergence, giving primarily the RG invariant condensate, $⟨\bar{q}q{⟩}_{RGI}^{1/3}(n_f=0)=-(0.840_{-0.016}^{+0.020}){\bar{\Lambda }}_0$, $⟨\bar{q}q{⟩}_{RGI}^{1/3}(n_f=2)=-(0.781_{-0.009}^{+0.019}){\bar{\Lambda }}_2$, $⟨\bar{q}q{⟩}_{RGI}^{1/3}(n_f=3)=-(0.751_{-.010}^{+0.019}){\bar{\Lambda }}_3$, where ${\bar{\Lambda }}_{n_f}$ is the basic QCD scale in the \overline{MS} scheme for $n_f$ quark flavors, and the range spanned is our rather conservative estimated theoretical error. This leads {\it e.g.} to $⟨\bar{q}q{⟩}_{n_f=3}^{1/3}(2\mathrm{GeV})=-(273_{-4}^{+7}\pm 13)$ MeV, using the latest ${\bar{\Lambda }}_3$ values giving the second uncertainties. We compare our results with some other recent determinations. As a by-product of our analysis we also provide complete five-loop and partial six-loop expressions of the perturbative QCD spectral density, that may be useful for other purposes.

28 pages, 4 figures

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