Chiral condensate from renormalization group optimized perturbation

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 qq 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 qq 1/3 n f =2 (2 GeV) = −(0.833 − 0.845)Λ2, and qq 1/3 n f =3 (2 GeV) = −(0.814 − 0.838)Λ3 where the range spanned by the first and second numbers (respectively fourand three-loop order results) defines our theoretical error, andΛn f is the basic QCD scale in the M S-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).

Λ MS ¯ n f = 2 $$ {\varLambda}_{\overline{\mathrm{MS}}}^{\left({n}_f=2\right)} $$ from a momentum space analysis of the quark-antiquark static potential

“…We note that the results (5.1) and (5.2) compare well with other determinations of Λ (n f =2) MS from the literature, e.g. [2,4,13,16,18,21,23,27] (cf. also the more detailed discussion and graphical summary in [5]).…”

“…the lattice computations [13][14][15][16][17][18][19][20][21] using the Schrödinger functional, vacuum polarization functions, ghost and gluon propagators, heavy quark correlators and the Dirac operator spectrum. Other works [22][23][24][25][26][27][28][29] focus e.g. on τ decays and electron-positron as well as electron-proton collisions.…”

Analytic structure of QCD propagators in Minkowski space

“…Among the many aspects that have not been addressed so far we mention a consistent use of RG equations [39,55], a variational estimate of the mass ratios by the effective potential (or by minimal sensitivity of physical observables [49][50][51]) and a formal extension to higher loops. The study of other covariant gauges [73] would not be straightforward or entirely analytical, but it would be very useful for addressing problems like the gauge dependence of the propagators or the role played by the Abelian dominance in the maximally Abelian gauge [74].…”

“…In the framework of optimized perturbation theory a RG improved expansion has been discussed in Ref. [55] for the chiral quark condensate. That work could provide the natural framework for an extension of the present method.…”

“…Thus, while based on a perturbative expansion, the method has a variational nature, with an accuracy that can be increased by tuning the mass ratio and the subtraction point to optimal values that minimize the effect of higher loops. In that sense, the method can be seen as a special case of the optimized perturbation theory that has been discussed by many authors in the past [49][50][51] and has been recently improved by RG methods [52][53][54][55].…”