QCD at Low Energies

TL;DR

This review synthesizes the status of basic low-energy QCD parameters, notably light-quark masses, quark and gluon condensates, and the strong coupling α_s, using QCD sum rules, tau-decay data, and charmonium/nucleon analyses. It employs the operator product expansion (OPE) with condensates up to high dimension and contour methods in the complex s-plane to connect experimental spectral data to nonperturbative QCD inputs, achieving cross-consistent determinations such as α_s(m_τ^2) ≈ 0.34, ⟨α_s/π G^2⟩ ≈ 0.005 GeV^4, and ⟨q̄q⟩_{1 GeV} ≈ −(254 MeV)^3, with m_c(m_c) ≈ 1.275 GeV. The results corroborate a coherent low-energy QCD picture and enable reliable predictions for hadron properties, while also addressing the role of instantons and the limits of perturbative methods at Q^2 ≈ 1 GeV^2. The work further shows that valence quark distributions in the nucleon at low Q^2 can be tied to condensate values within the QCD sum-rule framework, reinforcing the interconnectedness of vacuum structure and hadron phenomenology.

Abstract

The modern status of basic low energy QCD parameters is reviewed. It is demonstrated, that the recent data allows one to determine the light quark mass ratios with an accuracy 10-15%. The general analysis of vacuum condensates in QCD is presented, including those induced by external fields. The QCD coupling constant alpha_s is found from the tau-lepton hadronic decay rate. V-A spectral functions of tau-decay are used for construction of the V-A polarization operator Pi_{V-A}(s) in the complex s-plane. The operator product expansion (OPE) is used up to dimension D=10 and the sum rules along the rays in the complex s-plane are constructed. The best values of quark condensate and alpha_s<0|qq|0>^2 are found. The value of quark condensate is confirmed by considering the sum rules for baryon masses. Gluon condensate is found in four ways: by considering of V+A polarization operator based on the tau-decay data, by studying the sum rules for polarization operators momenta in charmonia in vector, pseudoscalar and axial channels. All of these determinations are in agreement and result in <(alpha_s/pi)G^2 > =0.005 \pm 0.004 GeV^4. Valence quark distributions in proton are calculated in QCD using the OPE in proton current virtuality. The quark distributions agree with those found from the deep inelastic scattering data. The same value of gluon condensate is favoured.