Chiral and flavor SU(2) and SU(3) symmetry breaking in quantum chromodynamics

We calculate light-quark mass differences in the framework of the Laplace-transform QCD sum rules using an improved parametrization of the hadronic spectral functions. Our results are (m${¯}_s$-m${¯}_u$)${\Vert }_1$ GeV=185±15 MeV and (m${¯}_d$-m${¯}_u$)${\Vert }_1$ GeV=4±1 MeV. Using an earlier determination of the quark-mass sums based on similar techniques, these results lead to: m${¯}_u$(1 GeV)=6±1 MeV, m${¯}_d$(1 GeV)=10±1 MeV, and m${¯}_s$(1 GeV)=192±15 MeV. Next, we estimate the difference of the light-quark vacuum condensates in the framework of the Laplace-transform QCD sum rules. Our results are ψ${\mathrm{}\left(0\right)\mathrm{}}_u^s$=-(0–3.5)×${10}^{\mathrm{-}4}$ ${\mathrm{GeV}}^4$ and ψ${\mathrm{}\left(0\right)\mathrm{}}_u^d$=-(0–2.4)×${10}^{\mathrm{-}7}$ ${\mathrm{GeV}}^4$, where ψ${\mathrm{}\left(0\right)\mathrm{}}_i^j$ are the renormalization-group-invariant quantities ψ${\mathrm{}\left(0\right)\mathrm{}}_i^j$=-(m${¯}_j$-m${¯}_i$) 〈ψ${¯}_j$ ${\psi }_j$-ψ${¯}_i$ ${\psi }_i$〉. These values imply a small flavor symmetry breaking in the QCD nonperturbative vacuum, i.e., 〈s¯s〉/〈ūu〉=0.9±0.1 and 1-〈d¯d〉/〈ūu〉=(0–6)×${10}^{\mathrm{-}3}$. .AE