We discuss in the spirit of the work of Shifman, Vainshtein, and Zakharov (SVZ) sum rules involving current-current vacuum correlation functions, whose Wilson expansions start off with the operators $\overline{q}q$ or ${\left(\overline{q}q\right)}^2$, and thus provide information about the chiral-symmetry-breaking parameters of QCD. We point out that under the type of crude approximations made by SVZ, a value of ${〈\overline{q}q〉}_{\mathrm{vac}}={\left(157\mathrm{} \mathrm{MeV}\right)}^3$ is obtained from one of these sum rules, somewhat smaller than expectations. Further, we show that a Borel-transformed version of the Weinberg sum rule for $VV-AA$ current products seems only to make sense for an $A_1$ mass close to 1.3 GeV and it makes little sense with the current-algebra mass $M_A=\sqrt{2}{M}_{\rho }$. We also give an estimate for the chiral-symmetry-breaking parameters ${{\mu }_1}^6=2\mathrm{}{〈g^2\left({\overline{q}}_L{\lambda }^a{\gamma }_{\mu }{q}_l\right)\left({\overline{q}}_R{\lambda }^q{\gamma }^{\mu }{q}_R\right)〉}_{\mathrm{vac}}$, entering in the Weinberg sum rules, and ${{\mu }_2}^6=g^2{〈\left({\overline{q}}_R{\lambda }^a{\sigma }_{\mu \nu }{q}_L\right)\left({\overline{q}}_R{\lambda }^a{\sigma }^{\mu \nu }{q}_L\right)〉}_{\mathrm{vac}}$, entering in a new sum rule we propose, involving antisymmetric tensor currents $J=\overline{q}{\sigma }_{\mu \nu }q$.

• Received 27 October 1981

DOI:https://doi.org/10.1103/PhysRevD.26.2430