Correlations between the strange quark condensate, strange quark mass, and kaon PCAC relation

Abstract

Correlations between the strange quark mass, strange quark condensate \(\left\langle\bar{s}s\right\rangle\), and the kaon partially conserved axial current (PCAC) relation are developed. The key dimensionless and renormalization-group invariant quantities in these correlations are the ratio of the strange to non-strange quark mass \(r_m=m_s/m_q\), the condensate ratio \(r_c=\left\langle\bar{s}s\right\rangle/\left\langle\bar{q}q\right\rangle\), and the kaon PCAC deviation parameter \(r_p=−m_s\left\langle\bar{s}s+\bar{q}q\right\rangle/2f^2_K m^2_K\). The correlations define a self-consistent trajectory in the \({r_m,r_c,r_p}\) parameter space constraining strange quark parameters that can be used to assess the compatibility of different predictions of these parameters. Combining the constraint with Particle Data Group (PDG) values of \(r_m\) results in \({r_c,r_p}\) constraint trajectories that are used to asses the self-consistency of various theoretical determinations of \({r_c,r_p}\). The most precise determinations of \(r_c\) and \(r_p\) are shown to be mutually consistent with the constraint trajectories and provide improved bounds on \(r_p\). In general, the constraint trajectories combined with \(r_c\) determinations tend to provide more accurate bounds on \(r_p\) than direct determinations. The \({r_c,r_p}\) correlations provide a natural identification of a self-consistent set of strange quark mass and strange quark condensate parameters.