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

Correlations between the strange quark mass, strange quark condensate $langle bar s srangle$, 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 $r_c=langle bar s srangle/langle bar q qrangle$, and the kaon PCAC deviation parameter $r_p=-m_slangle bar s s+bar q qrangle/2f_K^2m_K^2$. 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.