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

Correlations between the strange quark mass, strange quark condensate $⟨\overline{s}s⟩$, 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 nonstrange quark mass ${r}_{m}={m}_{s}/{m}_{q}$, the condensate ratio ${r}_{c}=⟨\overline{s}s⟩/⟨\overline{q}q⟩$, and the kaon PCAC deviation parameter ${r}_{p}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}{m}_{s}⟨\overline{s}s+\overline{q}q⟩/2{f}_{K}^{2}{m}_{K}^{2}$. 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 assess 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.

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