Kaons in hot and dense QCD

We present a systematic QCD sum-rule analysis of the in-medium properties of the charged kaon doublet $K^{\pm}$ over the full $(T,\rho)$ plane relevant to current and forthcoming heavy-ion experiments. Working within the QCD sum-rule framework and incorporating temperature-and density-dependent quark, gluon, and mixed condensates, we derive Borel-transformed sum rules for the effective masses $m_{K^{\pm}}$, the pseudoscalar decay constants $f_{K^{\pm}}$, and the vector self-energy $\Sigma_{v}$ of both charged states simultaneously. Our vacuum results, $m_{K^{-}} = 494.6^{+4.9}_{-6.9}$~MeV and $f_{K^{-}} = 157.3^{+4.1}_{-2.9}$~MeV (with near-degenerate $K^{+}$ values), are in excellent agreement with Particle Data Group values at the sub-percent level. In the medium, $m_{K^{\pm}}$ decreases monotonically with increasing baryon density and temperature, signalling progressive partial restoration of chiral symmetry. A pronounced mass splitting $\Delta m = m_{K^{-}} - m_{K^{+}}$ develops in baryonic matter, driven by the opposite sign of the Weinberg--Tomozawa vector interaction for the two charge states; it reaches $|\Delta m| \sim 0.35$~GeV near $\rho \simeq 3.2\,\rho_{\rm sat}$ at $T = 0$ and is partially quenched by thermal fluctuations. A central outcome of this study is the extraction of the critical onset density $\rho_c$, defined as the threshold beyond which the in-medium modifications of $K^{-}$ properties signal the onset of the transition toward the chirally restored phase. We stress that $\rho_c(T)$ should not be interpreted as a precise determination of the QCD critical point-a task beyond the reach of any current effective framework-but rather as an indicator ....