The assumption that the production of quark–antiquark pairs and their sequential string-breaking takes place, likely as a tunneling process, through the event horizon of the color confinement determines the freezeout temperature and gives a plausible interpretation for the thermal pattern of elementary and nucleus–nucleus collisions. When relating the black-hole electric charges to the baryon-chemical potentials, it was found that the phenomenologically deduced parameters from the ratios of various particle species and the higher-order moments of net-proton multiplicity in the statistical thermal models and Polyakov linear-sigma model agree well with the ones determined from the thermal radiation from charged black hole. Accordingly, the resulting freezeout conditions, such as normalized entropy density [Formula: see text] and average energy per particle [Formula: see text][Formula: see text]GeV, are confirmed at finite chemical potentials as well. Furthermore, the problem of strangeness production in elementary collisions can be interpreted by thermal particle production from the Hawking–Unruh radiation. Consequently, the freezeout temperature depends on the quark masses. This leads to a deviation from full equilibrium and thus a suppression of the strangeness production in the elementary collisions. But in nucleus–nucleus collisions, an average temperature should be introduced in order to dilute the quark masses. This nearly removes the strangeness suppression. An extension to finite chemical potentials is introduced. The particle ratios of kaon-to-pion ([Formula: see text]), phi-to-kaon ([Formula: see text]) and antilambda-to-pion ([Formula: see text]) are determined from Hawking–Unruh radiation and compared with the thermal calculations and the measurements in different experiments. We conclude that these particle ratios can be reproduced, at least qualitatively, as Hawking–Unruh radiation at finite chemical potential. With increasing energy, both [Formula: see text] and [Formula: see text] keep their maximum values at low SPS energies. But the further energy decrease rapidly reduces both ratios. For [Formula: see text], there is an increase with increasing [Formula: see text], i.e., no saturation is to be observed.
Residues, modularity, and the Cardy limit of the 4d $$ \mathcal{N} $$ = 4 superconformal index
