Measurement of beauty-strange meson production in Pb$-$Pb collisions at $\sqrt{s_{\mathrm{N}\mathrm{N}}}=5.02$ TeV via non-prompt ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ mesons

The production yields of non-prompt ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ mesons, namely ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ mesons from beauty-hadron decays, were measured for the first time as a function of the transverse momentum ($p_{\mathrm{T}}$) at midrapidity ($\mathrm{\mid }y\mathrm{\mid }<0.5$) in central and semi-central Pb$-$Pb collisions at a centre-of-mass energy per nucleon pair $\sqrt{s_{\mathrm{N}\mathrm{N}}}=5.02$ TeV with the ALICE experiment at the LHC. The ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ mesons and their charge conjugates were reconstructed from the hadronic decay channel $${\mathrm{D}}_{\mathrm{s}}$^{+} \to \phi\pi^+$,with$\phi \to {\rm K}^-{\rm K}^+$,inthe$4 < p_{\rm T}<36$GeV\mathrm{/}$c$and$2 < p_{\rm T}<24$GeV\mathrm{/}$c$intervalsforthe0$-$10$-$50% centrality classes, respectively. The measured yields of non-prompt ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ mesons are compared to those of prompt ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ and non-prompt ${\mathrm{D}}^0$ mesons by calculating the ratios of the production yields in Pb$-$Pb collisions and the nuclear modification factor $R_{\mathrm{A}\mathrm{A}}$. The ratio between the $R_{\mathrm{A}\mathrm{A}}$ of non-prompt ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ and prompt ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ mesons, and that between the $R_{\mathrm{A}\mathrm{A}}$ of non-prompt ${{\mathrm{D}}_{\mathrm{s}}}^{+}$ and non-prompt ${\mathrm{D}}^0$ mesons in central Pb$-$Pb collisions are found to be on average higher than unity in the $4<p_{\mathrm{T}}<12$ GeV/$c$ interval with a statistical significance of about $1.6\sigma$ and $1.7\sigma$, respectively. The measured $R_{\mathrm{A}\mathrm{A}}$ ratios are compared with the predictions of theoretical models of heavy-quark transport in a hydrodynamically expanding QGP that incorporate hadronisation via quark recombination.