The PHENIX collaboration presents first measurements of low-momentum ($0.4<p_T<3$ GeV/$c$) direct-photon yields from Au$+$Au collisions at $sqrt{s_{_{NN}}}$=39 and 62.4 GeV. For both beam energies the direct-photon yields are substantially enhanced with respect to expectations from prompt processes, similar to the yields observed in Au$+$Au collisions at $sqrt{s_{_{NN}}}$=200. Analyzing the photon yield as a function of the experimental observable $dN_{rm ch}/deta$ reveals that the low-momentum ($>$1,GeV/$c$) direct-photon yield $dN_{gamma}^{rm dir}/deta$ is a smooth function of $dN_{rm ch}/deta$ and can be well described as proportional to $(dN_{rm ch}/deta)^alpha$ with $alpha{approx}1.25$. This scaling behavior holds for a wide range of beam energies at the Relativistic Heavy Ion Collider and the Large Hadron Collider, for centrality selected samples, as well as for different, $A$$+$$A$ collision systems. At a given beam energy the scaling also holds for high $p_T$ ($>5$,GeV/$c$) but when results from different collision energies are compared, an additional $sqrt{s_{_{NN}}}$-dependent multiplicative factor is needed to describe the integrated-direct-photon yield.
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