We present a systematic theoretical analysis of the ALICE measurement of low-$p_T$ direct-photon production in central lead-lead collisions at the LHC with a centre-of-mass energy of $sqrt{s_{NN}}=2.76$ TeV. Using next-to-leading order of perturbative QCD, we compute the relative contributions to prompt-photon production from different initial and final states and the theoretical uncertainties coming from independent variations of the renormalisation and factorisation scales, the nuclear parton densities and the fragmentation functions. Based on different fits to the unsubtracted and prompt-photon subtracted ALICE data, we consistently find an exponential, possibly thermal, photon spectrum from the quark-gluon plasma (or hot medium) with slope $T=304pm 58$ MeV and $309pm64$ MeV at $p_Tin[0.8;2.2]$ GeV and $p_Tin[1.5;3.5]$ GeV as well as a power-law ($p_T^{-4}$) behavior for $p_T>4$ GeV as predicted by QCD hard scattering.
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