How robust is a thermal photon interpretation of the ALICE low-p_T data?

We present a rigorous 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 NLO 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 T = 304 \pm 58 MeV and 309 \pm 64 MeV for the effective temperature of the quark-gluon plasma (or hot medium) at p_T \in [0.8;2.2] GeV and p_T \in [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.