Pairing in quantum-critical systems: T_c, Δ and their ratio

We calcualte the ratio of the pairing gap Δ at T=0 and T_c for a set of quantum-critical models in which the pairing interaction is mediated by a gapless boson(the γ model, where different γ corresponds to different pairing models). The ratio 2Δ/T_c has been recently computed numerically for 0<γ<2 within Eliashberg theory and was found to increase with increasing γ [T-H Lee et al, arXiv:1805.10280]. We argue that the origin of the increase is the divergence of this ratio at γ = 3. We obtain an approximate analytical formula for the ratio for γ<3 and show that it fits numerical data well. We also consider in detail the opposite limit of small γ. Here we obtain the explicit expressions for T_c and Δ, including numerical prefactors. We show that these prefactors depend on fermionic self-energy in a rather non-trivial way. The dependence is, however, the same for T_c and Δ, and the ratio 2Δ/T_c approaches the BCS value 3.53 when γ approaches to 0.