Virasoro vacuum block at next-to-leading order in the heavy-light limit

We consider the semiclassical limit of the vacuum Virasoro block describing the diagonal 4-point correlation functions on the sphere. At large central charge c, after exponentiation, it depends on two fixed ratios h(H)/c and h(L)/c, where h(H,L) are the conformal dimensions of the 4-point function operators. The semiclassical block may be expanded in powers of the light ratio hL/c and the leading non-trivial (linear) order is known in closed form as a function of h(H/c). Recently, this contribution has been matched against AdS(3) gravity calculations where heavy operators build up a classical geometry corresponding to a BTZ black hole, while the light operators are described by a geodesic in this background. Here, we compute for the first time the next-to-leading quadratic correction O((h(L)/c)(2)), again in closed form for generic heavy operator ratio h(H/c). The result is a highly non-trivial extension of the leading order and may be relevant for further refined AdS(3)/CFT2 tests. Applications to the two-interval Renyi entropy are also presented.