A note on the large-$c$ conformal block asymptotics and $\alpha$-heavy operators

We consider $\alpha$-heavy conformal operators in CFT$_2$ which dimensions grow as $h = O(c^\alpha)$ with $\alpha$ being non-negative rational number and conjecture that the large-$c$ asymptotics of the respective 4-point Virasoro conformal block is exponentiated similar to the standard case of $\alpha=1$. It is shown that the leading exponent is given by a Puiseux polynomial which is a linear combination of power functions in the central charge with fractional powers decreasing from $\alpha$ to $0$ according to some pattern. Our analysis is limited by considering the first six explicit coefficients of the Virasoro block function in the coordinate. For simplicity, external primary operators are chosen to be of equal conformal dimensions that, therefore, includes the case of the vacuum conformal block. The consideration is also extended to the 4-point ${\cal W}_3$ conformal block of four semi-degenerate operators, in which case the exponentiation hypothesis works the same way. Here, only the first three block coefficients can be treated analytically.