How do you compute the OPE coefficients for the contribution from the identity in a 2D CFT?

Question

Using the OPE expansion:  where p indexes primary fields, h the weights of the fields, and k the level of the descendants.
I want to compute the coefficients that come from the identity operator:  up to and including level 2. (This is in 2D CFT)
I tried to compare the OPE expansion to the two point correlator: 
but the result seems trivial? Don't you just get 1 for the first coefficient, and then 0 for everything higher, since the identity operator has weight 0?

Sylvain Ribault · Answer

These computations are done in Section 2.2.4 of my review article https://arxiv.org/abs/1406.4290 . The coefficients you need can be extracted from Eqs. (2.2.58) - (2.2.60).
There are some subtleties due to the identity operator, whose $L_{-1}$ descendants vanish, and have ill-defined coefficients. Nevertheless, the coefficient of the energy-momentum tensor is unambiguous:
$$
f^{L_{-2}} = frac{2h_1}{c}  qquad text{with} qquad h_1=h_2
$$

How do you compute the OPE coefficients for the contribution from the identity in a 2D CFT?

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