and the KL formula was discussed at length in [@Kaplunovsky:1995jw]. The string theory formula takes the general form,

$$g_{a}^{-2}(\mu^{2};S,\bar{S},M,\bar{M})=k_{a}g_{string}^{-2}(S,\bar{S},M,\bar{M})+\frac{b_{a}}{16\pi^{2}}\ln\frac{M_{string}^{2}}{\mu^{2}}+\frac{\Delta_{a}(M,\bar{M})}{16\pi^{2}}.\label{eq:g-2string}$$

The term multiplying $k_{a}$ here is (the inverse square of) the gauge coupling at the string scale and is a sum of the classical term and a universal threshold correction $\Delta^{{\rm univ}}$. $M$ represents all the moduli. The last term is a possible non-universal threshold correction so that the total threshold correction is of the form. $$\tilde{\Delta}_{a}=\Delta_{a}+k_{a}\Delta^{{\rm univ}}.\label{eq:Deltatilde}$$ The main claim of [@Kaplunovsky:1995jw] is that the difference between $\tilde{\Delta}$ and the non-holomorphic terms in the KL formula, should be a holomorphic one loop correction to the gauge coupling. Thus the key relation between the two is $$\partial_{M}\partial_{\bar{M}}\tilde{\Delta}_{a}=\partial_{M}\partial_{\bar{M}}[c_{a}\hat{K}(M,\bar{M})-\sum_{r}T_{a}(r)\ln\det Z_{(r)}(M,\bar{M})].\label{eq:harmonic}$$ While much of the discussion concerns heterotic string theory the above is based on general arguments that apply to all string theories (or for that matter to any UV completion of field theory with heavy states). As pointed out in the introduction to this paper, “the non-harmonicity (of the physical couplings) is a purely low-energy effect and can be calculated from the low-energy EQFT (effective quantum field theory) without any knowledge of the superheavy particles; it is the harmonic terms in the moduli-dependent 