this is already a one-loop effect so this one-loop change in $K$ is not going to affect the KL formula to one loop.

Type IIB case
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Again we start with the 1-loop form of the KL formula but now we write $$K^{(0)}=-2\ln{\cal V}+\tilde{K}((S,\bar{S};z,\bar{z}).\label{eq:K0}$$ We have separated the Kaehler moduli dependence (through the internal volume ${\cal V}$) from the complex structure ($z$) dependence. Note that here in contrast to the heterotic case the appropriate axio-dilaton field in the orientifolded type IIB case is $$S=e^{-\phi}+ia$$ where $e^{\phi}$ is the 10 dimensional string coupling, and $a$ is the RR zero form field (see for example [@Polchinski:1998rr]). As in the LVS models discussed in [@Blumenhagen:2009gk][@deAlwis:2009fn] it is assumed that the MSSM is located on D3 branes at a singularity (or D7 branes wrapping a collapsing cycle). For these local models (or in fact for any no-scale like embedding of the matter sector valid even in the heterotic case) one can write $Z_{\bar{I}J}^{(0)}=\frac{1}{{\cal V}^{2/3}}\delta_{\bar{I}J}$ to leading order in the large volume expansion.

Then the one-loop coupling function becomes $$\begin{aligned}
\frac{1}{g_{a}^{2}(\Phi,\bar{\Phi};\mu^{2})} & = & \Re f_{a}(\Phi)+\frac{b_{a}}{16\pi^{2}}\ln\frac{\Lambda^{2}{\cal V}^{-2/3}}{\mu^{2}}+\frac{c_{a}}{16\pi^{2}}\tilde{K}((S,\bar{S};U,\bar{U})\nonumber \\
 &  & +\frac{T(G_{a})}{8\pi^{2}}\ln\Re f_{a}(\Phi).\label{eq:g_aIIB}\end{aligned}$$ The (holomorphic) gauge coupling functions are given by, $$f_{a}=k_{a}S+\alpha_{ai}T^{i}$$ In the case at 