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Applying $\partial_1\overline{\partial}_1$, we have the following. First term:$$\partial_1\overline\partial_1 G_1 = -\pi\alpha' \eta^{\mu_1\mu_2} \delta^2 (z_1-z_2,\overline z_1-\overline z_2) X^{\mu_3} (z_3,\overline z_3) X^{\mu_4} (z_4,\overline z_4)$$$$+\text{ } {\rm permutations~of~indices~} (2,3,4).$$In the second term, $z_1$ can be the logarithm or ...