Conventions:
$\bullet\ $ Everything is expressed in lightcone coordinates defined as $$\sigma_{\pm}=\frac{1}{\sqrt{2}}(\sigma_{1}\pm\sigma_{2})$$ $\bullet\ |\sigma_{12}|$ is the distance between two points.
Background:
We are considering string propagation in arbitrary backgrounds. The two-dimensional surface described by the string is the worldsheet whose coordinates are $\sigma$. I need to calculate beta functions when we consider background fields coupled to free theory. In order to do that, I need to extract out the singularity from the expectation values of vertex operators. These singularities are quantified by using a minimum distance cutoff $\epsilon$ in the worldsheet.
Problem
Consider the two-dimensional free field theory action :
$$S=\frac{1}{2\pi\alpha^{'}}\int{d^2\sigma\partial_{+}\eta^{\mu}\partial_{-}\eta_{\mu}}.$$
The propagator is given as $$\left<\eta^{\mu}{(\sigma_{1})}\eta^{\nu}(\sigma_{2})\right>=-\frac{\alpha^{'}}{2}\delta^{\mu\nu}\ln{|\sigma_{12}|^{2}}.$$
We will be using point-splitting regularization, where we introduce a minimum distance in the worldsheet, which is $\epsilon$ taken as constant. Therefore, the regularized propagator is given as
$$\left<\eta^{\mu}{(\sigma)}\eta^{\nu}(\sigma)\right>=\left<\eta^{\mu}{(\sigma)}\eta^{\nu}(\sigma+\epsilon)\right>=-\frac{\alpha^{'}}{2}\delta^{\mu\nu}\ln{(\epsilon^{2})}.$$
We also have the regularized operator $\left<e^{ik.\eta}\right>$: $$\left<e^{ik.\eta}\right>=\epsilon^{\frac{\alpha^{'}}{2}k^2}.$$
Here $\epsilon$ quantifies the singularity that occurs due to self-contraction.
Now I need to calculate $\left<\partial_{+}\eta^{\mu}(\sigma)\ e^{ik.\eta(\sigma)}\right>$. Here both operators are inserted at the same point.
First method: $$\left<\partial_{+}\eta^{\mu}\ e^{ik.\eta}\right>=\frac{1}{ik_{\mu}}\partial_{+}\left<e^{ik.\eta}\right>=\partial_{+}(\epsilon^{\frac{\alpha^{'}}{2}k^2})=0.$$
Second method: $$\left<\partial_{+}\eta^{\mu}(\sigma)\ e^{ik.\eta(\sigma)}\right>=\lim_{\sigma_{1}\rightarrow\sigma_{2}}\left<\partial_{+}\eta^{\mu}(\sigma_{1})\ e^{ik.\eta(\sigma_{2})}\right>$$ $$\Rightarrow \lim_{\sigma_{1}\rightarrow\sigma_{2}}(-\frac{\alpha^{'}}{2}\frac{ik^{\mu}}{(\sigma_{1}-\sigma_{2})_{+}}\epsilon^{\frac{\alpha^{'}}{2}k^2})=-\frac{\alpha^{'}}{2}\frac{ik^{\mu}}{\epsilon}\ \epsilon^{\frac{\alpha^{'}}{2}k^2}.$$
In last step we used the fact that minimum distance $\sigma_1$ and $\sigma_2$ can approach is $\epsilon$ and also since $\epsilon$ is infinitesimal, the component of the distance vector is approximated as the distance ($(\delta\vec{x})_{+}\approx|\delta\vec{x}|$)
Now, which of these is the correct result?
