On page 160-161 in Tong's notes on string, he calculates one-loop divergent diagram and finds that we need a counter term of $\frac{\alpha'}{3\epsilon}\mathcal{R}_{\mu\nu}\partial Y^{\mu}\partial Y^{\nu}$ (I have put in the coefficient of $\alpha'$). It seems to me that a wavefunction renormalization $Y^{\mu}\to Y^{\mu}+\frac{\alpha'}{6\epsilon}\mathcal{R}^\mu _{ \ \nu} Y^{\nu}$ is sufficient to absorb the counter term. However, Tong says that a renormalization of the metric $G_{\mu\nu}$ by $G_{\mu\nu}\to G_{\mu\nu}+\frac{\alpha'}{\epsilon}\mathcal{R}_{\mu\nu}$ is also needed.
Could some one explain to me why we need both the wavefunction renormalization and metric renormalization here? Does this have to do with the covariance?
Thank you.
