Beta function of the non-linear sigma model

Question

In chapter 7.1.1. in Tong's notes about String Theory could someone sketch how can I show the statements that he nmakes around eq. 7.5

• That the addition of the counterterm can be absorbed by renormalization the wavefunction and the metric

• How does he conclude from the renormalization $$G_{\mu \nu} \rightarrow G_{\mu \nu} + \dfrac{\alpha '}{\epsilon}\mathcal{R}_{\mu\nu}$$ that the beta function equals $$\beta_{\mu\nu}(G) = \alpha ' \mathcal{R}_{\mu \nu} \quad ?$$

This post imported from StackExchange Physics at 2014-10-01 22:39 (UTC), posted by SE-user Anne O'Nyme

Comments

Look up the computation of one-loop beta functions in dimensional regularisation.

This post imported from StackExchange Physics at 2014-10-01 22:39 (UTC), posted by SE-user suresh

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The calculation is done in Riemann normal coordinates in GSW, and it is easiest this way. The infinitesimal change in metric under integrating out high-frequency components of the string coordinates is the definition of the beta function of the nonlinear sigma model, although I found it obscure to do outside of Riemann coordinates. But the link you gave doesn't work, so I can't write an answer, but the answer is only correct to leading order in the inverse string tension, there are higher order corrections. You can understand everything but the coefficient from a "what else can it be" argument.

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I have now linked to the PDF which should work ...