# Beta function of the non-linear sigma model

In chapter 7.1.1. in Tong's notes about String Theory could someone sketch how can I show the statements that he makes 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 $$\beta_{\mu\nu}(G) = \alpha ' \mathcal{R}_{\mu \nu} \quad ?$$

• Look up the computation of one-loop beta functions in dimensional regularisation. – suresh Sep 25 '14 at 7:23

• In ε = d-2 dimensions, taking the "pion" fields φ to be dimensionless, but the bare metric to have dimension ε, to one loop rewrite your expression as $$(G_{\mu\nu}/\alpha')^{(0)}=M^\epsilon ( G_{\mu\nu}/\alpha' -\frac{1}{\epsilon}R^{(1)}_{\mu\nu} ).$$ But the bare α'-full metric must be independent of the RG scale M; so operating on this equation by $M \frac{d}{dM}$ at the pole $\epsilon \to 0$ nets $$0= M \frac{d}{dM} \frac{G_{\mu\nu}}{\alpha'} - R^{(1)}_{\mu\nu},$$ where the superscript (1) indicates the residue at the pole, and $$M \frac{d}{dM} \frac{G_{\mu\nu}}{\alpha'} = R^{(1)}_{\mu\nu}.$$
Thus, in our scaled conventions, in a hypersphere ($R_{\mu\nu}=2 G_{\mu\nu}$), whose inverse radius-squared, α', decreases with scale M , asymptotic freedom manifests itself: $d\alpha'/d\ln M =-2\alpha' ^2$. Asymptotically, the sphere flattens to a conformally invariant plane.