Renormalizability of the Polyakov Action

Question

I was told today that the Polyakov action for a $p$-brane is (superficially) re-normalizable iff $p\leq 1$. Of course, when I went to check for myself, I screwed up my power-counting, and I'm having trouble seeing why.

We work in units with $c=1=\hbar$, so that $L=T=M^{-1}$. In these units, any action must have dimension $1$, so from looking at the Nambu-Goto action, $$ S_{\text{NG}}:=-T_p\int d\sigma ^{1+p}\sqrt{-g}, $$ we see that $[T_p]=L^{-(1+p)}=M^{1+p}$. From the Polyakov action, $$ S_{\text{P}}:=-\frac{T_p}{2}\int d\sigma ^{1+p}\sqrt{-h}h^{\alpha \beta}\partial _\alpha X^\mu \partial _\beta X^\nu G_{\mu \nu}(X), $$ we see that the the coupling constant of the interaction of of the scalar fields $X^\mu$ with $h_{\alpha \beta}$ is precisely $\frac{T_p}{2}$, which has dimensions $M^{1+p}$, and so is going to be (superficially) re-normalizable iff $1+p\geq 0$ . . . But this, of course, is not the result I was looking for . . . Where is my mistake?

Answer 1

The problem with my analysis before was that $T_p/2$ is not the relevant coupling constant. To read off the coupling constant, we must put the action into a slightly different form, in which the kinetic and interaction terms are separate and apparent.

To do that, we assume that everything is (real) analytic, we work locally and pick coordinates $x^\mu$ on space-time (the codomain manifold) such that $G(0)=\eta _{\mu \nu}$ in these coordinates. Then, $G(x)=\eta _{\mu \nu}+\varepsilon f^{(1)}_{\mu \nu \rho}x^\rho +\cdots$, where we are taking $f^{(1)}$ to be dimensionless, so that $[\varepsilon ]=L^{-1}=M$. Plugging this into the action, we find $$ S_{\text{P}}=\int d\sigma ^{1+p}\, \left[ -\frac{T_p}{2}\partial _\alpha X^\mu \partial ^\alpha X_\mu -\varepsilon \frac{T_p}{2}f^{(1)}_{\mu \nu \rho}(\partial _\alpha X^\mu )(\partial ^\alpha X^\nu )X^\rho +\cdots \right] . $$ To get the kinetic term in the usual form so that we can read-off the appropriate coupling constant, we define $Y:=\sqrt{T_p}X$ and write the action in terms of $Y$: $$ S_{\text{P}}=\int d\sigma ^{1+p}\, \left[ -\frac{1}{2}\partial _\alpha Y^\mu \partial ^\alpha Y_\mu -\frac{\varepsilon}{2\sqrt{T_p}}f^{(1)}_{\mu \nu \rho}(\partial _\alpha Y^\mu )(\partial ^\alpha Y^\nu )Y^\rho +\cdots \right] . $$ From this, we see that the coupling constant of the lowest-order interaction is $$ \frac{\varepsilon}{2\sqrt{T_p}}f^{(1)}_{\mu \nu \rho}, $$ which has mass dimension $$ 1-1/2(1+p)=1/2-1/2p. $$ The theory will thus be superficially renormalizable iff $1/2-1/2p\geq 0$, that is, iff $p\leq 1$.

This is essentially user10001's answer given in the comments above, with further details added.