# Renormalization of the non-linear $\sigma$ model

1. Aug 8, 2014

### synoe

In section 3, they renormalize the bosonic non-linear $\sigma$ model at one loop level.
The action is given by
$$I_B[\phi]=\frac{1}{2}\int d^2xg_{ij}(\phi^k)\partial_\mu\phi^i\partial_\mu\phi^j.$$
Perturbation $\phi=\varphi+r$ of this action in Riemann normal coordinate is written by
\begin{align} I_B^{(2)}[\varphi+r]=I_B[\varphi]+\int d^2xg_{ij}\partial_\mu\varphi^iD_\mu\xi^j+\frac{1}{2}\int d^2x\left[g_{ij}D_\mu\xi^iD^\mu \xi^j+R_{ik_1k_2j}\xi^{k_1}\xi^{k_2}\partial_\mu\varphi^i\partial^\mu \partial^j\varphi\right]. \end{align}
The second term vanishes by using equation of motion. Then calculate the functional
$$\Omega_B[\phi]=\langle0|\exp i\int d^2xL_{\text{int}}(\phi,\xi)|0\rangle$$
where
$$\int d^2xL_{\text{int}}(\phi,\xi)\equiv I_B^{(2)}[\phi,\xi^a]-\frac{1}{4}\int d^2x\partial_\mu\xi^a\partial_\mu\xi^a.$$

According to this paper, the diagrams are like FIG2 and divergent one-loop diagrams are only these three types.

Why diagrams can be drawn like FIG2? I don't know how to treat the external field $\phi$.
Why divergent diagrams are the three types?
The definition of $L_\text{int}$ is correct? I think it should be $\int d^2xL_{\text{int}}(\phi,\xi)\equiv I_B^{(2)}[\phi,\xi^a]-\frac{1}{2}\int d^2x\partial_\mu\xi^a\partial_\mu\xi^a$

Last edited by a moderator: May 6, 2017
2. Aug 13, 2014