- #1
synoe
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I have some questions about this paper:http://users.phys.psu.edu/~radu/extra_strings/freedman_sigma_model.pdf
In section 3, they renormalize the bosonic non-linear [itex]\sigma[/itex] model at one loop level.
The action is given by
[tex]
I_B[\phi]=\frac{1}{2}\int d^2xg_{ij}(\phi^k)\partial_\mu\phi^i\partial_\mu\phi^j.
[/tex]
Perturbation [itex]\phi=\varphi+r[/itex] of this action in Riemann normal coordinate is written by
[tex]
\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}
[/tex]
The second term vanishes by using equation of motion. Then calculate the functional
[tex]
\Omega_B[\phi]=\langle0|\exp i\int d^2xL_{\text{int}}(\phi,\xi)|0\rangle
[/tex]
where
[tex]
\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.
[/tex]
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 [itex]\phi[/itex].
Why divergent diagrams are the three types?
The definition of [itex]L_\text{int}[/itex] is correct? I think it should be [itex]\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[/itex]
In section 3, they renormalize the bosonic non-linear [itex]\sigma[/itex] model at one loop level.
The action is given by
[tex]
I_B[\phi]=\frac{1}{2}\int d^2xg_{ij}(\phi^k)\partial_\mu\phi^i\partial_\mu\phi^j.
[/tex]
Perturbation [itex]\phi=\varphi+r[/itex] of this action in Riemann normal coordinate is written by
[tex]
\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}
[/tex]
The second term vanishes by using equation of motion. Then calculate the functional
[tex]
\Omega_B[\phi]=\langle0|\exp i\int d^2xL_{\text{int}}(\phi,\xi)|0\rangle
[/tex]
where
[tex]
\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.
[/tex]
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 [itex]\phi[/itex].
Why divergent diagrams are the three types?
The definition of [itex]L_\text{int}[/itex] is correct? I think it should be [itex]\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[/itex]
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