Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Renormalization of the non-linear [itex]\sigma[/itex] model

  1. Aug 8, 2014 #1
    I have some questions about this paper:http://users.phys.psu.edu/~radu/extra_strings/freedman_sigma_model.pdf [Broken]

    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]
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Aug 13, 2014 #2
    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Renormalization of the non-linear [itex]\sigma[/itex] model
Loading...