# What is application of nonlinear sigma model?

1. Sep 16, 2011

### ndung200790

I hear that linear sigma model has application in pion,nuclear physics.But do not understand what is the application of nonlinear sigma model in QFT?
Thank you very much in advance.

2. Sep 16, 2011

### tom.stoer

When compactifying the linear to the non-linear sigma model, i.e. when reducing the number of independent fields, it becomes possible to identify the three remaining fields with the pions. The nonlinearity is interpreted as pion-pion interaction, which is studied in chiral perturbation theory. One can add a gauge-invariant photon interaction and calculate e.g. pion-pion scattering or photon-pion scattering.

The non-linear sigma model allowes for topologically non-trivial soliton solutions by wrapping the SU(2) around S³ (compactified R³). Unfortunately these solitons are unstable; they would shrink to zero size by emitting pions. I studied an extension of the non-linear sigma model, the so-called Skyrme model where an order-4 term is added to the Lagrangian.This stabilizes the solition (the so-called Skyrmion) and allowes one to identify it with the nucleon. One can expand the Lagrangian into soliton + pion fluctuations around the nucleon. Based on this ansatz it's possible to study pion-nucleon scattering, photo-pion production etc. The Skyrmion itself allowes one to calculate the nucleon mass (rather good fit).

In addition one can quantize the pion fluctuations around the nucleon and calculate in second-order perturbation theory the (negative) energy shift of the nucleon mass with a remarkable good result (the calculation is terribly complicated b/c renormalization is required which is simple when using plane waves, but in the soliton background one has to use distorted waves for the pions)

Another application of the model is the calculation of electromagnetic form factors of the nucleon, including s-t- crossing. Again the agreement with the data is rather good. One idea I had in mind was to study inclusive deep inelastic scattering for which a small Q-x domain seems tobe available. My expectation was to find some non-perturbative results for nucleon structure functions, but the calculations became too complicated, so my decision was surrender ...

Hope this helps ...