# Lagrangian Taylor expansion

1. Jan 18, 2012

### Neitrino

Hi,
I have a following question....

Can it be that there is given some Lagrangian and instead of considering whole Lagrangian one makes its series expansion and considers only some orders of expansion? Can you bring some examples or why and when does this happen... ?

Thank you

2. Jan 18, 2012

### tom.stoer

One example are chiral effective theories like the skyrme model. The pion field appears as an SU(2) matrix

$$U(x)=e^{i\tau^a \phi^a(x)}$$

Nucleons are described by classical solitons

$$U_0(x)=e^{i\tau^a \phi_0^a(x)}$$

Now you can introduce fluctuations i.e. pions in a nucleon (soliton) background

$$U(x)=e^{i\tau^a (\phi_0^a + \pi^a)}$$

and expand the resulting Lagrangian in \pi(x) up to second order; you can couple this Lagrangian to el.-mag. fields and calculate photo-pion production; you can quantize the \pi(x) fluctuations and calculate quantum corrections to nucleon masses and form factors; ...

Have a look at (e.g.)

http://arxiv.org/abs/hep-ph/9602359
Quantum Corrections to Baryon Properties in Chiral Soliton ModelsAuthors: Frank Meier, Hans Walliser
(Fachbereich Physik, U-GH Siegen, Germany)
(Submitted on 21 Feb 1996 (v1), last revised 14 Aug 1996 (this version, v2))
Abstract: We present a procedure to calculate 1-loop graphs in the soliton sector of chiral Lagrangians and use it to calculate quantum corrections to certain baryon observables in Skyrme-type models. Results generally show an improvement over the values obtained in tree approximation except for the case of the axial coupling g_A.