Hi,(adsbygoogle = window.adsbygoogle || []).push({});

If I have the Lagrangian [itex] L=i\chi^{\dagger\alpha i}\bar{\sigma}^{\mu}(D_{\mu})_{\alpha}^{\beta}\chi_{\beta i}+i\xi^{\dagger}_{\bar{i}\alpha}\bar{\sigma}^{\mu}(\bar{D}_{\mu})^{\alpha}_{\beta}\xi^{\beta i}-1/4 F^{a\mu\nu}F_{\mu\nu}^{a} [/itex] where [itex]\alpha,\beta [/itex] are colour indices, and i=1,2 is a flavour index (the Lagrangian is for two massless quarks, approximating u,d quarks only), and spinor indices are supressed. chi and xi are both LH Weyl fiels. See Srednicki ch83 for more details, available free online.

Then it's obvious that this Lagrangian has global flavour symmetry [itex] \chi_{\alpha i}\to L_{i}^{j}\chi_{\alpha j} [/itex], [itex],\xi^{\alpha\bar{i}}\to (R*)^{\bar{i}}_{\bar{j}} \xi ^{\alpha\bar{j}} [/itex], where L and R* are constant unitary matrices and the c.c. of R just a notational convention. So we have [itex]U(2)_L \times U(2)_R [/itex] sym.

Then I can see that if we set [itex]L=R*=e^{i\alpha}I [/itex] , equivalent to [itex] \Psi\to e^{-i\alpha\gamma_5}\Psi [/itex] in terms of Dirac field then there is an anomaly in this axial U(1) sym, so I presume we just exclude this? then left over is the non-anomlous symmetry. Srednicki says this is [itex]SU(2)_L \times SU(2)_R \times U(1)_V [/itex], why is this the case? how has excluding this anomlous axial U(1) symmetry reduced [itex] U(2)_L\times U(2)_R [/itex] TO [itex]SU(2)_L\times SU(2)_R\times U(1)_V [/itex]?

thanks for any pointers

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Chiral Lagrangian symmetry

Loading...

Similar Threads - Chiral Lagrangian symmetry | Date |
---|---|

I Helicity and chirality | Jul 10, 2017 |

I Does the Chirality match the helicity? | May 28, 2017 |

A Trace in QCD lagrangian | Nov 23, 2016 |

Reading off masses of eight goldstone bosons from chiral Lagrangian mass term | Jan 12, 2012 |

Feynman Rules for Electroweak Effective Chiral Lagrangian - How to calculate them? | Aug 8, 2011 |

**Physics Forums - The Fusion of Science and Community**