- #1
LAHLH
- 405
- 2
Hi,
If I have the Lagrangian 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} where \alpha,\beta 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 \chi_{\alpha i}\to L_{i}^{j}\chi_{\alpha j}, ,\xi^{\alpha\bar{i}}\to (R*)^{\bar{i}}_{\bar{j}} \xi ^{\alpha\bar{j}}, where L and R* are constant unitary matrices and the c.c. of R just a notational convention. So we have U(2)_L \times U(2)_R sym.
Then I can see that if we set L=R*=e^{i\alpha}I , equivalent to \Psi\to e^{-i\alpha\gamma_5}\Psi 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 SU(2)_L \times SU(2)_R \times U(1)_V, why is this the case? how has excluding this anomlous axial U(1) symmetry reduced U(2)_L\times U(2)_R TO SU(2)_L\times SU(2)_R\times U(1)_V?
thanks for any pointers
If I have the Lagrangian 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} where \alpha,\beta 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 \chi_{\alpha i}\to L_{i}^{j}\chi_{\alpha j}, ,\xi^{\alpha\bar{i}}\to (R*)^{\bar{i}}_{\bar{j}} \xi ^{\alpha\bar{j}}, where L and R* are constant unitary matrices and the c.c. of R just a notational convention. So we have U(2)_L \times U(2)_R sym.
Then I can see that if we set L=R*=e^{i\alpha}I , equivalent to \Psi\to e^{-i\alpha\gamma_5}\Psi 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 SU(2)_L \times SU(2)_R \times U(1)_V, why is this the case? how has excluding this anomlous axial U(1) symmetry reduced U(2)_L\times U(2)_R TO SU(2)_L\times SU(2)_R\times U(1)_V?
thanks for any pointers