Chiral Symmetry and Charge Algebra
I'm following Section 5.1 in <PERSON> and <PERSON>'s Particle Physics book and I am having trouble reproducing some of the commutation relations.
The axial current is given by $$ (J_A^a)^\mu = \bar{\psi}\alpha\gamma^\mu\left(\frac{\lambda^a}{2}\right){\alpha\beta}\gamma_5\psi_\beta $$ where $\lambda^a$ are the Gell-Mann matrices. The associated charge is then $$ Q^{5a} = \int d^3 x\ \bar{\psi}{\alpha}(x)\gamma^0\left(\frac{\lambda^a}{2}\right){\alpha\beta}\gamma_5\psi_\beta(x) $$ The 'normal' current is given by $$ (J^a)^\mu = \bar{\psi}\alpha\gamma^\mu\left(\frac{\lambda^a}{2}\right){\alpha\beta}\psi_\beta $$ and has associated charge $$ Q^{a} = \int d^3 x\ \bar{\psi}{\alpha}(x)\gamma^0\left(\frac{\lambda^a}{2}\right){\alpha\beta}\psi_\beta(x) $$ I am trying to show that $$ [Q^a,Q^{5b}] = if^{abc}Q^{5c} $$ where $f^{abc}$ are the structure constants of SU(3). My work so far is, \begin{equation} \begin{split} [Q^a,Q^{5b}] &= \int d^3x\ d^3 y\ \left[\bar{\psi}\alpha(x)\gamma^0 (T^a){\alpha\beta}\psi_\beta(x),\bar{\psi}{\gamma}(y)\gamma^0(T^b){\gamma\delta}\gamma_5\psi_\delta(y)\right]\[0.25cm] &=\int d^3x\ d^3 y\ (T^a){\alpha\beta}(T^b){\gamma\delta} \left[\psi_\alpha^\dagger(x) \psi_\beta(x),\psi_{\gamma}^\dagger(y)\gamma_5\psi_\delta(y)\right] \end{split} \end{equation} Using, $$ [A B, C D]=A\left{B, C\right} D - AC\left{B,D\right} + \left{A,C\right}DB - C\left{A,D\right}B $$ and the normal anti-commutation relations for fermions, \begin{multline} \left[\psi_\alpha^\dagger(x) \psi_\beta(x),\psi_{\gamma}^\dagger(y)\gamma_5\psi_\delta(y)\right] =\ \psi_\alpha^\dagger(x)\left{\psi_\beta(x),\psi_\gamma^\dagger(y)\right}\gamma_5\psi_\delta(y) -\psi_\alpha^\dagger(x)\psi_\gamma^\dagger(y)\left{\psi_\beta(x),\gamma_5\psi_\delta(y)\right} -\psi_\gamma^\dagger(y)\left{\psi_\alpha^\dagger(x),\gamma_5\psi_\delta(y) \right}\psi_\beta(x) \end{multline} The first term is consistent what with I expect. However, I don't know how to deal with the anti-commutators that contain $\gamma_5$.
Any help would be appreciated! 656