Current divergence

Question

How do I calculate the current divergence  $$partial_mu A_{ud}^mu = partial_muleft (overline{u} gamma^mu gamma^5 d(x)right )?$$
I don't understand exactly how to manipulate gamma matrices.

Jurandi Leão · Answer

I did the following:
$$
partial_mu A_{ud}^mu =  (partial_mu overline{u}(x))gamma^mugamma^5 d(x) + overline{u}(x))(gamma^mu gamma^5)(partial_mu d(x))
= (i m_u overline{u}(x))gamma^5d(x) + overline{u}(x)(-gamma^5)(-i m_d d(x)
= i(m_u + m_d) overline{u}(x) gamma^5 d(x)
$$

Current divergence

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