Square of modified Dirac equation

[cbetanco]

If I take a modified Dirac Eq. of the form $(i\gamma^\mu \partial_\mu -M)\psi=0$ where $M=m+im_5 \gamma_5$, and whish to square it to get a Klein-Gordon like equation would I multiply on the left with $(i\gamma^\nu \partial_\nu +m+im_5\gamma_5)$ or $(i\gamma^\nu \partial_\nu +m-im_5\gamma_5)$?
I was under the impression that to take the square, you put a minus sign on the mass term and multiply with that expression on the left, but I am unsure if the $im_5\gamma_5$ term should also get the appropriate sign change, since its not the tradition mass term. Any thoughts?

 

[Bill_K]

I say use iγ_μ∂_μ + m - imγ₅. This will eliminate the cross terms.

 

[cbetanco]

Thanks. I was getting a little worried about the $m_5 \gamma^\nu\partial_\nu\gamma_5+m_5\gamma_5 \gamma^\mu \partial_\mu$ in my expression but $\gamma_5$ anti commutes with $\gamma^\mu$, so it goes away in the end. Thanks again.