Stress energy tensor in extended standard model

[cbetanco]

Maybe this is the wrong topic for this forum, but i am reading the following http://arxiv.org/abs/hep-ph/9703464
Which is on CPT violation and the standard model.

I do not understand how they get to equation (9), when they write down the stress energy tensor as
\Theta^\mu\nu=1/2 i \overline{\psi} \gamma^\mu\stackrel{\leftrightarrow}{\partial^\nu}\psi

But I get something more like
\Theta^\mu\nu=-1/2 i \overline{\psi}\gamma^\mu\stackrel{\leftrightarrow}{\partial^\nu}\psi + 1/4 (\sigma^{\mu\nu}+\sigma^{\nu\mu}) \overline{\psi} \gamma^\alpha \stackrel{\leftrightarrow}{\partial_\alpha}\psi+1/2 i (\sigma^{\mu\nu}+\sigma^{\nu\mu})(a_\alpha \overline{\psi} \gamma^\alpha \psi+ b_\alpha \overline{\psi} \gamma_5 \gamma^\alpha \psi)-m\overline{\psi} \psi

where \sigma^{\mu \nu}=-i \gamma^\mu \gamma^\nu

But then I get the same equation 10 as they do. Am I missing something? I know this requires you read and calculate this yourself to check (actually, you can just start from the lagrangian in equation 5 and use definition of the stress energy tensor). Thanks

 

[cbetanco]

sorry, should have said g^{\mu \nu} instead of -i/2 (\sigma^{\mu \nu}+\sigma^{\nu \mu})

 

[cbetanco]

never mind, I figured it out. Eq 9 is written neglecting the additional terms.