Stress energy tensor in extended standard model

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
[itex]\Theta[/itex]^{[itex]\mu\nu[/itex]}=1/2 i [itex]\overline{\psi}[/itex] [itex]\gamma[/itex]^{[itex]\mu[/itex]}[itex]\stackrel{\leftrightarrow}{\partial^\nu}\psi[/itex]

But I get something more like
[itex]\Theta[/itex]^{[itex]\mu\nu[/itex]}=[itex]-1/2 i \overline{\psi}\gamma[/itex]^{[itex]\mu[/itex]}[itex]\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[/itex]

where [itex]\sigma^{\mu \nu}=-i \gamma^\mu \gamma^\nu[/itex]

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

 

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

 

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