CP properties of field strength tensor

[d8586]

Hi,

I am trying to figure out why a term like

$L \sim i \bar \psi_L \sigma^{\mu\nu}G_{\mu\nu}^a t^a \psi_R + h.c=$
$= i \bar \psi_L \sigma^{\mu\nu}G_{\mu\nu}^a t^a \psi_R - i \bar \psi_R \sigma^{\mu\nu}G_{\mu\nu}^a t^a \psi_L$

violates CP by looking at all the terms composing the Lagrangian.

I made a calculation (with the guidance of http://www.physics.princeton.edu/~mcdonald/examples/EP/feinberg_pr_108_878_57.pdf) and I obtained that the fermionic part withouth the $\gamma^5$, which goes like $\bar \psi \gamma^\mu \gamma^\nu \psi$,
transforms as

$[(-1)] [(-1)^\mu (-1)^\nu]$, first square bracket for C and second for P and #(-1)^\mu=1# for #\mu=0# and -1 otherwise

The term with the $\gamma^5$ transforms as

$[(-1)] [(-1) (-1)^\mu (-1)^\nu ]$

and naively, by writing $G_{\mu\nu}=\partial_\mu G_\nu - \partial_\nu G_\mu$ and taking into account the vector properties of $G_\mu$ and that $C(G)=-1$ I assumed that the field strength transforms as

$[(-1)] [(-1)^\mu (-1)^\nu]$

In this way, under CP, the lagrangian goes into

$L \sim i \bar \psi_R \sigma^{\mu\nu}G_{\mu\nu}^a t^a \psi_L + h.c$ which implies CP violation, by comparing to the second term of the original lagrangian, since there is a sign difference. In case where there is no $\gamma^5$ CP is preserved.

However I was then thinking a bit more about this and immediately realized that

$G_{\mu\nu}=\partial_\mu G_\nu - \partial_\nu G_\mu + i [G_\mu,G_\nu]$

it seems to me that the derivative terms have the following CP properties

$\mu=0, \nu=i~~ P=-1, C=-1 \to CP=+1$
$\mu=i, \nu=j ~~ P=+1, C=-1 \to CP=-1$

whereas the $G^2$ proportional term seems to go

$\mu=0, \nu=i ~~ P=-1, C=+1 \to CP=-1$
$\mu=i, \nu=j ~~ P=+1, C=+1 \to CP=+1$

where $C=+1$ since I have two fields that have $C=-1$. This has opposite transformation property of the derivative part. Where I am mistaking?

Thanks a lot!

 

[saybrook1]

This is definitely an interesting question, I'm just a bit confused as to what exactly you're asking. You're trying to pull apart and manipulate that lagrangian to check for CP violation? It's been a little while for me but can you use a test function? Like Greg said, if you could reword it a little bit I would be interested in helping you get to the bottom of this.

 

[my2cts]

It could be that σ^μν is not hermitian.

 

[d8586]

Hi All,
Sorry for my silence but I've been offline for a bit.

So, to rephrase it a bit, I could say that I don't understand the CP properties transformation of the gluon fields strength tensor, since it seems to me that the derivative part and the pure non-abelian part (the G^2 term) have different transformation properties. Of course there is something I am missing here, but I don't know what...

To reply to my2cts, yes, $\sigma^{\mu\nu}$ is not hermitian, it's h.c. picks up a sign, but then you have a $i$ in the Lagrangian that fixes the issue.