# A CP properties of field strength tensor

1. Sep 30, 2016

### 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.

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

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!

2. Oct 5, 2016

### Greg Bernhardt

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Oct 7, 2016

### 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.

4. Oct 8, 2016

### my2cts

It could be that σμν is not hermitian.

5. Oct 24, 2016

### 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.