SU(5) model, gauge boson decay

[Accidently]

I am reading Mukhanov's 'Physical Foundations of Cosmology'. He claims that in the minimal SU(5) model, CP violation of a heavy SU(5) gauge boson X decay arises at the tenth order of perturbation theory.

Is that correct? The tenth order perturbation theory would lead to a very complicated diagram and why one loop correction doesn't contribute?

Btw, what confuses me is on page 212.

Thanks

 

[fzero]

The result is explained in http://inspirehep.net/record/11184?ln=en, section 6.3 (there's a scanned version available from KEK at the bottom of that page). It's fairly complicated, but I can try to give a bit of an outline. First of all, any CP violation will come from complex Yukawa couplings, since the gauge couplings are chosen to be real in minimal SU(5). So the diagrams we want are ones containing fundamental scalars coupled to quarks. In particular, we need diagrams that contribute a complex contribution to $S \leftrightarrow \bar{S}$. These are obtained by sewing the types of diagrams (tree+1-loop, but with the gauge field $X$ replaced by the scalar $S$) that Mukhanov displays together on the quark lines.

The lowest order diagram is therefore 2-loop order, but as they explain, it involves a purely real combination of Yukawa matrices. The 3-loop diagrams cannot contribute, since the addition of a single internal scalar line to the previous diagrams converts an external $S$ to an $\bar{S}$ (or vice versa). The 4-loop diagrams also give a real combination of Yukawas. Finally, the 5-loop graphs do in fact contribute a complex phase. Note, if you happen to look at the paper, the authors don't bother to draw in the external scalar lines in their diagrams. Putting these in makes the 5-loop diagrams 10th order in perturbation theory.

 

[Accidently]

The result is explained in http://inspirehep.net/record/11184?ln=en, section 6.3 (there's a scanned version available from KEK at the bottom of that page). It's fairly complicated, but I can try to give a bit of an outline. First of all, any CP violation will come from complex Yukawa couplings, since the gauge couplings are chosen to be real in minimal SU(5). So the diagrams we want are ones containing fundamental scalars coupled to quarks. In particular, we need diagrams that contribute a complex contribution to $S \leftrightarrow \bar{S}$. These are obtained by sewing the types of diagrams (tree+1-loop, but with the gauge field $X$ replaced by the scalar $S$) that Mukhanov displays together on the quark lines.

The lowest order diagram is therefore 2-loop order, but as they explain, it involves a purely real combination of Yukawa matrices. The 3-loop diagrams cannot contribute, since the addition of a single internal scalar line to the previous diagrams converts an external $S$ to an $\bar{S}$ (or vice versa). The 4-loop diagrams also give a real combination of Yukawas. Finally, the 5-loop graphs do in fact contribute a complex phase. Note, if you happen to look at the paper, the authors don't bother to draw in the external scalar lines in their diagrams. Putting these in makes the 5-loop diagrams 10th order in perturbation theory.

Thanks. I got it.