No CP-Violation for coinciding Quark-Masses

[Aigologist]

TL;DR Summary

How do hypothetically coinciding Quark-Masses affect CP-Symmetry

Hello everyone,
I know that if two of the quarks (e.g. strange & bottom) had coinciding masses, there would be no CP-violation in the standard model. Apparently the reason lies in the parameters of the CKM-matrix, but I don't understand how to show that. Can someone explain?

 

[malawi_glenn]

It lies in the (bi-)unitary transformations needed to diagnonalize the mass-matrix. In other words, the relation between weakly interacting egeinstates to mass eigenstates.

Good review by A. Pich https://cds.cern.ch/record/256553/files/9312297.pdf

 

[Orodruin]

TLDR; If two masses coincide then rotations in that flavor subspace become unphysical. You can therefore set one of the mixing angles to zero which makes the Jarlskog invariant zero. (Ie, no CP violation if any mixing angle is zero)

 

[malawi_glenn]

In terms of say Kaon decays, you would have perfect cancellation between loop diagrams containing say u and c quarks if they had same mass. But perhaps that is more related to FCNC in the SM now that I think about it...

 

[Vanadium 50]

Before discussing why a statement is true, we need to determine if a statement is true,

Your statement is not true. Quark masses do not have to be non-degenerate. The u-type quarks need to have unique masses, and the d-type quarks need to have unique masses, but a u-type and d-type quark can have the same masses.

If two u (or d)-type quarks have the same mass, the choice of definition of the quarks is arbitrary. One can always rotate them in such a way that the CP-violating amplitude is zero. And since this is just a convention, if it's true for any definition, it's true for any definition.

A slightly more mathematical and Swedish definition is that the CP violation is proportional to the following invariant:

$$J = \cos \theta_{12} \cos^2 \theta_{13} \sin \theta_{12} \sin \theta_{13} \sin \theta_{23} \sin \delta$$

If you have mass degeneracies, you can always define things so that J is zero by inspection.

 

[Aigologist]

TLDR; If two masses coincide then rotations in that flavor subspace become unphysical. You can therefore set one of the mixing angles to zero which makes the Jarlskog invariant zero. (Ie, no CP violation if any mixing angle is zero)

I can see that the Jarlskog invariant is zero, but the CKM-Matrix would still have one complex phase, right? How is that not conflicting?

 

[Vanadium 50]

CKM-Matrix would still have one complex phase, right? How is that not conflicting?

Write down an observable - any observable. You will find that it has no dependence on δ.

 

[malawi_glenn]

I can see that the Jarlskog invariant is zero, but the CKM-Matrix would still have one complex phase, right? How is that not conflicting?

If two quarks with same electric charge have same mass, there is an additional U(2) symmetry, which you can use to set that phase to zero.

 

[Orodruin]

I can see that the Jarlskog invariant is zero, but the CKM-Matrix would still have one complex phase, right? How is that not conflicting?

If two quarks with same electric charge have same mass, there is an additional U(2) symmetry, which you can use to set that phase to zero.

^^ What he said.

 

[malawi_glenn]

Swedish definition

I've actually met her one time when I visited Lund
Forgot to ask her to sign a printed copy of her famous paper. I think that was for the better, would have been pretty awkward.

 

[Orodruin]

I've actually met her one time when I visited Lund
Forgot to ask her to sign a printed copy of her famous paper. I think that was for the better, would have been pretty awkward.

With neutrino oscillations as my thesis topic, I met her on several occasions at seminars and conferences.

 

[Vanadium 50]

I've actually met her one time when I visited Lund

met her on several occasions at seminars and conferences.

She's very nice, no?

I was next to her on the #9 bus from CERN and I was advocating for Ray Davis to win the Nobel prize that year. He did. I tell myself that I was the deciding factor.

She also has a T-shirt with the Jarlskog invariant on it that her kids gave her.

Anyway, another way to say the same thing is that a general 3x3 CKM-like matrix has 3 (irremovable) angles and one phase. If you look at the special case where 2 generations can mix because they have the same mass and those states mix with the third, you only have two angles and no phase.

 

[malawi_glenn]

T-shirt with the Jarlskog invariant on it

And I only have those lame Higgs-potential and SM lagrangian ones :(

 

[Orodruin]

She's very nice, no?

Shi is. My wife (who has a PhD in particle physics too) met her at Nordita once and was absolutely charmed without knowing that it was Jarlskog.

Jarlskog’s talks are usually very full of flowers. I remember particularly one of her talks:
”I couldn’t make this equation beautiful no matter how hard I tried” *presses slide forward and flowers appear on slide*

 

[Orodruin]

and SM lagrangian one

The CERN one with the missing bar? :P

 

[malawi_glenn]

The CERN one with the missing bar? :P

No I got it elsewhere. I have not proof read it yet. But if the CERN t-shirt is good enough for John Ellis its good enough for me

 

[ohwilleke]

A slightly more mathematical and Swedish definition

I've never heard the term "Swedish definition" before. What does that mean?

 

[Vanadium 50]

I got it from Ikea.

 

[Vanadium 50]

J is the Jarlskog invariant. Cecelia Jarlskog is from Sweden.

 

[ohwilleke]

J is the Jarlskog invariant. Cecelia Jarlskog is from Sweden.

Ah! That makes sense. Thought it might be a term of art I wasn't familiar with or something.

 

[malawi_glenn]

Cecelia

Cecilia

The Greta Garbo of particle physics.