Feynman Diagrams: Exploring Electroweak Vertices

[Jodahr]

I have a question about Feynman Diagrams:

let's say we have a process: up antidown -> W+ -> up antidown...

the first vertex is like V_CKM G PL ( mixing, gamma, projector)
the second is the same..only with the complex conjugate CKM matrix...
but why?...

If I compute the M* I have to bar the vertices..and there I got the same vertex..with the same flow..but there I would change PL to PR and interchange PL and Gamma..why is that the case?

 

[francesco85]

I have a question about Feynman Diagrams:

let's say we have a process: up antidown -> W+ -> up antidown...

the first vertex is like V_CKM G PL ( mixing, gamma, projector)
the second is the same..only with the complex conjugate CKM matrix...
but why?...

If I compute the M* I have to bar the vertices..and there I got the same vertex..with the same flow..but there I would change PL to PR and interchange PL and Gamma..why is that the case?

Hello, in my opinion the answer is the following: the Feynman diagram you are considering is composed of two vertices: in the first an up is destroyed, an antidown is destroyed and a W+ is created; in the second vertex an up is created, an antidown is created and a W+ is destroyed; so, roughly speaking, the first is associated with a term in the lagrangian like (u dbar W-), while the second with (ubar d W+), that is its hermitian conjugate (of course I have forgot all the contraction matrices...); this is the origin of the conjugation of the CKM matrix paramters (and, of course, one should be careful with the imaginary units!)

Francesco

 

[AdrianTheRock]

I think another way of looking at it is like this.

That Feynman diagram also describes the processes

u → W⁺ + d
W⁺ + d → u​

as all I have done here is replace the incoming anti-d with an outgoing d, and the outgoing anti-d with an incoming d.

The CKM matrix, as defined, is the factor for 'converting' down-type quarks to up-type, eg

|u> = V_ud |d>​

Provided only the three known generations of quarks exist, the CKM matrix must be unitary, and hence

V^-1 = V^†​

so

|d> = V^*_ud |u>​