- #1
Siupa
- 30
- 5
When we try to see if the weak sector is CP invariant, we CP transform all the fields in the charged interactions terms and we get a condition involving the elements of the CKM matrix and the arbitrary phases of the CP transformed fields:
$$V_{ij} = V^*_{ij} \, e^{i(\xi_W + \phi_{d_j} - \phi_{u_i})}$$
Then, the argument goes: there are 9 parameters in V_CKM because it is a general 3x3 unitary matrix. These 9 parameters are split in 3 "angles" and 6 phases, the 3 angles being the ones you get if you restrict to an element of ##\text{SO}(3)##.
To make the above condition hold, we need to fix the CP phases to cancel the 6 CKM phases. We have 7 CP phases (1 from W, 3 from the downs and 3 from the ups), so it seems like we can do it.
But then we say "actually we have only 5 independent CP phases, because there are 2 residual global symmetries corresponding to baryon number and electric charge". Therefore, 1 phase remains in the CKM and the condition can never hold.
I don't understand the last point: why does the presence of ##\text{U}(1)_\text{B}## and ##\text{U}(1)_\text{Q}## global symmetries reduces the number of CP phases I can fix to cancel the CKM phases?
$$V_{ij} = V^*_{ij} \, e^{i(\xi_W + \phi_{d_j} - \phi_{u_i})}$$
Then, the argument goes: there are 9 parameters in V_CKM because it is a general 3x3 unitary matrix. These 9 parameters are split in 3 "angles" and 6 phases, the 3 angles being the ones you get if you restrict to an element of ##\text{SO}(3)##.
To make the above condition hold, we need to fix the CP phases to cancel the 6 CKM phases. We have 7 CP phases (1 from W, 3 from the downs and 3 from the ups), so it seems like we can do it.
But then we say "actually we have only 5 independent CP phases, because there are 2 residual global symmetries corresponding to baryon number and electric charge". Therefore, 1 phase remains in the CKM and the condition can never hold.
I don't understand the last point: why does the presence of ##\text{U}(1)_\text{B}## and ##\text{U}(1)_\text{Q}## global symmetries reduces the number of CP phases I can fix to cancel the CKM phases?
