# Fixing CP phases to cancel CKM phases

• I
• Siupa
In summary, when trying to determine if the weak sector is CP invariant, we CP transform all the fields in the charged interaction terms and obtain a condition involving the elements of the CKM matrix and the arbitrary phases of the CP transformed fields. This condition requires fixing the CP phases to cancel the 6 CKM phases, but there are only 5 independent CP phases due to the presence of 2 residual global symmetries (U(1)B and U(1)Q) corresponding to baryon number and electric charge. This means that there is always 1 phase remaining in the CKM matrix that cannot be canceled, making the condition impossible to hold.

#### Siupa

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?

You can use two of those phases to be included in your U(1)B and U(1)Q global transformations of your fields (one for each transformation)

Last edited:
• vanhees71