# Fixing $CP$ phases to cancel CKM phases

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 $$V_\text{CKM}$$ because it is a general $$3 \times 3$$ 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?

It is straightforward to see that only 5 of the 7 phases are independent, i.e. the unitary $$3\times 3$$ CKM matrix effectively only has a residual$$^1$$ $$\frac{U(1)^3\times U(1)^3}{U(1)}$$ flavor symmetry, cf. e.g. this Phys.SE post.
$$^1$$ The kinetic term for $$Q=\begin{pmatrix}u_L \cr u_d \end{pmatrix}$$ has a $$U(3)\times U(3)$$ flavor symmetry, which is broken by the Yukawa terms.