Fixing $CP$ phases to cancel CKM phases

Question

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?

Answer 1

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.

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$^1$ 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.