How to determine the number of physical parameters in Yukawa couplings? Suppose we have a Lagrangian containing a series of Yukawa couplings between fermions $\psi_i$ and scalars $\phi_i$,
$$\mathcal{L} \supset y^k_{ij} \phi_k \overline{\psi}_i\psi_j + \rm h.c., $$
where $y^k_{ij}$ is a complex Yukawa coupling (there is one matrix $y^k$ for each scalar $\phi_k$).
Assuming $n$ generations of fermions, $y^k$ is an  $n\times n$ unitary matrix, which a priori has $n^2$ independent parameters. However, I know one can change the phase of the fields to get rid of some of these parameters, namely the complex phases, such that only a few of the couplings actually end up being complex (i.e. their phases are "physical").

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*How do we determine the number of "physical" parameters (moduli and complex phases) for Yukawa couplings in general?


*And how does it change if the Yukawa couplings have a weird flavor structure (e.g. some generations decoupled from the others)? For instance, I read in a paper that in the following model with 3 fermions and 2 scalars $\phi_A,\phi_B$,
$$\mathcal{L}\supset y^A_{11} \phi_A \overline{\psi}_1 \psi_1 + y^A_{23} \phi_A \overline{\psi}_2 \psi_3 + y^B_{11} \phi_B \overline{\psi}_1 \psi_1 + y^B_{23} \phi_B \overline{\psi}_2 \psi_3 + \rm h.c.  $$
only has 1 physical phase. How does one reach that conclusion?
 A: It is in most QFT books and the PDG, but I'll just review the n-generation CKM result for one Higgs, as per your comment, and you might play and experiment with more Higgses, special forms, and neutrinos (so, then, Majoranas).
The crucial point is that the unitary × CKM matrix V connects an n-vector of Up type quarks to an n-vector of Down ones, starting out life as $n^2$ independent parameters/d.o.f. Now each flavor of quarks can absorb a phase and be redefined with that phase, so the Up and Down
quarks can all soak up 2n phases of V, almost. Almost, because one of the phases is not real, it's dross; an overall phase acts equally on the Ups and the downs, the left and the right, so it commutes with V, and can move from left to right with no effort. So you really only have 2n -1 phases to absorb.
The remaining $n^2-2n+1= (n-1)^2$ parameters of V are for real, and cannot be redefined/absorbed away.  How many are rotation angles and how many are phases?
The SU(n) matrix can be decomposed to an O(n) orthogonal matrix and phases. O(n) has $n(n-1)/2$ generators, and hence parameters (angles), and the rest must be phases;
$$
(n-1)^2-(n-1)n/2= (n-1)(n-2)/2.
$$
So you see the point of KM, that $n\geq 3$ is necessary for a CP-violating phase in V. Can you compute the angles and phases for 5 generations?
There are lots of different parameterizations in the PDG and Wikipedia, to gain familiarity with the makeup of V.
