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It is known that the masses of fermions in the Standard Model are represented in the form of singular values of complex Yukawa matrices (Yukawa couplings). The question is, are the values of the masses/couplings themselves real numbers? If so, what are the values of the imaginary part, what role do they play?

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Yes, the values of the masses are real positive numbers. Recall how you find them out of the complex matrix Y.

Note first that $Y Y^\dagger$ is hermitian, and has positive-eigenvalues, and so can be written as $$ Y Y^\dagger = U D ^2 U^\dagger $$ for some unitary U and diagonal real D with no zero entries, for simplicity.

Take the positive square roots. Define the hermitian matrix $$ H = UD U^\dagger ~~~~\leadsto ~~~ H^2= UD^2 U^\dagger ,\\ H^{-1}= U D^{-1} U^\dagger. $$ Define the unitary matrix $$ S\equiv H^{-1} Y ~~~ \leadsto ~~~ Y=H S, \\ Y= U D U^\dagger S\\ \equiv U D K^\dagger , $$ where K is unitary. Had we not taken the positive roots for the diagonal D, we could make it positive by incorporating the negative signs diagonal into $K^\dagger \equiv U^\dagger S$ preserving its unitarity.

So Y has been bidiagonalized to a real positive diagonal D, now safely identified with a diagonal M. Now you are ready to apply U or K, the one contiguous to the left-chiral fermion components, to produce the CKM matrix.

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