In the Standard Model, the Yukawa potential is given by Y(\overline{L}\phi R+\overline{R}\phi^{\dagger}L) and after symmetry breaking this term gives the fermion-Higgs interactions and at the lowest order the fermion mass term m\overline{\psi}\phi where m=Yv as known; on the other hand however, from the same fields it is also possible to construct the alternative Yukawa potential Gi(\overline{L}\phi R-\overline{R}\phi^{\dagger}L) which after symmetry breaking would give rise to additional fermion-Higgs interactions and at the lowest order a fermion mass-like term bi\overline{\psi}\gamma_{5}\psi where b=Gv as a new constant: this second term is parity-odd, but this is not a problem since the SM is already maximally parity-violating; aside from this both terms are real and they are Lorentz-invariant as well as gauge invariant for the SU(2)XU(1) group defining the SM... so why isn't this parity-odd Yukawa term added to the SM?
The Yukawa couplings already express parity violation, so the, supposedly, additional term, doesn't break parity any more than the other one. Just expand out the terms, recalling that there are, originally, four scalar fields and it's possible to find what are, just, phase choices and what aren't. Cf., for instance, http://master.particles.nl/LectureNotes/2011-PPII-Higgs.pdf
or
http://www-archive.ph.ed.ac.uk/sussp61/lectures/01_Sachradjda_StandardModel/standrews2.pdf
That the Yukawa couplings already express parity violation, so that the additional term doesn't break parity any more than the other one does, is true, and I already highlighted that, but I cannot really get your following thoughts about phase choices: could you be more specific, especially about the references?
If you denote the SU(2) doublet Higgs by φ, then Higgs potential contains φ2 and φ4 terms only. It is ensured (by SU(2) gauge invariance) that φ3 term is absent. This is because writing in terms of dimensions of irreducible representations, we get : 2x2=1+3, 1x2=2, 3x2=2+4. Therefore 2x2x2 does not contain a SU(2) singlet.
On the contrary, you may also consider an additional singlet scalar, say S. Then S3 term is allowed by SU(2) gauge invariance in general. To forbid this term you may impose a parity symmetry at the level of the Lagrangian by demanding that it is invariant under S->-S.
But there is no additional singlet scalar nor parity conservation in the standard model... so my question remains.
Mass terms for both scalars as well as gauge bosons are generated spontaneously after symmetry breaking. Adding a phenomenological mass term of fermions may spoil renormalizability. Sometimes soft mass terms of scalars (of mass dimension less than 4) are added by hand to break a symmetry softly. But we must admit that Yukawa couplings are unknown complex numbers. Only masses are experimentally measured. Therefore a model should generate experimentally measured quantities correctly with a minimum possible number of free parameters.
Please see the following course material from MIT by Frank Wilczek. Specially Eqn. 28.
http://web.mit.edu/8.701/www/Lecture%20Notes/8.701originsOfMass04FA13.pdf
That's exactly my point: Yukawa couplings are unknown complex numbers, so there can be both parity-even and parity-odd Yukawa terms; therefore my question about a possible reason to have only the parity-even and not the parity-odd. Weinberg once said that we have to put in the Standard Model all terms that are not forbidden by symmetries and renormalizability, and since those are certainly not then we have to include them: the parity-even terms are those that after symmetry breaking give the masses and on this I do agree, but this is a experimental reason. I am looking for a theoretical reason for which these terms should be forbidden as Yukawa terms from the beginning.
I think we write the Yukawa terms as Y \overline{L}\phi R+H.C. (Hermitian Conjugate). So if we write H.C. as Y\overline{R}\phi^{\dagger}L then perhaps we are assuming Y^{\dagger} = Y. On the other hand if we write Yukawa terms as Gi(\overline{L}\phi R-\overline{R}\phi^{\dagger}L) then probably we are assuming Gi^{\dagger} = -G. We can not make two different choices of phases in the Yukawa simultaneously, I guess.
For simplicity, we can assume that the VEV is real, and I guess also one generation :-) Then, if Y*=Y, we get the first case and on the contrary if Y*=-Y, we get the second case. Both are allowed because Yukawa couplings are arbitrary complex numbers. But not simultaneously.
If VEV is complex we have to absorb it's complex phase and redefine Y. Then go through the same argument.
The phase of Yukawa coupling can be absorbed into fermion spinor, after a field redefinition.
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