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I have the seen the following term used to describe Yukawa coupling for the lepton and Higgs field:

$$ \epsilon_{ij}\phi^{i}\bar{e}_{R}f_{L}^{j} $$

Under $SU(2) \otimes U(1)$ and so expected the following transformations: $$\phi \rightarrow \mathrm{e}^{i\alpha \cdot \tau/2}\mathrm{e}^{i\beta/2}\phi$$ $$ e_{R} \rightarrow \mathrm{e}^{-i\beta}e_{R}$$ $$f_{L} \rightarrow \mathrm{e}^{i\alpha \cdot \tau/2}\mathrm{e}^{-i\beta/2}f_{L}$$

but this does not give me an invariant term? Is there something obvious that I don't understand?

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  • $\begingroup$ Write the indices of the SU(2) matrices explicitly in the transformed term. You are skipping a step $\endgroup$
    – FrodCube
    May 14, 2021 at 22:12
  • $\begingroup$ Linked. $\endgroup$ May 15, 2021 at 0:26
  • $\begingroup$ Hi, sorry I didn't respond back immediately and thank you for you're answer it was very interesting. The only thing that surprised me is that my question was related to a question that I saw online which asked to prove that this term is invariant however, what you wrote contradictory. $\endgroup$ May 18, 2021 at 18:03
  • $\begingroup$ @CosmasZachos, I also believe this is the term (in question) that Sredniki uses in equation 88.5 of his textbook if I am not mistaken. $\endgroup$ May 20, 2021 at 19:17
  • $\begingroup$ Holy smoke! It is well known that Srednicki's conventions are upside down from mainstream usage, so the symbols you write are completely different than Wikipedia's, Schwarz's, and of course the original Weinberg model paper, etc.... I will detail the multiplets in my answer to remind you exactly what I mean! The neutral Higgs has its veg in the lower position of the isospinor. Check the charges! Phew.... $\endgroup$ May 20, 2021 at 19:53

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The transformation laws you wrote are correct, with $Y=2(Q-T_3)$ being 1 for $\phi$; -2 for $e_R$; and -1 for $f_L$. $$ f_L=\begin{pmatrix} \nu \\ e \end{pmatrix}, \qquad \phi= \begin{pmatrix} \phi^+\\ v+h + i\phi^0 \end{pmatrix}. $$

But the Yukawa you wrote is not invariant in mainstream (original) conventions, say, Peskin & Schroeder, Li & Cheng, Donoghue & Holstein, WP, and, and moreover, etc., and thus nonexistent!

First, recall the actual SM Yukawa term giving the electron its mass, $$ \bar e_R ~\phi^\dagger \cdot f_L = \bar e_R ~\phi ^- \nu_L +\bar e_R e_L ~(-i\phi^0 +v + h) , $$ +h.c.. The mass term comes from the v.e.v., and you should be able to tell it is invariant under the correct transformations of your question. Note how the positron component ensures this term has net hypercharge 0, whence net charge 0, much unlike what you wrote.

What you most probably meant to write is the alternate invariant Yukawa term which would give neutrinos a Dirac mass (analogous to how up-like quarks get their masses), $$ -\bar \nu_R ~\phi i\sigma_2 \cdot f_L= \bar \nu_R ~(i\phi^0+v+h)\nu_L -\bar \nu_R ~\phi^+ e_L , $$ which is also a hypercharge (and hence charge) singlet, given the null hypercharge of the sterile right-handed neutrino. Linked.


  • NB It became apparent from your comments that you are mixing mainstream conventions (transformation laws, except you redefined hypercharges by halving them all, the "modern", revisionist way) with Srednicki ones. He uses, by dint of his perverse notations, a Higgs doublet with the v.e.v. on the upper component, his (87.13), unlike the rest of the world. He is thus using the adjoint of the conjugate Higgs of the last paragraph above, calling it the Higgs field; which he can, but I did not have the patience to find where he discusses his left turn. Caveat lector. In any case, when in doubt in such checks, always run to the charges. The expressions I wrote conserve charge, and his only do provided you adopt the perverse choice (87.13). The advantages/disadvantages of his choices might be a good separate question.
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