How is this Yukawa coupling term invariant?

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?

• Write the indices of the SU(2) matrices explicitly in the transformed term. You are skipping a step May 14, 2021 at 22:12
• May 15, 2021 at 0:26
• 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. May 18, 2021 at 18:03
• @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. May 20, 2021 at 19:17
• 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.... May 20, 2021 at 19:53

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