# Is the right-handed electron really an $SU(2)$ singlet?

In the Standard Model, neutrinos are assumed to be massless, and the right-handed neutrinos thus do not exit. Is this the reason that the right-handed electron is regarded as an $SU(2)$ singlet? However, to be more precise (going beyond the Standard Model), neutrinos do have masses. This implies that right-handed neutrinos must exist. Then, is it more reasonable to think a right-handed electron and a right-handed neutrino to be an $SU(2)_{R}$ doublet? If this is true, the left-handed $SU(2)_{L}$ doublet $(\nu_{e}, e^{-})_{L}$ and the right-handed $SU(2)_{R}$ doublet $(\nu_{e}, e^{-})_{R}$ are symmetric, and the Yukawa interactions between the Higgs field and these lepton doublets will be entirely different from that of the Standard Model.

• Your first sentence contains an incorrect conclusion. The reason is that no reaction has ever been observed that would involve a right handed neutrino. Aug 5, 2018 at 21:28
• @my2cts - If the neutrino is massless and left-handed, then the right-handed neutrino is impossible to exist. This is also a theoretically valid reason.
– Shen
Aug 6, 2018 at 6:23
• This is tautology. Aug 6, 2018 at 10:24

Many of the above misconceptions would not have been enunciated and built up to an impossible crescendo if only a good summary of the SM were consulted before taking off on their avalanche.

The standard model is a tight gauge theory of charged (V-A) interactions and neutral current interactions of subtler chirality constraints. So it is simply not true that

In the Standard Model, neutrinos are assumed to be massless, and the right-handed neutrinos thus do not exit.

When the SM was written down, no neutrino masses were known, so right-handed neutrinos were not included in it. Nevertheless, the electron did get a mass by a suitable higgs coupling to the left-handed electron multiplied by a right-handed SU(2)L-singlet electron. This is the celebrated "second job of the higgs", and theorists cheered when they appreciated its genius.

As a result, instantly in 1968, any theorist would reassure you that, ipso facto, the neutrino could easily get a mass with a very analogous coupling of the neutrino to a conjugate higgs doublet, if there were a reason to accommodate such. (In fact, in the extension to quarks, both up and down type quarks got their masses via SU(2)L-singlet right-chiral components.) They simply stuck to Occam's razor and left neutrino masses an open question, skipping them until discovery. It is doubly important to understand that the discovery of neutrino masses did not upset or reverse absolutely anything about the structure and logic of the SM.

(Indeed, to hype up the surprising neutrino mass discovery, sensationalist self-promoting sides kept on insisting they went "beyond the SM", but this only betrayed they had never fully appreciated its symmetry structure and logic and had no compunction about confusing the impressionable: this question may be one of their victims. So "within" or "beyond" the SM are mere meaningless words expressly tossed in to confuse the confusable, by isolating those in the know. There comes a time when opinionatedness may well be less damaging than its avoidance.)

While the full nature of neutrino masses is still open, an SM Dirac mass for them analogous to the electron mass is not excluded.

Now, having reassured full left-chiral invariance at low, energies, one might dream about inter-connecting the right-handed leptons with fanciful hypothetical interactions, truly beyond the SM: there have been models putting the electron and the neutrino in right-handed doublets, but they involve superheavy W analogs in a different gauge group SU(2)R, and completely different exotic couplings visible at much, much higher energies than accessible at present, and I assume you are not asking about them: extant experimental bounds reassure us that no such things are faintly visible or even arguable at the standard energies of HEP today. The PDG provides exclusion limits for right-handed charged currents.

My strong urge is to suspend speculation of beyond-the-SM worlds unless a full admiring appreciation of the perfect-fitting-puzzle nature of the SM were achieved, fitting together tons of low (ours) energy data. To answer the question, then, the right-handed e is incontrovertibly an SM gauge singlet and there is absolutely no evidence or hint it might be part of an SU(2)R doublet.

• "Many of the above misconceptions would not have been enunciated and built up to an impossible crescendo if only a good summary of the SM were consulted before taking off on their avalanche." - How could one not smile when reading that. I believe I shall post this to my profile page. Aug 5, 2018 at 21:40
• Indeed, reading Zachos' answers is always a pleasure. One must remember to keep their Zachos to English dictionary at hand, though. Aug 5, 2018 at 21:44