I read 'The Standard Model Effective Field Theory at Work' by Isidor, Wilsch, and Wyler. In a footnote, they say that, in principle, right-handed neutrinos could be included in the Standard Model by extending the fermion content. But these would be completely neutral under the Group of Standard Model. Why is that a problem? Why is the chiral fermion such a necessity?
...right-handed neutrinos could be included in the Standard Model by extending the fermion content. But these would be completely neutral under the Group of Standard Model. Why is that a problem? Why is the chiral fermion such a necessity?
Most QFT texts covering the SM, such as M Schwartz's, deal with this.
It is not a problem. Modern texts could, or should, include the unresolved theoretical possibility of Right-chiral neutrinos. These are possible, completely inert under both the SU(2) and the hypercharge U(1) of the SM, and can couple to their left-chiral active mates and the higgs in a gauge invariant way, to produce conventional Dirac mass terms, exactly like up-like quarks do in the SM. (However, $u_R$ quarks do have non-vanishing hypercharge, so they are not fully inert/sterile in the SM, by contrast to R neutrinos.)
People are reluctant to introduce completely sterile/inert d.o.f., but, of course, they now speculate about such all the time. There is no logical necessity in sticking to purely L chiral fermions, and, as seen above, this does not obtain for quarks anyway. The historical reason bad science reporters made a hash of the issue about 20 years ago is because R neutrinos were unnecessary in the simplest version of the SM covered in books, with no indication around of neutrino masses.
So, texts left their logical possibility out, like zoology books leaving unicorns out. Upon the discovery of neutrino masses, R neutrinos became an almost attractive possibility, and clueless science reporters started perorating on particles "beyond the standard model", a misunderstanding your footnote attempts to moderate. R-chiral neutrinos do not violate anything about the SM, defined through its symmetries, not so much its particle content.