MichPod said:
I am following
https://www.rainerhauser.ch/public/scripts/StandardModel1.pdf, page 27 and 28.
They transform the electron field in the Dirac equation by multiplying the wave function by some arbitrary magnitude-one complex value different in each point in the space-time, i.e. by arbitrary changing the phase of the wave function, then to compensate for this (to promise local symmetry in respect to phase change), they introduce a spin 1 massless field which couples with electron and which they identify as electromagnetic field.
So practically my question was what would be the result of the same procedure if applied to other particles, not electron (don't they have the same sort of local symmetry?). Like what field would we need to introduce to promise the same kind of local symmetry for a neutrino? Would it be meaningful if we try to promise a local symmetry of the same kind for a photon or, alternatively to some other massive boson like Z boson and which field would we need to introduce as a result of that?
QED can be constructed by "gauging" the mentioned global U(1) symmetry, leading to the introdoction of a "gauge connection", i.e., the electromagnetic potential. In principle can do this for any fields of charged particles, e.g., for pions (although there a more sophisticated version, called "vector meson dominance model(s)" where beyond pions also the light vector mesons are involved leads to much better phenomenological results).
As already mentioned above, since neutrinos are electrically neutral you cannot use this approach of gauging the U(1) symmetry. Here you have to read about the extension of the gauge principle to non-Abelian gauge symmetries. Here the "flavor symmetry" is gauged. The symmetry group is a (chiral) ##\text{SU}(2) \otimes \text{U}(1)## symmetry (weak isospin and weak hyper-charged) which is "Higgsed" to the electromagnetic ##\text{U}(1)## symmetry, i.e., of the four gauge bosons three get massive by absorbing the would-be Goldstone field-degrees of freedom (and that's why there's in fact no spontaneous symmetry breaking of a local gauge symmetry although it's still called this way after more than 60 years after discovery of the Anderson-Higgs-Kibble-et-al mechanism). So you have three massive gauge bosons realized in a "hidden local gauge symmetry", which are the ##W^{\pm}## and ##Z^0## vector bosons of the weak interaction and a massless photon. In the physical spectrum you have in addition to the lepons, quarks, and gauge bosons (at least) one massive scalar boson, the Higgs boson(s).
Also note that this electroweak Glashow-Salam-Weinberg model (aka quantum-flavor dynamics, QFD) is only conistent taking into account leptons (consisting of a charged lepton and a neutrino) and the quarks (an "up" and "down" version, each with 3 color-degrees of freedom) in each generation. We have 3 generations in the Standard Model, i.e., the electron, electron neutrino, up- and down-quarks; muons muon neutrinos, charm- and strange-quarks; and the tauon, tau neutrino, and top- and botton-quarks.
A good textbook at "B-level" is
O. Nachtmann, Elementary Particle Physics - Concepts and
Phenomenology, Springer-Verlag, Berlin, Heidelberg, New
York, London, Paris, Tokyo (1990).
It's a bit outdated, because naturally it doesn't contain a discussion of neutrino masses and mixing, but it's an approach which introduces the Standard Model "as simple as possible but not simpler".
