Is neutrino mass really physics "beyond the Standard Model"? I took a postgraduate course in "Unification" during my MSc Physics degree. The lecture notes have things like "neutrinos are predicted to be massless" and "neutrino mass signals physics beyond the standard model".
However, aren't right handed neutrinos not included in the Standard Model because nobody ever observed them for sure? There's no neutrino mass term analogous to quarks or electrons because only left handed ones are in the SM, but if right handed ones were not included on purpose, how is this even a prediction? Isn't it a direct consequence of not including right handed ones...?
If there are right handed sterile neutrinos (that don't feel the weak force), can't we just add an analogous right handed singlet neutrino term, just like for the electron? Then we just "extend" the standard model - that's not new physics beyond it.
I understand that if they are Majorana particles, and if there's a Majorana mass term, then e.g. the Higgs field needs to be a triplet, or if we add both a Majorana & Dirac mass term there's the seesaw mechanism which predicts some new particles, but even then, that doesn't feel like "new Physics", and it feels like the extension of the SM, not something "beyond" it.
Based on trying to find similar questions to mine here, it really appears to me neutrinos aren't predicted to be massless, I guess I'm just looking for confirmation. So my real question is, do any of the extension of the SM to add neutrino mass signal new physics? What does even count as new physics? I think it would be something like a fifth force. Not sure what the general agreement is on "new physics".
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So my real question is, do any of the extension of the SM to add neutrino mass signal new physics? What does even count as new physics? I think it would be something like a fifth force. Not sure what the general agreement is on "new physics".

Your question is opinion based, and there is no solid answer to it. Lots of question on this site address it, and it is up to you to answer it. The SM is a loose term specifically designed to accommodate its systematic abuse. It was introduced to bypass pedantic questions as to whether something is part of the "Weinberg-Salam model" or not. Today, it represents the EW chiral gauge group and the Higgs Yukawa couplings giving mass to fermions consistently with that chiral group, notably both up and down-like quarks, and so charged leptons, and, if need be, neutrinos as well, via the very same mechanism. So, even if neutrinos were not known to be massive at the time, and so that Yukawa coupling had no good reason to be written down, any well-meaning learner was never in doubt about its legitimacy. Today's high-school wall charts include neutrino masses.
So, endless discussion about "part of the standard model" or not are essentially theological, and a subjective assessment of what "general agreement" means. The proper question to ask is "accommodated in the SM". If/when new elements such as Majorana masses, etc, are experimentally observed and accepted, they would ultimately enter a "new/improved" SM, that's the "standard" in the name;  and flesh out its ultimate development. Again, the term was invented by Pais and Treiman in 1975 precisely so people would  desist from arguing about this sort of thing.
A: Not including right neutrinos in the Standard Model is a prediction - it predicts that right neutrinos don't exist, or a more minimal statement would be that they do not interact with everything else we do include within the SM (including left neutrinos).
Having sterile right neutrinos that don't interact with the weak is beyond the SM, saying they only extend the SM means just the same. They are not part of it now.
A: The reason massive neutrinos is considered beyond-the-standard-model (BSM) is because even if you want to add neutrino mass in the most straight forward way -- add a right-handed neutrino current and having it couple to the Higgs, then just make that coupling arbitrarily small to match whatever the neutrino mass is -- it still leaves you with a problem.
There's no reason you can't still write down a Majorana mass term and say it contributes to the neutrino mass. So either the neutrino does still have a Majorana mass contribution (which is new physics), or it doesn't. If it doesn't have a Majarona term there must be a reason. That reason would be something like a new symmetry that forbids Majorana terms. That new symmetry would be BSM too. So right now there's no way to add neutrino mass without it being new physics.
