In layman's terms, what is a Lagrange
Why does the Standard Model predict that neutrinos are massless?
I agree with the quantum physicist's answer that the zero mass for neutrinos was an input to the Standard Model, not a prediction, as measurements showed a mass compatible with zero.
But I will add that the discovery that neutrinos must have mass does not destroy the Standard Model, it just has to contain a different Lagrangian for the neutrinos.
The experimentally established phenomenon of neutrino oscillation, in which neutrino taste states are mixed with neutrino mass states (analogous to CKM mixing), however, requires that neutrinos have non-zero masses. Massive neutrinos were originally conceived by Bruno Pontecorvo in the 1950s. Adding the backbone to match their mass is easy by adding a right-handed Lagrangian. This can be done in two ways. If, as with other basic standard model particles, mass is generated by the Dirac mechanism, the scaffold would require an SU (2) singlet. This particle would have no other Standard Model interactions (apart from the Yukawa interactions with the neutral component of the Higgs doublet), so it is called a sterile neutrino. Or mass can be created by the Majorana mechanism, in which neutrino and antineutrino must be the same particle.
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