It is conserved (only terms that are overall weak-hypercharge neutral are allowed in the Lagrangian). However, one of the interactions is with the Higgs field. Since the Higgs field vacuum expectation value is nonzero, particles interact with this field all the time even in vacuum. This changes their weak hypercharge (and weak isospin T3). Only a specific combination of them, (electric charge), is conserved.
Mathematically, weak hypercharge appears similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions (which is not conserved in weak interactions and is zero for leptons).
In the electroweak theory SU(2) transformations commute with U(1) transformations by definition and therefore U(1) charges for the elements of the SU(2) doublet (for example lefthanded up and down quarks) have to be equal. This is why U(1) cannot be identified with U(1)em and weak hypercharge has to be introduced.[3][4]
Weak hypercharge is the generator of the U(1) component of the electroweak gauge group, SU(2)×U(1) and its associated quantum fieldB mixes with the W3 electroweak quantum field to produce the observed Z gauge boson and the photon of quantum electrodynamics.
The weak hypercharge satisfies the relation
where Q is the electric charge (in elementary charge units) and T3 is the third component of weak isospin (the SU(2) component).
Rearranging, the weak hypercharge can be explicitly defined as:
More information Fermion family, Left-chiral fermions ...
where "left"- and "right"-handed here are left and right chirality, respectively (distinct from helicity).
The weak hypercharge for an anti-fermion is the opposite of that of the corresponding fermion because the electric charge and the third component of the weak isospin reverse sign under charge conjugation.
where X is a conserved quantum number in GUT. Since weak hypercharge is always conserved within the Standard Model and most extensions, this implies that baryon number minus lepton number is also always conserved.
Hence this hypothetical proton decay would conserve B − L, even though it would individually violate conservation of both lepton number and baryon number.
Anderson, M.R. (2003). The Mathematical Theory of Cosmic Strings. CRC Press. p.12. ISBN0-7503-0160-0.
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