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In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted YW and corresponds to the gauge symmetry U(1).
Contents
It is conserved (only terms that are overall weak-hypercharge neutral are allowed in the Lagrangian). However, one of the interactions is with Higgs field. Since 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). Only a specific combination of them,
Mathematically, weak hypercharge is similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions (which is not conserved in weak interactions).
Definition
Weak hypercharge is the generator of the U(1) component of the electroweak gauge group, SU(2)×U(1) and its associated quantum field B 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 electrical charge (in elementary charge units) and T3 is the third component of weak isospin. Rearranging, the weak hypercharge can be explicitly defined as:
where "left" and "right"-handed here has to be understood as left and right chirality, not helicity.
Note: sometimes weak hypercharge is scaled so that
although this is a minority usage.
Hypercharge assignments in the Standard Model are determined up to a twofold ambiguity by demanding cancellation of all anomalies.
Baryon and lepton number
Weak hypercharge is related to baryon number minus lepton number via:
where X is a GUT-associated conserved quantum number. Since weak hypercharge is always conserved this implies that baryon number minus lepton number is also always conserved, within the Standard Model and most extensions.
Neutron decay
n
→
p
+
e−
+
ν
e
Hence neutron decay conserves baryon number B and lepton number L separately, so also the difference B − L is conserved.
Proton decay
Proton decay is a prediction of many grand unification theories.
p+
→
e+
+
π0
→
e+
+ 2
γ
Hence proton decay conserves B − L, even though it violates both lepton number and baryon number conservation.