Weak hypercharge

The weak hypercharge in particle physics is a conserved quantum number relating the electrical charge and the third component of weak isospin, and is similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions (which is not conserved). It is frequently denoted Y_W and corresponds to the gauge symmetry U(1).^[1]

Definition

It is the generator of the U(1) component of the electroweak gauge group, SU(2)xU(1) and its associated quantum field B mixes with the W³ electroweak quantum field to produce the observed Z⁰ gauge boson and the photon of quantum electrodynamics.

Weak hypercharge, usually written as Y_W, is defined as:

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 expressed as:

Note: sometimes weak hypercharge is scaled so that

although this is a minority usage.^[2]

Baryon and lepton number

Weak hypercharge is related to baryon number - lepton number via:

where X is a GUT-associated conserved quantum number. Since weak hypercharge is also conserved this implies that baryon number - lepton number is also 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⁺ + 2γ

Hence proton decay conserves B-L, even though it violates both lepton number and baryon number conservation.

See also

• Standard Model (mathematical formulation)

Notes

1. ^ J. F. Donoghue, E. Golowich, B. R. Holstein (1994). Dynamics of the standard model. Cambridge University Press. pp. 52. ISBN 0521476526. 
2. ^ M. R. Anderson (2003). The mathematical theory of cosmic strings. CRC Press. pp. 12. ISBN 0750301600.