Weak hypercharge

The weak hypercharge in particle physics is a quantum number relating the electric charge and the third component of weak isospin. It is conserved (only terms that are overall weak-hypercharge neutral are allowed in the Lagrangian) and is similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions (which is not conserved in weak interactions). It is frequently denoted YW and corresponds to the gauge symmetry U(1).[1]

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.

Weak hypercharge, usually written as YW, satisfies the equality:

$\qquad Q = T_3 + {Y_{\rm W} \over 2}$

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:

$\qquad Y_{\rm W} = 2(Q - T_3)$
left-handed el. charge
Q
weak isospin
T3
weak
hyper-
charge
YW
right-handed el. charge
Q
weak isospin
T3
weak
hyper-
charge
YW
Leptons ν
e
, ν
μ
, ν
τ
0 +1/2 −1 Do not interact (if exist at all)
e, μ, τ −1 −1/2 −1 e
R
, μ
R
, τ
R
−1 0 −2
Quarks u, c, t +2/3 +1/2 +1/3 u
R
, c
R
, t
R
+2/3 0 +4/3
d, s, b
−1/3 −1/2 +1/3 d
R
, s
R
, b
R
−1/3 0 −2/3

Note: sometimes weak hypercharge is scaled so that

$\qquad Y_{\rm W} = Q - T_3$

although this is a minority usage.[2]

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:

$X + 2Y_{\rm W} = 5(B - L) \,$

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

np + 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+ + π0e+ + 2γ

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