# 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 frequently denoted YW and corresponds to the gauge symmetry U(1).[1]

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, ${\displaystyle Q=T_{3}+{\frac {Y_{\rm {W}}}{2}}}$ (electric charge), is conserved.

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

Weinberg angle θW, and relation between coupling constants g, g', and e. Adapted from T D Lee's book Particle Physics and Introduction to Field Theory (1981).

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

${\displaystyle \qquad Q=T_{3}+{Y_{\rm {W}} \over 2}}$

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:

${\displaystyle \qquad Y_{\rm {W}}=2(Q-T_{3})}$
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

where "left" and "right"-handed here has to be understood as left and right chirality, not helicity.

el. charge
Q
weak isospin
T3
weak
hyper-
charge
YW
Electroweak
W
±1 ±1 0

Z
0 0 0

γ
0 0 0
Higgs
H0
0 -1/2 +1
The pattern of weak isospin, T3, and weak hypercharge, YW, of the known elementary particles, showing electric charge, Q, along the Weinberg angle. The neutral Higgs field (circled) breaks the electroweak symmetry and interacts with other particles to give them mass. Three components of the Higgs field become part of the massive W and Z bosons.

Note: sometimes weak hypercharge is scaled so that

${\displaystyle \qquad Y_{\rm {W}}=Q-T_{3}}$

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:

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

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

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.