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 Y_W 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

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 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

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

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

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.

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. p. 52. ISBN 0-521-47652-6.
2. M. R. Anderson (2003). The mathematical theory of cosmic strings. CRC Press. p. 12. ISBN 0-7503-0160-0.