Weak hypercharge

 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 Y_W and corresponds to the gauge symmetry U(1).^[1][2]

It is conserved (only terms that are overall weak-hypercharge neutral are allowed in the Lagrangian). However, one of the interactions is with the Higgs field. Since the 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 T₃). Only a specific combination of them, Q = T₃ + 1/2 Y_W (electric charge), is conserved.

Mathematically, weak hypercharge appears similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions (which is not conserved in weak interactions and is zero for leptons).

In the electroweak theory SU(2) transformations commute with U(1) transformations by definition and therefore U(1) charges for the elements of the SU(2) doublet (for example lefthanded up and down quarks) have to be equal. This is why U(1) cannot be identified with U(1)_em and weak hypercharge has to be introduced.^[3][4]

Weak hypercharge was first introduced by Sheldon Lee Glashow in 1961.^[5][6][7]

DefinitionEdit

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

See also: Weinberg angle

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 W ³ electroweak quantum field to produce the observed 
Z
 gauge boson and the photon of quantum electrodynamics.

The weak hypercharge satisfies the relation

${\displaystyle \qquad Q=T_{3}+{\tfrac {1}{2}}Y_{\rm {W}}~,}$

where Q is the electric charge (in elementary charge units) and T₃ is the third component of weak isospin (the SU(2) component).

Rearranging, the weak hypercharge can be explicitly defined as:

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

Fermion family	Left-chiral fermions				Right-chiral fermions
		Electric charge Q	Weak isospin T3	Weak hyper- charge YW		Electric charge Q	Weak isospin T3	Weak hyper- charge YW
Leptons	ν e,  ν μ,  ν τ	0	+1/2	−1	No interaction, if existent			0
	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 are left and right chirality, respectively (distinct from helicity).

Mediated interaction	Boson	Electric charge Q	Weak isospin T3	Weak hypercharge YW
Weak	W±	±1	±1	0
	Z0	0	0	0
Electromagnetic	γ	0	0	0
Higgs	H0	0	−1/2	+1

The pattern of weak isospin, T₃, and weak hypercharge, Y_W, 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.

The sum of −isospin and +charge is zero for each of the gauge bosons; consequently, all the electroweak gauge bosons have ${\displaystyle Y_{\text{W}}=0}$.

Hypercharge assignments in the Standard Model are determined up to a twofold ambiguity by requiring cancellation of all anomalies.

Alternative half-scaleEdit

For convenience, weak hypercharge is often represented at half-scale, so that

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

which is equal to just the average electric charge of the particles in the isospin multiplet.^[8]

Baryon and lepton numberEdit

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

${\displaystyle {\tfrac {1}{2}}X+Y_{\rm {W}}={\tfrac {5}{2}}(B-L)\,}$

where X is a conserved quantum number in GUT. 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 decayEdit

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 decayEdit

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.

This article uses material from the Wikipedia article  Metasyntactic variable, which is released under the  Creative Commons Attribution-ShareAlike 3.0 Unported License.