Weak hypercharge

 ( Nuclear Physics ) 

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 $Y\_\backslash mathsf\{W\}$ and corresponds to the gauge symmetry U(1).

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 ). Only a specific combination of them, $\backslash \; Q\; =\; T\_3\; +\; \backslash tfrac\{1\}\{2\}\backslash ,\; Y\_\backslash mathsf\{W\}\backslash$ (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 Commutator 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.

Weak hypercharge was first introduced by Sheldon Glashow in 1961.

The weak hypercharge satisfies the relation

$Q\; =\; T\_3\; +\; \backslash tfrac\{1\}\{2\}\; Y\_\backslash text\{W\}\; ~,$

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

Rearranging, the weak hypercharge can be explicitly defined as:

$Y\_\{\backslash rm\; W\}\; =\; 2(Q\; -\; T\_3)$

! Interaction
 mediated
! [[Boson]]
! Electric 
 charge 

! Weak 
 isospin 

! Weak 
 hypercharge 

The sum of −isospin and +charge is zero for each of the gauge bosons; consequently, all the electroweak gauge bosons have

$\backslash ,\; Y\_\backslash text\{W\}\; =\; 0\; ~.$

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

$\backslash ,\; Y\_\{\backslash rm\; W\}\; =\; Q\; -\; T\_3\; ~,$

which is equal to just the average electric charge of the particles in the isospin multiplet.

$\backslash tfrac\{1\}\{2\}X\; +\; Y\_\{\backslash rm\; W\}\; =\; \backslash tfrac\{5\}\{2\}(B\; -\; L)\; \backslash ,$

where X is a conserved quantum number in GUT. Since weak hypercharge is always conserved within the Standard Model and most extensions, this implies that baryon number minus lepton number is also always conserved.

 \rm p^+ \longrightarrow e^+ + & \ \pi^0 \\
 & \downarrow \\
 & 2\gamma
     

Hence this hypothetical proton decay would conserve , even though it would individually violate conservation of both lepton number and baryon number.

• Standard Model (mathematical formulation)
• Weak charge