Weak hypercharge

Flavour in particle physics

Flavour quantum numbers: Isospin: I or I3 Charm: C Strangeness: S Topness: T Bottomness: B′ Related quantum numbers: Baryon number: B Lepton number: L Weak isospin: T or T3 Electric charge: Q X-charge: X Combinations: Hypercharge: Y Y = (B + S + C + B′ + T) Y = 2 (Q − I3) Weak hypercharge: YW YW = 2 (Q − T3) X + 2YW = 5 (B − L) Flavour mixing CKM matrix PMNS matrix Flavour complementarity

The weak hypercharge in particle physics is a conserved quantum number relating the electrical charge and the third component of weak isospin, and is similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions (which is not conserved). It is frequently denoted Y_W and corresponds to the gauge symmetry U(1).

Definition

It is the generator of the U(1) component of the electroweak gauge group, SU(2)xU(1) and its associated quantum field B mixes with the W³ electroweak quantum field to produce the observed Z⁰ gauge boson and the photon of quantum electrodynamics.

Weak hypercharge, usually written as Y_W, is defined as:

$\qquad Q = T_3 + {Y_W \over 2}$

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 expressed as:

$\qquad Y_W = 2(Q - T_3)$

Note: sometimes weak hypercharge is scaled so that

$\qquad Y_W = Q - T_3$

although this is a minority usage.

Baryon and lepton number

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

$X + 2Y_W = 5(B - L) \,$

where X is a GUT-associated conserved quantum number. Since weak hypercharge is also conserved this implies that baryon number minus lepton number is also 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+
+ 2γ

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

See also

• Standard Model (mathematical formulation)