Weak hypercharge

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Flavour in particle physics
Flavour quantum numbers:

Related quantum numbers:


Combinations:

  • Hypercharge: Y
    • Y = (B + S + C + B′ + T)
    • Y = 2 (QI3)
  • Weak hypercharge: YW
    • YW = 2 (QT3)
    • X + 2YW = 5 (BL)

Flavour mixing

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

Definition[edit]

It 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, usually written as YW, 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

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[edit]

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 also conserved[clarification needed] this implies that baryon number minus lepton number is also conserved, within the Standard Model and most extensions.[clarification needed]

Neutron decay[edit]

np + 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[edit]

Proton decay is a prediction of many grand unification theories.

p+e+ + π0e+ + 2γ

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

See also[edit]

Notes[edit]

  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.