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 $T 3$ ). Only a specific combination of them, $Q=T_{3}+{\tfrac {1}{2}}Y_{\rm {W}}$ (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 which does not apply to leptons.

Definition

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).

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.

The Weak hypercharge satisfies the relation

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

where $Q$ is the electric charge (in elementary charge units) and $T$ ₃ is the third component of weak isospin (the SU(2) component).

Rearranging, the weak hypercharge can be explicitly defined as:

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

Fermion Family	Left- chiral Fermions	Electric Charge Q	Weak Isospin T₃	Weak Hyper- Charge Y_W	Right- chiral Fermions	Electric Charge Q	Weak isospin T₃	Weak Hyper- Charge Y_W
Leptons	ν _e, ν _μ, ν _τ	0	+¹⁄₂	−1	No interaction (if they exist)
Leptons	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
Quarks	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 Fundamental Interaction	Boson	Electric Charge Q	Weak Isospin T₃	Weak Hyper- charge Y_W
Weak	W	±1	±1	0
Weak	Z	0	0	0
Electric	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.

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

Alternative scale

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

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

which is equal to the average electric charge of the particles in the isospin multiplet, although this is a minority usage.^[3]

Baryon and lepton number

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

{\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 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.

References

↑ J. F. Donoghue; E. Golowich; B. R. Holstein (1994). Dynamics of the standard model. Cambridge University Press. p. 52. ISBN 0-521-47652-6.
↑ T. P. Cheng; L. F. Li (2006). Gauge theory of elementary particle physics. Oxford University Press. ISBN 0-19-851961-3.
↑ Anderson, M. R. (2003). The mathematical theory of cosmic strings. CRC Press. p. 12. ISBN 0-7503-0160-0.

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[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] T. P. Cheng; L. F. Li (2006). Gauge theory of elementary particle physics. Oxford University Press. ISBN 0-19-851961-3.

[3] Anderson, M. R. (2003). The mathematical theory of cosmic strings. CRC Press. p. 12. ISBN 0-7503-0160-0.

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: Y_W Y_W = 2 (Q − T₃) X + 2Y_W = 5 (B − L)
Flavour mixing
CKM matrix PMNS matrix Flavour complementarity