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 YW 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
T3). Only a specific combination of them,
Q
=
T
3
+
1
2
Y
W
{\displaystyle 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 ==
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
Q
=
T
3
+
1
2
Y
W
,
{\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:
Y
W
=
2
(
Q
−
T
3
)
{\displaystyle \qquad Y_{\rm {W}}=2(Q-T_{3})}
where "left"- and "right"-handed here are
left and right chirality, respectively (distinct
from helicity).
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
Y
W
=
Q
−
T
3
,
{\displaystyle \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.
== Baryon and lepton number ==
Weak hypercharge is related to baryon number
minus lepton number via:
1
2
X
+
Y
W
=
5
2
(
B
−
L
)
{\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 decay ===
n → p + e− + νeHence 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+ + 2γHence proton decay
conserves B − L, even though it violates
both lepton number and baryon number conservation.
== See also ==
Standard Model (mathematical formulation
