weak hypercharge

Weak hypercharge is the generator of the U(1) component of the electroweak gauge group, {{gaps|SU(2)|×|U(1)}} and its associated quantum field {{math|B}} mixes with the {{math|W}}{{sup|3}} electroweak quantum field to produce the observed {{math| {{subatomic particle|Z boson|link=yes}} }} gauge boson and the photon of quantum electrodynamics.

The weak hypercharge satisfies the relation

{{in5}}$Q\; =\; T\_3\; +\; \backslash tfrac\{1\}\{2\}\; Y\_\backslash text\{W\}\; ~,$

where {{mvar|Q}} is the electric charge (in elementary charge units) and {{mvar|T}}{{sub|3}} is the third component of weak isospin (the SU(2) component).

Rearranging, the weak hypercharge can be explicitly defined as:

{{in5}}$Y\_\{\backslash rm\; W\}\; =\; 2(Q\; -\; T\_3)$

where "left"- and "right"-handed here are left and right chirality, respectively (distinct from helicity).

The weak hypercharge for an anti-fermion is the opposite of that of the corresponding fermion because the electric charge and the third component of the weak isospin reverse sign under charge conjugation.

File:Weinberg angle (relation between coupling constants).svg $~\backslash theta\_\backslash mathsf\{W\}~,$ and relation between coupling constants g, g′, and e. Adapted from Lee (1981).]]

File:Electroweak.svg, {{mvar| T}}{{sub|3}}, and weak hypercharge, {{mvar| Y}}{{sub|W}}, of the known elementary particles, showing electric charge, {{mvar| 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.]]

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

{{in5}}$\backslash ,\; Y\_\backslash text\{W\}\; =\; 0\; ~.$

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

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

{{in5}}$\backslash ,\; Y\_\{\backslash rm\; W\}\; =\; Q\; -\; T\_3\; ~,$

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

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

{{in5}}$\backslash tfrac\{1\}\{2\}X\; +\; Y\_\{\backslash rm\; W\}\; =\; \backslash tfrac\{5\}\{2\}(B\; -\; L)\; \backslash ,$

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.

{{in5}}{{math|{{SubatomicParticle|neutron|link=yes}} → {{SubatomicParticle|proton|link=yes}} + {{SubatomicParticle|electron}} + {{SubatomicParticle|electron antineutrino}}}}

Hence neutron decay conserves baryon number {{mvar|B}} and lepton number {{mvar|L}} separately, so also the difference {{nobr| {{mvar|B}} − {{mvar|L}} }} is conserved.

Proton decay is a prediction of many grand unification theories.

$$\backslash begin\{align\}$$

\rm p^+ \longrightarrow e^+ + & \ \pi^0 \\

& \downarrow \\

& 2\gamma

\end{align}

Hence this hypothetical proton decay would conserve {{nobr| {{mvar|B}} − {{mvar|L}} }}, even though it would individually violate conservation of both lepton number and baryon number.

• Standard Model (mathematical formulation)
• Weak charge

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{{Standard model of physics}}

Category:Nuclear physics

Category:Standard Model

Category:Electroweak theory

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