View source for Weak hypercharge

{{confuse|Hypercharge}}
{{Flavour quantum numbers}}
In the [[Standard Model (mathematical formulation)|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 {{math|''Y''<sub>W</sub>}} and corresponds to the [[gauge symmetry]] [[U(1)]].<ref>{{cite book|title=Dynamics of the standard model|pages=52|author1=J. F. Donoghue |author2=E. Golowich |author3=B. R. Holstein |publisher=Cambridge University Press|year=1994|isbn=0-521-47652-6}}</ref><ref>T. P. Cheng; L. F. Li (2006). ''Gauge theory of elementary particle physics''. [[Oxford University Press]]. {{ISBN| 0-19-851961-3}}. </ref>

It is [[Conservation law (physics)|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 {{math|''T''<sub>3</sub>}}). Only a specific combination of them, {{nowrap|1=''Q'' = ''T''<sub>3</sub> + {{sfrac|1|2}} ''Y''<sub>W</sub>}} (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 is zero for leptons).

==Definition==
[[File:Weinberg angle (relation between coupling constants).svg|300px|thumb|Weinberg angle {{math|''θ<sub>W</sub>''}}, and relation between coupling constants ''g'', ''g&prime;'', and ''e''. Adapted from T. D. Lee's book ''Particle Physics and Introduction to Field Theory'' (1981).]]
{{see also|Weinberg angle}}
Weak hypercharge is the [[charge (physics)|generator]] of the U(1) component of the [[electroweak]] gauge group, {{gaps|SU(2)|×|U(1)}} and its associated [[quantum field]] ''B'' mixes with the ''W''<sup>&nbsp;3</sup> electroweak quantum field to produce the observed {{subatomic particle|Z boson|link=yes}} gauge boson and the [[photon]] of [[quantum electrodynamics]].

The Weak hypercharge satisfies the relation
:<math>\qquad Q = T_3 + \tfrac{1}{2} Y_{\rm W}~,</math>
where {{mvar|Q}} is the electric charge (in [[elementary charge]] units) and {{mvar|T}}<sub>3</sub> is the third component of [[weak isospin]] (the SU(2) component). 

Rearranging, the weak hypercharge can be explicitly defined as:
:<math>\qquad Y_{\rm W} = 2(Q - T_3)</math>
{| class = "wikitable" style = "text-align center;"
 |-
 ! rowspan=2 style="border-right:medium #aaa solid;" | [[Fermion|Fermion <br/> family]]
 ! colspan=4 style="border-right:medium #aaa solid;" | [[Chirality (physics)|Left-chiral]] [[fermion]]s
 ! colspan=4 | [[Chirality (physics)|Right-chiral]] [[fermion]]s
 |-
 !
 ! Electric <br/> [[charge (physics)|charge]] <br/> ''Q''
 ! [[Weak isospin|Weak <br/> isospin]] <br/> ''T''<sub>3</sub>
 ! style="border-right:medium #aaa solid;" | Weak <br/> hyper-<br/>charge <br/> ''Y''<sub>W</sub>
 !
 ! Electric <br/> [[charge (physics)|charge]] <br/> ''Q''
 ! [[Weak isospin|Weak <br/> isospin]] <br/> ''T''<sub>3</sub>
 ! Weak <br/> hyper-<br/>charge <br/> ''Y''<sub>W</sub>
 |- style="border-top:medium #aaa solid;"
 | rowspan = "2" style="border-right:medium #aaa solid;" | Leptons
 | {{subatomic particle|electron neutrino|link=yes}}, {{subatomic particle|muon neutrino|link=yes}}, {{subatomic particle|tau neutrino|link=yes}}
 | 0
 | +{{sfrac|1|2}}
 | style="border-right:medium #aaa solid" | &minus;1
 | colspan=4 {{n/a|No interaction, if existent}}
 |- style="border-bottom:medium #aaa solid;"
 | {{subatomic particle|electron|link=yes}}, {{subatomic particle|muon|link=yes}}, {{subatomic particle|tau|link=yes}}
 | &minus;1
 | &minus;{{sfrac|1|2}}
 | style="border-right:medium #aaa solid" | −1
 | {{physics particle|e|TR=−|BR=R}}, {{physics particle|μ|TR=−|BR=R}}, {{physics particle|τ|TR=−|BR=R}}
 | &minus;1
 | 0
 | &minus;2
 |-
 | rowspan = "2" style="border-right:medium #aaa solid;" | Quarks
 | {{subatomic particle|up quark|link=yes}}, {{subatomic particle|charm quark|link=yes}}, {{subatomic particle|top quark|link=yes}}
 | +{{sfrac|2|3}}
 | +{{sfrac|1|2}}
 | style="border-right:medium #aaa solid;" | +{{sfrac|1|3}}
 | {{physics particle|u|BR=R}}, {{physics particle|c|BR=R}}, {{physics particle|t|BR=R}}
 | +{{sfrac|2|3}}
 | 0
 | +{{sfrac|4|3}}
 |-
 | [[down quark|d]], [[strange quark|s]], [[bottom quark|b]]
 | &minus;{{sfrac|1|3}}
 | &minus;{{sfrac|1|2}}
 | style="border-right:medium #aaa solid;" | +{{sfrac|1|3}}
 | {{physics particle|d|BR=R}}, {{physics particle|s|BR=R}}, {{physics particle|b|BR=R}}
 | &minus;{{sfrac|1|3}}
 | 0
 | &minus;{{sfrac|2|3}}
 |}
where "left"- and "right"-handed here are left and right [[Chirality (physics)|chirality]], respectively (distinct from [[Helicity (particle physics)|helicity]]).

{| class = "wikitable" style = "text-align: center"
 ! [[Fundamental interaction|Mediated <br/> fundamental <br/> interaction]]
 ! [[Boson]]
 ! Electric <br/> charge <br/> ''Q''
 ! Weak <br/> isospin <br/> ''T''<sub>3</sub>
 ! Weak <br/> hypercharge <br/> ''Y''<sub>W</sub>
 |- style="border-top: 2pt #aaa solid;"
 | rowspan = "2" | Weak
 | {{subatomic particle|W boson+-|link=yes}}
 | ±1
 | ±1
 | 0
 |-
 | {{subatomic particle|Z boson0|link=yes}}
 | 0
 | 0
 | 0
 |- style="border-bottom: 2pt #aaa solid;"
 | Electric
 | {{subatomic particle|Photon|link=yes}}
 | 0
 | 0
 | 0
 |-
 | Higgs
 | {{subatomic particle|Higgs boson|link=yes}}
 | 0
 | &minus;{{sfrac|1|2}}
 | +1
 |}
[[File:Electroweak.svg|300px|right|thumb|The pattern of [[weak isospin]], T<sub>3</sub>, and weak hypercharge, Y<sub>W</sub>, 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 often represented at half-scale, so that
:<math>\qquad Y_{\rm W} = Q - T_3 \,,</math>
which is equal to just ''the average electric charge of the particles in the isospin multiplet''.<ref>{{cite book|title= An Introduction to Quantum Field Theory|url= https://archive.org/details/introductiontoqu0000pesk|url-access= registration|authors=Peskin, Michael E. &  Schroeder, Daniel V. |year=1995|publisher=Addison-Wesley Publishing Company| isbn=978-0-201-50397-5}} ;
{{cite book |title=The mathematical theory of cosmic strings |author=Anderson, M. R. |pages=12 |publisher=CRC Press |year=2003 |isbn=0-7503-0160-0}}</ref>

==Baryon and lepton number==
Weak hypercharge is related to [[B − L|baryon number minus lepton number]] via:

:<math>\tfrac{1}{2}X + Y_{\rm W} = \tfrac{5}{2}(B - L) \,</math>

where ''[[X (charge)|X]]'' is a conserved quantum number in [[Grand Unification Theory|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===
:{{SubatomicParticle|neutron|link=yes}} &rarr; {{SubatomicParticle|proton|link=yes}} + {{SubatomicParticle|electron}} + {{SubatomicParticle|electron antineutrino}}

Hence neutron decay conserves [[baryon number]] ''B'' and [[lepton number]] ''L'' separately, so also the difference ''B''&nbsp;−&nbsp;''L'' is conserved.

===Proton decay===
[[Proton decay]] is a prediction of many [[Grand unification theory|grand unification theories]].
:{{SubatomicParticle|proton+}} &rarr; {{SubatomicParticle|antielectron|link=yes}} + {{SubatomicParticle|pion0|link=yes}} &rarr; {{SubatomicParticle|antielectron}} + 2{{SubatomicParticle|photon|link=yes}}

Hence proton decay conserves ''B''&nbsp;&minus;&nbsp;''L'', even though it violates both [[lepton number]] and [[baryon number]] conservation.

==See also==
*[[Standard Model (mathematical formulation)]]

==References==
<references/>

{{Standard model of physics}}

[[Category:Particle physics]]
[[Category:Nuclear physics]]
[[Category:Standard Model]]
[[Category:Electroweak theory]]

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