View source for Weak hypercharge

{{Short description|Abelian charge found in electroweak theory}}
{{distinguish|Hypercharge|Weak charge}}
{{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_\mathsf{W}</math> and corresponds to the [[gauge symmetry]] [[U(1)]].<ref name=Donoghue-Golowich-Holstein-1994-DynSM/><ref name=Cheng-Li-2006-GaThElPP/>

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, <math>\ Q = T_3 + \tfrac{1}{2}\, Y_\mathsf{W}\ </math> (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).

In the electroweak theory SU(2) transformations [[Commutator|commute]] with U(1) transformations by definition and therefore U(1) charges for the elements of the SU(2) doublet (for example lefthanded up and down quarks) have to be equal. This is why U(1) cannot be identified with U(1)<sub>em</sub> and weak hypercharge has to be introduced.<ref name=Tully-2012-Nutsh/><ref name=Glashow-1961-02-NucPh/>

Weak hypercharge was first introduced by [[Sheldon Glashow]] in 1961.<ref name=Glashow-1961-02-NucPh/><ref name=Hoddeson-Brown-Riordan-etal-1997/><ref name=Quigg-2015-10-19-ARevNuPaSc/>

==Definition==
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]] {{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}}<math> Q = T_3 + \tfrac{1}{2} Y_\text{W} ~,</math>

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}}<math>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/> {{mvar|Q}}
 ! [[Weak isospin|Weak <br/> isospin]] <br/> {{mvar|T}}{{sub|3}}
 ! style="border-right:medium #aaa solid;" | Weak <br/> hyper-<br/>charge <br/> {{mvar|Y}}{{sub|W}}
 !
 ! Electric <br/> [[charge (physics)|charge]] <br/> {{mvar|Q}}
 ! [[Weak isospin|Weak <br/> isospin]] <br/> {{mvar|T}}{{sub|3}}
 ! Weak <br/> hyper-<br/>charge <br/> {{mvar|Y}}{{sub|W}}
 |- style="border-top:medium #aaa solid;"
 | rowspan = "2"; style="border-right:medium #aaa solid;" | Leptons
 | {{math|{{subatomic particle|electron neutrino|link=yes}}, {{subatomic particle|muon neutrino|link=yes}}, {{subatomic particle|tau neutrino|link=yes}}}}
 | align="center"; | 0
 | align="center"; | +{{sfrac|1|2}}
 | align="center"; style="border-right:medium #aaa solid" | &minus;1
 | align="center"; {{n/a|{{math|&nu;}}{{sub|{{sc|r}} <br/> {{small|May not exist}} }}}}
 | align="center"; {{n/a| 0 }}
 | align="center"; {{n/a| 0 }}
 | align="center"; {{n/a| 0 }}
 |- style="border-bottom:medium #aaa solid;"
 | {{subatomic particle|electron|link=yes}}, {{subatomic particle|muon|link=yes}}, {{subatomic particle|tau|link=yes}}
 | align="center"; | &minus;1
 | align="center"; | &minus;{{sfrac|1|2}}
 | align="center"; style="border-right:medium #aaa solid" | &minus;1
 | {{physics particle|e|TR=−|BR=R}}, {{physics particle|μ|TR=−|BR=R}}, {{physics particle|τ|TR=−|BR=R}}
 | align="center"; | &minus;1
 | align="center"; | 0
 | align="center"; | &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}}
 | align="center"; | +{{sfrac|2|3}}
 | align="center"; | +{{sfrac|1|2}}
 | align="center"; style="border-right:medium #aaa solid;" | +{{sfrac|1|3}}
 | {{physics particle|u|BR=R}}, {{physics particle|c|BR=R}}, {{physics particle|t|BR=R}}
 | align="center"; | +{{sfrac|2|3}}
 | align="center"; | 0
 | align="center"; | +{{sfrac|4|3}}
 |-
 | [[down quark|d]], [[strange quark|s]], [[bottom quark|b]]
 | align="center"; | &minus;{{sfrac|1|3}}
 | align="center"; | &minus;{{sfrac|1|2}}
 | align="center"; style="border-right:medium #aaa solid;" | +{{sfrac|1|3}}
 | {{physics particle|d|BR=R}}, {{physics particle|s|BR=R}}, {{physics particle|b|BR=R}}
 | align="center"; | &minus;{{sfrac|1|3}}
 | align="center"; | 0
 | align="center"; | &minus;{{sfrac|2|3}}
 |}
where "left"- and "right"-handed here are left and right [[Chirality (physics)|chirality]], respectively (distinct from [[Helicity (particle physics)|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|250px|right|thumb| [[Weinberg angle]] <math>~\theta_\mathsf{W}~,</math> and relation between coupling constants ''g'', ''g&prime;'', and ''e''. Adapted from Lee (1981).<ref name=Lee-1981-PPhIntFTh/>]]

{| class = "wikitable" style = "text-align:center"
 ! Interaction<br/> mediated
 ! [[Boson]]
 ! Electric <br/> charge <br/> {{mvar|Q}}
 ! Weak <br/> isospin <br/> {{mvar|T}}{{sub|3}}
 ! Weak <br/> hypercharge <br/> {{mvar|Y}}{{sub|W}}
 |- style="border-top: 2pt #aaa solid;"
 | rowspan = "2" | Weak
 | {{math| {{subatomic particle|W boson+-|link=yes}} }}
 | ±1
 | ±1
 | 0
 |-
 | {{math| {{subatomic particle|Z boson0|link=yes}} }}
 | 0
 | 0
 | 0
 |- style="border-bottom: 2pt #aaa solid;"
 | Electromagnetic
 | {{math| {{subatomic particle|Photon0|link=yes}} }}
 | 0
 | 0
 | 0
 |-
 | Strong
 | {{math| {{subatomic particle|Gluon|link=yes}} }}
 | 0
 | 0
 | 0
 |-
 | Higgs
 | {{math| {{subatomic particle|Higgs boson|link=yes}} }}
 | 0
 | &minus;{{sfrac|1|2}}
 | +1
 |}

[[File:Electroweak.svg|300px|right|thumb|The pattern of [[weak isospin]], {{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 boson]]s.]]

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

{{in5}}<math>\, Y_\text{W} = 0 ~.</math> 

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

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

{{in5}}<math>\, Y_{\rm W} = Q - T_3 ~,</math>

which is equal to just ''the average electric charge of the particles in the [[isospin multiplet]]''.<ref name=Peskin-Schroeder-1995-IntQFT/><ref name=Anderson-2003-MThCoStr/>

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

{{in5}}<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 within the [[Standard Model]] and most extensions, this implies that baryon number minus lepton number is also always conserved.

===Neutron decay===
{{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===
[[Proton decay]] is a prediction of many [[Grand unification theory|grand unification theories]].
<math display=block> \begin{align} 
  \rm p^+ \longrightarrow e^+ + & \ \pi^0 \\
  & \downarrow \\
  & 2\gamma 
\end{align}</math>

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]].

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

==References==
{{reflist|25em|refs=

<ref name=Anderson-2003-MThCoStr>
{{cite book
 |author=Anderson, M.R.
 |year=2003
 |title=The Mathematical Theory of Cosmic Strings
 |page=12
 |publisher=CRC Press
 |isbn=0-7503-0160-0
}}
</ref>

<ref name=Cheng-Li-2006-GaThElPP>
{{cite book
 |first1=T.P. |last1=Cheng
 |first2=L.F. |last2=Li  
 |year=2006
 |title=Gauge Theory of Elementary Particle Physics
 |publisher=[[Oxford University Press]]
 |isbn=0-19-851961-3
}}
</ref>

<ref name=Donoghue-Golowich-Holstein-1994-DynSM>
{{cite book
 |first1=J.F. |last1=Donoghue
 |first2=E.   |last2=Golowich
 |first3=B.R. |last3=Holstein
 |year=1994
 |title=Dynamics of the Standard Model
 |page=[https://archive.org/details/dynamicsstandard00jfdo/page/n69 52]
 |publisher=Cambridge University Press
 |isbn=0-521-47652-6
 |url=https://archive.org/details/dynamicsstandard00jfdo
 |url-access=limited
}}
</ref>

<ref name=Glashow-1961-02-NucPh>
{{Cite journal
 |last=Glashow |first=Sheldon L.
 |date=February 1961
 |title=Partial-symmetries of weak interactions
 |journal=Nuclear Physics
 |volume=22 |issue=4 |pages=579–588
 |doi=10.1016/0029-5582(61)90469-2
 |bibcode=1961NucPh..22..579G
 |url=https://linkinghub.elsevier.com/retrieve/pii/0029558261904692
 |lang=en
}}
</ref>

<ref name=Hoddeson-Brown-Riordan-etal-1997>
{{Cite book
 |editor1-last=Hoddeson |editor1-first=Lillian |editor1-link=Lillian Hoddeson
 |editor2-last=Brown    |editor2-first=Laurie |editor2-link=Laurie Brown (physicist)
 |editor3-last=Riordan  |editor3-first=Michael |editor3-link=Michael Riordan (physicist)
 |editor4-last=Dresden  |editor4-first=Max
 |date=1997-11-13
 |title=The rise of the Standard Model: A history of particle physics from 1964 to 1979
 |edition=1st  |page=14
 |publisher=Cambridge University Press
 |isbn=978-0-521-57082-4
 |doi=10.1017/cbo9780511471094
 |url=https://www.cambridge.org/core/product/identifier/9780511471094/type/book
}}
</ref>

<ref name=Lee-1981-PPhIntFTh>
{{cite book
 |last=Lee |first=T.D.
 |year=1981
 |title=Particle Physics and Introduction to Field Theory
 |place=Boca Raton, FL / New York, NY
 |publisher=CRC Press / Harwood Academic Publishers
 |isbn=978-3718600335
 |url=https://archive.org/details/particlephysicsi0000leet
 |url-access=limited
 |via=Archive.org
}}
</ref>

<ref name=Peskin-Schroeder-1995-IntQFT>
{{cite book
 |last1=Peskin    |first1=Michael E.
 |last2=Schroeder |first2=Daniel V.
 |year=1995
 |title= An Introduction to Quantum Field Theory
 |publisher=Addison-Wesley Publishing Company
 |isbn=978-0-201-50397-5
 |url=https://archive.org/details/introductiontoqu0000pesk
 |url-access=registration
}}
</ref>

<ref name=Quigg-2015-10-19-ARevNuPaSc>
{{cite journal
 |last=Quigg  |first=Chris
 |date=2015-10-19
 |title=Electroweak symmetry breaking in historical perspective
 |journal=Annual Review of Nuclear and Particle Science
 |volume=65  |issue=1  |pages=25–42
 |lang=en  |issn=0163-8998
 |doi=10.1146/annurev-nucl-102313-025537 |arxiv=1503.01756
 |bibcode=2015ARNPS..65...25Q
 |doi-access=free
 }}
</ref>

<ref name=Tully-2012-Nutsh>
{{cite book
 |last=Tully |first=Christopher G.
 |date=2012
 |title=Elementary Particle Physics in a Nutshell
 |publisher=Princeton University Press
 |isbn=978-1-4008-3935-3
 |doi=10.1515/9781400839353
 |page=87
 |url=https://princetonup.degruyter.com/view/title/563410
}}
</ref>


}} <!-- end "refs=" -->

{{Standard model of physics}}

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[[Category:Standard Model]]
[[Category:Electroweak theory]]

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