View source for Weak charge

{{Short description|Type of weak interaction in nuclear and atomic physics}}
{{distinguish|weak hypercharge|weak isospin}}
{{use dmy dates|date=September 2021}}

In [[nuclear physics]] and [[atomic physics]], '''weak charge''', or rarely '''neutral weak charge''', refers to the [[Standard Model]] [[weak interaction]] coupling of a particle to the [[Z boson]]. For example, for any given nuclear isotope, the total weak charge is approximately −0.99 per [[neutron]], and +0.07 per [[proton]].<ref name=Hagen-Ekström-Forssén-etal-2016-NatPhys/> It also shows an effect of [[parity violation]] during [[electron scattering]].

This same term is sometimes also used to refer to other, different quantities, such as ''[[weak isospin]]''<ref name=U-Nürnberg-prop-Z0/> or ''[[weak hypercharge]]''; this article concerns the use of ''weak charge'' for a quantity that measures the degree of vector coupling of a [[fermion]] to the [[Z boson]] (i.e. the coupling strength of [[weak neutral current]]s).<ref name=Woods-SLAC-E158/>

==Empirical formulas==
Measurements in 2017 give the weak charge of the proton as {{val|0.0719|0.0045}}&nbsp;.<ref name=Androić-Armstrong-Asaturyan-Gweak-clbn-2018-Natr/>

The weak charge may be summed in atomic nuclei, so that the predicted weak charge for {{sup|133}}[[Cesium|Cs]] (55&nbsp;protons, 78&nbsp;neutrons) is 55×(+0.0719) + 78×(−0.989) {{=}} −73.19, while the value determined experimentally, from measurements of [[parity violation|parity violating]] electron scattering, was −72.58&nbsp;.<ref name=Dzuba-Berengut-Flambaum-Roberts-2012-PRL/>

A recent study used four even-numbered isotopes of [[ytterbium]] to test the formula {{nobr| {{mvar|Q}}{{sub|w}} {{=}} −0.989 {{mvar|N}} + 0.071 {{mvar|Z}} ,}} for weak charge, with {{mvar|N}} corresponding to the number of neutrons and {{mvar|Z}} to the number of protons. The formula was found consistent to 0.1%&nbsp;accuracy using the {{sup|170}}[[ytterbium|Yb]], {{sup|172}}[[ytterbium|Yb]], {{sup|174}}[[ytterbium|Yb]], and {{sup|176}}[[ytterbium|Yb]] [[isotopes of ytterbium]].<ref name=Antypas-Fabricant-Stalnaker-etal-2018-NatPhys/>

In the [[ytterbium]] test, atoms were excited by laser light in the presence of electric and magnetic fields, and the resulting parity violation was observed.<ref name=Phys-org-2018-11-12-PR/> The specific transition observed was the [[forbidden transition]] from 6s{{sup|2}} [[Term symbol|{{sup|1}}S{{sub|0}}]] to 5d6s [[Term symbol|{{sup|3}}D{{sub|1}}]] (24489&nbsp;cm{{sup|−1}}). The latter state was mixed, due to weak interaction, with 6s6p [[Term symbol|{{sup|1}}P{{sub|1}}]] (25068&nbsp;cm{{sup|−1}}) to a degree proportional to the nuclear weak charge.<ref name=Antypas-Fabricant-Stalnaker-etal-2018-NatPhys/>

== Particle values ==
This table gives the values of the electric charge (the coupling to the photon, referred to in this article as {{nobr| <math>Q_\epsilon</math>{{efn|name=Q_w_electric_charge_note}}).}} Also listed are the approximate '''weak charge''' <math>Q_\mathsf{w}</math> (the ''vector'' part of the Z&nbsp;boson coupling to fermions), ''[[weak isospin]]'' <math>T_3</math> (the coupling to the [[W bosons]]), ''[[weak hypercharge]]'' <math>Y_\mathsf{w}</math> (the coupling to the B&nbsp;boson) and the approximate Z&nbsp;boson coupling factors (<math>Q_\boldsymbol\mathsf{L}</math> and <math>Q_\boldsymbol\mathsf{R}</math> in the "Theoretical" section, below).

The table's values are approximate: They happen to be exact for particles whose energies make the [[Weinberg angle|weak mixing angle]] <math>\ \theta_\mathsf{w} = 30^\circ\ ,</math> with <math>\ \sin^2 \theta_\mathsf{w} = \tfrac{1}{4} ~.</math> This value is very close to the typical {{nobr|[[Weinberg angle|approximately 29°]] }} angle observed in particle accelerators. The embedded formulas give the (more) exact values for when the [[Weinberg angle]], <math>\ \theta_\mathsf{w}\ ,</math> is known.

{|  class="wikitable"  style="text-align:center;"
|+  Electroweak charges of Standard Model particles
!rowspan="3"| Spin<br/>{{mvar|J}}
!rowspan="3"| Particle(s)
!rowspan="2"| Weak charge <br/><math>Q_\mathsf{w}</math>
!rowspan="3"| Electric<br/>charge<br/><math>Q ~ \mathsf{ or } ~ Q_{\epsilon}</math>
!colspan="2" rowspan="2"| Weak isospin <br/> <math>T_3</math>
!colspan="2" rowspan="2"| Weak hypercharge <br/> <math>Y_\mathsf{w}</math>
!colspan="2"| Z boson<br/>coupling
|-
!rowspan="2"| <math>2\ Q_\boldsymbol\mathsf{L}</math><br/> {{small|&nbsp;{{sc|Left}}&nbsp;}}
!rowspan="2"| <math>2\ Q_\boldsymbol\mathsf{R}</math><br/> {{small| {{sc|Right}} }}
|-
|style="background:#EBEBEF;"|{{nobr|{{small| {{=}} {{math|2}} {{mvar|Q}}{{sub|'''{{sc|L}}'''}} {{math|+ 2}} {{mvar|Q}}{{sub|'''{{sc|R}}'''}} }} }}
! {{small|&nbsp;{{sc|Left}}&nbsp;}}
! {{small| {{sc|Right}} }}
! {{small|&nbsp;{{sc|Left}}&nbsp;}}
! {{small| {{sc|Right}} }}
|-
| {{math| {{sfrac| 1 |2}} }}
| e{{sup|−}},{{math| μ{{sup|−}} }}, {{math|τ{{sup|−}} }} <br/> [[electron]], [[muon]], [[tau]]{{efn-lr|name=antifermion_note}}
| {{small| {{nobr| −1 + 4 sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;0 }}
| {{math| −1 }}
| {{math| {{sfrac|−| 1 |2}} }}
| {{math|  0 }}
| {{math| −1 }}
| {{math| −2 }}
| {{small| {{nobr| −1 + 2 sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|−| 1 |2}} }}
| {{small| {{nobr| 2 sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|+| 1 |2}}  }}
|-
| {{math| {{sfrac| 1 |2}} }}
| u, c, t <br/> [[Up quark|up]], [[Charm quark|charm]], [[Top quark|top]]{{efn-lr|name=antifermion_note}}
| {{small| {{nobr| +1 − {{sfrac| 8 |3}} sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|+| 1 |3}} }}
| {{math| {{sfrac|+| 2 |3}} }}
| {{math| {{sfrac|+| 1 |2}} }}
| {{math| 0 }}
| {{math| {{sfrac|+| 1 |3}} }}
| {{math| {{sfrac|+| 4 |3}} }}
| {{small| {{nobr| 1 − {{sfrac| 4 |3}} sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|+| 2 |3}} }}
| {{small| {{nobr| {{sfrac|−| 4 |3}} sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|−| 1 |3}} }}
|-
| {{math| {{sfrac| 1 |2}} }}
| d, s, b <br/> [[Down quark|down]], [[Strange quark|strange]], [[Bottom quark|bottom]]{{efn-lr|name=antifermion_note}}
| {{small| {{nobr| −1 + {{sfrac| 4 |3}} sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|−| 2 |3}} }}
| {{math| {{sfrac|−| 1 |3}} }}
| {{math| {{sfrac|−| 1 |2}} }}
| {{math| 0 }}
| {{math| {{sfrac|+| 1 |3}} }}
| {{math| {{sfrac|−| 2 |3}} }}
| {{small| {{nobr| −1 + {{sfrac| 2 |3}} sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|−| 5 |6}} }}
| {{small| {{nobr| {{sfrac|+| 2 |3}} sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;{{sfrac|+| 1 |6}} }}
|-
| {{math| {{sfrac| 1 |2}} }}
| {{math|&nu;{{sub|e}}}}, {{math|&nu;{{sub|μ}}}}, {{math|&nu;{{sub|τ}} }} <br/> [[neutrino]]s{{efn-lr|name=antifermion_note}}
| {{math| +1 }}
| {{math| &nbsp; 0 }}
| {{math| {{sfrac|+|1|2}} }}
| {{nobr| &emsp;&emsp; {{math|0}} {{efn-lr|name=sterile_neutrinos_note}}{{efn|name=neutrino_property_options}} }}
| {{math| −1 }}
| {{nobr| &emsp;  &emsp; {{math|0}} {{efn-lr|name=sterile_neutrinos_note}}{{efn|name=neutrino_property_options}} }}
| {{math| +1 }}
| {{nobr| &emsp;  &emsp; {{math|0}} {{efn-lr|name=sterile_neutrinos_note}}{{efn|name=neutrino_property_options}} }}
|-
| {{math| 1 }}
| {{math|g}}, {{math|&gamma;}}, {{math|Z}}{{sup|0}},<br/> [[gluon]]{{efn-lr|name=out_of_context_gluons_note}}, [[photon]], and {{nobr| [[Z boson]],{{efn-lr|
name=anti_particle_neutral_boson_note}} }}
|colspan="8"| {{big| {{math|0}} }}{{efn-lr|name=anti_particle_neutral_boson_note}} 
|-
| {{math| 1 }}
| W{{sup|+}} <br/> [[W boson]]{{efn-lr|name=anti_particle_W_boson_note}}
| {{small| {{nobr| +2 − 4 sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;+1 }}
| {{math| +1 }}
|colspan="2"| {{math| +1 }}
|colspan="2"| {{math|  0 }}
|colspan="2"| {{small| {{nobr| +2 − 4 sin{{sup|2}} {{mvar|θ}}{{sub|w}} }} <br/>}}{{math|≈&nbsp;+1 }}
|-
| {{math|  0 }}
| H{{sup|0}} <br/> [[Higgs boson]]
| {{math| −1 }}
| {{math|  0 }}
|colspan="2"| {{math| {{sfrac|−| 1 |2}} }}
|colspan="2"| {{math| +1 }}
|colspan="2"| {{math| −1 }}
|}

{{notelist-lr|refs=

{{efn-lr|
name=antifermion_note|
Only (regular) fermion charges are listed. For the matching antifermions, the electric charge, {{mvar| Q{{sub|ϵ}} }}, has the same magnitude, but opposite sign; other charges, such as [[weak isospin]], {{mvar|T}}{{sub|3}}, and [[weak hypercharge]], {{mvar|Y}}{{sub|w}}, that have columns subtitled {{sc|LEFT}} and {{sc|RIGHT}}, are left-right swapped as well as sign-reversed.
}}

{{efn-lr|
name=anti_particle_neutral_boson_note|
The [[additive quantum number|quantum charges]] of every kind for [[photons]] and [[Z boson]]s are all zero, so the [[photon]] and [[Z boson]] are their own antiparticles: They are "[[truly neutral particle]]s"; in particular, they are truly neutral [[vector boson]]s.
{{main|Two-photon physics}}
Whilst not having charge themselves, [[photon]]s and {{nobr|[[Z boson]]s}} none the less ''do'' interact with particles carrying the relevant quantum charge: [[electrical charge]] ({{mvar| Q{{sub|ϵ}} }}) for [[photon]]s ({{mvar|γ}}), and left and right weak charges ({{mvar|Q}}{{sub|'''{{sc|L}}'''}}, {{mvar|Q}}{{sub|'''{{sc|R}}'''}}) for [[Z boson]]s ({{math|Z{{sup|0}}}}). They cannot interact with other {{mvar|γ}} or {{math|Z{{sup|0}}}} directly, and except at extremely high energies, ''usually'' do not interact with other {{mvar|γ}} or {{math|Z{{sup|0}}}} at all. However, because of [[Heisenberg's uncertainty principle|quantum uncertainty]] even low energy versions of either particle can briefly [[pair creation|split into a particle-antiparticle pair]], each of which happens to have the [[electrical charge]] needed to interact with a [[photon|{{mvar|γ}}]], or the left or right weak charge needed to interact with {{math|Z{{sup|0}}}}, or both. After that interaction has happened, the particle-antiparticle pair recombines into the same {{mvar|γ}} or {{math|Z{{sup|0}}}} particle that originally split, precluding the intermediate pair – whatever it may have been – from ever being observed: The only observed effect is the change in the recombined particle's momentum. This disappearing-act makes it appear the same as if a direct {{math|Z{{sup|0}}}}-{{math|Z{{sup|0}}}} or {{math|Z{{sup|0}}-{{var|γ}}}} or [[Two-photon physics|{{math|{{var|γ}}-{{var|γ}}}}]] interaction had occurred. {{pb}} Because at normal, low energies, it depends on a fortuitous and ephemeral [[pair creation]] event, this kind of interaction of a neutral vector boson with another neutral vector boson is so rare that even though technically very slightly possible, it is treated as ''effectively'' impossible and ignored. Hence the blanket zero value for the neutral weak bosons' ({{mvar|γ}}, {{math|Z{{sup|0}}}}) row in the table are all ''almost'' exactly zero, but some are not ''precisely'' zero as shown.
}}

{{efn-lr|
name=anti_particle_W_boson_note|
Only the {{math|W{{sup|+}}}} boson's charges are listed; values for its antiparticle {{math|W{{sup|−}}}} have reversed sign (or remain zero). The same rule applies as for all particle-antiparticle pairs: Their [[additive quantum number|"charge"-like quantum numbers]] are equal and opposite. {{pb}} [[W bosons]] can interact with both [[photon]]s and [[Z boson]]s, since they carry both electric charge and weak charge; for the same reason, they can also self-interact.
}}

{{efn-lr|
name=out_of_context_gluons_note|
Gluons only have [[color charge]]s of the [[strong force]] and spin: Their electroweak charges are all zero, although their color charges give them distinct antiparticles (see [[Gluon]] for details). {{pb}} Strictly speaking, [[gluon]]s are out-of-context among of the [[electroweak interaction|electroweak-interacting]] particles described by this table. However, since each of the three electrically neutral [[elementary particle|elementary]] [[vector boson]]s' [[electroweak]] charges all are zero, they can all be accommodated by the same row in this table, hence allowing the table to show a complete list of all elementary particles currently incorporated in the [[Standard Model]].
}}

{{efn-lr|
name=sterile_neutrinos_note|
Although "[[sterile neutrinos]]" are ''not'' included in the [[Standard Model]] and have ''not'' been confirmed experimentally, if they did actually exist, giving the value zero for electric charge and weak isospin, as shown, is a simple way to annotate their non-participation in any electroweak interaction, and does so in a manner consistent with all the other elementary fermions.
}}

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

For brevity, the table omits antiparticles. Every particle listed (except for the uncharged bosons the [[photon]], [[Z boson]], [[gluon]], and [[Higgs boson]]{{efn|See [[Higgs mechanism]].}} which are their own antiparticles) has an antiparticle with identical mass and opposite charge. All non-zero signs in the table have to be reversed for antiparticles. The paired columns labeled {{sc|left}} and {{sc|right}} for fermions (top four rows), have to be swapped in addition to their signs being flipped.

All left-handed (regular) fermions and right-handed antifermions have <math>\ T_3 = \pm\tfrac{1}{2}\ ,</math> and therefore interact with the [[W boson]]. They could be referred to as ''"proper"-handed''. Right-handed fermions and left-handed antifermions, on the other hand, have zero weak isospin and therefore do not interact with the [[W boson]] (except for electrical interaction); they could therefore be referred to as ''"wrong"-handed'' (i.e. they are ''"wrong handed"'' for [[W boson|W{{sup|±}}]] interactions). "Proper"-handed fermions are organized into isospin doublets, while "wrong"-handed fermions are represented as isospin singlets. While "wrong"-handed particles do not interact with the [[W boson]] (no ''charged current interactions''), all "wrong"-handed fermions known to exist ''do'' interact with the [[Z boson]] (''neutral current interactions'').

"Wrong"-handed neutrinos ([[sterile neutrinos]]) have never been observed, but may still exist since they would be invisible to existing detectors.<ref name=Fermilab-sterile-ν/> Sterile neutrinos play a role in speculations about the way [[Neutrino#Mass|neutrinos have masses]] (see [[Seesaw mechanism]]). The above statement that the {{math|Z{{sup|0}}}} interacts with ''all'' fermions will need an exception for sterile neutrinos inserted, if they are ever detected experimentally.

Massive fermions – except (perhaps) neutrinos{{efn|
The exception stated for neutrinos, implying that neutrinos ''do not'' exist as left- and right-chiral superpositions might be wrong: It presumes that there are no [[sterile neutrino]]s. Whether there are or aren't any sterile neutrinos is not known; it's a question still being investigated by current particle research.
}}{{efn|
name=neutrino_property_options|
The whole matter of revised neutrino physics can be crudely summarized by stating that there are two generic options for new physics that make different changes to the relation between neutrinos and their antiparticles. The baseline [[Standard Model]] supposes that there are three [[neutrino flavor]]s, and each has only left-handed neutrinos and only right-handed anti-neutrinos, making six total neutrinos. Revisions to the current Standard Model either propose that there is only one neutrino, which is the same particle as its antineutrino, for each [[neutrino flavor|flavor]], making a total of three, not six; or propose that just like every other elementary fermion, there are both left and right handed neutrinos and antineutrinos of each flavor, making twelve, not six. (The second option would make neutrinos naturally match all the other [[elementary particle|elementary]] [[fermion|fermion-antifermion]] pairs.)
:
If right-handed antineutrinos are actually the same as the left-handed neutrinos, just observed traveling in opposite directions relative to their spin, then there would only be one kind of neutrino per [[neutrino flavor|lepton flavor]] rather than two – only three particles, not six; the only difference between a neutrino and its antineutrino would be a flipped orientation of the neutrino's spin relative to its direction travel. In the case of identical neutrinos and antineutrinos, the unique physical properties of neutrinos – unlike any other elementary fermion – would be explained as due to their zero electrical charge (and also being their own antiparticles) which makes them different from all the other fermions, which ''do'' interact with the [[Higgs field]]. The explanation for the current supposedly mistaken presumption that neutrinos and antineutrinos are different particles would then be because all observable neutrinos (those with [[collision cross-section]]s large enough to feasibly detect) travel at such high speeds that they seem to be the same as the [[speed of light]]. At those high speeds, it would be infeasible to detect a confirmed neutrino interacting as an antineutrino in any terrestrial laboratory; however, it can be indirectly confirmed if during two simultaneous [[beta decay]]s in the same atomic nucleus, the two antineutrinos produced happened to collide and anihilate – acting as each other's antiparticles – hence current research seeking evidence of [[neutrinoless double beta decay]].
:
The other possibility is that the known neutrinos and antineutrinos are indeed distinct particles, but are not the strangely exceptional single-handed fermions that the Standard Model supposes: That they are like every other type of elementary fermion, both neutrinos and antineutrinos ({{=}}2) exist as both left-handed and right-handed particle (×2) for each of the three flavors (×3), making a total of twelve neutrinos, not six. If that were the case, then naïvely applying the rules for quantum charges to the "wrong"-handed neutrinos (left handed antineutrinos and right handed regular neutrinos) leaves the "wrong" handed neutrinos with coupling of zero to every other elementary particle in the Standard Model. Having no interactions ([[sterile neutrino|"sterile"]]) renders them completely invisible – undetectable – hence the Standard Model supposes they aren't there because have we never seen them, but we couldn't see them if they ''did'' exist. (''Absence of evidence is not evidence of absence.'') If sterile neutrinos do exist, then all neutrinos could / should interact with the Higgs field and consequently change handedness and switch between "active" and "sterile". Hence current searches for unexplained disappearances of active neutrinos in [[Deep Underground Neutrino Experiment|long-baseline beams of neutrinos]]. But the problem with that is the apparently minuscule [[neutrino mass|masses of the active neutrinos]]: Coupling to the Higgs field is proportional to mass; minuscule mass means minuscule interactions, so neutrinos vanishing by transitions from active to sterile would also have to be exceedingly rare, perhaps rare enough to make detection infeasible for current experiments.
}} – always exist in a [[Quantum superposition|superposition]] of left-handed and right-handed states, and never in pure chiral states. This mixing is caused by interaction with the [[Higgs field]], which acts as an infinite source and sink of [[weak isospin]] and / or hypercharge, due to its non-zero [[vacuum expectation value]] (for further information see [[Higgs mechanism]]).

==Theoretical basis==
{{See also|Electroweak interaction}}
The formula for the weak charge is derived from the Standard Model, and is given by<ref name=Playfer-ElectWk-Lect16-U-Edinburgh/><ref name=Kumar-Moller-clbrn-2014-08-25--29/>

<math display="block">~ Q_\mathsf{w} ~=~ 2 \, T_3 - Q_\epsilon \, 4 \, \sin^2 \theta_\mathsf{w}  ~\approx~ 2 \, T_3 - Q_\epsilon \; , \qquad \mathsf{ or } \qquad  ~ Q_\mathsf{w} + Q_\epsilon ~\approx~ 2 \, T_3 ~=~ \pm 1 ; ~</math>

where <math>~ Q_\mathsf{w} ~</math> is the weak charge,{{efn|
Other Wikipedia articles use the weak vector coupling, <math>g_\mathsf{V},</math> a different version of <math>~ Q_\mathsf{w} ~</math> which is exactly half the size given here.
}}
<math>T_3</math> is the weak isospin,{{efn|
Specifically, the weak isospin for ''left-handed'' fermions, and ''right-handed'' anti-fermions (both are "proper"-handed). Weak isospin is always zero for right-handed fermions and left-handed anti-fermions (both are "wrong"-handed, that is, "wrong" for the {{subatomic particle|W boson+-|link=yes}}).
}}
<math>\theta_\mathsf{w}</math> is the [[weak mixing angle]], and <math>\, Q_\epsilon \,</math> is the [[electric charge]].{{efn|
name=Q_w_electric_charge_note|
<math>\, Q \,</math> is conventionally used as the symbol for electric charge. The subscript <math>\ \epsilon\ </math> is added in this article to keep the several symbols for weak charge <math>\ Q_\boldsymbol\mathsf{L}\ ,</math> <math>\ Q_\boldsymbol\mathsf{R}\ ,</math> and <math>\ Q_\mathsf{w}\ ,</math> and for electric charge <math>\ Q_\epsilon \ </math> from being easily confused.
}}
The approximation for the weak charge is usually valid, since the weak mixing angle typically is {{nobr| 29° ≈ 30° ,}} and <math>\ 4 \sin^2 30^\circ = 1\ ,</math> and <math>\; 4 \sin^2 29^\circ \approx 0.940\ ,</math> a discrepancy of only a little more than {{nobr| 1 in 17 .}}

=== Extension to larger, composite protons and neutrons ===
This relation only directly applies to [[quark]]s and [[lepton]]s ([[fundamental particles]]), since [[weak isospin]] is not clearly defined for [[composite particles]], such as protons and neutrons, partly due to weak isospin not being conserved. One can set the weak isospin of the proton to {{sfrac|+|1|2}} and of the neutron to {{sfrac|−|1|2}},<ref name=Rosen-1978-05-PhRev/><ref name=Robson-2004-04-12-JMP/> in order to obtain approximate value for the weak charge. Equivalently, one can sum up the weak charges of the constituent quarks to get the same result.

Thus the calculated weak charge for the neutron is

<math display="block"> Q_\mathsf{w} = 2 \, T_3 - 4 \, Q_\epsilon \, \sin^2 \theta_\mathsf{w} = 2 \cdot \left( -\tfrac{1}{2} \right) = -1 ~\approx~ -0.99 ~.</math>

The weak charge for the proton calculated using the above formula and a weak mixing angle of 29° is

<math display="block"> Q_\mathsf{w} = 2 \, T_3 - 4\, Q_\epsilon \, \sin^2 \theta_\mathsf{w} ~=~ 2 \; \tfrac{1}{2} -4 \, \sin^2 29^\circ ~\approx~ 1 - 0.94016 ~=~ 0.05983 \approx 0.06 \approx 0.07 ~,</math>

a very small value, similar to the nearly zero total weak charge of charged leptons (see the table above).

Corrections arise when doing the full theoretical calculation for nucleons, however. Specifically, when evaluating [[Feynman diagrams]] beyond the tree level (i.e. diagrams containing loops), the weak mixing angle becomes dependent on the momentum scale due to the [[renormalization group|running]] of [[coupling constants]],<ref name=Kumar-Moller-clbrn-2014-08-25--29/> and due to the fact that nucleons are composite particles.

=== Relation to weak hypercharge {{mvar|Y}}{{sub|w}} ===
Because [[weak hypercharge]] {{mvar|Y}}{{sub|w}} is given by

<math display="block">Y_\mathsf{w} = 2\, ( Q_\epsilon - T_3 ) ~</math>

the [[weak hypercharge]]  {{mvar|Y}}{{sub|w}} , weak charge  {{mvar|Q}}{{sub|w}} , and electric charge <math>\, Q \equiv Q_\epsilon \,</math> are related by

<math display="block">Q_\mathsf{w} + Y_\mathsf{w} = 2\,Q_\epsilon \left( 1 - 2\ \sin^2\theta_\mathsf{w} \right) = 2\,Q_\epsilon \, \cos\left( 2\, \theta_\mathsf{w} \right)\ ,</math>
or equivalently
<math display="block">Q_\mathsf{w} + Y_\mathsf{w} =  Q_\epsilon + Q_\epsilon \left( 1 - 4\ \sin^2\theta_\mathsf{w} \right) \approx Q_\epsilon + 0\ ,</math>

where <math> ~ Y_\mathsf{w} ~</math> is the weak hypercharge for ''left-handed'' fermions and ''right-handed'' antifermions, hence

<math display="block">Q_\mathsf{w} + Y_\mathsf{w} \approx Q_\epsilon ~,</math>

in the typical case, when the weak mixing angle is approximately 30°.

=== Derivation ===
The Standard Model coupling of [[fermions]] to the Z boson and [[photon]] is given by:<ref name=Buchmüller-Lüdeling-CERN-984122/>

<math display="block"> \mathcal{L}_\mathrm{int} ~ = ~ -\bar{\Psi}_\boldsymbol{\mathsf{L}} \, \left[ \left( Q_\epsilon \, - \, T_3 \right) \, \frac{e}{\; \cos \theta_\mathsf{w} } \, B_\mu ~ + ~ T_3 \, \frac{e}{\; \sin \theta_\mathsf{w} \,} W^3_\mu \;\right] \, \bar{\sigma}^\mu \, \Psi_\boldsymbol{\mathsf{L}} ~ - ~ \bar{\Psi}_\boldsymbol{\mathsf{R}} \, \left[ \, Q_\epsilon \frac{e}{\; \cos\theta_\mathsf{w} \;} \, B_\mu \, \right] \, \sigma^\mu \, \Psi_\boldsymbol{\mathsf{R}} ~ ,</math>

where
*<math>~\Psi_\mathsf{L}~</math> and <math>~\Psi_\boldsymbol{\mathsf{R}}~</math> are a left-handed and right-handed fermion field respectively,
*<math>~ B_\mu ~</math> is the [[B boson]] field, <math>~ W^3_\mu ~</math> is the W{{sub|3}} boson field, and
*<math>~e = \sqrt{4\pi\alpha}~</math> is the [[elementary charge]] expressed as [[rationalized Planck units]],

and the expansion uses for its [[basis (linear algebra)|basis vectors]] the (mostly implicit) [[Pauli matrices]] from the [[Weyl equation]]:{{clarify|date=September 2021}}

<math display="block">\sigma^\mu = \Bigl(\, I\,,\; ~~\sigma^1\,,\; ~~\sigma^2\,,\; ~~\sigma^3 \, \Bigr)~</math>

and

<math display="block">\bar{\sigma}^\mu = \Bigl(\, I\,,\; -\sigma^1 \,,\; -\sigma^2 \,,\; -\sigma^3 \, \Bigr) ~</math>

The fields for B and W{{sub|3}} boson are related to the Z&nbsp;boson field <math>Z_\mu,</math> and [[electromagnetic field]] <math>A_\mu</math> (photons) by

<math display="block">~B_\mu = \left( \, \cos \theta_\mathsf{w} \, \right) \, A_\mu - \left( \, \sin \theta_\mathsf{w} \, \right) Z_\mu ~</math>

and

<math display="block">W^3_\mu = \left( \, \cos \theta_\mathsf{w} \, \right) Z_\mu ~ + ~ \left( \, \sin \theta_\mathsf{w} \, \right) \, A_\mu ~.</math>

By combining these relations with the above equation and separating by <math>Z_\mu</math> and <math>~A_\mu~,</math> one obtains:

<math display="block">
\begin{align}
\mathcal{L}_\mathrm{int} ~=~ -\bar{\Psi}_\boldsymbol{\mathsf{L}}\left[\;\left(\, Q_\epsilon \,-\, T_3 \,\right) \frac{e}{\; \cos \theta_\mathsf{w} \;}\left(\; \cos \theta_\mathsf{w} \, A_\mu - \sin \theta_\mathsf{w} \, Z_\mu \;\right) \,+\, T_3 \frac{ e }{\; \sin\theta_\mathsf{w} \;} \left(\; \cos \theta_\mathsf{w} Z_\mu \,+\, \sin \theta_\mathsf{w} \, A_\mu \;\right)\right] \bar{\sigma}^\mu \Psi_\boldsymbol{\mathsf{L}} \\
- \bar{\Psi}_\boldsymbol{\mathsf{R}} \biggl[ Q_\epsilon \, \frac{ e }{\; \cos\theta_\mathsf{w} \;}\left(\, \cos \theta_\mathsf{w} \, A_\mu \,-\, \sin \theta_\mathsf{w} \, Z_\mu \,\right) \; \biggr] \sigma^\mu  \Psi_\boldsymbol{\mathsf{R}} \\
\\
~ = ~ - ~ e \, \bar{\Psi}_\boldsymbol{\mathsf{L}} \left[\; Q_\epsilon \, A_\mu \, + \, \left(\; T_3 \, - \, Q_\epsilon \sin^2 \theta_\mathsf{w} \;\right) \frac{ 1 }{\; \cos \theta_\mathsf{w} \sin \theta_\mathsf{w} \;} \; Z_\mu \;\right] \bar{\sigma}^\mu \Psi_\boldsymbol{\mathsf{L}} \\
~ - ~ e \, \bar{\Psi}_\boldsymbol{\mathsf{R}} \left[\; Q_\epsilon \, A_\mu \, - \, Q_\epsilon \sin^2 \theta_\mathsf{w} \; \frac{ 1 }{\;\cos \theta_\mathsf{w} \, \sin \theta_\mathsf{w} \;} \; Z_\mu \;\right] \sigma^\mu \Psi_\boldsymbol{\mathsf{R}} ~ .
\end{align}
</math>

The <math>Q_\epsilon\,A_\mu</math> term that is present for both left- and right-handed fermions represents the familiar [[electromagnetic interaction]]. The terms involving the [[Z boson]] depend on the [[chirality]] of the fermion, thus there are two different coupling strengths:

<math display="block">~ Q_\boldsymbol{\mathsf{L}} = T_3 - Q_\epsilon \sin^2 \theta_\mathsf{w} \quad </math>
and
<math display="block">\quad Q_\boldsymbol{\mathsf{R}} = -Q_\epsilon \sin^2 \theta_\mathsf{w} ~.</math>

It is however more convenient to treat fermions as a single particle instead of treating left- and right-handed fermions separately. The [[Weyl basis]] is chosen for this derivation:<ref name=Tong-2009-DiracEq/>

<math display="block">\boldsymbol{\Psi} \equiv \begin{pmatrix}\Psi_{\boldsymbol\mathsf{L}} \\ \Psi_\boldsymbol{\mathsf{R}} \end{pmatrix} ~, \qquad \gamma^\mu \equiv \begin{pmatrix}0 & \sigma^\mu \\ \bar{\sigma}^\mu & 0 \end{pmatrix} \quad \text{ for } ~ \mu = 0, 1, 2, 3 ~;</math> <math>\qquad \gamma^5 \equiv \begin{pmatrix} -I & 0 \\ ~~0 & I \end{pmatrix} ~ .</math>

Thus the above expression can be written fairly compactly as:

<math display="block"> \mathcal{L}_\mathrm{int} = -e \ \boldsymbol{\bar{\Psi}} \ \gamma^\mu\ \left[\ Q_\epsilon\ A_\mu\; + \; \frac{ \left(\ Q_\mathsf{w} - 2\ T_3\ \gamma^5\ \right) }{\ 2\ \sin\left(\ 2\ \theta_\mathsf{w}\ \right)\ }\; Z_\mu\ \right]\ \boldsymbol{\Psi} ~ ,</math>

where

<math display="block"> Q_\mathsf{w} \; \equiv \; 2 \,\left(\, Q_\boldsymbol{\mathsf{L}} + Q_\boldsymbol{\mathsf{R}} \,\right) \; = \; 2 \, T_3 - 4 \, Q_\epsilon \sin^2 \theta_\mathsf{w} ~ .</math>

== See also ==
* [[Weak hypercharge]]
* [[Weak isospin]]
* [[Z boson]]
* [[Weak interaction]]
* [[Neutral current]]

== Notes ==
{{notelist}}

== References ==
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{{Standard model of physics}}

[[Category:Nuclear physics]]