Weak charge

Measurements in 2017 give the weak charge of the proton as {{val|0.0719|0.0045}} .

The weak charge may be summed in atomic nuclei, so that the predicted weak charge for {{sup|133}}Cs (55 protons, 78 neutrons) is 55×(+0.0719) + 78×(−0.989) {{=}} −73.19, while the value determined experimentally, from measurements of parity violating electron scattering, was −72.58 .

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% accuracy using the {{sup|170}}Yb, {{sup|172}}Yb, {{sup|174}}Yb, and {{sup|176}}Yb isotopes of ytterbium.

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. The specific transition observed was the forbidden transition from 6s{{sup|2}} Term symbol to 5d6s Term symbol (24489 cm{{sup|−1}}). The latter state was mixed, due to weak interaction, with 6s6p Term symbol (25068 cm{{sup|−1}}) to a degree proportional to the nuclear weak charge.

This table gives the values of the electric charge (the coupling to the photon, referred to in this article as {{nobr| $Q\_\backslash epsilon${{efn|name=Q_w_electric_charge_note}}).}} Also listed are the approximate weak charge $Q\_\backslash mathsf\{w\}$ (the vector part of the Z boson coupling to fermions), weak isospin $T\_3$ (the coupling to the W bosons), weak hypercharge $Y\_\backslash mathsf\{w\}$ (the coupling to the B boson) and the approximate Z boson coupling factors ($Q\_\backslash boldsymbol\backslash mathsf\{L\}$ and $Q\_\backslash boldsymbol\backslash mathsf\{R\}$ in the "Theoretical" section, below).

If the variable correction terms shown for different $\backslash \; \backslash theta\_\backslash mathsf\{w\}\backslash$ values are not added in, then the table's constant values for weak charge are only approximate: They happen to be exact for particles whose energies make the weak mixing angle $\backslash \; \backslash theta\_\backslash mathsf\{w\}\; =\; 30^\backslash circ\backslash \; ,$ with $\backslash \; \backslash sin^2\; \backslash theta\_\backslash mathsf\{w\}\; =\; \backslash tfrac\{1\}\{4\}\; ~.$ This value is very close to the typical {{nobr|approximately 29° }} angle observed in particle accelerators. The embedded formulas give the (more) exact values for when the Weinberg angle, $\backslash \; \backslash theta\_\backslash mathsf\{w\}\backslash \; ,$ is known.

{{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 quantum charges of every kind for photons and Z bosons are all zero, so the photon and Z boson are their own antiparticles: They are "truly neutral particles"; in particular, they are truly neutral vector bosons.

{{main|Two-photon physics}}

Whilst not having charge themselves, photons and {{nobr|Z bosons}} none the less do interact with particles carrying the relevant quantum charge: electrical charge ({{mvar| Q{{sub|ϵ}} }}) for photons ({{mvar|γ}}), and left and right weak charges ({{mvar|Q}}{{sub|{{sc|L}}}}, {{mvar|Q}}{{sub|{{sc|R}}}}) for Z bosons ({{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 quantum uncertainty even low energy versions of either particle can briefly split into a particle-antiparticle pair, each of which happens to have the electrical charge needed to interact with a photon, 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 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 "charge"-like quantum numbers are equal and opposite. {{pb}} W bosons can interact with both photons and Z bosons, 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 charges 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, gluons are out-of-context among of the electroweak-interacting particles described by this table. However, since each of the three electrically neutral elementary vector bosons' 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.

}}

}}

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 $\backslash \; T\_3\; =\; \backslash pm\backslash tfrac\{1\}\{2\}\backslash \; ,$ and therefore interact with the W boson. They could be referred to as "proper"-handed (that is, they have the "proper" handedness for a W{{sup|±}} interaction). 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 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. Sterile neutrinos play a role in speculations about the way 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 neutrinos. 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 flavors, 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 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 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 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-sections 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 decays 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") 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 long-baseline beams of neutrinos. But the problem with that is the apparently minuscule 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 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).

{{See also|Electroweak interaction}}

The formula for the weak charge is derived from the Standard Model, and is given by

$$~\; Q\_\backslash mathsf\{w\}\; ~=~\; 2\; \backslash ,\; T\_3\; -\; Q\_\backslash epsilon\; \backslash ,\; 4\; \backslash ,\; \backslash sin^2\; \backslash theta\_\backslash mathsf\{w\}\; ~\backslash approx~\; 2\; \backslash ,\; T\_3\; -\; Q\_\backslash epsilon\; \backslash ;\; ,\; \backslash qquad\; \backslash mathsf\{\; or\; \}\; \backslash qquad\; ~\; Q\_\backslash mathsf\{w\}\; +\; Q\_\backslash epsilon\; ~\backslash approx~\; 2\; \backslash ,\; T\_3\; ~=~\; \backslash pm\; 1\; ;\; ~$$

where $~\; Q\_\backslash mathsf\{w\}\; ~$ is the weak charge,{{efn|

Other Wikipedia articles use the weak vector coupling, $g\_\backslash mathsf\{V\},$ a different version of $~\; Q\_\backslash mathsf\{w\}\; ~$ which is exactly half the size given here.

}}

$T\_3$ 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}}).

}}

$\backslash theta\_\backslash mathsf\{w\}$ is the weak mixing angle, and $\backslash ,\; Q\_\backslash epsilon\; \backslash ,$ is the electric charge.{{efn|

name=Q_w_electric_charge_note|

$\backslash ,\; Q\; \backslash ,$ is conventionally used as the symbol for electric charge. The subscript $\backslash \; \backslash epsilon\backslash$ is added in this article to keep the several symbols for weak charge $\backslash \; Q\_\backslash boldsymbol\backslash mathsf\{L\}\backslash \; ,$ $\backslash \; Q\_\backslash boldsymbol\backslash mathsf\{R\}\backslash \; ,$ and $\backslash \; Q\_\backslash mathsf\{w\}\backslash \; ,$ and for electric charge $\backslash \; Q\_\backslash epsilon\; \backslash$ from being easily confused.

}}

The approximation for the weak charge is usually valid, since the weak mixing angle typically is {{nobr| 29° ≈ 30° ,}} and $\backslash \; 4\; \backslash sin^2\; 30^\backslash circ\; =\; 1\backslash \; ,$ and $\backslash ;\; 4\; \backslash sin^2\; 29^\backslash circ\; \backslash approx\; 0.940\backslash \; ,$ a discrepancy of only a little more than {{nobr| 1 in 17 .}}

This relation only directly applies to quarks and leptons (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}}, 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

$$Q\_\backslash mathsf\{w\}\; =\; 2\; \backslash ,\; T\_3\; -\; 4\; \backslash ,\; Q\_\backslash epsilon\; \backslash ,\; \backslash sin^2\; \backslash theta\_\backslash mathsf\{w\}\; =\; 2\; \backslash cdot\; \backslash left(\; -\backslash tfrac\{1\}\{2\}\; \backslash right)\; =\; -1\; ~\backslash approx~\; -0.99\; ~.$$

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

$$Q\_\backslash mathsf\{w\}\; =\; 2\; \backslash ,\; T\_3\; -\; 4\backslash ,\; Q\_\backslash epsilon\; \backslash ,\; \backslash sin^2\; \backslash theta\_\backslash mathsf\{w\}\; ~=~\; 2\; \backslash ;\; \backslash tfrac\{1\}\{2\}\; -4\; \backslash ,\; \backslash sin^2\; 29^\backslash circ\; ~\backslash approx~\; 1\; -\; 0.94016\; ~=~\; 0.05983\; \backslash approx\; 0.06\; \backslash approx\; 0.07\; ~,$$

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 running of coupling constants, and due to the fact that nucleons are composite particles.

Because weak hypercharge {{mvar|Y}}{{sub|w}} is given by

$$Y\_\backslash mathsf\{w\}\; =\; 2\backslash ,\; (\; Q\_\backslash epsilon\; -\; T\_3\; )\; ~$$

the weak hypercharge  {{mvar|Y}}{{sub|w}} , weak charge  {{mvar|Q}}{{sub|w}} , and electric charge $\backslash ,\; Q\; \backslash equiv\; Q\_\backslash epsilon\; \backslash ,$ are related by

$$Q\_\backslash mathsf\{w\}\; +\; Y\_\backslash mathsf\{w\}\; =\; 2\backslash ,Q\_\backslash epsilon\; \backslash left(\; 1\; -\; 2\backslash \; \backslash sin^2\backslash theta\_\backslash mathsf\{w\}\; \backslash right)\; =\; 2\backslash ,Q\_\backslash epsilon\; \backslash ,\; \backslash cos\backslash left(\; 2\backslash ,\; \backslash theta\_\backslash mathsf\{w\}\; \backslash right)\backslash \; ,$$

or equivalently

$$Q\_\backslash mathsf\{w\}\; +\; Y\_\backslash mathsf\{w\}\; =\; Q\_\backslash epsilon\; +\; Q\_\backslash epsilon\; \backslash left(\; 1\; -\; 4\backslash \; \backslash sin^2\backslash theta\_\backslash mathsf\{w\}\; \backslash right)\; \backslash approx\; Q\_\backslash epsilon\; +\; 0\backslash \; ,$$

where $~\; Y\_\backslash mathsf\{w\}\; ~$ is the weak hypercharge for left-handed fermions and right-handed antifermions, hence

$$Q\_\backslash mathsf\{w\}\; +\; Y\_\backslash mathsf\{w\}\; \backslash approx\; Q\_\backslash epsilon\; ~,$$

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

The Standard Model coupling of fermions to the Z boson and photon is given by:

$$\backslash mathcal\{L\}\_\backslash mathrm\{int\}\; ~\; =\; ~\; -\backslash bar\{\backslash Psi\}\_\backslash boldsymbol\{\backslash mathsf\{L\}\}\; \backslash ,\; \backslash left[\; \backslash left(\; Q\_\backslash epsilon\; \backslash ,\; -\; \backslash ,\; T\_3\; \backslash right)\; \backslash ,\; \backslash frac\{e\}\{\backslash ;\; \backslash cos\; \backslash theta\_\backslash mathsf\{w\}\; \}\; \backslash ,\; B\_\backslash mu\; ~\; +\; ~\; T\_3\; \backslash ,\; \backslash frac\{e\}\{\backslash ;\; \backslash sin\; \backslash theta\_\backslash mathsf\{w\}\; \backslash ,\}\; W^3\_\backslash mu\; \backslash ;\backslash right]\; \backslash ,\; \backslash bar\{\backslash sigma\}^\backslash mu\; \backslash ,\; \backslash Psi\_\backslash boldsymbol\{\backslash mathsf\{L\}\}\; ~\; -\; ~\; \backslash bar\{\backslash Psi\}\_\backslash boldsymbol\{\backslash mathsf\{R\}\}\; \backslash ,\; \backslash left[\; \backslash ,\; Q\_\backslash epsilon\; \backslash frac\{e\}\{\backslash ;\; \backslash cos\backslash theta\_\backslash mathsf\{w\}\; \backslash ;\}\; \backslash ,\; B\_\backslash mu\; \backslash ,\; \backslash right]\; \backslash ,\; \backslash sigma^\backslash mu\; \backslash ,\; \backslash Psi\_\backslash boldsymbol\{\backslash mathsf\{R\}\}\; ~\; ,$$

where

• $~\backslash Psi\_\backslash mathsf\{L\}~$ and $~\backslash Psi\_\backslash boldsymbol\{\backslash mathsf\{R\}\}~$ are a left-handed and right-handed fermion field respectively,
• $~\; B\_\backslash mu\; ~$ is the B boson field, $~\; W^3\_\backslash mu\; ~$ is the W{{sub|3}} boson field, and
• $~e\; =\; \backslash sqrt\{4\backslash pi\backslash alpha\}~$ is the elementary charge expressed as rationalized Planck units,

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

$$\backslash sigma^\backslash mu\; =\; \backslash Bigl(\backslash ,\; I\backslash ,,\backslash ;\; ~~\backslash sigma^1\backslash ,,\backslash ;\; ~~\backslash sigma^2\backslash ,,\backslash ;\; ~~\backslash sigma^3\; \backslash ,\; \backslash Bigr)~$$

and

$$\backslash bar\{\backslash sigma\}^\backslash mu\; =\; \backslash Bigl(\backslash ,\; I\backslash ,,\backslash ;\; -\backslash sigma^1\; \backslash ,,\backslash ;\; -\backslash sigma^2\; \backslash ,,\backslash ;\; -\backslash sigma^3\; \backslash ,\; \backslash Bigr)\; ~$$

The fields for B and W{{sub|3}} boson are related to the Z boson field $Z\_\backslash mu,$ and electromagnetic field $A\_\backslash mu$ (photons) by

$$~B\_\backslash mu\; =\; \backslash left(\; \backslash ,\; \backslash cos\; \backslash theta\_\backslash mathsf\{w\}\; \backslash ,\; \backslash right)\; \backslash ,\; A\_\backslash mu\; -\; \backslash left(\; \backslash ,\; \backslash sin\; \backslash theta\_\backslash mathsf\{w\}\; \backslash ,\; \backslash right)\; Z\_\backslash mu\; ~$$

and

$$W^3\_\backslash mu\; =\; \backslash left(\; \backslash ,\; \backslash cos\; \backslash theta\_\backslash mathsf\{w\}\; \backslash ,\; \backslash right)\; Z\_\backslash mu\; ~\; +\; ~\; \backslash left(\; \backslash ,\; \backslash sin\; \backslash theta\_\backslash mathsf\{w\}\; \backslash ,\; \backslash right)\; \backslash ,\; A\_\backslash mu\; ~.$$

By combining these relations with the above equation and separating by $Z\_\backslash mu$ and $~A\_\backslash mu~,$ one obtains:

\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}

The $Q\_\backslash epsilon\backslash ,A\_\backslash mu$ 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:

$$~\; Q\_\backslash boldsymbol\{\backslash mathsf\{L\}\}\; =\; T\_3\; -\; Q\_\backslash epsilon\; \backslash sin^2\; \backslash theta\_\backslash mathsf\{w\}\; \backslash quad$$

and

$$\backslash quad\; Q\_\backslash boldsymbol\{\backslash mathsf\{R\}\}\; =\; -Q\_\backslash epsilon\; \backslash sin^2\; \backslash theta\_\backslash mathsf\{w\}\; ~.$$

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:

Thus the above expression can be written fairly compactly as:

$$\backslash mathcal\{L\}\_\backslash mathrm\{int\}\; =\; -e\; \backslash \; \backslash boldsymbol\{\backslash bar\{\backslash Psi\}\}\; \backslash \; \backslash gamma^\backslash mu\backslash \; \backslash left[\backslash \; Q\_\backslash epsilon\backslash \; A\_\backslash mu\backslash ;\; +\; \backslash ;\; \backslash frac\{\; \backslash left(\backslash \; Q\_\backslash mathsf\{w\}\; -\; 2\backslash \; T\_3\backslash \; \backslash gamma^5\backslash \; \backslash right)\; \}\{\backslash \; 2\backslash \; \backslash sin\backslash left(\backslash \; 2\backslash \; \backslash theta\_\backslash mathsf\{w\}\backslash \; \backslash right)\backslash \; \}\backslash ;\; Z\_\backslash mu\backslash \; \backslash right]\backslash \; \backslash boldsymbol\{\backslash Psi\}\; ~\; ,$$

where

$$Q\_\backslash mathsf\{w\}\; \backslash ;\; \backslash equiv\; \backslash ;\; 2\; \backslash ,\backslash left(\backslash ,\; Q\_\backslash boldsymbol\{\backslash mathsf\{L\}\}\; +\; Q\_\backslash boldsymbol\{\backslash mathsf\{R\}\}\; \backslash ,\backslash right)\; \backslash ;\; =\; \backslash ;\; 2\; \backslash ,\; T\_3\; -\; 4\; \backslash ,\; Q\_\backslash epsilon\; \backslash sin^2\; \backslash theta\_\backslash mathsf\{w\}\; ~\; .$$

• Weak hypercharge
• Weak isospin
• Z boson
• Weak interaction
• Neutral current

{{notelist}}

{{reflist|25em|refs=

{{cite journal

|last1=Androić |first1=D.

|last2=Armstrong |first2=D.S.

|last3=Asaturyan |first3=A.

|display-authors=etal

|collaboration=The Jefferson Lab. Qweak Collaboration

|year=2018

|title=Precision measurement of the weak charge of the proton

|journal=Nature

|volume=557 |issue= 7704|pages=207–211

|arxiv=1905.08283 |doi=10.1038/s41586-018-0096-0

}}

{{cite journal

|first1=D. |last1=Antypas |first2=A. |last2=Fabricant

|first3=J.E. |last3=Stalnaker |first4=K. |last4=Tsigutkin

|first5=V.V. |last5=Flambaum |first6=D. |last6=Budker

|year=2018

|title=Isotopic variation of parity violation in atomic ytterbium

|journal=Nature Physics

|volume=15 |issue=2 |pages=120–123 |arxiv=1804.05747 |doi=10.1038/s41567-018-0312-8

}}

{{cite web

|first1=W. |last1=Buchmüller

|first2=C. |last2=Lüdeling

|title=Field Theory and the Standard Model

|publisher=CERN

|url=https://cds.cern.ch/record/984122/files/p1.pdf

|access-date=May 14, 2021

}}

{{cite journal

|first1=V.A. |last1=Dzuba |first2=J.C. |last2=Berengut

|first3=V.V. |last3=Flambaum |first4=B. |last4=Roberts

|year=2012

|title=Revisiting parity non-conservation in Cesium

|journal=Physical Review Letters

|volume=109 |issue=20 |page=203003

|doi=10.1103/PhysRevLett.109.203003

|arxiv=1207.5864 |pmid=23215482

}}

{{cite web

|title=Sterile neutrinos

|website=All Things Neutrino

|publisher=Fermilab

|url=https://neutrinos.fnal.gov/types/sterile-neutrinos

|access-date=May 18, 2021

}}

{{cite journal

|first1=G. |last1=Hagen |first2=A. |last2=Ekström

|first3=C. |last3=Forssén |first4=G.R. |last4=Jansen

|first5=W. |last5=Nazarewicz |first6=T. |last6=Papenbrock

|first7=K.A. |last7=Wendt

|first8=S. |last8=Bacca |author8-link=Sonia Bacca

|first9=N. |last9=Barnea |first10=B. |last10=Carlsson

|first11=C. |last11=Drischler |first12=K. |last12=Hebeler

|first13=M. |last13=Hjorth-Jensen |first14=M. |last14=Miorelli

|first15=G. |last15=Orlandini |first16=A. |last16=Schwenk

|first17=J. |last17=Simonis |display-authors=6

|year=2016

|title=Charge, neutron, and weak size of the atomic nucleus

|journal=Nature Physics

|volume=12 |issue=2 |pages=186–190

|doi=10.1038/nphys3529 |arxiv=1509.07169

}}

{{cite conference

|first=Krishna S. |last=Kumar

|collaboration=MOLLER collaboration

|date=25–29 August 2014

|title=Parity-violating electron scattering

|conference=20th International Conference on Particles and Nuclei (PANIC 2014)

|place=Hamburg, Germany

|editor1=Schmidt, A.

|editor2=Sander, C.

|book-title=Proceedings, 20th International Conference on Particles and Nuclei (PANIC 14)

|id=DESY-PROC-2014-04

|doi=10.3204/DESY-PROC-2014-04/255 |doi-access=free

|publisher=Deutsches Elektronen-Synchrotron (DESY)

|url=http://www-library.desy.de/preparch/desy/proc/proc14-04/255.pdf

|access-date=2021-06-20 }}

{{cite press release

|title=Atomic parity violation research reaches new milestone

|date=2018-11-12

|publisher=Universität Mainz

|website=phys.org

|url=https://phys.org/news/2018-11-atomic-parity-violation-milestone.html

|access-date=2018-11-13

}}

{{cite web

|title=Lecture 16 - Electroweak Theory

|publisher=University of Edinburgh

|page =7

|url=https://www2.ph.ed.ac.uk/~playfer/PPlect16.pdf

|access-date=May 11, 2021

}}

{{cite journal

|first=S.P. |last=Rosen

|date=1 May 1978

|title=Universality and the weak isospin of leptons, nucleons, and quarks

|journal=Physical Review

|volume=17

|issue=9

|pages=2471–2474

|doi=10.1103/PhysRevD.17.2471

}}

{{cite journal

|first=B.A. |last=Robson

|date=12 April 2004

|title=Relation between strong and weak isospin

|journal=International Journal of Modern Physics

|volume=13 |issue=5 |pages=999–1018

|doi=10.1142/S0218301304002521

}}

{{cite web

|first=David |last=Tong

|date=2009

|title=Dirac Equation

|publisher=University of Cambridge

|page =11

|url=https://www.damtp.cam.ac.uk/user/tong/qft/four.pdf

|access-date=May 15, 2021

}}

{{cite web

|title=Properties of the Z{{sup|0}} boson

|date=August 2015

|publisher=Friedrich-Alexander-Universität Erlangen-Nürnberg

|page=7

|url=https://www.fp.fkp.uni-erlangen.de/advanced-laboratory-course/selection-of-experiments/MSc-Versuchsanleitungen-englisch/M48E.pdf

|access-date=11 May 2021

}}

{{cite press release

|first=Michael B. |last=Woods

|date=28 June 2005

|title=Measuring the electron's WEAK charge

|id=SLAC E158

|publisher=SLAC, Stanford University

|page=34

|quote=Studying electron-electron scattering in mirror worlds to search for new phenomena at the energy frontier

|url=https://www-project.slac.stanford.edu/e158/weakforce.html

|access-date=2021-09-02 }}

}}

{{Standard model of physics}}

Category:Nuclear physics