Weak charge

Not to be confused with weak hypercharge or weak isospin.

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.^[1] 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,^[2] weak hypercharge, or the vector coupling of a fermion to the Z boson (i.e. the coupling strength of weak neutral currents).^[3]

Empirical formulas

Measurements in 2017 give the weak charge of the proton as 0.0719±0.0045 .^[4]

The weak charge may be summed in atomic nuclei, so that the predicted weak charge for ¹³³Cs (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 .^[5]

A recent study used four even-numbered isotopes of ytterbium to test the formula Q_w = −0.989 N + 0.071 Z , for weak charge, with N corresponding to the number of neutrons and Z to the number of protons. The formula was found consistent to 0.1% accuracy using the ¹⁷⁰Yb, ¹⁷²Yb, ¹⁷⁴Yb, and ¹⁷⁶Yb isotopes of ytterbium.^[6]

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.^[7] The specific transition observed was the forbidden transition from 6s^{2 1}S₀ to 5d6s ³D₁ (24489 cm⁻¹). The latter state was mixed, due to weak interaction, with 6s6p ¹P₁ (25068 cm⁻¹) to a degree proportional to the nuclear weak charge.^[6]

Particle values

This table gives the values of the electric charge (the coupling to the photon, referred to in this article as ${\displaystyle Q_{\epsilon }}$^{[lower-alpha 1]}). Also listed are the approximate weak charge ${\displaystyle Q_{\mathsf {w}}}$ (the vector part of the Z boson coupling to fermions), weak isospin ${\displaystyle T_{3}}$ (the coupling to the W bosons), weak hypercharge ${\displaystyle Y_{\mathsf {w}}}$ (the coupling to the B boson) and the approximate Z boson coupling factors (${\displaystyle Q_{\boldsymbol {\mathsf {L}}}}$ and ${\displaystyle Q_{\boldsymbol {\mathsf {R}}}}$ in the "Theoretical" section, below).

The table's values are approximate: They happen to be exact for particles whose energies make the weak mixing angle ${\displaystyle \ \theta _{\mathsf {w}}=30^{\circ }\ ,}$ with ${\displaystyle \ \sin ^{2}\theta _{\mathsf {w}}={\tfrac {1}{4}}~.}$ This value is very close to the typical approximately 29° angle observed in particle accelerators.

Electroweak charges of Standard Model particles

Spin J	Particle(s)	Weak charge ${\displaystyle Q_{\mathsf {w}}}$	Electric charge ${\displaystyle Q~{\mathsf {or}}~Q_{\epsilon }}$	Weak isospin ${\displaystyle T_{3}}$		Weak hypercharge ${\displaystyle Y_{\mathsf {w}}}$		Z boson coupling
								${\displaystyle 2\ Q_{\boldsymbol {\mathsf {L}}}}$  LEFT	${\displaystyle 2\ Q_{\boldsymbol {\mathsf {R}}}}$ RIGHT
		= 2 QL + 2 QR		LEFT	RIGHT	LEFT	RIGHT
⁠ 1 /2⁠	e−, μ− , τ− electron, muon, tau[lower-roman 1]	−1 + 4 sin2 θw ≈ 0	−1	⁠−+ 1 /2⁠	0	−1	−2	−1 + 2 sin2 θw ≈ ⁠−+ 1 /2⁠	2 sin2 θw ≈ ⁠++ 1 /2⁠
⁠ 1 /2⁠	u, c, t up, charm, top[lower-roman 1]	+1 − ⁠ 8 /3⁠ sin2 θw ≈ ⁠++ 1 /3⁠	⁠++ 2 /3⁠	⁠++ 1 /2⁠	0	⁠++ 1 /3⁠	⁠++ 4 /3⁠	1 − ⁠ 4 /3⁠ sin2 θw ≈ ⁠++ 2 /3⁠	⁠−+ 4 /3⁠ sin2 θw ≈ ⁠−+ 1 /3⁠
⁠ 1 /2⁠	d, s, b down, strange, bottom[lower-roman 1]	−1 + ⁠ 4 /3⁠ sin2 θw ≈ ⁠−+ 2 /3⁠	⁠−+ 1 /3⁠	⁠−+ 1 /2⁠	0	⁠++ 1 /3⁠	⁠−+ 2 /3⁠	−1 + ⁠ 2 /3⁠ sin2 θw ≈ ⁠−+ 5 /6⁠	⁠++ 2 /3⁠ sin2 θw ≈ ⁠++ 1 /6⁠
⁠ 1 /2⁠	νe, νμ, ντ neutrinos[lower-roman 1]	+1	0	⁠++1/2⁠	0 [lower-roman 2]	−1	0 [lower-roman 2]	+1	0 [lower-roman 2]
1	g, γ, Z0, gluon[lower-roman 3], photon, and Z boson,[lower-roman 4]	0 [lower-roman 4]
1	W+ W boson[lower-roman 5]	+2 − 4 sin2 θw ≈ +1	+1	+1		0		+2 − 4 sin2 θw ≈ +1
0	H0 Higgs boson	−1	0	⁠−+ 1 /2⁠		+1		−1

1. Only (regular) fermion charges are listed. For the matching antifermions, the electric charge, Q_ϵ , has the same magnitude, but opposite sign; other charges, such as weak isospin, T₃, and weak hypercharge, Y_w, that have columns subtitled LEFT and RIGHT, are left-right swapped as well as sign-reversed.
2. 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.
3. Gluons carry only color charges of the strong force: Their electroweak charges are all zero, although their color charges give them distinct antiparticles (see Gluon for details).

   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.
4. 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 article: Two-photon physics

   Whilst not having charge themselves, photons and Z bosons, none the less do interact with particles carrying the relevant quantum charge: electrical charge ( Q_ϵ ) for photons (γ), and left and right weak charges (Q_L, Q_R) for Z bosons (Z⁰). They cannot interact with other γ or Z⁰ directly, and except at extremely high energies, usually do not interact with other γ or Z⁰ at all. However, because of quantum uncertainty even low energy versions of either particle can briefly split into a particle-antiparticle pairs each of which happens to have the electrical charge needed to interact with a γ, or the left or right weak charge needed to interact with Z⁰, or both. After that interaction has happened, the particle-antiparticle pair recombines into the same γ or Z⁰ 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 Z⁰-Z⁰ or Z⁰-γ or γ-γ interaction had occurred.

   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' (γ, Z⁰) row in the table are all almost exactly zero, but some are not precisely zero as shown.
5. Only the W⁺ boson's charges are listed; values for its antiparticle W⁻ 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.

   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.

For brevity, the table omits antiparticles. Every particle listed (except for the uncharged bosons the photon, Z boson, gluon, and Higgs boson^{[lower-alpha 2]} 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 LEFT and 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 ${\displaystyle \ T_{3}=\pm {\tfrac {1}{2}}\ ,}$ 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^± 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.^[8] Sterile neutrinos play a role in speculations about the way neutrinos have masses (see Seesaw mechanism). The above statement that the Z⁰ interacts with all fermions will need an exception for sterile neutrinos inserted, if they are ever detected experimentally.

Massive fermions – except (perhaps) neutrinos^{[lower-alpha 3]} – 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).

Theoretical basis

See also: Electroweak interaction

The formula for the weak charge is derived from the Standard Model, and is given by^[9][10]

$${\displaystyle ~Q_{\mathsf {w}}~=~2\,T_{3}-Q_{\epsilon }\,4\,\sin ^{2}\theta _{\mathsf {w}}~\approx ~2\,T_{3}-Q_{\epsilon }\;,\qquad {\mathsf {or}}\qquad ~Q_{\mathsf {w}}+Q_{\epsilon }~\approx ~2\,T_{3}~=~\pm 1;~}$$

where ${\displaystyle ~Q_{\mathsf {w}}~}$ is the weak charge,^{[lower-alpha 4]} ${\displaystyle T_{3}}$ is the weak isospin,^{[lower-alpha 5]} ${\displaystyle \theta _{\mathsf {w}}}$ is the weak mixing angle, and ${\displaystyle \,Q_{\epsilon }\,}$ is the electric charge.^{[lower-alpha 1]} The approximation for the weak charge is usually valid, since the weak mixing angle typically is 29° ≈ 30° , and ${\displaystyle \ 4\sin ^{2}30^{\circ }=1\ ,}$ and ${\displaystyle \;4\sin ^{2}29^{\circ }\approx 0.940\ ,}$ a discrepancy of only a little more than 1 in 17 .

Extension to larger, composite protons and neutrons

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 ⁠++1/2⁠ and of the neutron to ⁠−+1/2⁠,^[11][12] 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

$${\displaystyle Q_{\mathsf {w}}=2\,T_{3}-4\,Q_{\epsilon }\,\sin ^{2}\theta _{\mathsf {w}}=2\cdot \left(-{\tfrac {1}{2}}\right)=-1~\approx ~-0.99~.}$$

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

$${\displaystyle Q_{\mathsf {w}}=2\,T_{3}-4\,Q_{\epsilon }\,\sin ^{2}\theta _{\mathsf {w}}~=~2\;{\tfrac {1}{2}}-4\,\sin ^{2}29^{\circ }~\approx ~1-0.94016~=~0.05983\approx 0.06\approx 0.07~,}$$

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,^[10] and due to the fact that nucleons are composite particles.

Relation to weak hypercharge Y_w

Because weak hypercharge Y_w is given by

$${\displaystyle Y_{\mathsf {w}}=2\,(Q_{\epsilon }-T_{3})~}$$

the weak hypercharge  Y_w , weak charge  Q_w , and electric charge ${\displaystyle \,Q\equiv Q_{\epsilon }\,}$ are related by

$${\displaystyle Q_{\mathsf {w}}+Y_{\mathsf {w}}=2\,Q_{\epsilon }\,(1-2\,\sin ^{2}\theta _{\mathsf {w}})=2\,Q_{\epsilon }\,\cos \left(2\,\theta _{\mathsf {w}}\right)~,}$$

where ${\displaystyle ~Y_{\mathsf {w}}~}$ is the weak hypercharge for left-handed fermions and right-handed antifermions, or

$${\displaystyle Q_{\mathsf {w}}+Y_{\mathsf {w}}\approx Q_{\epsilon }~,}$$

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:^[13]

$${\displaystyle {\mathcal {L}}_{\mathrm {int} }~=~-{\bar {\Psi }}_{\boldsymbol {\mathsf {L}}}\,\left[\left(Q_{\epsilon }\,-\,T_{3}\right)\,{\frac {e}{\;\cos \theta _{\mathsf {w}}}}\,B_{\mu }~+~T_{3}\,{\frac {e}{\;\sin \theta _{\mathsf {w}}\,}}W_{\mu }^{3}\;\right]\,{\bar {\sigma }}^{\mu }\,\Psi _{\boldsymbol {\mathsf {L}}}~-~{\bar {\Psi }}_{\boldsymbol {\mathsf {R}}}\,\left[\,Q_{\epsilon }{\frac {e}{\;\cos \theta _{\mathsf {w}}\;}}\,B_{\mu }\,\sigma ^{\mu }\,\right]\,\Psi _{\boldsymbol {\mathsf {R}}}~,}$$

where

• ${\displaystyle ~\Psi _{\mathsf {L}}~}$ and ${\displaystyle ~\Psi _{\boldsymbol {\mathsf {R}}}~}$ are a left-handed and right-handed fermion field respectively,
• ${\displaystyle ~B_{\mu }~}$ is the B boson field, ${\displaystyle ~W_{\mu }^{3}~}$ is the W₃ boson field, and
• ${\displaystyle ~e={\sqrt {4\pi \alpha }}~}$ 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:^{[clarification needed]}

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

and

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

The fields for B and W₃ boson are related to the Z boson field ${\displaystyle Z_{\mu },}$ and electromagnetic field ${\displaystyle A_{\mu }}$ (photons) by

$${\displaystyle ~B_{\mu }=\left(\,\cos \theta _{\mathsf {w}}\,\right)\,A_{\mu }-\left(\,\sin \theta _{\mathsf {w}}\,\right)Z_{\mu }~}$$

and

$${\displaystyle W_{\mu }^{3}=\left(\,\cos \theta _{\mathsf {w}}\,\right)Z_{\mu }~+~\left(\,\sin \theta _{\mathsf {w}}\,\right)\,A_{\mu }~.}$$

By combining these relations with the above equation and separating by ${\displaystyle Z_{\mu }}$ and ${\displaystyle ~A_{\mu }~,}$ one obtains:

$${\displaystyle {\begin{aligned}{\mathcal {L}}_{\mathrm {int} }~=~-{\bar {\Psi }}_{\boldsymbol {\mathsf {L}}}\left[\;\left(\,Q_{\epsilon }\,-\,T_{3}\,\right){\frac {e}{\;\cos \theta _{\mathsf {w}}\;}}\left(\;\cos \theta _{\mathsf {w}}\,A_{\mu }-\sin \theta _{\mathsf {w}}\,Z_{\mu }\;\right)\,+\,T_{3}{\frac {e}{\;\sin \theta _{\mathsf {w}}\;}}\left(\;\cos \theta _{\mathsf {w}}Z_{\mu }\,+\,\sin \theta _{\mathsf {w}}\,A_{\mu }\;\right)\right]{\bar {\sigma }}^{\mu }\Psi _{\boldsymbol {\mathsf {L}}}\\-{\bar {\Psi }}_{\boldsymbol {\mathsf {R}}}{\biggl [}Q_{\epsilon }\,{\frac {e}{\;\cos \theta _{\mathsf {w}}\;}}\left(\,\cos \theta _{\mathsf {w}}\,A_{\mu }\,-\,\sin \theta _{\mathsf {w}}\,Z_{\mu }\,\right)\;{\biggr ]}\sigma ^{\mu }\Psi _{\boldsymbol {\mathsf {R}}}\\\\~=~-~e\,{\bar {\Psi }}_{\boldsymbol {\mathsf {L}}}\left[\;Q_{\epsilon }\,A_{\mu }\,+\,\left(\;T_{3}\,-\,Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}\;\right){\frac {1}{\;\cos \theta _{\mathsf {w}}\sin \theta _{\mathsf {w}}\;}}\;Z_{\mu }\;\right]{\bar {\sigma }}^{\mu }\Psi _{\boldsymbol {\mathsf {L}}}\\~-~e\,{\bar {\Psi }}_{\boldsymbol {\mathsf {R}}}\left[\;Q_{\epsilon }\,A_{\mu }\,-\,Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}\;{\frac {1}{\;\cos \theta _{\mathsf {w}}\,\sin \theta _{\mathsf {w}}\;}}\;Z_{\mu }\;\right]\sigma ^{\mu }\Psi _{\boldsymbol {\mathsf {R}}}~.\end{aligned}}}$$

The ${\displaystyle Q_{\epsilon }\,A_{\mu }}$ 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:

$${\displaystyle ~Q_{\boldsymbol {\mathsf {L}}}=T_{3}-Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}\quad }$$ and $${\displaystyle \quad Q_{\boldsymbol {\mathsf {R}}}=-Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}~.}$$

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:^[14]

$${\displaystyle {\boldsymbol {\Psi }}\equiv {\begin{pmatrix}\Psi _{\boldsymbol {\mathsf {L}}}\\\Psi _{\boldsymbol {\mathsf {R}}}\end{pmatrix}}~,\qquad \gamma ^{\mu }\equiv {\begin{pmatrix}0&\sigma ^{\mu }\\{\bar {\sigma }}^{\mu }&0\end{pmatrix}}\quad {\text{ for }}~\mu =0,1,2,3~;}$$ ${\displaystyle \qquad \gamma ^{5}\equiv {\begin{pmatrix}-I&0\\~~0&I\end{pmatrix}}~.}$

Thus the above expression can be written fairly compactly as:

$${\displaystyle {\mathcal {L}}_{\mathrm {int} }=-e\ {\boldsymbol {\bar {\Psi }}}\ \gamma ^{\mu }\ \left[\ Q_{\epsilon }\ A_{\mu }\;+\;{\frac {\left(\ Q_{\mathsf {w}}-2\ T_{3}\ \gamma ^{5}\ \right)}{\ 2\ \sin \left(\ 2\ \theta _{\mathsf {w}}\ \right)\ }}\;Z_{\mu }\ \right]\ {\boldsymbol {\Psi }}~,}$$

where

$${\displaystyle Q_{\mathsf {w}}\;\equiv \;2\,\left(\,Q_{\boldsymbol {\mathsf {L}}}+Q_{\boldsymbol {\mathsf {R}}}\,\right)\;=\;2\,T_{3}-4\,Q_{\epsilon }\sin ^{2}\theta _{\mathsf {w}}~.}$$

See also

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

Notes

1. ${\displaystyle \,Q\,}$ is conventionally used as the symbol for electric charge. The subscript ${\displaystyle \ \epsilon \ }$ is added in this article to keep the several symbols for weak charge ${\displaystyle \ Q_{\boldsymbol {\mathsf {L}}}\ ,}$ ${\displaystyle \ Q_{\boldsymbol {\mathsf {R}}}\ ,}$ and ${\displaystyle \ Q_{\mathsf {w}}\ ,}$ and for electric charge ${\displaystyle \ Q_{\epsilon }\ }$ from being easily confused.
2. See Higgs mechanism.
3. 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.
4. Other Wikipedia articles use the weak vector coupling, ${\displaystyle g_{\mathsf {V}},}$ a different version of ${\displaystyle ~Q_{\mathsf {w}}~}$ which is exactly half the size given here.
5. 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
   W^±
   ).

References

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2. "Properties of the Z⁰ boson" (PDF). Friedrich-Alexander-Universität Erlangen-Nürnberg. August 2015. p. 7. Retrieved 11 May 2021.
3. Woods, Michael B. (28 June 2005). "Measuring the electron's WEAK charge" (Press release). SLAC, Stanford University. p. 34. SLAC E158. Retrieved 2 September 2021. Studying electron-electron scattering in mirror worlds to search for new phenomena at the energy frontier
4. Androić, D.; Armstrong, D.S.; Asaturyan, A.; et al. (The Jefferson Lab. Qweak Collaboration) (2018). "Precision measurement of the weak charge of the proton". Nature. 557 (7704): 207–211. arXiv:1905.08283. doi:10.1038/s41586-018-0096-0.
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6. Antypas, D.; Fabricant, A.; Stalnaker, J.E.; Tsigutkin, K.; Flambaum, V.V.; Budker, D. (2018). "Isotopic variation of parity violation in atomic ytterbium". Nature Physics. 15 (2): 120–123. arXiv:1804.05747. doi:10.1038/s41567-018-0312-8.
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8. "Sterile neutrinos". All Things Neutrino. Fermilab. Retrieved 18 May 2021.
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10. Kumar, Krishna S.; et al. (MOLLER collaboration) (25–29 August 2014). "Parity-violating electron scattering" (PDF). In Schmidt, A.; Sander, C. (eds.). Proceedings, 20th International Conference on Particles and Nuclei (PANIC 14). 20th International Conference on Particles and Nuclei (PANIC 2014). Hamburg, Germany: Deutsches Elektronen-Synchrotron (DESY). doi:10.3204/DESY-PROC-2014-04/255. DESY-PROC-2014-04. Retrieved 20 June 2021.
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