Weak charge

In nuclear physics and atomic physics, 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. 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, weak hypercharge, or the vector coupling of a fermion to the Z boson (i.e. the coupling strength of weak neutral currents).

Empirical formulas
Measurements in 2017 give the weak charge of the proton as $0.072$.

The weak charge may be summed in atomic nuclei, so that the predicted weak charge for $133$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 $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 $170$Yb, $172$Yb, $174$Yb, and $176$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$2$ $1$S$0$ to 5d6s $3$D$1$ (24489 cm$−1$). The latter state was mixed, due to weak interaction, with 6s6p $1$P$1$ (25068 cm$−1$) to a degree proportional to the nuclear weak charge.

Particle values
This table gives the values of the electric charge (the coupling to the photon, referred to in this article as $Q_\epsilon$). Also listed are the approximate weak charge $$Q_\mathsf{w}$$ (the vector part of the Z boson coupling to fermions), weak isospin $$T_3$$ (the coupling to the W bosons), weak hypercharge $$Y_\mathsf{w}$$ (the coupling to the B boson) and the approximate Z boson coupling factors ($$Q_\boldsymbol\mathsf{L}$$ and $$Q_\boldsymbol\mathsf{R}$$ 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 $$\ \theta_\mathsf{w} = 30^\circ\ ,$$ with $$\ \sin^2 \theta_\mathsf{w} = \tfrac{1}{4} ~.$$ This value is very close to the typical approximately 29° angle observed in particle accelerators.


 * {| class="wikitable"  style="text-align:center;"

!rowspan="3"| Spin $J$ !rowspan="3"| Particle(s) !rowspan="2"| Weak charge $$Q_\mathsf{w}$$ !rowspan="3"| Electric charge $$Q ~ \mathsf{ or } ~ Q_{\epsilon}$$ !colspan="2" rowspan="2"| Weak isospin $$T_3$$ !colspan="2" rowspan="2"| Weak hypercharge $$Y_\mathsf{w}$$ !colspan="2"| Z boson coupling !rowspan="2"| $$2\ Q_\boldsymbol\mathsf{L}$$ !rowspan="2"| $$2\ Q_\boldsymbol\mathsf{R}$$ ! ! ! !
 * + Electroweak charges of Standard Model particles
 * style="background:#EBEBEF;"|= $2$ $Q$$$ $+ 2$ $Q$$$
 * style="background:#EBEBEF;"|= $1⁄2$ $−$$2$ $μ−$ $θ$$w$
 * e$2$,$τ−$, $≈ 0$ electron, muon, tau
 * −1 + 4 sin$θ$ $w$$2$$−1$
 * −1 + 2 sin$θ$ $w$$8⁄3$$− 1⁄2$
 * 2 sin$2$ $θ$$w$$0$
 * u, c, t up, charm, top
 * +1 − $4⁄3$ sin$2$ $θ$$w$$−1$
 * 1 − $− 4⁄3$ sin$2$ $θ$$w$$−2$
 * $4⁄3$ sin$2$ $θ$$w$$≈ − 1⁄2$
 * d, s, b down, strange, bottom
 * −1 + $2⁄3$ sin$2$ $θ$$w$$≈ + 1⁄2$
 * −1 + $+ 2⁄3$ sin$2$ $θ$$w$$1⁄2$
 * $0$ sin$+$ $2$$θ$$≈ + 1⁄3$
 * $+ 2⁄3$, $+ 1⁄2$, $0$ neutrinos
 * &emsp; $+ 1⁄3$
 * &emsp; $+ 4⁄3$
 * $≈ + 2⁄3$, $≈ − 1⁄3$, $1⁄2$$w$, gluon, photon, and Z boson,
 * colspan="8"| $≈ − 2⁄3$
 * W$2$ W boson
 * +2 − 4 sin$θ$ $w$$0$$− 1⁄3$
 * colspan="2"| $− 1⁄2$
 * colspan="2"| $0$
 * colspan="2"| +2 − 4 sin$Qϵ$ $T$$3$$+ 1⁄3$
 * H$Y$ Higgs boson
 * colspan="2"| $− 2⁄3$
 * colspan="2"| $≈ − 5⁄6$
 * colspan="2"| $≈ + 1⁄6$
 * }
 * −1 + $w$ sin$Qϵ$ $γ$$Q$$1⁄2$
 * $$ sin$Q$ $$$γ$$&nu;e$
 * $&nu;μ$, $&nu;τ$, $+1$ neutrinos
 * &emsp; $0$
 * &emsp; $+ 1⁄2$
 * $0$, $−1$, $0$$γ$, gluon, photon, and Z boson,
 * colspan="8"| $+1$
 * W$γ$ W boson
 * +2 − 4 sin$γ$ $±$$+ 1⁄2$$0$
 * colspan="2"| $1$
 * colspan="2"| $g$
 * colspan="2"| +2 − 4 sin$− 1⁄2$ $Y$$w$$&gamma;$
 * H$Y$ Higgs boson
 * colspan="2"| $Z$
 * colspan="2"| $0$
 * colspan="2"| $1$
 * }
 * $≈ +1$, $+1$, $+1$$w$, gluon, photon, and Z boson,
 * colspan="8"| $0$
 * W$Y$ W boson
 * +2 − 4 sin$w$ $Q$$w$$≈ +1$
 * colspan="2"| $0$
 * colspan="2"| $−1$
 * colspan="2"| +2 − 4 sin$3$ $3$$0$$0$
 * H⇭⇭⇭ Higgs boson
 * colspan="2"| $− 1⁄2$
 * colspan="2"| $+1$
 * colspan="2"| $−1$
 * }
 * H⇭⇭⇭ Higgs boson
 * colspan="2"| $Z0$
 * colspan="2"| $Z0$
 * colspan="2"| $Z0$
 * }
 * colspan="2"| $Z0$
 * colspan="2"| $Z0$
 * colspan="2"| $Z0$
 * }

For brevity, the table omits antiparticles. Every particle listed (except for the uncharged bosons the photon, Z boson, gluon, and Higgs boson 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 and  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 $$\ T_3 = \pm\tfrac{1}{2}\ ,$$ 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. 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 – 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
The formula for the weak charge is derived from the Standard Model, and is given by

$$~ Q_\mathsf{w} ~=~ 2 \, T_3 - Q_\epsilon \, 4 \, \sin^2 \theta_\mathsf{w} ~\approx~ 2 \, T_3 - Q_\epsilon \;, \qquad \mathsf{ or } \qquad  ~ Q_\mathsf{w} + Q_\epsilon ~\approx~ 2 \, T_3 ~=~ \pm 1 ; ~$$

where $$~ Q_\mathsf{w} ~$$ is the weak charge, $$T_3$$ is the weak isospin, $$\theta_\mathsf{w}$$ is the weak mixing angle, and $$\, Q_\epsilon \,$$ is the electric charge. The approximation for the weak charge is usually valid, since the weak mixing angle typically is 29° ≈ 30°, and $$\ 4 \sin^2 30^\circ = 1\ ,$$ and $$\; 4 \sin^2 29^\circ \approx 0.940\ ,$$ 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 ⇭⇭⇭ and of the neutron to ⇭⇭⇭, 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

$$ Q_\mathsf{w} = 2 \, T_3 - 4 \, Q_\epsilon \, \sin^2 \theta_\mathsf{w} = 2 \cdot \left( -\tfrac{1}{2} \right) = -1 ~\approx~ -0.99 ~ .$$

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

$$ Q_\mathsf{w} = 2 \, T_3 - 4\, Q_\epsilon \, \sin^2 \theta_\mathsf{w} ~=~ 2 \; \tfrac{1}{2} -4 \, \sin^2 29^\circ ~\approx~ 1 - 0.94016 ~=~ 0.05983 \approx 0.06 \approx 0.07 ~ ,$$

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

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.

Relation to weak hypercharge ⇭⇭⇭⇭⇭⇭
Because weak hypercharge ⇭⇭⇭⇭⇭⇭ is given by

$$Y_\mathsf{w} = 2\, ( Q_\epsilon - T_3 ) ~$$

the weak hypercharge  ⇭⇭⇭⇭⇭⇭, weak charge  ⇭⇭⇭⇭⇭⇭ , and electric charge $$\, Q \equiv Q_\epsilon \,$$ are related by

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

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

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

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:

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

where
 * $$~\Psi_\mathsf{L}~$$ and $$~\Psi_\boldsymbol{\mathsf{R}}~$$ are a left-handed and right-handed fermion field respectively,
 * $$~ B_\mu ~$$ is the B boson field, $$~ W^3_\mu ~$$ is the W⇭⇭⇭ boson field, and
 * $$~e = \sqrt{4\pi\alpha}~$$ 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:

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

and

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

The fields for B and W⇭⇭⇭ boson are related to the Z boson field $$Z_\mu,$$ and electromagnetic field $$A_\mu$$ (photons) by

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

and

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

By combining these relations with the above equation and separating by $$Z_\mu$$ and $$~A_\mu~,$$ 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 $$Q_\epsilon\,A_\mu$$ 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:

$$~ Q_\boldsymbol{\mathsf{L}} = T_3 - Q_\epsilon \sin^2 \theta_\mathsf{w} \quad $$ and $$\quad Q_\boldsymbol{\mathsf{R}} = -Q_\epsilon \sin^2 \theta_\mathsf{w} ~.$$

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:

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

Thus the above expression can be written fairly compactly as:

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

where

$$ Q_\mathsf{w} \; \equiv \; 2 \,\left(\, Q_\boldsymbol{\mathsf{L}} + Q_\boldsymbol{\mathsf{R}} \,\right) \; = \; 2 \, T_3 - 4 \, Q_\epsilon \sin^2 \theta_\mathsf{w} ~ .$$