Weak charge

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. 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 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 currents).

Empirical formulas

Measurements in 2017 give the weak charge of the proton as Template:Val .

The weak charge may be summed in atomic nuclei, so that the predicted weak charge for Template:SupCs (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 Template:Nobr for weak charge, with Template:Mvar corresponding to the number of neutrons and Template:Mvar to the number of protons. The formula was found consistent to 0.1% accuracy using the Template:SupYb, Template:SupYb, Template:SupYb, and Template:SupYb 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 6sTemplate:Sup [[Term symbol|Template:SupSTemplate:Sub]] to 5d6s [[Term symbol|Template:SupDTemplate:Sub]] (24489 cmTemplate:Sup). The latter state was mixed, due to weak interaction, with 6s6p [[Term symbol|Template:SupPTemplate:Sub]] (25068 cmTemplate:Sup) 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 Template:Nobr Also listed are the approximate weak charge $Q_{𝗐}$ (the vector part of the Z boson coupling to fermions), weak isospin $T_3$ (the coupling to the W bosons), weak hypercharge $Y_{𝗐}$ (the coupling to the B boson) and the approximate Z boson coupling factors ($Q_{𝖫}$ and $Q_{𝖱}$ 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 ${\theta }_{𝗐}=30^{\circ },$ with ${\sin }^2{\theta }_{𝗐}=\frac{1}{4}.$ This value is very close to the typical Template:Nobr angle observed in particle accelerators. The embedded formulas give the (more) exact values for when the Weinberg angle, ${\theta }_{𝗐},$ is known.

Electroweak charges of Standard Model particles

Spin Template:Mvar	Particle(s)	Weak charge $Q_{𝗐}$	Electric charge $Q𝗈𝗋Q_{ϵ}$	Weak isospin $T_3$		Weak hypercharge $Y_{𝗐}$		Z boson coupling
								$2Q_{𝖫}$ Template:Small	$2Q_{𝖱}$ Template:Small
		Template:Nobr		Template:Small	Template:Small	Template:Small	Template:Small
Template:Math	eTemplate:Sup,Template:Math, Template:Math electron, muon, tauTemplate:Efn-lr	Template:SmallTemplate:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:SmallTemplate:Math	Template:SmallTemplate:Math
Template:Math	u, c, t up, charm, topTemplate:Efn-lr	Template:SmallTemplate:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:SmallTemplate:Math	Template:SmallTemplate:Math
Template:Math	d, s, b down, strange, bottomTemplate:Efn-lr	Template:SmallTemplate:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math	Template:SmallTemplate:Math	Template:SmallTemplate:Math
Template:Math	Template:Math, Template:Math, Template:Math neutrinosTemplate:Efn-lr	Template:Math	Template:Math	Template:Math	Template:Nobr	Template:Math	Template:Nobr	Template:Math	Template:Nobr
Template:Math	Template:Math, Template:Math, Template:MathTemplate:Sup, gluonTemplate:Efn-lr, photon, and Template:Nobr	Template:BigTemplate:Efn-lr
Template:Math	WTemplate:Sup W bosonTemplate:Efn-lr	Template:SmallTemplate:Math	Template:Math	Template:Math		Template:Math		Template:SmallTemplate:Math
Template:Math	HTemplate:Sup Higgs boson	Template:Math	Template:Math	Template:Math		Template:Math		Template:Math

For brevity, the table omits antiparticles. Every particle listed (except for the uncharged bosons the photon, Z boson, gluon, and Higgs bosonTemplate:Efn 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 Template:Sc and Template:Sc 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 $T_3=\pm \frac{1}{2},$ 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 boson|WTemplate: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. Sterile neutrinos play a role in speculations about the way neutrinos have masses (see Seesaw mechanism). The above statement that the Template:Math interacts with all fermions will need an exception for sterile neutrinos inserted, if they are ever detected experimentally.

Massive fermions – except (perhaps) neutrinosTemplate:EfnTemplate:Efn – 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

Template:See also The formula for the weak charge is derived from the Standard Model, and is given by

$$Q_{𝗐}=2T_3-Q_{ϵ}4{\sin }^2{\theta }_{𝗐}\approx 2T_3-Q_{ϵ},𝗈𝗋Q_{𝗐}+Q_{ϵ}\approx 2T_3=\pm 1;$$

where $Q_{𝗐}$ is the weak charge,Template:Efn $T_3$ is the weak isospin,Template:Efn ${\theta }_{𝗐}$ is the weak mixing angle, and $Q_{ϵ}$ is the electric charge.Template:Efn The approximation for the weak charge is usually valid, since the weak mixing angle typically is Template:Nobr and $4{\sin }^{2}30^{\circ }=1,$ and $4{\sin }^{2}29^{\circ }\approx 0.940,$ a discrepancy of only a little more than Template:Nobr

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 Template:Sfrac and of the neutron to Template:Sfrac, 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

$$Q_{𝗐}=2T_3-4Q_{ϵ}{\sin }^2{\theta }_{𝗐}=2\cdot (-\frac{1}{2})=-1\approx -0.99.$$

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

$$Q_{𝗐}=2T_3-4Q_{ϵ}{\sin }^2{\theta }_{𝗐}=2\frac{1}{2}-4{\sin }^{2}29^{\circ }\approx 1-0.94016=0.05983\approx 0.06\approx 0.07,$$

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, and due to the fact that nucleons are composite particles.

Relation to weak hypercharge Template:MvarTemplate:Sub

Because weak hypercharge Template:MvarTemplate:Sub is given by

$$Y_{𝗐}=2(Q_{ϵ}-T_3)$$

the weak hypercharge  Template:MvarTemplate:Sub , weak charge  Template:MvarTemplate:Sub , and electric charge $Q\equiv Q_{ϵ}$ are related by

$$Q_{𝗐}+Y_{𝗐}=2Q_{ϵ}(1-2{\sin }^2{\theta }_{𝗐})=2Q_{ϵ}\cos \left(2{\theta }_{𝗐}\right),$$ or equivalently $$Q_{𝗐}+Y_{𝗐}=Q_{ϵ}+Q_{ϵ}(1-4{\sin }^2{\theta }_{𝗐})\approx Q_{ϵ}+0,$$

where $Y_{𝗐}$ is the weak hypercharge for left-handed fermions and right-handed antifermions, hence

$$Q_{𝗐}+Y_{𝗐}\approx Q_{ϵ},$$

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:

$${ℒ}_{\mathrm{i}\mathrm{n}\mathrm{t}}=-{\overline{\mathrm{\Psi }}}_{𝖫}[(Q_{ϵ}-T_3)\frac{e}{\cos {\theta }_{𝗐}}{B}_{\mu }+T_3\frac{e}{\sin {\theta }_{𝗐}}{W}_{\mu }^3]{\overline{\sigma }}^{\mu }{\mathrm{\Psi }}_{𝖫}-{\overline{\mathrm{\Psi }}}_{𝖱}\left[Q_{ϵ}\frac{e}{\cos {\theta }_{𝗐}}{B}_{\mu }\right]{\sigma }^{\mu }{\mathrm{\Psi }}_{𝖱},$$

where

• ${\mathrm{\Psi }}_{𝖫}$ and ${\mathrm{\Psi }}_{𝖱}$ are a left-handed and right-handed fermion field respectively,
• $B_{\mu }$ is the B boson field, $W_{\mu }^3$ is the WTemplate:Sub 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:Template:Clarify

$${\sigma }^{\mu }=(I,{\sigma }^1,{\sigma }^2,{\sigma }^3)$$

and

$${\overline{\sigma }}^{\mu }=(I,-{\sigma }^1,-{\sigma }^2,-{\sigma }^3)$$

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

$$B_{\mu }=(\cos {\theta }_{𝗐})A_{\mu }-(\sin {\theta }_{𝗐})Z_{\mu }$$

and

$$W_{\mu }^3=(\cos {\theta }_{𝗐})Z_{\mu }+(\sin {\theta }_{𝗐})A_{\mu }.$$

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

$$\begin{array}{c}{ℒ}_{\mathrm{i}\mathrm{n}\mathrm{t}}=-{\overline{\mathrm{\Psi }}}_{𝖫}[(Q_{ϵ}-T_3)\frac{e}{\cos {\theta }_{𝗐}}(\cos {\theta }_{𝗐}{A}_{\mu }-\sin {\theta }_{𝗐}{Z}_{\mu })+T_3\frac{e}{\sin {\theta }_{𝗐}}(\cos {\theta }_{𝗐}{Z}_{\mu }+\sin {\theta }_{𝗐}{A}_{\mu })]{\overline{\sigma }}^{\mu }{\mathrm{\Psi }}_{𝖫}\\ -{\overline{\mathrm{\Psi }}}_{𝖱}\left[Q_{ϵ}\frac{e}{\cos {\theta }_{𝗐}}(\cos {\theta }_{𝗐}{A}_{\mu }-\sin {\theta }_{𝗐}{Z}_{\mu })\right]{\sigma }^{\mu }{\mathrm{\Psi }}_{𝖱}\\ \\ =-e{\overline{\mathrm{\Psi }}}_{𝖫}[Q_{ϵ}{A}_{\mu }+(T_3-Q_{ϵ}{\sin }^2{\theta }_{𝗐})\frac{1}{\cos {\theta }_{𝗐}\sin {\theta }_{𝗐}}{Z}_{\mu }]{\overline{\sigma }}^{\mu }{\mathrm{\Psi }}_{𝖫}\\ -e{\overline{\mathrm{\Psi }}}_{𝖱}[Q_{ϵ}{A}_{\mu }-Q_{ϵ}{\sin }^2{\theta }_{𝗐}\frac{1}{\cos {\theta }_{𝗐}\sin {\theta }_{𝗐}}{Z}_{\mu }]{\sigma }^{\mu }{\mathrm{\Psi }}_{𝖱}.\end{array}$$

The $Q_{ϵ}{A}_{\mu }$ 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:

$$Q_{𝖫}=T_3-Q_{ϵ}{\sin }^2{\theta }_{𝗐}$$ and $$Q_{𝖱}=-Q_{ϵ}{\sin }^2{\theta }_{𝗐}.$$

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:

$$\mathrm{\Psi }\equiv \left(\begin{array}{c}{\mathrm{\Psi }}_{𝖫}\\ {\mathrm{\Psi }}_{𝖱}\end{array}\right),{\gamma }^{\mu }\equiv \left(\begin{array}{cc}0& {\sigma }^{\mu }\\ {\overline{\sigma }}^{\mu }& 0\end{array}\right)\text{ for }\mu =0,1,2,3;$$ ${\gamma }^5\equiv \left(\begin{array}{cc}-I& 0\\ 0& I\end{array}\right).$

Thus the above expression can be written fairly compactly as:

$${ℒ}_{\mathrm{i}\mathrm{n}\mathrm{t}}=-e\overline{\mathrm{\Psi }}{\gamma }^{\mu }[Q_{ϵ}{A}_{\mu }+\frac{(Q_{𝗐}-2T_3{\gamma }^5)}{2\sin \left(2{\theta }_{𝗐}\right)}{Z}_{\mu }]\mathrm{\Psi },$$

where

$$Q_{𝗐}\equiv 2(Q_{𝖫}+Q_{𝖱})=2T_3-4Q_{ϵ}{\sin }^2{\theta }_{𝗐}.$$

See also

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