Weak charge

The formula for the weak charge is derived from the Standard Model, and is given by[4][5]

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

where ${\displaystyle ~Q_{\mathsf {w}}~}$ is the weak charge,[lower-alpha 1] ${\displaystyle T_{3}}$ is the weak isospin,[lower-alpha 2] ${\displaystyle \theta _{\mathsf {w}}}$ is the weak mixing angle, and ${\displaystyle \,Q_{\epsilon }\,}$ is the electric charge.[lower-alpha 3] 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,[6][7] 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 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,[5] 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:[8]

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

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:

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

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:[9]

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

${\displaystyle \qquad \gamma ^{5}\equiv {\begin{pmatrix}-I&0\\~~0&I\end{pmatrix}}~.}$

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

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

This table gives the values of the electric charge (the coupling to the photon, referred to in the previous section as ${\displaystyle Q_{\epsilon }}$), approximate weak charge ${\displaystyle Q_{\mathsf {w}}}$ (the vector part of the Z boson coupling for fermions), weak isopsin ${\displaystyle T_{3}}$ (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 ${\displaystyle Q_{\boldsymbol {\mathsf {R}}}}$ from the previous section).

The table's values are approximate: They are exact for particle energies which make the weak mixing angle ${\displaystyle \ \theta _{\mathsf {w}}=30^{\circ }\ }$ exactly, with ${\displaystyle \ \sin ^{2}\theta _{\mathsf {w}}={\tfrac {1}{4}}\ ,}$ which is close to the typical approximate 29° .

Electroweak charges of Standard Model particles

Spin J	Particle(s)	Electric charge ${\displaystyle Q~{\mathsf {or}}~Q_{\epsilon }}$	Weak charge ${\displaystyle Q_{\mathsf {w}}}$	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	−1 + 4 sin2 θw ≈ 0	−+ 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]	++ 2 /3	+1 −  8 /3 sin2 θw ≈ ++ 1 /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 /3	−1 +  4 /3 sin2 θw ≈ −+ 2 /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]	0	+1	++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
1	W+ W boson[lower-roman 5]	+1	+2 − 4 sin2 θw ≈ +1	+1		0		+2 − 4 sin2 θw ≈ +1
0	H0 Higgs boson	0	−1	−+ 1 /2		+1		−1

For brevity, the table omits antiparticles. Every particle listed (except for the uncharged bosons the photon, Z boson, gluon, and Higgs boson[lower-alpha 4] 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 ("wrong" for W^±). "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 known "wrong"-handed fermions 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.[10] 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 found to exist.

Massive fermions – except neutrinos – always exist in a superposition of left-handed and right-handed states, and never in pure chiral states. (The exception for neutrinos presumes the non-existence of sterile neutrinos, which is a matter of current research.) 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).

Measurements in 2017 give the weak charge of the proton as 0.0719±0.0045 .[11]

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 .[12]

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.[13]

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.[14] 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.[13]

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