Weak charge

In nuclear physics and atomic physics, weak charge refers to the Standard Model weak interaction coupling of the vector Z boson; it indicates an effect on parity violating electron scattering. For any given nuclear isotope, the total weak charge is approximately −0.99 per neutron, and +0.07 per proton.^[1] This same term is sometimes also used to refer to weak isospin,^[2] weak hypercharge, or the vector coupling of any fermion to the Z boson (i.e. the coupling strength of weak neutral currents), all of which are distinct.^[3]

Theoretical basis[]

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

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

where ${\displaystyle ~Q_{\mathsf {w}}~}$ is the weak charge,^[a] ${\displaystyle T_{3}}$ is the weak isospin,^[b] ${\displaystyle \theta _{\mathsf {w}}}$ is the weak mixing angle, and ${\displaystyle \,Q_{\epsilon }\,}$ is the electric charge.^[c]

Since ${\displaystyle \,4\sin ^{2}30^{\circ }=1\,}$ and a typical weak mixing angle is 29° ≈ 30° , so one can approximate the weak charge by:

${\displaystyle ~Q_{\mathsf {w}}~\approx ~2\,T_{3}-Q_{\epsilon }~.}$

However this relation is only applicable to fundamental particles, since the weak isospin is not clearly defined for composite particles, such as nucleons, partly due to weak isospin not being conserved. However 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 also sum up the weak charges of the constituent quarks to get the same results.

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 near vanishing of the weak charge for charged leptons (see the table below).

However corrections arise when doing the full theoretical calculation. Specifically when evaluating Feynman diagrams beyond the tree level (i.e. diagrams containing loops), where 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.

Because weak hypercharge Y_W is given by

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

the weak hypercharge ${\displaystyle Y_{\mathsf {w}}}$, weak charge ${\displaystyle Q_{\mathsf {w}}}$, and electric charge ${\displaystyle Q_{\epsilon }}$ 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, or

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

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 Lorentz-Heaviside 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:

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

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]

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

Thus the above expression can be written fairly compactly as:

${\displaystyle {\mathcal {L}}_{\mathrm {int} }=-e\,{\boldsymbol {\bar {\Psi }}}\,\gamma ^{\mu }\left[\;Q_{\epsilon }\,A_{\mu }\;+\;{\frac {1}{2}}\;\left({\frac {\;Q_{\mathsf {w}}\;}{2}}-T_{3}\gamma ^{5}\right){\frac {1}{\;\cos \theta _{\mathsf {w}}\sin \theta _{\mathsf {w}}\;}}\,Z_{\mu }\;\right]{\boldsymbol {\Psi }}~,}$

where

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

Particle values[]

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 ${\displaystyle Q_{W}}$ (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 (${\displaystyle Q_{\mathsf {L}}}$ and ${\displaystyle Q_{\mathsf {R}}}$ from the previous section). Approximate values are for the weak mixing angle θ_w = 30° ≈ 29° , actual.

Electroweak charges of Standard Model particles

Particle(s)	Electric charge	Weak charge	Weak isospin		Weak hypercharge		Z boson coupling
			LEFT	RIGHT	LEFT	RIGHT	LEFT	RIGHT
u, c, t up, charm, top	++2/3	+1 − 8/3 sin² θw ≈ ++1/3	++1/2	0	++1/3	++4/3	++1/2 − 2/3 sin² θw ≈ ++1/3	−+2/3 sin² θw ≈ −+1/6
d, s, b down, strange, bottom	−+1/3	−1 + 4/3 sin² θw ≈ −+2/3	−+1/2	0	++1/3	−+2/3	−+1/2 + 1/3 sin² θw ≈ −+5/12	++1/3 sin² θw ≈ ++1/12
νe, νμ, ντ neutrinos	0	+1	++1/2	0 [i]	−1	0 [i]	++1/2	0 [i]
e−, μ− , τ− electron, muon, tauon	−1	−1 + 4 sin² θw ≈ 0	−+1/2	0	−1	−2	−+1/2 + sin² θw ≈ −+1/4	sin² θw ≈ ++1/4
Z0, γ, g photon, Z boson, and gluon[ii]	0 [iii]
W+ W boson[iv]	+1	+2 − 4 sin² θw ≈ +1	+1		0		+1 − sin² θw ≈ ++3/4
H0 Higgs boson	0	-1	−+1/2		+1		−+1/2

The table omits antiparticles. Every particle listed (except for the uncharged bosons the photon, Z boson, gluon, and Higgs boson^[d] which are their own antiparticles) has an antiparticle with identical mass and opposite charge. All signs in the table have to be flipped 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 fermions and right-handed antifermions have ${\displaystyle T_{3}=\pm {\tfrac {1}{2}},}$ and therefore interact with the W boson. They are referred to as proper-handed. Right-handed fermions and left-handed antifermions, on the other hand, do not carry weak isospin and therefore do not interact with the W boson. They are referred to as "wrong"-handed. While wrong-handed particles do not participate in charged current interactions (interactions involving the W boson), they do interact through neutral current interactions (interactions involving the Z boson). Proper-handed fermions form a isospin doublet, while "wrong"-handed fermions are isospin singlets.

"Wrong"-handed neutrinos (sterile neutrinos) have never been observed, but may still exist, due to the fact that they are predicted to only interact gravitationally, and therefore be invisible to existing detectors.^[10] Sterile neutrinos play a role in speculations about the way neutrinos have masses (see seesaw mechanism for discussion).

Massive fermions 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 a infinite source and sink of weak isospin and / or hypercharge, due to its non-zero vacuum expectation value.^[11]

Empirical formulas[]

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]

Measurements in 2017 give the weak charge of the proton as 0.0719±0.0045 .^[13]

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.^[14]

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.^[15] 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.^[14]

See also[]

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

Notes[]

References[]