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; it is named by analogy to electric charge, which measures coupling to the photon of electromagnetism. 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 0.0719±0.0045 . 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 . A recent study used four even-numbered isotopes of ytterbium to test the formula Qw = −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. 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 6s2 1S0 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.
Particle values This table gives the values of the electric charge (the coupling to the photon, referred to in this article as Q ϵ {\displaystyle Q_{\epsilon }} ). Also listed are the approximate weak charge Q w {\displaystyle Q_{\mathsf {w}}} (the vector part of the Z boson coupling to fermions), weak isospin T 3 {\displaystyle T_{3}} (the coupling to the W bosons), weak hypercharge Y w {\displaystyle Y_{\mathsf {w}}} (the coupling to the theoretical B boson) and the approximate Z boson coupling factors ( Q L {\displaystyle Q_{\boldsymbol {\mathsf {L}}}} and Q R {\displaystyle Q_{\boldsymbol {\mathsf {R}}}} in the "Theoretical" section, below). If the variable correction terms shown for different θ w {\displaystyle \ \theta _{\mathsf {w}}\ } values are omitted, then the table's constant values for weak charge are only approximate: They happen to be exact for particles whose energies make the weak mixing angle θ w = 30 ∘ , {\displaystyle \ \theta _{\mathsf {w}}=30^{\circ }\ ,} with sin 2 θ w = 1 4 . {\displaystyle \ \sin ^{2}\theta _{\mathsf {w}}={\tfrac {1}{4}}~.} This value is very close to the typical approximately 29° angle observed in particle accelerators. The embedded formulas give more accurate values for those cases when the Weinberg angle, θ w , {\displaystyle \ \theta _{\mathsf {w}}\ ,} is known.
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 "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 interact with the W boson, and so are assigned T 3 = ± 1 2 . {\displaystyle \ T_{3}=\pm {\tfrac {1}{2}}~.} They could be referred to as "proper"-handed (that is, they have the "proper" handedness for a W± interaction). Right-handed fermions and left-handed antifermions, on the other hand, do not interact with the W boson (except for electrical interaction) and therefore are assigned zero weak isospin; for that reason, they could be referred to as "wrong"-handed (i.e. they have the wrong handedness to interact with the W± boson). "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), excepting perhaps sterile neutrinos, if they exist. "Wrong"-handed neutrinos (sterile neutrinos) have never been observed, but may still exist since they would be invisible to existing detectors. Sterile neutrinos, assuming they exist, are speculated to play a role in a few theoretical mechanisms that might provide neutrinos with mass (see Seesaw mechanism). The statement above, that the Z0 interacts with all fermions, will need an exception inserted for sterile neutrinos, 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 w = 2 T 3 − Q ϵ 4 sin 2 θ w ≈ 2 T 3 − Q ϵ , o r Q w + Q ϵ ≈ 2 T 3 = ± 1 ; {\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 Q w {\displaystyle ~Q_{\mathsf {w}}~} is the weak charge,
T 3 {\displaystyle T_{3}} is the weak isospin,
θ w {\displaystyle \theta _{\mathsf {w}}} is the weak mixing angle, and Q ϵ {\displaystyle \,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 ∘ = 1 , {\displaystyle \ 4\sin ^{2}30^{\circ }=1\ ,} and 4 sin 2 29 ∘ ≈ 0.940 , {\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, 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 = 2 T 3 − 4 Q ϵ sin 2 θ w = 2 ⋅ ( − 1 2 ) = − 1 ≈ − 0.99 . {\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
Q w = 2 T 3 − 4 Q ϵ sin 2 θ w = 2 1 2 − 4 sin 2 29 ∘ ≈ 1 − 0.94016 = 0.05983 ≈ 0.06 ≈ 0.07 , {\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, and due to the fact that nucleons are composite particles.
Relation to weak hypercharge Yw Because weak hypercharge Yw is given by
Y w = 2 ( Q ϵ − T 3 ) {\displaystyle Y_{\mathsf {w}}=2\,(Q_{\epsilon }-T_{3})~}
the weak hypercharge Yw , weak charge Qw , and electric charge Q ≡ Q ϵ {\displaystyle \,Q\equiv Q_{\epsilon }\,} are related by
Q w + Y w = 2 Q ϵ ( 1 − 2 sin 2 θ w ) = 2 Q ϵ cos ( 2 θ w ) , {\displaystyle Q_{\mathsf {w}}+Y_{\mathsf {w}}=2\,Q_{\epsilon }\left(1-2\ \sin ^{2}\theta _{\mathsf {w}}\right)=2\,Q_{\epsilon }\,\cos \left(2\,\theta _{\mathsf {w}}\right)\ ,}
or equivalently
Q w + Y w = Q ϵ + Q ϵ ( 1 − 4 sin 2 θ w ) ≈ Q ϵ + 0 , {\displaystyle Q_{\mathsf {w}}+Y_{\mathsf {w}}=Q_{\epsilon }+Q_{\epsilon }\left(1-4\ \sin ^{2}\theta _{\mathsf {w}}\right)\approx Q_{\epsilon }+0\ ,}
where Y w {\displaystyle ~Y_{\mathsf {w}}~} is the weak hypercharge for left-handed fermions and right-handed antifermions, hence
Q w + Y w ≈ Q ϵ , {\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:
L i n t = − Ψ ¯ L [ ( Q ϵ − T 3 ) e cos θ w B μ + T 3 e sin θ w W μ 3 ] σ ¯ μ Ψ L − Ψ ¯ R [ Q ϵ e cos θ w B μ ] σ μ Ψ R , {\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 }\,\right]\,\sigma ^{\mu }\,\Psi _{\boldsymbol {\mathsf {R}}}~,}
where
Ψ L {\displaystyle ~\Psi _{\mathsf {L}}~} and Ψ R {\displaystyle ~\Psi _{\boldsymbol {\mathsf {R}}}~} are a left-handed and right-handed fermion field respectively,
B μ {\displaystyle ~B_{\mu }~} is the B boson field, W μ 3 {\displaystyle ~W_{\mu }^{3}~} is the W3 boson field, and
e = 4 π α {\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:
σ μ = ( I , σ 1 , σ 2 , σ 3 ) {\displaystyle \sigma ^{\mu }={\Bigl (}\,I\,,\;~~\sigma ^{1}\,,\;~~\sigma ^{2}\,,\;~~\sigma ^{3}\,{\Bigr )}~}
and
σ ¯ μ = ( I , − σ 1 , − σ 2 , − σ 3 ) {\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 Z μ , {\displaystyle Z_{\mu },} and electromagnetic field A μ {\displaystyle A_{\mu }} (photons) by
B μ = ( cos θ w ) A μ − ( sin θ w ) Z μ {\displaystyle ~B_{\mu }=\left(\,\cos \theta _{\mathsf {w}}\,\right)\,A_{\mu }-\left(\,\sin \theta _{\mathsf {w}}\,\right)Z_{\mu }~}
and
W μ 3 = ( cos θ w ) Z μ + ( sin θ w ) A μ . {\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 Z μ {\displaystyle Z_{\mu }} and A μ , {\displaystyle ~A_{\mu }~,} one obtains:
L i n t = − Ψ ¯ L [ ( Q ϵ − T 3 ) e cos θ w ( cos θ w A μ − sin θ w Z μ ) + T 3 e sin θ w ( cos θ w Z μ + sin θ w A μ ) ] σ ¯ μ Ψ L − Ψ ¯ R [ Q ϵ e cos θ w ( cos θ w A μ − sin θ w Z μ ) ] σ μ Ψ R = − e Ψ ¯ L [ Q ϵ A μ + ( T 3 − Q ϵ sin 2 θ w ) 1 cos θ w sin θ w Z μ ] σ ¯ μ Ψ L − e Ψ ¯ R [ Q ϵ A μ − Q ϵ sin 2 θ w 1 cos θ w sin θ w