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In
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
and
atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned w ...
, weak charge refers to the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It w ...
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
coupling of a particle to the
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
. For example, for any given nuclear isotope, the total weak charge is approximately −0.99 per
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...
, and +0.07 per
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
. It also shows an effect of
parity violation In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
during
electron scattering Electron scattering occurs when electrons are deviated from their original trajectory. This is due to the electrostatic forces within matter interaction or, if an external magnetic field is present, the electron may be deflected by the Lorentz fo ...
. This same term is sometimes also used to refer to other, distinct quantites, such as
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It can ...
,
weak hypercharge In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted Y_\mathsf and corresponds to the gauge ...
, or the ''vector'' coupling of a
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
to the
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
(i.e. the coupling strength of
weak neutral current Weak neutral current interactions are one of the ways in which subatomic particles can interact by means of the weak force. These interactions are mediated by the Z boson. The discovery of weak neutral currents was a significant step towar ...
s).


Theoretical basis

The formula for the weak charge is derived from the Standard Model, and is given by :~Q_\mathsf ~=~ 2 \, T_3 - Q_\epsilon \, 4 \, \sin^2 \theta_\mathsf ~\approx~ 2 \, T_3 - Q_\epsilon \; , where ~ Q_\mathsf ~ is the weak charge, T_3 is the weak isospin, \theta_\mathsf is the weak mixing angle, and \, Q_\epsilon \, is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectiv ...
. The approximation for the weak charge is usually valid, since the weak mixing angle typically is and \ 4 \sin^2 30^\circ = 1\ , and \; 4 \sin^2 29^\circ \approx 0.940\ , a discrepancy of only a little more than


Extension to larger, composite protons and neutrons

This relation only directly applies to
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s and
lepton In particle physics, a lepton is an elementary particle of half-integer spin ( spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neut ...
s (
fundamental particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, anti ...
), since
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It can ...
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 = 2 \, T_3 - 4 \, Q_\epsilon \, \sin^2 \theta_\mathsf = 2 \cdot \left( -\tfrac \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 = 2 \, T_3 - 4\, Q_\epsilon \, \sin^2 \theta_\mathsf ~=~ 2 \; \tfrac -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 In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...
beyond the tree level (i.e. diagrams containing loops), the weak mixing angle becomes dependent on the momentum scale due to the
running Running is a method of terrestrial locomotion allowing humans and other animals to move rapidly on foot. Running is a type of gait characterized by an aerial phase in which all feet are above the ground (though there are exceptions). This i ...
of coupling constants, and due to the fact that nucleons are composite particles.


Relation to weak hypercharge

Because
weak hypercharge In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted Y_\mathsf and corresponds to the gauge ...
is given by :Y_\mathsf = 2\, ( Q_\epsilon - T_3 ) ~ the
weak hypercharge In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted Y_\mathsf and corresponds to the gauge ...
  , weak charge   , and electric charge \, Q \equiv Q_\epsilon \, are related by :Q_\mathsf + Y_\mathsf = 2\,Q_\epsilon\,( 1 - 2 \, \sin^2\theta_\mathsf ) = 2\,Q_\epsilon \, \cos\left( 2\, \theta_\mathsf \right) ~ , where ~ Y_\mathsf ~ is the weak hypercharge for ''left-handed'' fermions and ''right-handed'' antifermions, or :Q_\mathsf + Y_\mathsf \approx Q_\epsilon ~, in the typical case, when the weak mixing angle is approximately 30°.


Derivation

The Standard Model coupling of
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
to the Z boson and
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always m ...
is given by: :\mathcal_\mathrm ~ = ~ -\bar_\boldsymbol \, \left \left( Q_\epsilon \, - \, T_3 \right) \, \frac \, B_\mu ~ + ~ T_3 \, \frac W^3_\mu \;\right\, \bar^\mu \, \Psi_\boldsymbol ~ - ~ \bar_\boldsymbol \, \left \, Q_\epsilon \frac \, B_\mu \, \sigma^\mu \, \right\, \Psi_\boldsymbol ~ , where : ~\Psi_\mathsf~ and ~\Psi_\boldsymbol~ 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~ is the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
expressed as rationalized Planck units, and the expansion uses for its
basis vectors In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as component ...
the (mostly implicit)
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
from the Weyl equation: : \sigma^\mu = \Bigl(\, I\,,\; ~~\sigma^1\,,\; ~~\sigma^2\,,\; ~~\sigma^3 \, \Bigr)~ and : \bar^\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 An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...
A_\mu (photons) by :~B_\mu = \left( \, \cos \theta_\mathsf \, \right) \, A_\mu - \left( \, \sin\theta_\mathsf \, \right) Z_\mu ~ and :W^3_\mu = \left( \, \cos \theta_\mathsf \, \right) Z_\mu ~ + ~ \left( \, \sin \theta_\mathsf \, \right) \, A_\mu ~. By combining these relations with the above equation and separating by Z_\mu and ~A_\mu~, one obtains: : \begin \mathcal_\mathrm ~=~ -\bar_\boldsymbol\left[\;\left(\, Q_\epsilon \,-\, T_3 \,\right) \frac\left(\; \cos \theta_\mathsf \, A_\mu - \sin \theta_\mathsf \, Z_\mu \;\right) \,+\, T_3 \frac \left(\; \cos \theta_\mathsf Z_\mu \,+\, \sin \theta_\mathsf \, A_\mu \;\right)\right] \bar^\mu \Psi_\boldsymbol \\ - \bar_\boldsymbol \biggl[ Q_\epsilon \, \frac\left(\, \cos \theta_\mathsf \, A_\mu \,-\, \sin \theta_\mathsf \, Z_\mu \,\right) \; \biggr] \sigma^\mu \Psi_\boldsymbol \\ \\ ~ = ~ - ~ e \, \bar_\boldsymbol \left ; Q_\epsilon \, A_\mu \, + \, \left(\; T_3 \, - \, Q_\epsilon \sin^2 \theta_\mathsf \;\right) \frac \; Z_\mu \;\right\bar^\mu \Psi_\mathsf \\ ~ - ~ e \, \bar_\boldsymbol \left ; Q_\epsilon \, A_\mu \, - \, Q_\epsilon \sin^2 \theta_\mathsf \; \frac \; Z_\mu \;\right\sigma^\mu \Psi_\boldsymbol ~ . \end The Q_\epsilon\,A_\mu term that is present for both left- and right-handed fermions represents the familiar
electromagnetic interaction In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
. The terms involving the
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
depend on the
chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
of the fermion, thus there are two different coupling strengths: :~ Q_\boldsymbol = T_3 - Q_\epsilon \sin^2 \theta_\mathsf \quad and \quad Q_\boldsymbol = -Q_\epsilon \sin^2 \theta_\mathsf ~. 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 = \begin\Psi_ \\ \Psi_\boldsymbol \end ~, \qquad \gamma^\mu = \begin0 & \sigma^\mu \\ \bar^\mu & 0 \end \quad \text ~ \mu = 0, 1, 2, 3 ~; \qquad \gamma^5 = \begin -I & 0 \\ ~~0 & I \end ~ . Thus the above expression can be written fairly compactly as: : \mathcal_\mathrm = -e \, \boldsymbol \, \gamma^\mu \left \; Q_\epsilon \, A_\mu \; + \; \frac \left( \, Q_\mathsf - 2 \, T_3 \, \gamma^5 \right) \frac \, Z_\mu \;\right\boldsymbol ~ , where : Q_\mathsf \; \equiv \; 2 \,\left(\, Q_\boldsymbol + Q_\boldsymbol \,\right) \; = \; 2 \, T_3 - 4 \, Q_\epsilon \sin^2 \theta_\mathsf ~ .


Particle values

This table gives the values of the electric charge (the coupling to the photon, referred to in the previous section as approximate weak charge Q_\mathsf (the vector part of the Z boson coupling for fermions), weak isopsin T_3 (the coupling to the
W bosons In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
), weak hypercharge Y_\mathsf (the coupling to the B boson) and the approximate Z boson coupling factors (Q_\mathsf and Q_\mathsf from the previous section). The table's values are approximate: They are exact for particle energies which make the weak mixing angle \ \theta_\mathsf = 30^\circ\ exactly, with \ \sin^2 \theta_\mathsf = \tfrac\ , which is close to the typical : The table omits antiparticles. Every particle listed (except for the uncharged bosons the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always m ...
,
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
,
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...
, and
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
which are their own antiparticles) has an antiparticle with identical mass and opposite charge. All 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 fermions and right-handed antifermions have T_3 = \pm\tfrac, and therefore interact with the
W boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
. They could be referred to as ''"proper"-handed''. Right-handed fermions and left-handed antifermions, on the other hand, do not carry weak isospin and therefore (except for electrical interaction) do not interact with the
W boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
, hence they could be referred to as ''"wrong"-handed''. While "wrong"-handed particles do not participate in charged current interactions (interactions involving the
W boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
), they ''do'' interact through neutral current interactions (interactions involving the
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
). "Proper"-handed fermions form isospin doublets, while "wrong"-handed fermions are isospin singlets. "Wrong"-handed neutrinos (
sterile neutrinos Sterile neutrinos (or inert neutrinos) are hypothetical particles (neutral leptons – neutrinos) that are believed to interact only via gravity and not via any of the other fundamental interactions of the Standard Model. The term ''sterile neutrin ...
) have never been observed, but may still exist, due to the fact that they are predicted to only interact gravitationally, and therefore would be invisible to existing detectors. Sterile neutrinos play a role in speculations about the way neutrinos have masses (''see''
seesaw mechanism In the theory of grand unification of particle physics, and, in particular, in theories of neutrino masses and neutrino oscillation, the seesaw mechanism is a generic model used to understand the relative sizes of observed neutrino masses, of the ...
''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 The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Standa ...
, which acts as an infinite source and sink of
weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It can ...
and / or hypercharge, due to its non-zero
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
.''Further information'':
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...


Empirical formulas

Measurements in 2017 give the weak charge of the proton as  . The weak charge may be summed in atomic nuclei, so that the predicted weak charge for 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 Ytterbium is a chemical element with the symbol Yb and atomic number 70. It is a metal, the fourteenth and penultimate element in the lanthanide series, which is the basis of the relative stability of its +2 oxidation state. However, like the othe ...
to test the formula for weak charge, with corresponding to the number of neutrons and to the number of protons. The formula was found consistent to 0.1% accuracy using the Yb, Yb, Yb, and Yb isotopes of ytterbium. In the
ytterbium Ytterbium is a chemical element with the symbol Yb and atomic number 70. It is a metal, the fourteenth and penultimate element in the lanthanide series, which is the basis of the relative stability of its +2 oxidation state. However, like the othe ...
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 In spectroscopy, a forbidden mechanism (forbidden transition or forbidden line) is a spectral line associated with absorption or emission of photons by atomic nuclei, atoms, or molecules which undergo a transition that is not allowed by a particul ...
from 6s S to 5d6s D (24489 cm). The latter state was mixed, due to weak interaction, with 6s6p P (25068 cm) to a degree proportional to the nuclear weak charge.


See also

*
Weak hypercharge In the Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently denoted Y_\mathsf and corresponds to the gauge ...
*
Weak isospin In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol or , with the third component written as or . It can ...
*
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
*
Weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
*
Neutral current Weak neutral current interactions are one of the ways in which subatomic particles can interact by means of the weak force. These interactions are mediated by the Z boson. The discovery of weak neutral currents was a significant step towar ...


Notes


References

{{Standard model of physics Nuclear physics