Weak interaction

The beta decay of an atomic nucleus occurs through weak interaction. A neutron is transformed into a proton , an electron and an electron antineutrino .

The weak interaction (also called weak nuclear force, occasionally also β-interaction ) is one of the four basic forces in physics . In contrast to the interactions of gravity and electromagnetism known from everyday life, however, it only acts over very short distances. Like other forces, it can ensure the exchange of energy and momentum , but it is primarily effective in the decay or transformation of the particles involved, such as the beta decay of certain radioactive atomic nuclei . Because of the weak interaction, no bound states can be formed, which distinguishes them from the other three interactions.

The weak interaction is of decisive importance for life on earth due to its role in the fusion of hydrogen to helium in the sun ( proton-proton reaction ), since it is the only way to convert protons into neutrons . In this way, a stable helium nucleus with two protons and two neutrons is created from four protons (the hydrogen nuclei) through several intermediate steps. Through this process, the sun releases energy. Due to the low strength of the weak interaction, this process takes place so slowly that the sun has been shining steadily for 4.5 billion years and will probably continue to do so for another five to six billion years.

overview

The weak interaction can be divided into charged currents and uncharged currents. Charged currents act between all ( left-handed ) quarks and (left-handed) leptons as well as the ( right-handed ) antiquarks and (right-handed) anti-leptons. Uncharged currents act between the same particles that interact with charged currents, but also between all charged (anti-) quarks and (anti-) leptons regardless of their chirality .

The electromagnetic is approx. 10 ¹¹ times, the strong interaction approx. 10 ¹³ times stronger than the weak interaction. Like the strong and the electromagnetic interaction, it is described by the exchange of gauge bosons . These exchange particles of the weak interaction are the neutral Z boson and the two positively and negatively charged W bosons . Since these are massive , the weak force only has an extremely short range below the atomic nucleus radius .

The easiest way to observe the weak interaction is with the decays of quarks or leptons. In scattering experiments, on the other hand, this is rather difficult to access, since it is superimposed by the strong or electromagnetic interaction with charged leptons or hadrons . Particles that are not subject to either the strong or the electromagnetic interaction (no color charge and no electrical charge ) are the uncharged leptons, i.e. the neutrinos , but which in scattering experiments have extremely small cross- sections.

The weak interaction violates parity conservation , as demonstrated in the Wu experiment . It also violates the CP conservation about the decay of the uncharged K ⁰ - meson ( kaon ).

A quantum field theory that describes the weak interaction along with the electromagnetic interaction is the glass show-Weinberg-Salam model . This formulation also speaks of two aspects of the electroweak interaction that are unified by the Higgs mechanism .

Exchange particles

The exchange particles of the weak interaction are massive vector bosons. You have spin 1. Your behavior can be described by the Proca equation .

The following table gives an overview of the properties of the exchange particles (mass and resonance width according to Particle Data Group , service life calculated using the energy-time uncertainty relation):

Boson	Mass (GeV / c 2 )	Resonance width (GeV)	Lifetime (s)
${\displaystyle W^{\pm }}$	${\displaystyle 80{,}379\;\;\pm 0{,}012\;\;}$	${\displaystyle 2{,}085\;\;\pm 0{,}042\;\;}$	${\displaystyle 3{,}16\cdot 10^{-25}}$
${\displaystyle Z^{0}}$	${\displaystyle 91{,}1876\pm 0{,}0021}$	${\displaystyle 2{,}4952\pm 0{,}0023}$	${\displaystyle 2{,}64\cdot 10^{-25}}$

The range can be estimated roughly, by assuming that the particles during their lifetime (in the rest system of the particle) at 71% the speed of light in the laboratory system scroll ( Lorentz factor :) . For a lifetime of 3 · 10 ⁻²⁵  s, this results in a range of around 0.09  femtometers - the smallest atomic nucleus, the proton, has a diameter of around 1.7 femtometers. ${\displaystyle r}$ ${\displaystyle \tau }$ ${\displaystyle c}$ ${\displaystyle \gamma =1{,}41}$ ${\displaystyle r\approx \gamma v\tau \approx c\tau }$

In the electroweak theory, the mass ratio of W and Z bosons is linked to the Weinberg angle ( CODATA 2014) ${\displaystyle \theta _{W}\approx 28{,}18^{\circ }}$

${\displaystyle {\frac {m_{W}}{m_{Z}}}=\cos \theta _{W}\approx 0{,}8788}$ .

As a consequence of the Weinberg mixture, it follows that the coupling strength of the Z bosons is not identical to that of the W bosons. The coupling strength of the W boson to a left-handed fermion is given by

${\displaystyle Q_{W}=g\,T_{z}\;}$ ,

the coupling strength of the to a fermion is through ${\displaystyle Z^{0}}$

${\displaystyle Q_{Z}={\frac {g}{\cos \theta _{W}}}\left(T_{z}-z_{f}\,\sin ^{2}\theta _{W}\right)}$ ,

where is the charge of the fermion in units of elementary charge . denotes the third component of the weak isospin . For example, for left-handed neutrinos . ${\displaystyle z_{f}}$ ${\displaystyle e}$ ${\displaystyle T_{z}}$ ${\displaystyle T_{z}=1/2}$

The coupling strengths of weak and electromagnetic interaction overhang

${\displaystyle e=g\,\sin \theta _{W}\approx 0{,}481\,g}$ .

Reactions, crossing symmetry, reaction probability

To describe a weak process one usually uses the notation of a reaction equation, such as

${\displaystyle a+b\rightarrow c+d}$

The particles a and b are converted into the particles c and d in one process. When this process is possible, as well as all other possible that after the commutation of cruising (Engl. Crossing ) arise. A particle can therefore be written on the other side of the reaction equation by noting its corresponding antiparticle :

${\displaystyle b\rightarrow c+d+{\bar {a}}}$

In addition, the reverse processes are possible.

${\displaystyle c+d\rightarrow a+b}$

${\displaystyle c+d+{\bar {a}}\rightarrow b}$

Whether these processes are actually observed in nature (i.e. their probability, which can differ by many orders of magnitude ), depends not only on the strength of the weak interaction, but also, among other things, on the energy , mass and momentum of the particles involved.

For every reaction, the well-known principles of conservation of energy , conservation of momentum and conservation of angular momentum apply , which, according to Noether's theorem, are associated with the invariances of temporal and spatial translation and rotations in space.

If the sums of the masses of the particles involved are greater on the right than on the left, then it is an endothermic reaction that is only possible if the particles on the left carry sufficient kinetic energy . If there is only one particle on the left, then the reaction is forbidden in this case, because there is always a reference system for a massive particle in which this particle is at rest (that is, that mass would have to be generated from nothing, which not possible). On the other hand, there is never a system of rest for a massless particle on the left side, so that in the center of gravity of the particle on the right side the conservation of momentum would be violated in this case.

If the masses of the incoming particles are greater than the masses of the generated particles, the reaction is exothermic , and the difference in masses is found as the difference in the kinetic energies between the initial particles and the generated particles.

Processes

Weak processes, a distinction is both according to whether leptons and / or quark of them are involved, as well as by whether the process by an electrically charged - or boson (charged currents or currents charged: CC) or the neutral boson (neutral Currents or neutral currents: NC). The names of weak processes are as follows: ${\displaystyle W^{+}}$ ${\displaystyle W^{-}}$ ${\displaystyle Z^{0}}$

involved	mediated by
	${\displaystyle W^{+}}$, ${\displaystyle W^{-}}$	${\displaystyle Z^{0}}$
only quarks	" Hadronically charged"	"Hadronic neutral"
Quarks and leptons	"Semileptonically charged"	"Semileptonically neutral"
only leptons	"Leptonically charged"	"Leptonically neutral"

All reactions in which neutrinos are involved take place exclusively via the weak interaction (neglecting gravitation). Conversely, there are also weak reactions without the participation of neutrinos.

Similar to the photon and in contrast to the W bosons, the Z boson mediates an interaction between particles without changing the particle type (more precisely: flavor ). While the photon only mediates forces between electrically charged particles, the Z boson also interacts with the uncharged neutrinos. In the case of neutral processes, the fermions involved remain unchanged (no change in mass or charge). The Z ⁰ boson acts on all left-handed fermions and, thanks to the Weinberg mix, also on the right-handed parts of charged fermions. Like the W bosons, it is not maximally parity-violating because it contains a part of the B ⁰ boson (see: Electroweak interaction ).

Examples of neutral processes are: The scattering of two electrons against each other (but is superimposed for low energies by the stronger electromagnetic interaction and only at high energies do the interactions become comparable in strength). The scattering of muon neutrinos by electrons (no competing processes, first experimental proof of the neutral currents in 1973 at CERN ).

Leptonic process

An elementary charged leptonic process is a decay process of a lepton L into a lepton L 'with the participation of their respective neutrinos or antineutrinos ( ): ${\displaystyle \nu _{L},{\bar {\nu }}_{L}}$

${\displaystyle L\rightarrow \nu _{L}+L^{\prime }+{\bar {\nu }}_{L^{\prime }}}$

An example of this is the decay of muons :

${\displaystyle \mu ^{-}\to e^{-}+{\bar {\nu }}_{e}+\nu _{\mu }}$

as well as the associated scattering processes

${\displaystyle \mu ^{-}+{\bar {\nu }}_{\mu }\to e^{-}+{\bar {\nu }}_{e}}$

${\displaystyle \mu ^{-}+\nu _{e}\to e^{-}+\nu _{\mu }}$

Semileptonic process

Beta decay of the neutron

In an elementary charged semileptonic process, not only leptons but also quarks or antiquarks ( ) are involved: ${\displaystyle q_{1},{\bar {q}}_{2}}$

${\displaystyle q_{1}+{\bar {q}}_{2}\rightarrow L+{\bar {\nu }}_{L}}$

An example of a semileptonic process is the aforementioned β ^- decay of the neutron , in which a down quark of the neutron is converted into an up quark:

${\displaystyle d^{-{\frac {1}{3}}}\rightarrow u^{+{\frac {2}{3}}}+e^{-}+{\bar {\nu }}_{e}}$ (Quark representation)

This turns a neutron n = udd into a proton p = uud :

${\displaystyle n\rightarrow p+e^{-}+{\bar {\nu }}_{e}}$ (Hadron representation)

A down quark and an up quark are not involved. They are called "spectator quarks".

This process is mediated by a boson, whereby the negatively charged down quark is converted into a positively charged up quark - the negative charge is "carried away" by a boson. and must therefore be quarks whose charge difference is even . ${\displaystyle W^{-}}$ ${\displaystyle W^{-}}$ ${\displaystyle d^{-1/3}}$ ${\displaystyle u^{+2/3}}$ ${\displaystyle -e}$

Further examples of semileptonic processes are:

${\displaystyle \pi ^{-}\equiv d+{\bar {u}}\to \mu ^{-}+{\bar {\nu }}_{\mu }}$

${\displaystyle K^{-}\equiv s+{\bar {u}}\to \mu ^{-}+{\bar {\nu }}_{\mu }}$

Hadronic process

Kaon decay

In an elementary charged hadronic (or nonleptonic) process, only quarks or antiquarks are involved:

${\displaystyle q_{1}+{\bar {q}}_{2}\rightarrow q_{3}+{\bar {q}}_{4}}$

The kaon decay is a good example of a hadronic process

Quark display: ${\displaystyle {\bar {s}}^{+{\frac {1}{3}}}\rightarrow u^{+{\frac {2}{3}}}+{\bar {u}}^{-{\frac {2}{3}}}+{\bar {d}}^{+{\frac {1}{3}}}}$

Hadron representation: ${\displaystyle K^{+}\rightarrow \pi ^{+}+\pi ^{0}}$

The particles involved are structured as follows: and as well as . In this process, the up-quark of the kaon is again an uninvolved bystander. The positive charge of the strange antiquark is carried away by a boson. With this exchange, the quark changes its flavor to an anti-up quark. ${\displaystyle K^{+}=u{\bar {s}}}$ ${\displaystyle \pi ^{+}=u{\bar {d}}}$ ${\displaystyle \pi ^{0}=u{\bar {u}}}$ ${\displaystyle W^{+}}$

Further examples of hadronic processes are two decay channels of the Λ-baryon :

${\displaystyle \Lambda ^{0}\equiv (u,d,s)\to (u,d,u)+({\bar {u}},d)\equiv p+\pi ^{-}}$

${\displaystyle \Lambda ^{0}\equiv (u,d,s)\to (u,d,d)+({\bar {u}},u)\equiv n+\pi ^{0}}$

Particle conversions

In the case of charged currents of the weak interaction, only particles from the same doublet can transform into one another:

${\displaystyle {\begin{pmatrix}\nu _{e}\\e^{-}\end{pmatrix}}_{L}\ ,\quad {\begin{pmatrix}\nu _{\mu }\\\mu ^{-}\end{pmatrix}}_{L}\ ,\quad {\begin{pmatrix}\nu _{\tau }\\\tau ^{-}\end{pmatrix}}_{L}\ ,\quad {\begin{pmatrix}u^{\frac {2}{3}}\\(d^{\prime })^{-{\frac {1}{3}}}\end{pmatrix}}_{L}\ ,\quad {\begin{pmatrix}c^{\frac {2}{3}}\\(s^{\prime })^{-{\frac {1}{3}}}\end{pmatrix}}_{L}\ ,\quad {\begin{pmatrix}t^{\frac {2}{3}}\\(b^{\prime })^{-{\frac {1}{3}}}\end{pmatrix}}_{L}}$

They are only left-handed fermions. These have a weak isospin , the third component of the weak isospin being for the upper particles and the lower . The weak hypercharge , i.e. twice the difference between the electrical charge and the third weak isospin component, is constant within a doublet. It amounts to for the lepton doublets and for the quark doublets . ${\displaystyle T=1/2}$ ${\displaystyle T_{3}=+1/2}$ ${\displaystyle T_{3}=-1/2}$ ${\displaystyle Y_{W}=2(Q-T_{3})}$ ${\displaystyle Y_{W}=-1}$ ${\displaystyle Y_{W}=1/3}$

Right-handed fermions do not couple to W bosons and therefore do not carry a weak isospin. It is also found that neutrinos only occur left-handed in nature ( Goldhaber experiment ). Thus, right-handed fermions are described as singlets . Since the charged streams only couple to the left-handed doublets, a maximum violation of parity occurs during these processes. This was investigated experimentally in the Wu experiment and explained by the VA theory . ${\displaystyle T=T_{3}=0}$

For quarks, the doublets (u, d '), (c, s'), (t, b') are eigenstates of the weak interaction and not (u, d), (c, s), (t, b). The states of the canceled particles are each a linear combination of three states. That is, the canceled quark states are rotated as follows compared to the quark states : ${\displaystyle d^{\prime },s^{\prime },b^{\prime }}$ ${\displaystyle d,s,b}$

${\displaystyle {\begin{pmatrix}|d^{\prime }\rangle \\|s^{\prime }\rangle \\|b^{\prime }\rangle \end{pmatrix}}=\mathbf {V} {\begin{pmatrix}|d\rangle \\|s\rangle \\|b\rangle \end{pmatrix}}\quad {\text{mit}}\quad |\mathbf {V} |\approx {\begin{pmatrix}0{,}9743&0{,}2253&0{,}0035\\0{,}2252&0{,}9734&0{,}0412\\0{,}0087&0{,}0404&0{,}9991\end{pmatrix}}}$

Mass-charge diagram of the quarks and their decay possibilities under weak interaction (the finer the dashed arrows, the less likely the process is.)

This is the so-called CKM matrix . This is unitary and has four independent parameters. The squares of the elements of the given matrix are proportional to the transition probabilities between the quarks. ${\displaystyle \mathbf {V} }$ ${\displaystyle |\mathbf {V} |}$

${\displaystyle |V_{ij}|^{2}}$	d	s	b
u	0.9492	0.0508	0.00001
c	0.0507	0.9476	0.0017
t	0.00007	0.0016	0.9983

The transitions within the same quark family (u, d), (c, s), (t, b) take place most frequently, since the diagonal elements indicate the greatest transition probabilities. There is also a lower chance that the generation of the particle will change. This behavior is caused by the fact that the mass eigenstates do not match the so-called interaction eigenstates.

The decay of quarks or leptons by neutral currents, e.g. B. the transitions c → u or s → d or μ → e have not yet been observed.

Neutrino oscillations

The neutrino eigenstates of weak interaction , , (Flavor states are eigenstates of the weakly interacting part of the Hamiltonian) are not identical to the eigenstates of the mass operator , , (eigenstates of the kinematic part of the Hamiltonian). Analogous to the CKM matrix, the so-called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix can be introduced here ${\displaystyle |\nu _{e}\rangle }$ ${\displaystyle |\nu _{\mu }\rangle }$ ${\displaystyle |\nu _{\tau }\rangle }$ ${\displaystyle |\nu _{1}\rangle }$ ${\displaystyle |\nu _{2}\rangle }$ ${\displaystyle |\nu _{3}\rangle }$

${\displaystyle {\begin{pmatrix}|\nu _{e}\rangle \\|\nu _{\mu }\rangle \\|\nu _{\tau }\rangle \end{pmatrix}}=\mathbf {U} {\begin{pmatrix}|\nu _{1}\rangle \\|\nu _{2}\rangle \\|\nu _{3}\rangle \end{pmatrix}}}$

Current values ​​are:

${\displaystyle \mathbf {U} \approx {\begin{pmatrix}0{,}82&0{,}55&-0{,}15+0{,}045\,\mathrm {i} \\-0{,}35+0{,}023\,\mathrm {i} &0{,}70+0{,}015\,\mathrm {i} &0{,}62\\0{,}45+0{,}028\,\mathrm {i} &-0{,}46+0{,}019\,\mathrm {i} &0{,}77\end{pmatrix}}}$

${\displaystyle |\mathbf {U} |^{2}={\begin{pmatrix}0{,}68&0{,}30&0{,}024\\0{,}13&0{,}49&0{,}38\\0{,}20&0{,}21&0{,}59\end{pmatrix}}}$

The matrix has large values ​​also outside the diagonal. This differentiates it from the CKM matrix and leads to a strong mix of neutrino families over time.

${\displaystyle {\begin{pmatrix}|\nu _{e}(t)\rangle \\|\nu _{\mu }(t)\rangle \\|\nu _{\tau }(t)\rangle \end{pmatrix}}=\mathbf {U} {\begin{pmatrix}\exp(-iE_{\nu _{1}}t/\hbar )\,|\nu _{1}\rangle \\\exp(-iE_{\nu _{2}}t/\hbar )\,|\nu _{2}\rangle \\\exp(-iE_{\nu _{3}}t/\hbar )\,|\nu _{3}\rangle \end{pmatrix}}\quad {\text{mit}}\quad E_{\nu _{i}}={\sqrt {p^{2}c^{2}+m_{\nu _{i}}^{2}c^{4}}}}$

If a neutrino was originally created with one of these three flavors , a later quantum measurement can result in a different flavor (preservation of the lepton family numbers is violated). Since the probabilities for every flavor change periodically with the propagation of the neutrino, one speaks of neutrino oscillations .

When a (left-handed) lepton decays due to the weak interaction, the flavor does not change during the interaction (preservation of the lepton family number in each interaction vertex ), but neutrinos that arise can transform into one another in the further time evolution, whereby the flavor changes and thus the lepton family number. Conservation is violated. However, the lepton number is always preserved in this oscillation.

If the neutrinos had no mass, then every flavor state would also be an eigenstate of the mass operator. Consequently, no flavor oscillations could be observed.

Lagrangian density

In the following, the interaction components between fermions and gauge bosons are analyzed for the Lagrange density of the weak interaction. ${\displaystyle {\mathcal {L}}}$

In order to better classify the description of the weak interaction, the electromagnetic interaction is described first. All quantities given below with Greek indices are four-vectors .

Electromagnetic interaction

In quantum electrodynamics, the interaction energy is the coupling of (four) currents of charged particles to photons, represented by the electromagnetic (four) potential , given by: ${\displaystyle J_{\mu }^{em}}$ ${\displaystyle A^{\mu }}$

${\displaystyle {\mathcal {L}}_{em}=eJ_{\mu }^{em}A^{\mu }}$

The coupling constant is the elementary charge . The current density is given by ${\displaystyle e}$

${\displaystyle J_{\mu }^{em}=\sum _{u}Q{\overline {u}}(p_{f})\gamma _{\mu }u(p_{i})}$

where is the charge quantum number (the electrical charge of the particles in units of elementary charge) are the Dirac matrices . is the field of the incoming fermion (or outgoing antifermion) with (four) momentum and that of the outgoing fermion (or incoming antifermion) with momentum . In a Feynman diagram describe the spinors and the outer solid lines. ${\displaystyle Q}$ ${\displaystyle \gamma _{\mu }}$ ${\displaystyle u(p_{i})}$ ${\displaystyle p_{i}}$ ${\displaystyle {\overline {u}}(p_{f})}$ ${\displaystyle p_{f}}$ ${\displaystyle u(p_{i})}$ ${\displaystyle {\overline {u}}(p_{f})}$

Electron-electron scattering , , , ${\displaystyle p_{1}=p_{i}}$ ${\displaystyle p_{2}=k_{i}}$ ${\displaystyle p_{3}=p_{f}}$ ${\displaystyle p_{4}=k_{f}}$

The scattering of two charged particles is described in the Born approximation (lowest order perturbation theory ) by the Feynman diagram opposite . The associated scattering amplitude is

${\displaystyle {\begin{aligned}T_{fi}&={\bar {u}}(p_{f})\left[-\mathrm {i} Qe\gamma ^{\mu }\right]u(p_{i})\left[-\mathrm {i} {\frac {\eta _{\mu \nu }}{q^{2}}}\right]{\bar {u}}(k_{f})\left[-\mathrm {i} Qe\gamma ^{\nu }\right]u(k_{i})\\&=\underbrace {Q{\bar {u}}(p_{f})\gamma ^{\mu }u(p_{i})} _{j^{\mu }}\left[{\frac {e^{2}}{q^{2}}}\right]\underbrace {Q{\bar {u}}(k_{f})\gamma _{\mu }u(k_{i})} _{j_{\mu }}\end{aligned}}}$

A factor must be multiplied at each vertex of the charge . At the vertex, the four-vector of the photon applies because of the conservation of energy and momentum . ${\displaystyle Qe}$ ${\displaystyle -\mathrm {i} Qe\gamma ^{\mu }}$ ${\displaystyle q^{\mu }=(p_{f}-p_{i})^{\mu }=(k_{i}-k_{f})^{\mu }}$

Inner lines of the Feynman diagram are the so-called propagators, here the photon propagator, where the (four) momentum transfer and the metric tensor of the special theory of relativity are. ${\displaystyle -\mathrm {i} {\frac {\eta _{\mu \nu }}{q^{2}}}}$ ${\displaystyle q}$ ${\displaystyle \eta _{\mu \nu }}$

Weak interaction

In the case of the weak interaction, (neutral current) and (charged current) describe the summands of the Lagrange density that contain the interaction between fermions and the gauge bosons. ${\displaystyle {\mathcal {L}}_{NC}}$ ${\displaystyle {\mathcal {L}}_{CC}}$

Charged currents

The weakly charged currents are described by the following interaction component:

${\displaystyle {\mathcal {L}}_{CC}=-{\frac {g}{\sqrt {2}}}\left[\left(J_{d'u}^{\mu }\right)_{L}^{+}+\left(J_{e\nu }^{\mu }\right)_{L}^{+}\right]W_{\mu }^{+}-{\frac {g}{\sqrt {2}}}\left[\left(J_{ud'}^{\mu }\right)_{L}^{-}+\left(J_{\nu e}^{\mu }\right)_{L}^{-}\right]W_{\mu }^{-}}$

The bosons couple with the same coupling constant to all left-handed leptons and quarks. ${\displaystyle W^{\pm }}$ ${\displaystyle g/{\sqrt {2}}}$

The chirality operator appears when describing the individual currents (this transforms a polar into an axial vector ). In the case of massive particles, this converts particle spinors of positive helicity into antiparticle spinors of negative helicity and vice versa ( ). The left-handed operator can be constructed from this: ${\displaystyle \gamma ^{5}}$ ${\displaystyle \gamma ^{5}\gamma ^{\mu }=-\gamma ^{\mu }\gamma ^{5}}$

${\displaystyle L={\frac {1-\gamma ^{5}}{2}}}$

This operator applied to a spinor projected onto the left-handed part: ${\displaystyle u=u_{R}+u_{L}}$

${\displaystyle u_{L}=Lu}$

Because of the appearance of this operator, the weak interaction is a chiral theory. The left-handed stream

${\displaystyle J_{L}^{\mu }={\overline {u}}\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}u={\frac {1}{2}}\left({\overline {u}}\gamma ^{\mu }u-{\overline {u}}\gamma ^{\mu }\gamma ^{5}u\right)={\frac {1}{2}}\left(J_{V}^{\mu }-J_{A}^{\mu }\right)}$

is the (half) difference between the vector current and the axial vector current , therefore V minus A (see: VA theory ). ${\displaystyle J_{V}^{\mu }}$ ${\displaystyle J_{A}^{\mu }}$

Weak charged left-handed quark currents with , , , is the CKM mixing matrix: ${\displaystyle u_{i}=\{u,c,t\}}$ ${\displaystyle d_{i}=\{d,s,b\}}$ ${\displaystyle d_{i}^{\prime }=\{d^{\prime },s^{\prime },b^{\prime }\}}$ ${\displaystyle V_{ij}^{\mathrm {CKM} }}$

${\displaystyle {\begin{aligned}\left(J_{d'u}^{\mu }\right)_{L}^{+}&={\overline {d}}_{i,L}^{\prime }\gamma ^{\mu }u_{i,L}={\overline {d}}_{i}\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}V_{ij}^{\mathrm {CKM} }u_{j}\\\left(J_{ud'}^{\mu }\right)_{L}^{-}&={\overline {u}}_{i,L}\gamma ^{\mu }d_{i,L}^{\prime }={\overline {u}}_{i}\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}V_{ij}^{\mathrm {CKM} }d_{j}\end{aligned}}}$

Weak charged left-handed Leptonenströme with , : ${\displaystyle e_{i}=\{e,\mu ,\tau \}}$ ${\displaystyle \nu _{i}=\{\nu _{e},\nu _{\mu },\nu _{\tau }\}}$

${\displaystyle {\begin{aligned}\left(J_{e\nu }^{\mu }\right)_{L}^{+}&={\overline {\nu }}_{i,L}\gamma ^{\mu }e_{i,L}={\overline {\nu }}_{i}\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}e_{i}\\\left(J_{\nu e}^{\mu }\right)_{L}^{-}&={\overline {e}}_{i,L}\gamma ^{\mu }\nu _{i,L}={\overline {e}}_{i}\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}\nu _{i}\end{aligned}}}$

The following factor must be multiplied at a vertex: ${\displaystyle W}$

${\displaystyle -\mathrm {i} {\frac {g}{\sqrt {2}}}\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}}$

The propagator for massive (mass ) spin-1 particles, such as the W and Z bosons, is: ${\displaystyle M}$

${\displaystyle -\mathrm {i} {\frac {\eta _{\mu \nu }-q_{\mu }q_{\nu }/M^{2}}{q^{2}-M^{2}}}\approx \mathrm {i} {\frac {\eta _{\mu \nu }}{M^{2}}}}$

Since the following applies in most cases , the propagator can be approximated. In contrast to the photon propagator, the propagator is constant for small momentum transfers. ${\displaystyle q^{2}\ll M^{2}}$ ${\displaystyle -{\tfrac {\mathrm {i} \eta _{\mu \nu }}{q^{2}}}}$ ${\displaystyle {\tfrac {\mathrm {i} \eta _{\mu \nu }}{M^{2}}}}$

With small values, the weak interaction is much weaker than the electromagnetic one. This is not due to the coupling constant of the weak interaction, because the coupling strength is in the same order of magnitude as the electric charge . The reason for the weakness of the interaction lies in the shape of the propagator of the exchange particles, since the huge boson mass is in the denominator and thus reduces the interaction term. ${\displaystyle q^{2}}$ ${\displaystyle g}$ ${\displaystyle e}$

The scattering of two leptons mediated by a W boson has a scattering amplitude (in the lowest order) of:

${\displaystyle T_{fi}={\bar {u}}(p_{f})\left[-\mathrm {i} {\frac {g}{\sqrt {2}}}\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}\right]u(p_{i})\left[-\mathrm {i} {\frac {\eta _{\mu \nu }-q_{\mu }q_{\nu }/M_{W}^{2}}{q^{2}-M_{W}^{2}}}\right]{\bar {u}}(k_{f})\left[-\mathrm {i} {\frac {g}{\sqrt {2}}}\gamma ^{\nu }{\frac {1-\gamma ^{5}}{2}}\right]u(k_{i})}$

In the approximated form

${\displaystyle {\begin{aligned}T_{fi}&\approx -\mathrm {i} \underbrace {{\bar {u}}(p_{f})\left[\gamma ^{\mu }{\frac {1-\gamma ^{5}}{2}}\right]u(p_{i})} _{(j_{L})^{\mu }}\underbrace {\left[{\frac {g^{2}}{2M_{W}^{2}}}\right]} _{4G_{F}/{\sqrt {2}}}\underbrace {{\bar {u}}(k_{f})\left[\gamma _{\mu }{\frac {1-\gamma ^{5}}{2}}\right]u(k_{i})} _{(j_{L})_{\mu }}\end{aligned}}}$

the scattering amplitude is described by the coupling of two left-handed currents by means of a coupling constant. This was described by Enrico Fermi through the Fermi interaction , namely the interaction of four particles involved at a space-time point. The Fermi constant has the value . ${\displaystyle {\frac {G_{\text{F}}}{(\hbar c)^{3}}}=1{,}1663787(6)\cdot 10^{-5}\,\mathrm {GeV} ^{-2}}$

Neutral currents

The weak neutral currents are described by the following interaction component:

${\displaystyle {\mathcal {L}}_{NC}={\frac {g}{\cos \theta _{W}}}J_{\mu }^{NC}Z^{\mu }={\frac {g}{\cos \theta _{W}}}\left(J_{\mu }^{3}-\sin ^{2}\theta _{W}J_{\mu }^{em}\right)Z^{\mu }}$

The bosons couple with the coupling constant to the neutral current . This is composed of an isospin current and the electromagnetic current . ${\displaystyle Z}$ ${\displaystyle g/\cos \theta _{W}}$ ${\displaystyle J_{\mu }^{NC}=J_{\mu }^{3}-\sin ^{2}\theta _{W}J_{\mu }^{em}}$ ${\displaystyle J_{\mu }^{3}}$ ${\displaystyle J_{\mu }^{em}}$

The isospin current is calculated using

${\displaystyle J_{\mu }^{3}=\sum _{f}I_{f}^{3}{\overline {f}}\gamma _{\mu }\left({\frac {1-\gamma ^{5}}{2}}\right)f}$

f stands for the spinor wave function of the fermion. is the third component of the weak isospin. It is calculated as follows: ${\displaystyle I_{f}^{3}}$

• ${\displaystyle +{\frac {1}{2}}}$   for and ${\displaystyle \{u_{i}\}}$ ${\displaystyle \{\nu _{i}\}}$
• ${\displaystyle -{\frac {1}{2}}}$   for and ${\displaystyle \{d_{i}\}}$ ${\displaystyle \{e_{i}\}}$

Because of the left-handed operator , the boson only couples to the left-handed parts of fermions via the isospin current. ${\displaystyle {\tfrac {1-\gamma ^{5}}{2}}}$ ${\displaystyle Z}$

The electromagnetic current is calculated according to

${\displaystyle J_{\mu }^{em}=\sum _{f}q_{f}{\overline {f}}\gamma _{\mu }f}$

where denotes the electrical charge of the fermion involved. ${\displaystyle q_{f}}$

When calculating scattering cross-sections with the help of Feynman diagrams, the factor must be multiplied by another factor for each vertex , depending on the type of particle involved. This is ${\displaystyle Z}$ ${\displaystyle -{\frac {\mathrm {i} g}{\cos \theta _{W}}}\gamma ^{\mu }}$

• ${\displaystyle +{\frac {1}{2}}\left({\frac {1-\gamma ^{5}}{2}}\right)\pm \,0\,\sin ^{2}\theta _{W}}$   for uncharged leptons (neutrinos) with the charge , d. H. for , ${\displaystyle \pm 0\;e}$ ${\displaystyle \{\nu _{i}\}}$
• ${\displaystyle -{\frac {1}{2}}\left({\frac {1-\gamma ^{5}}{2}}\right)+\,1\,\sin ^{2}\theta _{W}}$   for charged leptons (electron, muon, tauon) with the charge , d. H. for , ${\displaystyle -1\;e}$ ${\displaystyle \{e_{i}\}}$
• ${\displaystyle +{\frac {1}{2}}\left({\frac {1-\gamma ^{5}}{2}}\right)-{\frac {2}{3}}\sin ^{2}\theta _{W}}$   for charged quarks , d. H. for and ${\displaystyle +{\frac {2}{3}}\;e}$ ${\displaystyle \{u_{i}\}}$
• ${\displaystyle -{\frac {1}{2}}\left({\frac {1-\gamma ^{5}}{2}}\right)+{\frac {1}{3}}\sin ^{2}\theta _{W}}$   for charged quarks , d. H. for . ${\displaystyle -{\frac {1}{3}}\;e}$ ${\displaystyle \{d_{i}^{\prime }\}}$

The last three factors have summands without the left-handed operator. These Z-couplings thus act on both left- and right-handed parts of the fermions involved.

In the case of neutrinos , only the left-handed parts couple to the Z boson. In the case of the charged fermions , and, on the other hand , couple right-handed and left-handed parts to the Z boson. In the scattering of charged fermions, in addition to the interaction via an electromagnetic field, an interaction via the field of the uncharged Z boson can also take place. If the particle energies involved are small compared to the rest energy of the Z boson, however, the electromagnetic interaction predominates in scattering processes. ${\displaystyle \{\nu _{i}\}}$ ${\displaystyle \{u_{i}\}}$ ${\displaystyle \{d_{i}\}}$ ${\displaystyle \{e_{i}\}}$

Combination of electromagnetic and neutral currents

In the electroweak theory, electromagnetic and weak neutral currents can be combined. Instead of electromagnetic currents of photons and weak neutral currents of Z bosons

${\displaystyle {\mathcal {L}}_{em,NC}=eJ_{\mu }^{em}A^{\mu }+{\frac {g}{\cos \theta _{W}}}\left(J_{\mu }^{3}-\sin ^{2}\theta _{W}J_{\mu }^{em}\right)Z^{\mu }}$

now couple isospin currents - and hypercharge currents - bosons: ${\displaystyle W_{3}}$ ${\displaystyle B_{0}}$

${\displaystyle {\mathcal {L}}_{ew,n}=gJ_{\mu }^{3}W_{3}+{\frac {1}{2}}g^{\prime }J_{\mu }^{Y}B_{0}}$

Whereby a hypercharge current based on the hypercharge of a fermion was introduced: ${\displaystyle Y_{f}=2(q_{f}-I_{f}^{3})}$

${\displaystyle J_{\mu }^{Y}=2\left(J_{\mu }^{em}-J_{\mu }^{3}\right)=\sum _{f}Y_{f}{\overline {f}}\gamma _{\mu }f}$

The connection of the gauge bosons is given over the Weinberg angle with (where the photon is) and and and the connection of the two weak coupling constants with the elementary charge is given over . ${\displaystyle \gamma =B^{0}\cos \theta _{W}+W^{0}\sin \theta _{W}}$ ${\displaystyle \gamma }$ ${\displaystyle Z^{0}=-B^{0}\sin \theta _{W}+W^{0}\cos \theta _{W}}$ ${\displaystyle W^{\pm }=-(W^{1}\pm iW^{2})/{\sqrt {2}}}$ ${\displaystyle e=g\,\sin \theta _{W}=g^{\prime }\,\cos \theta _{W}}$

history

The weak interaction was first discovered in beta decay (for its history see the article beta decay). The discovery that the beta decay showed a continuous spectrum and apparently violated the conservation of energy led Wolfgang Pauli in 1930 to postulate the neutrino as the third decay partner. Building on this, after the neutron had also been discovered in 1932, Enrico Fermi formulated the first theory of beta decay in 1934. It had a similar structure to quantum electrodynamics (QED), but in the form of a current coupling with a vanishing range and with a dimensional coupling constant. In contrast to the QED, it could not be renormalized . Further advances in the 1930s were the selection rules of George Gamow and Edward Teller (Gamow-Teller transitions, 1936) and the discovery of the role of the weak interaction in nucleosynthesis in stars by Gamow and Hans Bethe (1938) and in the formation of Neutron stars in supernovae ( Robert Oppenheimer , Lew Landau ). In addition, new weak processes such as the decay of muons , pions , kaons and hyperons were discovered up to the 1950s .

In the 1950s, the parity violation of the weak interaction was discovered (theoretically proposed by Tsung-Dao Lee , Chen Ning Yang 1956, experimentally discovered by Chien-Shiung Wu 1957). This was built into the V − A weak interaction theory by Richard Feynman and Murray Gell-Mann on the one hand and Robert Marshak and George Sudarshan on the other in 1958, an important step towards the modern theory of weak interaction in the Standard Model . Contributed Sheldon Lee Glashow , Abdus Salam and Steven Weinberg with the Association of electromagnetic and weak interactions in the late 1960s at (massive introduction vector bosons whose exchange replaced the point-like interaction in the Fermi theory), and Makoto Kobayashi and Toshihide Maskawa with the incorporation of the CP violation discovered by James Cronin and Val Fitch in 1964 into the theory of their KM matrix or CKM matrix (additionally after Nicola Cabibbo , who introduced the Cabibbo angle in 1963 to describe weak decays of strange particles).

Classification of the weak interaction

Fundamental interactions and their descriptions
	Strong interaction	Electromagnetic interaction	Weak interaction	Gravity
classic		Electrostatics & magnetostatics , electrodynamics		Newton's law of gravitation , general relativity
quantum theory	Quantum ( standard model )	Quantum electrodynamics	Fermi theory	Quantum gravity  ?
		Electroweak Interaction ( Standard Model )
	Big Unified Theory  ?
	World formula ("theory of everything")?
Theories at an early stage of development are grayed out.

literature

• B. Povh , K. Rith , C. Scholz, F. Zetsche: Particles and nuclei. 8th edition. Springer, Berlin 2009, ISBN 978-3-540-68075-8
• C. Berger: Elementary Particle Physics. 2nd Edition. Springer, Berlin 2006, ISBN 978-3-540-23143-1
• EA Paschos: Electroweak Theory. 1st edition. Cambridge University Press, Cambridge 2007, ISBN 978-0-521-86098-7

Individual evidence

1. ↑ J. Beringer et al., Particle Data Group , PR D86, 010001 (2012), THE CKM QUARK-MIXING MATRIX
2. ↑ Fogli et al. (2012): Global analysis of neutrino masses, mixings and phases: entering the era of leptonic CP violation searches , arxiv : 1205.5254v3
3. ↑ Paul Langacker also gives an overview in this lecture, STIAS, January 2011
4. ↑ Fermi, Attempting a Theory of Beta Rays. I, Zeitschrift für Physik, Volume 88, 1934, p. 161, published in Italian as: Tentativo di una teoria dei raggi β, Il Nuovo Cimento, Volume 11, 1934, pp. 1-19.