VA theory

The VA interaction (abbreviation for vector - axial vector interaction , pronounced "V minus A theory") is a field theoretical model for the weak interaction . It is an extension of Fermi's current-current interaction to explain the parity violation discovered in the Wu experiment in 1956 . It is a low-energy approximation for electrostatic weak interaction , which interaction by exchange of bosons ( W ⁺ - W ^- - , and Z ⁰ boson describes).

background

In 1936, Fermi postulated his theory to describe beta decay in the form of the Hamilton operator

${\displaystyle {\mathcal {\hat {H}}}_{\mathrm {Fermi} }={\frac {G_{\mathrm {F} }}{\sqrt {2}}}\int \mathrm {d} ^{3}x\cdot j_{H}^{\mu }(x)\cdot j_{L\mu }(x)}$

• ${\displaystyle G_{\mathrm {F} }}$is the Fermi constant ,
• ${\displaystyle j_{H}^{\mu }(x)}$and are the 4-dimensional current densities for hadrons or leptons (in Fermi's theory only for the proton , neutron , electron and electron-neutrino particles involved in beta decay ). ${\displaystyle j_{L}^{\mu }(x)}$
• The current density is related to the fields . ${\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi }$

In 1956, Tsung-Dao Lee and Chen Ning Yang published the hypothesis according to which parity is not maintained in the weak interaction, in contrast to the strong and electromagnetic interaction . They had also suggested several special experiments. The observation was made by Chien-Shiung Wu , which made it necessary to adjust the current, which had previously been parity-preserving.

A year later Richard Feynman , Murray Gell-Mann, and independently George Sudarshan and Robert Marshak developed the VA interaction. To do this, an axial vector current must be introduced into the Hamilton operator, on which the parity operator has a different effect than on polar currents. Such an axial stream with a parity violating component can be written as

${\displaystyle j^{\mu }={\bar {\psi }}\gamma ^{\mu }(g_{v}-g_{a}\gamma _{5})\psi }$

• ${\displaystyle \gamma ^{\mu }}$are Dirac matrices .
• ${\displaystyle \gamma _{5}}$ transforms a polar into an axial vector.
• ${\displaystyle g_{v}}$and are coefficients that indicate the ratio between polar and axial vector currents. ${\displaystyle g_{a}}$

Leptons are point particles , hence is . In contrast, hadrons are extended particles, they are made up of quarks , so one finds experimentally and . ${\displaystyle g_{v}=1=g_{a}}$ ${\displaystyle g_{v}=1}$ ${\displaystyle g_{a}\approx 1{,}25}$

The Hamilton operator for the VA interaction results from

${\displaystyle {\mathcal {\hat {H}}}_{\mathrm {V-A} }={\frac {G_{\mathrm {F} }}{\sqrt {2}}}\int \mathrm {d} ^{3}x\cdot {\bar {\psi }}_{p}\gamma ^{\mu }(1-g_{a}\gamma _{5})\psi _{n}\cdot {\bar {\psi }}_{e}\gamma _{\mu }(1-\gamma _{5})\psi _{\nu }}$

• ${\displaystyle \psi _{p}}$, , , The proton, neutron, electron and electron-neutrino fields. ${\displaystyle \psi _{n}}$ ${\displaystyle \psi _{e}}$ ${\displaystyle \psi _{\nu }}$

In order to make the theory applicable to all three generations of elementary particles , one has to expand the currents with the remaining particle fields and introduce the CKM matrix .

credentials

• Walter Greiner , Berndt Müller : Calibration theory of the weak interaction . 2nd edition, Harri Deutsch, 1995, p. 184, ISBN 3-8171-1427-3 .

1. ^ RP Feynman, M. Gell-Mann: Theory of the Fermi Interaction . In: Physical Review . tape 109 , no. 1 , 1958, p. 193-198 , doi : 10.1103 / PhysRev.109.193 .
2. ^ ECG Sudarshan, RE Marshak: Chirality Invariance and the Universal Fermi Interaction . In: Physical Review . tape 109 , no. 5 , 1958, pp. 1860–1862 , doi : 10.1103 / PhysRev.109.1860.2 .
3. ^ ECG Sudarshan, RE Marshak: The Nature of the Four-Fermion Interaction . In: Padua Conference on Mesons and Recently Discovered Particles . 1957, p. 1860–1862 ( article on the lecture ( Memento from June 16, 2012 in the Internet Archive ) (PDF; 94 kB) - lecture).