Fermi's idea was to compare the four-fermion point interaction as an interaction of two currents in analogy to the electromagnetic interaction. For proton-electron scattering, for instance, the matrix-element can be written as
J represents the current densities for proton and electron, respectively, and q the momentum transfer. Here the proton couples to the electron with the strength a which is known as the fine structure parameter . Fermi consequently introduced the weak current that decomposes into a hadronic and a leptonic current. Using the picture of second quantization (field quantization) the transition operator can be pictured as annihilator of the particle of the right hand side and creator of the particle on the left hand side. In the theory of four-fermion point interaction two particles are transformed simultaneously and therefore one writes the matrix element [Fey58]
In order to describe observables (which must be real expressions) in the Dirac theory, bilinear forms like with W being a 4x4 hermitean matrix, have to be generated. This results in 16 linear independent basic matrices which are be classified as follows
This set of matrices guarantees that solutions of the Dirac equation also will fulfil Lorentz-invariance. The parity operator can be introduced by building the chiral projection operators:
(and analogous for the right-handed component).
The chiral symmetry was introduced by Feynman and Gell-Mann to account for parity violation, which was not known to Fermi. It can be shown that only V- and A-coupling correctly describe the maximal parity violation, since
Using this, the matrix element becomes
A textbook example to verify (V-A) theory is the calculation of pion-decay into leptons, since parity violation in pion and muon decays was observed along with the nuclear b -decay measurements mentioned above. Following the four fermion interaction picture with replacing the pion by a hadronic current of an up-quark changing into a down quark in the single-point, which demands universality of the weak interaction. It claims an universal coupling constant for all lepton decay and scattering processes (the equivalence of the coupling strength for weak interacting hadronic particles is expressed in the CVC hypothesis). In this example of a pion decay into leptons the conservation of angular momentum then determines the spin direction of the charged leptons, which have to be opposite to the (anti-)neutrino since pions do not carry spin. Finally, regarding conservation of momentum, the charged leptons have the same chirality as the neutrinos. The helicity of the neutrino must be -1. Hence, neutrino helicity determines the chirality of the charged lepton. Due to the fact that the probability to find a massive particle with unfavourable helicity goes with 1- b , the rate of the decay p +->e+ n e should be lower than for p +->µ+ n µ. With the approximation me2/m p 2 <<1 the ratio of decay rates turns out to be
The function f comprises the kinematic terms of the integrated state density and the electromagnetic interaction of the electrons and t represents the half-life. The connection of [florin]t-values, which are used to classify the b -decays, to the coupling constant one obtains after calculating the matrix element. Here the simplest case is a transition without a flip of the nucleon spin (I) and thus with anti-parallel electron and neutrino spins. An example of this pure Fermi transition is the inverse b -decay . In order to calculate the matrix element one forms an isospin T=1 triplet of isotopes with the atomic number A=14. The isospin positions Tz are calculated assuming a core with additional nucleons
since terms other than of vector type do not contribute Nuclear b -decays are classified according to their log([florin]t)-values:
|| D I
Before one can extract the exact coupling constant from the [florin]t-value, corrections have to be taken into account (see [Wil94,97] for a comprehensive discussion. Those are
* radiative corrections d r (from QED and QCD) of 3-4%
* nuclear correction d c of 0.2-1%
* screening effects due to Coulomb interaction with the core
Thus one writes f't = [florin]t (1+ d r) (1- d c). The rate of the pure Fermi nuclear b -decay is directly related to the vector coupling constant assuming the CVC hypothesis that claims the coupling to the hadronic current independent from the participating particles. Pions and nucleons therefore can be treated equally within this theory. Furthermore the standard model assumes the existence of an universal coupling constant for all weak processes. Hence, measuring the decay rate of a b -decay, GV can be obtained or, taking Gµ from the muon decay, universality of the weak interaction can be tested.
(A more recent calculation by Sirlin [Sir92] gives (0.3996±0.0006) s-1).
Here the function f' is representing the integrated state density including electroweak corrections, d p the radiative corrections for the p of 0.012, and d µ.represents the radiative correction for the µ+->e+ n edecay. With the pion lifetime of 26.03 ns this is equivalent to a branching ratio (BR) of 1.031*10-8 or a ratio of 0.838*10-4 compared to our calibration decay p +->e+ n e .