Pion beta decay () is one of the fundamental semileptonic weak interaction processes. It is a transition between two spin-zero members of an isospin triplet and is therefore analogous to the superallowed pure Fermi transitions in nuclear beta decay. At tree level, the rate of pure Fermi nuclear beta decay is directly related to the vector coupling constant and the nuclear Fermi matrix element . The Conserved Vector Current hypothesis (CVC) predicts that the product ft of the Fermi integral function f and the decay half live is a universal constant given by:
where K is given by equation (). By comparing the vector coupling constant of the nucleon beta decay to that of the muon decay, it is possible to determine , the mixing matrix element between the u and d quarks. Cabibbo universality relates to the muon weak coupling constant via the CKM matrix element [Shr-78]
is the muon decay coupling constant (see equation ()); and are the inner radiative corrections for and decays respectively. The inner radiative corrections do not depend on the electron energy. They do depend however on the details of the strong interaction involved and the structure of the decay. In addition, they are model-dependent and tend to diverge theoretically. They are relevant when comparing different types of beta decay but act only to renormalize the coupling constant when comparing the rates of beta decays of the same type. In such cases, they are directly defined into the coupling constants themselves. Some nuclear beta decay processes, characterized by the single parameter are as follows:
For instance, the third reaction represents the transition from the ground state of with to the first excited state of with .
Two types of nucleus-dependent corrections must be applied to equation () in order to extract from the measured decay rates: the phase space dependent radiative corrections to the statistical rate function f and the nuclear structure dependent corrections to the matrix element . The uncorrected Fermi matrix element is given by equation (). Taking into account the corrections and and the matrix element of equation (), the nucleus independent ft value becomes
where the uncorrected Fermi integral function f and the decay half live t are given by equations () and (). The empirical values of are determined from the values.
The outer radiative corrections are model-independent and uniquely specified.
Figure: Feynman diagrams for the beta decay of a point nucleon: (1) no radiative correction; (2)--(6) first order radiative corrections.
They include the inner bremsstrahlung and contain a dependence of the beta particle energy. In addition, they are discussed in terms of orders of , the fine structure constant. They originate from, for instance, the emission of virtual photons by the final state charged particles. The zeroth and the first order outer radiative corrections are shown in figure . In the course of the past two decades, radiative corrections of O() have been calculated as well as the leading terms of O(Z) [Sir-87a]. The resulting radiative corrections (inner plus outer) amount to with theoretical uncertainty of arising mainly from short-distance axial-vector-induced contributions [Mar-86].
The nucleus structure dependent corrections are conventionally factorized into two components: isospin impurity and radial overlap corrections. The fact that is given by equation () rests on the assumption that the states involved are pure isospin states and eigenstates of a charge independent hamiltonian. Even after accounting for Coulomb interactions and spin-orbit coupling of the nucleus, the nuclear force is still slightly charge dependent [Bli-73]. As a result, the nuclear states involved in the decay are not pure isospin states. The radial overlap corrections are due to small differences in the single-particle radial wave functions of the neutron and the proton. Due to these differences, the radial overlap integral of the parent and the daughter nucleus deviates from unity. The evaluation of the isospin impurity correction is performed by adding an isospin nonconserving (INC) interaction term to the standard shell-model hamiltonian while the radial overlap correction is based on the radial wave functions obtained from a suitable parametrization of the mean field. There have been three systematic evaluations of by Tower, Hardy and Harvey (THH) [Tow-77], Wilkinson (W) [Wil-78] and Ormand and Brown (OB) [Orm-85] and [Orm-89]. The THH calculations of were carried out as follows: the radial wave functions were obtained from a Woods-Saxon plus Coulomb potential and the INC interaction term was determined by:
As shown in table , both OB and THH values of yield essentially constant but inconsistent averaged values. The THH and OB nuclear corrections yield and respectively for ; the errors quoted are dominated by the uncertainties in mentioned above. Furthermore, considering the accepted value for and ( confidence level), one gets for values of and for THH and OB calculations respectively. The OB calculations imply violation of CKM unitarity at the confidence level.
Recently, Drukarev and Strikman discovered that the electron final state interaction effects (Coulomb screening corrections) are more pronounced than previously thought. This effect moves the central value of away from unitarity.
Hardy et al. [Har-90] studied the spread of the THH and OB's , did a critical survey of available superallowed beta decay data and took unweighted averages. As a result, Hardy et al. removed the discrepancy of THH and OB calculations and reduced the overall uncertainties, bringing the unitarity test to
a value which is lower than the Standard Model prediction.
From neutron beta decay, can be extracted knowing the neutron lifetime and , the ratio of the weak axial vector to the vector coupling constants. A new precision measurement of the electron asymmetry in neutron decay [Ero-90] yields a new value for the ratio . This result, combined with the average value of the neutron lifetime, gives which leads to the unitarity test of
This value is higher than the Standard Model prediction by .
The pion beta decay, proceeding solely via the weak vector current interaction, constitutes a superallowed transition from which can be determined. Given the discrepancy between the nuclear beta decay and the neutron beta decay results discussed above, the determination of the pion beta decay rate at the level of half a percent is of a clear importance and constitutes an independent test of universality in the meson sector.
Contrary to nuclear beta decay, pion beta decay is free of the nuclear overlap correction as well as the screening corrections. Sirlin [Sir-78] has calculated the decay rate taking into account the effects of radiative corrections, which involve only an axial-vector contribution [Mar-86].
This result may be compared to the one obtained in chapter 4 without radiative corrections.
This theoretical advantage of the pion beta decay is offset by experimental difficulties. In fact, the main disavantage of the pion beta decay is its small branching ratio which makes it a very difficult process to study with precision. The most precise measurements of the pion beta decay rate to date are as follow:
Using a stopped pion method, Depommier et al. [Dep-68] measured the branching ratio for the pion beta decay at CERN. They observed pion beta events for stopped in their apparatus. This led to a decay rate of . Their result is in good agreement with theory but the experimental error is too large for a significant test of CVC.
The most recent and most precise measurement of the decay rate is due to McFarlane et al. [McF-85] who performed an in-flight decay measurement at LAMPF. Their result (), though in good agreement with theoretical predictions (() is not precise enough ( overall uncertainties) to test the full extent of the radiative corrections which stand at .
A precision determination of the pion beta decay rate is being carried out in a two stage program at the Paul Scherrer Institute (PSI) in Switzerland. The first phase of the experiment is expected to yield a decay rate with an overall uncertainty of . At this level of precision, a test of CVC and the radiative corrections will be possible. In addition, the superallowed Fermi decay data could be compare to the neutron beta decay. The second stage of the experiment will reduce the uncertainties still further to the level of which will test the CKM unitarity and constrain the following extensions to the minimal Standard Model: