Subtraction Point Research Articles

Electromagnetic Corrections to Weak Interactions. The Beta Decays of the Muon, Neutron, andO14

The momentum-dependent radiative corrections to the beta decays of the muon, the neutron, and ${\mathrm{O}}^{14}$ have been calculated to order $\ensuremath{\alpha}$ using the techniques of dispersion theory. The transition matrix elements can be expressed to this order (neglecting some effects of strong interactions) in terms of sets of vertex functions which satisfy once-subtracted dispersion relations. The absorptive parts of the vertex functions can be expressed to the appropriate order in terms of the vertex functions themselves and the amplitudes for electromagnetic scattering of the charged particles. It is a curious feature of the present calculation that the choice of the subtraction points is not arbitrary, but is determined uniquely by the requirement that such physically significant quantities as decay rates and the momentum spectra of the leptons should contain no infrared divergences when calculated including the contributions of processes in which soft photons are emitted [inner bremsstrahlung]. The subtraction constants play the role of renormalized weak coupling constants. The significance of this electromagnetic renormalization, and the connection between the choice of the subtraction point and the infrared divergence is examined in detail in the case of the muon. Two models for the beta decay of ${\mathrm{O}}^{14}$ have been considered. In one model, the nucleon involved in the transition is treated as a free particle insofar as the calculation of radiative corrections is concerned; in the other, the ${\mathrm{O}}^{14}$ and ${\mathrm{N}}^{14*}$ nuclei are treated as point particles, and the effects of the nuclear structure are ignored. The results obtained from the two models differ only slightly. Because of the appearance in the absorptive parts of the vertex functions of the form factors of the charged particles, evaluated for the particles on the mass shell, we are able to study analytically the effects on the transition amplitude of the finite electromagnetic structure of the nuclei. The effects of the finite spacial distribution of the decaying matter are treated using the usual multipole expansion of the nuclear matrix element. The leading electromagnetic structure correction is of the same form as the familiar $Z\ensuremath{\alpha}\mathrm{RW}$ in the correction for finite nuclear structure (finite deBroglie wavelength effect), but is of a different origin, and leads to a near doubling of the total structure corrections. The known theoretical corrections to the deay rates for the 0+ \ensuremath{\rightarrow} 0+ transitions ${\mathrm{O}}^{14}({\ensuremath{\beta}}^{+}){\mathrm{N}}^{14*}$, ${\mathrm{Al}}^{26*}({\ensuremath{\beta}}^{+}){\mathrm{Mg}}^{26}$, and ${\mathrm{Cl}}^{34}({\ensuremath{\beta}}^{+}){\mathrm{S}}^{34}$ are summarized. Using the recent, very accurate data on the decays of the muon, ${\mathrm{O}}^{14}$ and ${\mathrm{Al}}^{26*}$, we obtain the values ${G}_{\ensuremath{\mu}}=(1.436\ifmmode\pm\else\textpm\fi{}0.001)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$, ${G}_{\ensuremath{\beta}}({\mathrm{O}}^{14})=(1.419\ifmmode\pm\else\textpm\fi{}0.002)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$, and ${G}_{\ensuremath{\beta}}({\mathrm{Al}}^{26*})=(1.430\ifmmode\pm\else\textpm\fi{}0.002)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$ for the renormalized vector coupling constants for these transitions. The less accurate data on the neutron yield ${G}_{V}=(1.356\ifmmode\pm\else\textpm\fi{}0.068)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$. The results for ${G}_{\ensuremath{\mu}}$ and ${G}_{\ensuremath{\beta}}$ are not directly comparable in the present theory, but the different values of ${G}_{\ensuremath{\beta}}$ should be. The discrepancy of (0.8\ifmmode\pm\else\textpm\fi{}0.5)% between the effective coupling constants for the decays of ${\mathrm{O}}^{14}$ and ${\mathrm{Al}}^{26*}$ could, therefore, be significant, and may yield information about the still uncertain Coulomb corrections to the nuclear matrix elements. If the validity of the cutoff-dependent results of perturbation theory is assumed, the renormalization constants can be evaluated, and one obtains ${G}_{\ensuremath{\mu},\mathrm{bare}}=(1.431\ifmmode\pm\else\textpm\fi{}0.001)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$ and ${G}_{\ensuremath{\beta},\mathrm{bare}}({\mathrm{O}}^{14})=(1.404\ifmmode\pm\else\textpm\fi{}0.002\ifmmode\pm\else\textpm\fi{}0.007)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$. Those coupling constants differ by 1.9\ifmmode\pm\else\textpm\fi{}0.2%, but because of uncertainties regarding the nuclear matrix element for ${\mathrm{O}}^{14}$, the effects of strong interactions, and the possible existence of an intermediate vector meson which mediates the weak interactions, a direct comparison of these numbers may not be relevant to the possible universality of the Fermi interaction.