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The theory of the order-a radiative correction to the B decay of baryons is briefly reviewed. An effective procedure for the calculation of the real-photonic part of the order-a correction is outlined. Numerical results are tabulated for the radiative corrections to the branching ratio, the electron-antineutrino correlation parameter, the electron-energy (E,) spectrum, the final-baryonenergy (El) spectrum, and the electron-antineutrino angular-correlation (case,,) distribution, and to the (E,,El) and the (E,,cosQ,,.) two-variable distributions in the cases of the decays B --n e v , Z -+ A e G , ZP-.AeV, and A + p e v .

The "model independent" part of the order a radiative correction due to virtual photon exchanges and inner bremsstrahlung is studied for semileptonic decays of hyperons. Numerical results of high accuracy are given for the rela tive correction to the branching ratio, the electron energy spectrum and the (Ee ,Ef) Dalitz distribution in case of four different decays: I" •> n e v , l~-* ■ Aev,-* • Aev and Л-* • pev. АННОТАЦИЯ Изучена "независимая от модели" часть радиационных поправок в порядке а, которая соответствует обмену виртуальных фотонов и тормозного излучения. Для распадов Е-*nev, £-* ■ Aev, = Aev и Л-* • pev получены точные численные резуль таты для поправок к "branching ratio", спектру энергии электронов и диаграмме Далица (Е£ , E f). KIVONAT A virtuális foton csere és fékezési sugárzás következtében fellépő a ren di! sugárzási korrekciók "modell független" részét tanulmányozzuk hyperonok szemileptonos bomlásaiban. Nagy pontosságú numerikus eredményeket adunk a l~-* ■ nev, l~ ■ * Aev, =" ■ + r^eV és A ■ + pev bomlások esetén az elágazási arány, az elektron energia spektrum és az (Ee ,Ef) Dalitz eloszlás relativ korrekci óira .

Recently we developed a Monte Carlo-type program for the numerical integration of one-photon bremsstrahlung final states ͓1͔. In order to check this program we recalculated the numbers we published in 11 Tables of Ref. ͓2͔ for baryon semileptonic decays. In most cases we found convincing agreement, but, unfortunately, we also found exceptions in the cases of Table VI ͑first three lines referring to the decays ͚ Ϫ →ne, ͚ Ϫ →⌳e, ⌶ Ϫ →⌳e͒ and of Table X ͑in the last two lines referring to the decays ⌳→pe, n→pe, the coefficients of f 1 2 and g 1 2 ͒. In order to resolve this discrepancy we reconstructed our earlier program, which was based upon a mixed, analytical and numerical method for the integration of bremsstrahlung events. By means of this program we could perfectly reproduce the numbers we published in Tables I-XI of Ref. ͓2͔, let alone the ominous parts of Tables VI and X. In these cases we found complete agreement with the results obtained by means of the recently developed Monte Carlo program. This convinces us that the new numbers are correct, therefore we present here these two tables after the necessary corrections.In our effort to follow up the origin of this error we could not find anything that could be a direct explanation. As our late calculation is now available only in the form of building blocks ͑subroutines, etc.͒, we cannot say that an error in the original program is excluded. We think, nevertheless, that, most probably, a confusion of results happened, which were obtained in some intermediate stage of our work.

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The order-a radiative correction for the P-decay process A-pe? is studied in order to meet the needs of high-precision experiments which measure the electron-neutrino angular distribution. The relative correction is tabulated for the decay distribution in terms of the electron energy E, and cosO,,. The one-variable distribution with respect to cosOe, is decomposed in terms of the formfactor combinations f:, f , f , , etc., and the radiative correction is evaluated for each term. For the electron-neutrino correlation parameter a,,,, a huge relative correction is found due to its small value in the lowest order.The A+peG process is of particular importance among the hyperon semileptonic decays. It is possible to produce A-particle beams of large intensity, and the two charged particles in the final state can be well identified. For this reason the form factors of the strangenesschanging weak current can be measured with an accuracy which is matched, at present, only by the neutron+decay experiments measuring the A S = O weak form factors. Therefore the A-+peV data, combined with the neutron-decay results, offer a chance for a precise test of the Cabibbo theory.We are motivated to present the results which are included in this paper by the experiment of Wise et al. ' In that experiment the ratio of the axial-vector and the vector form factor, g l / f , and the vector form factor f were determined by measuring the electron-neutrino correlation function d a /d (cosO,,) and the semileptonic branching ratio RA-pev, respectively. They obtained2 lgl/fll=0.715k0.025 and fl=1.238+0.024, which is a noteworthy result, because the errors are much smaller than in the previous experiments. At such an accuracy fine details of the theory, especially the radiative corrections, must be carefully incorporated in the experimental analysis. The present paper is devoted to the radiative corrections which are relevant to electron-neutrino correlation when it is measured in an experiment like that of Wise et al.While the radiative correction to the branching ratio has been reliably calculated by a number of authors, it is not the case with the electron-neutrino correlation. We know of two papers reporting on the calculation of the radiative correction to the latter quantity.3,4 (The paper by Yokoo, Suzuki, and ~o r i t a~ is incorrectly cited in Ref.1. It contains no information on the radiative correction to the electron-neutrino correlation.) In Refs. 3 and 4 the definition is used, where p, and p, denote the three-momenta of the electron and the antineutrino, respectively, in the rest frame of the decaying A. However, the antineutrino is not detected in the experiment of Wise et al. where, for case,,, the quantity was used, where pf is the three-momentum of the final proton. Apparently, these definitions are not equivalent, if the radiative corrections are important, as in the case of bremsstrahlung events A-pevy, there is one more undetected particle, the photon, in the final state. For this reason the results of Fujikawa and ~~a r a ...

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Unitary and nonunitary representations of the SL(2, C) group are investigated in such a basis, in which the subgroup diagonalized is that one which in the four-dimensional representation leaves invariant the 4-vector pμ = (½(1 + v), 0, 0, ½(1 − v)) for an arbitrary real value of Pμ2=v. The split of the representation space into irreducible subspaces changes smoothly when varying the value of v. The formalism is of importance in physical theories which postulate analyticity requirements and Lorentz invariance simultaneously (e.g., Regge and Lorentz pole theory). In this paper we construct explicit basis functions of the representation spaces.