The $\mathrm{np}\to \mathrm{pn}$ amplitudes are written in simple Veneziano form. Only an $M=0\mathrm{}$ pion pole and an $M=1\mathrm{}$ pion parity doublet with corresponding exchange-degenerate $B$ are considered in our case to give rise to a sharp peak in $\frac{d\sigma }{\mathrm{dt}}$ at $t=0\mathrm{}$, giving qualitative agreement with experimental data.

Combining conserved vector current, partially conserved axial-vector current, and the meson-dominance approximation, we develop a theory of current algebra ($S{U}_3\otimes S{U}_3$) "on the mass shell" by means of Ward-Takahashi-type identities for three-point functions. Models of mixing are presented which are compatible with the spectral-function sum rules derived from broken chiral $S{U}_3\otimes S{U}_3$ symmetry and with CVC to the first order of symmetry breaking. Further, requiring absence of divergences from radiative corrections to pion $\beta$ decays, we deduce an asymptotic sum rule which relates two different types of three-point functions and thus enables us to correlate two different meson processes: radiative and strong decays. In addition to these decay processes, applications are made to electromagnetic form factors of mesons, $K_{l3\mathrm{}}$ form factors, and $\pi \mathrm{}\left(K\right)\mathrm{}\to l\nu \gamma$ decays. Agreement with experiments is very reasonable.

Using the Fierz-Bargmann-Wigner basis, we construct independent covariants, free of kinematic singularities and zeros, for vertices in which two external particles carry arbitrary spins, with one off-mass-shell leg corresponding to a totally symmetric Lorentz tensor. Results for (pseudo) scalars, (pseudo) vectors, and symmetric tensors of second rank are tabulated explicitly.

A previously derived sum rule, based on $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ equal-time commutation relations for the space components of the electromagnetic current, implies mean polarization asymmetries of greater than 20% throughout most of the inelastic continuum.

An estimate of the strength of the magnetic radiation term in $K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$ decays is made by relating it to the ${\tau }^{+}$ decay rate and using the Veneziano model to give the off-mass-shell dependence of the amplitudes. We find this strength to be small ($\left|x_m\right|\le 0.14$) and practically constant in the kinetic-energy range $55 \mathrm{MeV}\le T_{{\pi }^{+}}\le 80 \mathrm{MeV}$. The results are discussed in the light of $K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$ data, particularly for possible $\mathrm{CP}$-violating effects.

A previous calculation of the radiative corrections to the rate for pure Fermi $\beta$ decay in the $V-A$ theory, computed to first order in $\alpha$ and to zero order in the lepton momenta, is extended here to include, in addition, the corrections linear in the product of a lepton momentum and a nuclear charge radius. These additional contributions are shown to modify the Coulomb correction to the positron wave function to account for the finite distribution of electric and isotopic charge within the nucleus. They are evaluated and taken together with the other electromagnetic corrections, the origin of which are reviewed in the text, to reassess the question of the universality of the weak vector coupling constant. The conclusions obtained, although basically in agreement with the results of others, differ in detail from previous surveys of universality, generally in a direction to strengthen the concept of universality in the sense of Cabibbo.

Starting from the equal-time commutator of the vector currents and the assumption that the contributions from "exotic channels" are small, we find bounds on the asymptotic behaviors of various form factors. For instance, the pion electromagnetic form factor and both of the $K_{l3\mathrm{}}$ form factors are found to be bounded by $\frac{1}{\surd \mathrm{}\left|t\right|\mathrm{}}$ as $\mathrm{}\left|t\right|\mathrm{}\to \infty$. Implications of this result for $K_{l3\mathrm{}}$ decay are analyzed.

The electroproduction of $\rho$, $\omega$, and $\phi$ mesons is discussed. The spin-density matrix elements are shown to satisfy a number of conditions if and where the production is dominated by Pomeranchuk exchange. Constraints are also given for the case that a single spin-parity (of type $0^{\pm }$, $1^{\pm }$) dominates the $t$ channel in some kinematic region. The extension to the production of a $2\pi$ continuum is straightforward. Vector-meson and $2\pi$ production by neutrinos is also treated.

A phenomenological amplitude for proton Compton scattering is obtained, and fits are made to the differential cross-section data. The amplitude is used to calculate the cross sections for the Compton scattering of polarized photons and the polarization of the recoil proton. It is found that the polarized-photon cross sections are sensitive to the amount of $P_{11}$ resonance in the scattering amplitude.

Cross sections for pion production were calculated using several different chiral-dynamics Lagrangians including those of Schwinger and Weinberg. In order to compare the results with cross sections above threshold all tree diagrams were retained and integrations over final state variables were performed by a Monte Carlo method. A model in which the symmetry-breaking term transforms as a rank-2 chiral tensor was found to give the best fit to the total cross-section data for incident pion energies of less than 330 MeV, but none of the models gave good fits to the differential cross-section data.

An analysis of the recently measured photon-proton total cross sections is performed. Smooth fits to the cross sections are obtained and used to calculate, by means of the forward dispersion relation, the real part of the spin-averaged forward amplitude. The resulting predictions for the real part are given. At high energies, the fits to the present total cross-section data, together with the calculated real part, suggest the presence in the high-energy behavoir of an extra real constant in addition to what one would have predicted from Regge theory and the high-energy behavior of the imaginary part. This extra real constant, which is consistent in sign and magnitude with the Thomson limit, $-\frac{\alpha }{M_N}$, could correspond to a fixed pole at $J=0\mathrm{}$ in Regge-pole language. Possible ways to test the forward dispersion relation are discussed.

A model is constructed as an alternative to, and for examination of, various extrapolation procedures used in determinations of $\pi \pi$ phase shifts from the reaction $\pi +N\to \pi +\pi +N$. This model is similar to that suggested by Ross and Kane in that it incorporates absorption effects approximately.

Interference effects in the leptonic decay modes of photoproduced $\rho$, $\omega$, and $\varphi$ mesons are considered. Two experimental methods for determining the relative phase angles of the final-state muons are discussed. The phase-angle difference between the $\rho$ and $\omega$ is estimated from existing data and appears to be in disagreement with the value of 0°.

The Dalitz plot density of $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ is examined. We find the asymmetry on the ${\pi }^{\pm }$ energies to be $A=\frac{(N^{+}-N^{-})}{(N^{+}+N^{-})}=-0.014\mathrm{}\pm 0.030$, where $N^{\pm }$ is the number of events for which $T_{\mp }>T_{\mp }$. The density is well fitted with a linear dependence on $T_0:{\mathrm{}\left|M\right|\mathrm{}}^2\propto 1+2\alpha \left[\right(\frac{3T_0}{Q})-1]$, with $\alpha =-0.47\mathrm{}\pm 0.04$.

Electrons have been scattered inelastically from a hydrocarbon target containing protons polarized normal to the scattering plane. Scattered electrons with energies corresponding to the production of the $\Delta \mathrm{}\left(1236\right)\mathrm{}$, $N\left(1512\right)\mathrm{}$, and $N\left(1688\right)\mathrm{}$ pion-nucleon resonances were observed. A search was made for changes in the cross section as the target polarization was reversed. Any changes would have been evidence of a violation of time-reversal invariance in the electromagnetic interactions of the hadrons. No such changes were observed. With the maximum time-reversal-violating effect possible, the asymmetry would be 10 times the upper limit of this experiment. Early attempts at a coincidence-polarization experiment are described in an appendix.

The energy spectrum of the decay pions in ${\tau }^{+}$ decay was obtained from a sample of 17 898 events. It is found that the spectrum is adequately fitted by a squared matrix element of linear form, ${\mathrm{}\left|M\right|\mathrm{}}^2\propto 1+{\alpha }_{\tau }\times \left(\frac{M_{K}Q}{{m_{\pi }}^2}\right)Y$, with ${\alpha }_{\tau }=0.131\mathrm{}\pm 0.008$. No evidence for $\mathrm{CP}$ noninvariance is found for ${\tau }^{\pm }$ decays. A significant violation of the $\Delta I=\frac{1}{2}$ rule is observed for $K^{+}\to 3\pi$ decays. Comparison with several theoretical models for $K\to 3\pi$ decays shows fair agreement between experiment and theory.

Branching ratios have been measured for reactions initiated by negative kaons coming to rest in a helium bubble chamber. The data were obtained from an analysis of ∼1000 events found in a careful scan using a very limited fiducial volume. Charged-$\Sigma$ emission accounts for (30.8±3.2)% of the absorptions and $\frac{\Lambda }{{\Sigma }^0}$ emission for (69.2±6.6)%. Evidence for $\Sigma$-to-$\Lambda$ conversion is clearly seen in the events with a visible $\Lambda +{\pi }^{-}$, and (66±9)% of this final state results from such a process. Multinucleon (i.e., nonpionic) absorption is found to occur in (16.5±2.6)% of the events, which is similar to the results reported from studies using heavy-liquid bubble chambers and nuclear emulsion. In general, the kinematic spectra for single-nucleon absorption events agree well with the predictions of a simple impulse approximation. The ratio of $K^{-}$ capture on neutrons to that on protons is examined using the $\Lambda {\pi }^{-}{\mathrm{He}}^3$ and the $\Lambda {\pi }^0{\mathrm{H}}^3$ final states. The data for charged- and neutral-particle emission are consistent with the predictions of isospin invariance for kaon absorption in ${\mathrm{He}}^4$.

As a test of the validity of quantum electrodynamics (QED) for heavy virtual leptons, cross sections for wide-angle electron-pair photoproduction have been measured using hydrogen and carbon targets. Electron pairs were detected in two independent mirror-image spectrometers which identified production angles and momenta. A symmetrical arrangement was used in which interference between Bethe-Heitler and virtual Compton graphs is identically zero. Data were obtained at electron production angles from 5° to 7° and momenta from 1 to 2 GeV/c, and were compared with theory by means of a Monte Carlo calculation. Up to the maximum invariant mass of the electron pair 489 MeV/${\mathit{c}}^2$, the data are consistent with the predictions of QED for both targets, confirming the results of other recent wide-angle pair-production and bremsstrahlung experiments in this energy range. Expressing the results in terms of a cutoff parameter $\Lambda$, where any positive deviation from QED is proportional to ${\Lambda }^{-4\mathrm{}}{Q_l}^4$ (where $Q_l$ is the mass of the virtual lepton), one obtains $\Lambda >0.76\mathrm{}$ GeV/${\mathit{c}}^2$ with 95% confidence. For a similar negative deviation, $\Lambda >0.55\mathrm{}$ GeV/${\mathit{c}}^2$.

We have determined an upper limit to the branching ratio for the so-called structure-dependent radiation in the radiative decay mode $K^{+}\to e^{+}+{\nu }^e+\gamma$. The measurement was made by searching for moderate-energy photons in large-angle coincidence with positrons in the momentum region above the maximum allowed in ${K_{e3\mathrm{}}}^{+}$ decay and below the momentum of ${K_{e2\mathrm{}}}^{+}$. We obtain with 90% confidence $\frac{{{\Gamma }_{e\nu \gamma }}^{\mathrm{SD}}}{\Gamma \left(\mathrm{all}\right)}<7.1\mathrm{}\times {10}^{-5\mathrm{}}$, assuming ${\gamma }_K\equiv \frac{A_K\mathrm{}\left(0\right)\mathrm{}}{V_K\mathrm{}\left(0\right)\mathrm{}}\ge 0$. This in turn yields $\left|V_K\mathrm{}\right(0\left)\right|<\mathrm{}0.24\mathrm{}{M_K}^{-1\mathrm{}}$, which is consistent with $K^{*}$ dominance.

The unitarity relation of quantum scattering theory is studied to show how it can be utilized to obtain the phase of the scattering amplitude that corresponds to a given differential cross section. The main results are the following: (i) Provided the cross section is sufficiently small, repeated iterations of the unitarity equation yield a series which converges uniformly to the correct phase of the scattering amplitude. (ii) A modified iteration procedure can be introduced which provides a sequence of alternating upper and lower bounds on the phase of the scattering amplitude. Subject to a certain restriction on the magnitude of the cross section, the upper and the lower bounds so generated converge, respectively, from above and from below to the true value of the phase. (iii) A high-energy approximation to the unitarity relation, referred to as the small-phase approximation (SPA), is introduced. It is shown to be solvable by means of an iteration sequence, and upper and lower bounds to its solutions are constructed. (iv) Using some very crude approximations, we solve the SPA equation for the phase of the scattering amplitude explicitly in terms of the cross section. Since extremely oversimplifying assumptions are employed, this can be regarded only as a primitive model for a possible behavior of the exact solutions. The outline of these results is followed by a discussion of the practical applicability of the above methods for obtaining the phase of the scattering amplitude from the experimental cross section, in which their potential significance for atom-atom collision theory is stressed.