Evidence is presented from a proton-proton bubble-chamber experiment at 6.6-GeV/c beam momentum that the cross-section ratio $\frac{\sigma (pp\to {\pi }^{+}p{\pi }^{+}{\pi }^{-}n)}{\sigma (pp\to {\pi }^{+}pn)}$ for low momentum transfer to the neutron ($t<\mathrm{}0.2\mathrm{}$ Ge${\mathrm{V}}^2$) is approximately the same as the cross-section ratio $\frac{\sigma ({\pi }^{+}p\to {\pi }^{+}p{\pi }^{+}{\pi }^{-})}{\sigma ({\pi }^{+}p\to {\pi }^{+}p)}$ when compared as a function of invariant mass of the non-neutron system in the $\mathrm{pp}$ reactions. These results imply that (a) pion exchange dominates $\mathrm{pp}\to \left({\pi }^{+}p{\pi }^{+}{\pi }^{-}\right)n$ for low momentum transfer to the neutron, in view of the recent results of Ma et al. on $\mathrm{pp}\to \left({\pi }^{+}p\right)n$, and (b) the off-shell corrections to the pion pole equation for $\mathrm{pp}\to \left({\pi }^{+}p{\pi }^{+}{\pi }^{-}\right)n$ are similar to those for $\mathrm{pp}\to \left({\pi }^{+}p\right)n$; the results presented here complement those presented earlier by Colton et al. for the reactions $\mathrm{pp}\to \left(p{\pi }^{-}\right){\Delta }^{++}$, $\left(p{\pi }^{-}{\pi }^0\right){\Delta }^{++}$, and $\left(n{\pi }^{-}{\pi }^{+}\right){\Delta }^{++}$.

The present data for the reaction $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ are analyzed with the aid of a general effective-range resonance formula for the $\rho$ meson. It is concluded that (a) $m_{\rho }=769\mathrm{}\pm 5$ MeV and ${\Gamma }_{\rho }=109\mathrm{}\pm 10$ MeV provide a reasonable explanation of the present experimental points; (b) the full width at half-maximum regardless of the peak height should give an excellent estimate of the $\rho$ width; (c) it is hard to escape ${\left|F_{\pi }\right({m_{\rho }}^2|}^2=\frac{{m_{\rho }}^2}{{{\Gamma }_{\rho }}^2}$; (d) the effects of final-state interactions on resonance shape ought to be investigated.

By using a previously developed dispersion technique, we investigate the Chew-Frautschi and the $\mathrm{Im}\alpha$-versus-$s$ plots for the cascade resonances. The eight resonances are found to lie on three nearly linear plots on both graphs as was found to be the case previously for the baryon $N_{\gamma }$ and ${\Delta }_{\delta }$ sequences. Of the two types of resonances, probable and doubtful, all of the former are fitted (with no gaps on the trajectories) and none of the latter is needed. Severe self-consistency requirements are well satisfied by the two plots.

Observed angular distributions for the processes ${\pi }^{+}n\to {\pi }^{-}+{\Delta }^{++}\mathrm{}\left(1236\right)\mathrm{}$ and $K^{-}+p\to {Y_1}^{*-}\mathrm{}\left(1385\right)+{\pi }^{+}$ have been explained both in shape and in magnitude using the rescattering model.

A simple proof of a theorem derived recently by Dashen and Weinstein on $K_{l3\mathrm{}}$ form factors is presented. With the further assumption that $D\left(t\right)\mathrm{}\equiv 〈\pi \left|{\partial }_{\lambda }{V_{\lambda }}^1\right|K〉$ is dominated by the $0^{+}$ $\kappa$ meson, a relation between $\kappa$-meson mass and $\frac{f_K}{f_{\pi }{F}_{+}\mathrm{}\left(0\right)\mathrm{}}$ is obtained.

With the notable exception of the $S$ waves, reasonably good fits to the $\pi N$ phase-shift analyses had been obtained with a phenomenological one-pole, multichannel $N{D}^{-1\mathrm{}}$ model. We show that a two-channel ($\pi N, \pi \Delta$) model with a second pole in only the diagonal $\pi N$ potential yields a good fit to the detailed behavior of the $S_{31}$ partial wave.

The rescattering model has been applied to explain the observed angular distributions at various energies for the process ${\pi }^{-}p\to {\Sigma }^{-}{K}^{+}$. The theoretical calculations are in reasonable agreement with experiments regarding magnitude, shape, and energy dependence.

The phenomenological prescriptions outlined in the preceding paper (I) for meson-baryon couplings in a quark model of broken $\mathrm{SU}\left(6\right)\mathrm{}\times O\left(3\right)\mathrm{}$ are employed for the construction of meson couplings in an almost identical spirit in respect of relativistic generalization as well as the inclusion of the recoil contribution. The multiplying form factor for a supermultiplet transition from an initial $Q\overline{Q}$ state (mass $M$) to a final $Q\overline{Q}$ state (mass $m$) is assumed to have the analogous structure $f_L\left(k^2\right)={\overline{g}}_L{\mu }^{-L-1\mathrm{}}{\left(\frac{\mu }{m_{\pi }}\right)}^{\frac{1}{2}}{\left(\frac{\mu }{{\omega }_k}\right)}^{L\pm 1}\mathrm{}\left(2M\right)\mathrm{}$, where ${\overline{g}}_L$ is a free parameter different from $g_L$ for all cases except $L=0\mathrm{}$ (when both are equal to the $\overline{Q}\mathrm{QP}$ coupling constant $f_q$). The agreement of this meson coupling scheme with experiment is found to be extremely good for a large number and variety of decays from $L^P=0^{-}, 1^{+}, \mathrm{and} 2^{-}$ supermultiplets. These results not only provide a sensitive test of the (rapidly varying) structure of the form factor, but also serve to emphasize the essential unity of description of both meson and baryon couplings through a common set of assumptions. A more sensitive test of this model is provided by its polarization predictions for $B\to \omega \pi$ and $A_1\to \rho \pi$ decays which are appreciably different from, and in somewhat better experimental accord than, those of the simple quark model.

A new form of parametrization is proposed for meson-baryon couplings in a phenomenological quark ($Q$) model of broken $\mathrm{SU}\left(6\right)\mathrm{}\times O\left(3\right)\mathrm{}$ for the decays of baryons (mass $M$) belonging to the representations (56, $2l^{+}$) or [70, ${\mathrm{}(2l+1)\mathrm{}}^{-}$] to baryons (mass $m$) belonging to 56, together with the emission of pseudoscalar ($P$) mesons (mass $\mu$). The starting point is the use of the direct term in $\overline{Q}\mathrm{QP}$ coupling for the evaluation of the meson-baryon coupling structures which are reexpressed in terms of nonrelativistic Rarita-Schwinger fields together with multiple derivative structures in the meson field. A relativistic generalization of the latter is then proposed through a simple extension of the index structures in the ($L+1\mathrm{}$) partial-wave coupling terms. For the ($L-1\mathrm{}$)-wave coupling terms, which appear with an extra multiplying factor ${\mathbf{k}}^2$ in the meson three-momentum k, an additional ansatz ${\mathrm{k}}^2\to k_{\mu }{k}_{\mu } (\equiv -{\mu }^2)$ is used in order to include the contributions of the recoil terms for ($L-1\mathrm{}$)-wave transitions in a certain special combination, so as to incorporate the experimental feature of enhanced heavy-meson decays in the $s$ wave. Finally, all these coupling terms are assumed to be multiplied by the following form factor, the plausibility of whose structure is defended on physical grounds: $f_L\left(k^2\right)=g_L{\mu }^{-L-1\mathrm{}}{\left(\frac{\mu }{m_{\pi }}\right)}^{\frac{1}{2}}{\left(\frac{\mu }{w_k}\right)}^{L\pm 1}{\left(\frac{M}{m}\right)}^{\frac{1}{2}}$, where the exponents ($L\pm 1$) are used for emissions in the corresponding waves, and $g_L$ is a single free parameter governing the entire supermultiplet transition. The scheme, which is subjected to a detailed experimental test in respect to a large number of baryonic transitions from (70, $1^{-}$) and (56, $2^{+}$) states involving a wide range of masses and momenta, is found to provide an impressive number and quality of agreements with experiment. It is also shown to yield almost equal values of the coupling constants $g_L$ in respect to the couplings of several Regge recurrences of the $\Delta$ resonances, in conformity with the general expectation of a universal coupling for the Regge trajectory of a given particle.

We propose a classical optical model for high-energy elastic scattering. The model predicts that the logarithmic slopes for forward differential cross sections should be either the same or twice as large as the backward ones. The backward peak is due to the contribution from the discontinuity in the boundary of the optical medium. The model has the property that the mechanisms responsible for the forward and backward scatterings are not independent of each other. This is contrasted with conventional models which view the forward and backward scatterings as arising from independent mechanisms. For example, in absorptive and Regge-pole models the main contribution to the forward scattering comes from $t$-channel exchange, while backward scattering arises from $u$-channel exchange. A fit is made to the high-energy ${\pi }^{-}p$ data in both forward and backward directions, and good results are obtained.

An alternative method for deriving Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin- (KSRF-) type relations, which uses a low-energy theorem in quantum electrodynamics, is presented. The method is applied to processes of the kind $V\to P+P^{\prime }+\gamma$. The amplitude for this process is exactly calculable to order zero in the photon momentum $k$ in terms of the strong vertex $f_{\mathrm{VP}{P}^{\prime }}({m_V}^2, {m_P}^2, {m_P}^2)$, with all particles on the mass shell. KSRF-type relations are obtained when the momentum-independent term from the series expansion (in powers of the photon momentum) of the radiative amplitude is equated with the parallel term obtained from an explicit calculation of the amplitude. The latter calculation involves the use of current-algebra, partial conservation of axial-vector current, and hard-pion techniques. The derivation is worked out in detail for the $\rho \to {\pi }^{+}{\pi }^{-}\gamma$ process, and the method is further considered in connection with the appropriate radiative decays of ${\phi }^0$ and $K^{*0\mathrm{}}$ mesons.

A symmetry-breaking mechanism is used to derive a sum rule between $f_K$, $f_{\pi }$, and $f_{\kappa }$, the decay constants for $K$, $\pi$, $\kappa \to l+\nu$. By means of the sum rule $\frac{f_K}{f_{\pi }}$, $\frac{f_{\kappa }}{f_{\pi }}$ and $f_{+}\mathrm{}\left(0\right)\mathrm{}$ (the form factor which enters in $K_{l3\mathrm{}}$ decay) are expressed in terms of the $\kappa$-meson mass and the experimentally known number $\frac{f_K}{f_{\pi }{f}_{+}\mathrm{}\left(0\right)\mathrm{}}$. The second-order $\mathrm{SU}\left(3\right)\mathrm{}$-symmetry-breaking corrections to $f_{+}\mathrm{}\left(0\right)\mathrm{}$ are found to be small. The $K_{l3\mathrm{}}$ form factors are determined. The $\kappa \to K\pi$ width is determined and in certain cases is found to be the same as in the Veneziano model.

A configuration mixing of the states $|56, W=\frac{1}{2}, L=0^{+}〉$ and $|70, W=\frac{3}{2}, L=1^{-}〉$ in $\mathrm{SU}\left(6\right)\mathrm{}\otimes O\left(3\right)\mathrm{}$ is shown to reproduce experiments on $G_A$, $G^{*}$, and $\alpha =\frac{d}{\mathrm{}(d+f)\mathrm{}}$ more accurately than the $|56, W=\frac{1}{2}, L=0^{+}〉$ and $|20, W=\frac{1}{2}, L=1^{-}〉$. This is indeed desirable, since it ties up neatly with the quark model, phase-shift analysis, and experiments—that the (56,$0^{+}$) and (70,$1^{-}$) are the lowest-lying supermultiplets which accommodate the physical baryons. At the same time, it raises the question of the dynamical origin of representation mixing.

The finite-energy sum rule, and a class of sum rules which can be used to probe the existence of fixed poles, are obtained for amplitudes whose left- and right-hand cuts are not related by crossing symmetry. The finite-energy sum rule is evaluated for each of four independent $\pi N$ amplitudes with $u$ fixed at ${\mathrm{}\left(1236\mathrm{M}\mathrm{e}\mathrm{V}\right)\mathrm{}}^2$, both sides of the resultant four sum rules being obtained from the properties of the low-energy $\pi N$ resonances. Results are presented for three choices of end point: ${\mathrm{}\left(1356\mathrm{M}\mathrm{e}\mathrm{V}\right)\mathrm{}}^2$, ${\mathrm{}\left(1808\mathrm{M}\mathrm{e}\mathrm{V}\right)\mathrm{}}^2$, and ${\mathrm{}\left(2313\mathrm{M}\mathrm{e}\mathrm{V}\right)\mathrm{}}^2$. For the intermediate end point, all four sum rules work. For the highest one, however, they all fail. These results, while pointing to a failure of the resonance dominance approximation above 1800 MeV, give us a new confirmation of Regge high-energy behavior on the basis of low-energy data alone. In particular, they verify in some detail the relation predicted by Reggeism between the high-energy, fixed-$u$ behavior of the amplitudes and the low-energy $u$-channel resonances. They also show that for $u={\left(1236\mathrm{} \mathrm{MeV}\right)}^2$, all the $\pi N$ amplitudes have Regge behavior on the average (duality) above 1800 MeV. The finite-energy sum rules are shown to be violated in a fictitious universe where the lowest particle on each of the leading $\pi N$ Regge trajectories is accompanied by a degenerate partner of opposite parity.

While the measured charge asymmetry in the semileptonic decays of ${K_L}^0$ is clear-cut evidence for $\mathrm{CP}$ violation in these processes, the detailed theoretical interpretation of this result is somewhat complicated by the presence of electromagnetic effects. It is shown, however, that whereas these electromagnetic effects can produce sizable differences in decay distributions for decays into charge-conjugate states, their contribution to the asymmetry parameter ${{\delta }_L}^{\pi }$ is not greater than $\frac{\alpha \Gamma \left({K_{l4\mathrm{}}}^0\right)}{\Gamma \left({K_{l3\mathrm{}}}^0\right)}$.

The ${\Omega }^{-}$ nonleptonic decays are investigated in the framework of a chiral Lagrangian model. In this way, ambiguities of extension of meson momenta to zero which are present in the current-algebra approach are made explicit. We find that these decays may possess a rather complicated structure and a discussion of approximation schemes is given which may help to explain the present experimental data. The approximation of the $\Xi 2\pi$ decay mode by ${\Xi }^{*}\pi$ is also discussed.

Including $\rho$ contributions in the crossed channels of the $\pi \pi$ scattering amplitude, we use the $\frac{N}{D}$ method to obtain an expression for the pion electromagnetic form factor. The correction to the formula based on a direct-channel $\rho$ pole is somewhat larger than generally believed, but still consistent with the recent colliding-beam experiments.

We present evidence that current divergences belong to the (3,3*)+(3*,3) representation of $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$. We also give various estimates of the parameter $c$ that describes the strength of the $\mathrm{SU}\left(3\right)\mathrm{}$-breaking term relative to the chiral-symmetry-breaking term, using the Veneziano model for $\pi -\pi$ and $\pi -K$ scattering. The ratio $\frac{f_K}{f_{\pi }}$ is also estimated.