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.

This paper presents and discusses two sum rules for the axial-vector form factor for the ${\mathrm{He}}^3$-${\mathrm{H}}^3$ isodoublet, the conventional Weisberger-Adler sum rule, and a new plane-wave sum rule. Both sum rules are well satisfied. A method of improving the impulse-approximation estimate of pion-cross-section differences is presented, and the various disintegration cuts and anomalous thresholds inherent in a nuclear dispersion relation are discussed.

Photoproduction cross sections for neutral $\pi$, $\eta$, $\rho$, and $\phi$ mesons have been measured at the Stanford Linear Accelerator Center for photon energies between 5 and 17.8 GeV, and $t$ (four-momentum transfer squared) between -0.12 and -1.4 ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$, using a missing-mass technique. The pion production at lower energies is characterized by a fast falloff with increasing $\mathrm{}\left|t\right|\mathrm{}$ at small $\mathrm{}\left|t\right|\mathrm{}$ values, with a dip at $t=-0.5\mathrm{}$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$ followed by a secondary maximum around $t=-0.9\mathrm{}$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$ and a smooth falloff at larger $\mathrm{}\left|t\right|\mathrm{}$ values. As the incident photon energy increases, the dip becomes less pronounced, in contradiction to the expectations of simple Regge theories based only on the exchange of the $\omega$ and $B$ trajectories. $\eta$ photo-production was measured around 6 GeV and at 9 GeV. The cross section decreases smoothly with $t$ and shows no dip at $t=-0.5\mathrm{}$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$, in disagreement with predictions based on Reggeized $\rho$ exchange. $\rho$ production rates agree well with predictions assuming diffraction production. The differential cross section varies approximately as $e^{8.5t}$. The total cross section decreases from 16.0 μb at 5.5 GeV to 12.3 μb at 17.8 GeV incident photon energy. A quark-model relation between $\pi -p$ elastic scattering and $\rho$ photoproduction gives a good representation of the data. $\phi$ production also appears consistent with the predictions of the diffraction-dissociation model. We also searched for evidence of photoproduction of other particles with mass up to 2 GeV. Production of one particle of mass 1240±20 MeV and width around 100 MeV was observed. No particles with mass between 1300 and 2000 MeV were found. Any particle with cross section larger than 4% of the $\rho$ cross section would have been visible.

The decuplet parity-conserving weak spurion has been evaluated in the Sugawara-Suzuki model by assuming that the relevant matrix elements, in a currentXcurrent model, are saturated by baryon octet and decuplet single-particle intermediate states. $\mathrm{SU}\left(6\right)\mathrm{}$ symmetry has been used to determine the decuplet form factors at zero momentum transfer. In this approach, octet dominance for the decuplet spurion emerges in a natural manner. The $P$-wave ${\Omega }^{-}$ decay amplitudes are obtained with the baryon octet and decuplet spurion strengths as input. By assuming that the $D$-wave amplitudes for ${\Omega }^{-}\to \Lambda K, \Xi \pi$ decays are given by the $K^{*}$-pole model, we predict a zero asymmetry parameter for ${\Omega }^{-}\to \Lambda K$ and a maximal one ($\alpha \approx 1$) for ${\Omega }^{-}\to \Xi \pi$ decays. The total decay rate of the ${\Omega }^{-}$ is estimated to be $\Gamma \left({\Omega }^{-}\right)\cong 8\times {10}^9$ ${\sec }^{-1\mathrm{}}$, which is in good agreement with experiment.

Inelastic effects in high-energy pion-deuteron scattering are investigated using a model in which the pion is a two-particle composite system. The inelastic pion form factors required in calculating inelastic pion-nucleon scattering are related to the elastic form factor by assuming that the low-lying states of the pion system are identical with the corresponding states of a nonrelativistic isotropic harmonic oscillator. In the model, pion-deuteron scattering can be viewed in two ways which are shown to be completely equivalent: as the scattering of two two-particle composites, or as pion-deuteron scattering in which excited pions can occur in intermediate states. Inelastic contributions are found to be negligible near the forward direction, but important at large angles. Including corrections from the lowest-lying excited states of the pion considerably improves the accuracy of the calculations at intermediate angles.

The hypothesis is made that the sum of the imaginary parts of the baryonic Regge contributions to exotic meson-baryon scattering amplitudes must vanish. This hypothesis is combined with the assumptions that all coupling constants satisfy exact $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry and that the $\Sigma -\Lambda$ mass splitting cannot be neglected, so that the $\Sigma$ and $\Lambda$ lie on different Regge trajectories. All possible partitions of the spin-parity-${\frac{3}{2}}^{-}$ $\Lambda \text{'}\mathrm{s}$ and $\Sigma \text{'}\mathrm{s}$ (belonging to an octet, singlet, and decuplet) into the two trajectories are considered. A set of consistency equations is derived, and all solutions are obtained. If the $\Lambda \mathrm{}\left(1520\right)\mathrm{}$ is the only odd-parity hyperon on the bottom [$\Lambda \mathrm{}\left(1115\right)\mathrm{}$] trajectory, and the $\Lambda \mathrm{}\left(1690\right)\mathrm{}$ and one or two $\Sigma \text{'}\mathrm{s}$ lie on the top trajectory, there is only one solution. In this solution, there is no odd-parity decuplet, the $\frac{F}{D}$ ratios of the couplings to mesonbaryon states of the $j^P={\frac{1}{2}}^{+} \mathrm{and} {\frac{3}{2}}^{-}$ trajectories are 1, and the singlet-octet mixing angle of the ${\frac{3}{2}}^{-}$ $\Lambda \text{'}\mathrm{s}$ is $\theta =invtan\surd \frac{1}{2}$. This solution is in approximate agreement with experiment.

A simple mass-averaging device is described for the investigation of the effect of the pion mass difference on the nucleon-nucleon interaction. While the mass-difference correction in the one-pion-exchange contribution is known to be significant, we find that the corresponding correction plays a less significant role in the two-pion-exchange contribution. The effect of difference in the pion coupling constants is also discussed, and found to be equally significant in the one- and two-pion-exchange contributions. The applicability of our approach to the $\rho$-exchange contribution is pointed out.

Assuming that the hadronic electromagnetic current contains an isoscalar $C$-even part, we use the softpion technique and pole dominance to determine the isospin of the final state in the $\mathrm{CP}$-violating part of the $K^0\to 2\pi$ amplitude. If the usual dimensional arguments for the order of magnitude of the amplitude hold, as they appear to do for the observed value of ${\eta }_{+-}$, then values of ${\eta }_{00}$ close to ${\eta }_{+-}$ are excluded.

The hadronic decays of hyperons are discussed in the context of broken chiral symmetry, the breaking being introduced according to the prescription of Gell-Mann, Oakes, and Renner. The corrections due to symmetry breaking lead to contributions to the $p$-wave amplitudes, giving better agreement with the experimental data, whereas the $s$-wave amplitudes remain almost unchanged.

We have calculated the radiative corrections to the Dalitz plot for ${K_{\mu 3}}^{\pm }$ and ${K_{\mu 3}}^0$ decays, assuming a phenomenological weak $K-\pi$ vertex and using perturbation theory. The answer depends logarithmically on a cutoff. We have evaluated all terms which contribute to order $\alpha$ analytically, without any approximations concerning the smallness of the muon mass or the "real inner bremsstrahlung." Thus, the dependence on the parameter $\xi$ (the ratio of the form factor $\frac{f_{-}}{f_{+}}$) is exact. The radiative corrections to the ${K_{\mu 3}}^0$ Dalitz plot, muon spectrum, and lifetime average around 2% over most of their respective ranges and are not especially sensitive to the cutoff. The radiative corrections to ${K_{\mu 3}}^{\pm }$ decays is a fraction of a percent over most of the Dalitz plot and is sensitive to the cutoff. The radiative correction to the $\Delta I=\frac{1}{2}$ rule prediction for the ratio of the charged and neutral decay rates is approximately 2%. The final-state Coulomb correction accounts for most of this numerical result, the rest being model-dependent noise.

The $\mathrm{CP}$-conserving decay mode ${\pi }^{+}{\pi }^{-}\gamma$ of the ${K_2}^0$ is calculated in the pion-pole model (with $\eta -{\pi }^0$ mixing) with the strong-interaction part of the decay amplitude described by the Veneziano model. Because of sign ambiguities introduced by a model of physical $\mathrm{PVV}$ couplings (which fits both ${\pi }^0\to 2\gamma$ and $\eta \to 2\gamma$ rates) and the unknown sign of $\eta -{\pi }^0$ mixing relative to the weak $K-\pi$ mixing, values of the branching ratio $r\left(\frac{({K_2}^0\to {\pi }^{+}{\pi }^{-}\gamma )}{({K_2}^0\to \mathrm{all}\mathrm{modes})}\right)$ are obtained ranging from 7.6×${10}^{-5\mathrm{}}$ to 3.1×${10}^{-4\mathrm{}}$, but all are consistent with the present experimental upper limit of 4×${10}^{-4\mathrm{}}$.

The $\pi -\pi$ amplitude in the low-energy region is parametrized in a crossing-symmetric way as the sum of the $\rho$ and $f^0$ resonance poles plus a polynomial background. The parametrization is flexible and capable of producing amplitudes having quite different features in the energy region below 1 GeV. The parameters are then varied so as to minimize the deviation from elastic unitarity on a set of closely spaced points. In addition, negative-moment finite-energy sum rules are used to connect the low-energy region with assumed Regge asymptotic behavior in the $I=1, 2$ amplitudes. With the mass and width of the $f^0$, the mass of the $\rho$, and the slope of the $\rho$ trajectory fixed, an approximate solution satisfying the constraints is found, yielding a $\rho$ width of 80± 30 MeV. The solution displays the usual characteristics of the low-energy $\pi -\pi$ amplitudes suggested by other analyses, namely, small scattering lengths and a large $I=0\mathrm{}$ $S$-wave phase shift near the mass of the $\rho$. This resonantlike behavior is found without introducing an $S$-wave pole in the parametrization, while the small scattering lengths are obtained as results of the numerical bootstrap, although no current-algebra constraints are included. Our proposed solution is also found to satisfy various inequalities proposed by Martin for the ${\pi }^0-{\pi }^0$ scattering amplitude.

Compton scattering of pions is investigated in the Veneziano model. It is shown that the removal of parity doubling causes undersirable features, such as inconsistency with the low-energy theorem or the appearance of an unwanted pole in the energy variable ($s=0\mathrm{}$) in one of the amplitudes. If we ignore the problem of parity doubling or overlook the pole at $s=0\mathrm{}$, we get a sum rule which closely resembles that obtained from superconvergence relations or the quark model, or current algebra.

The relation between the processes $\gamma +N\to \pi +N$ and $\pi +N\to \rho +N$, according to the vector-dominance model (VDM), is reexamined. The VDM is defined in terms of the behavior of the invariants in the matrix elements of the vector current. The consequences of the VDM, and assumptions about asymptotic properties, for measurable quantities are discussed. The inability of the VDM to account for the results of polarized-photoproduction experiments is confirmed. An appendix provides a detailed discussion of the relation between invariants and density matrix elements in various frames.

It is shown that the suppression of $\Delta S=1\mathrm{}$ leptonic decay modes of hadrons relative to the $\Delta S=0\mathrm{}$ modes can be successfully explained within the framework of the universal Fermi interaction by suitable assumptions regarding hadron states and weak hadron currents in the presence of symmetry-breaking effects.

We have measured the differential cross section for elastic scattering of $K^{-}$ mesons on ${\mathrm{He}}^4$ in the momentum intervals 100-150 MeV/c and 150-200 MeV/c, and the total inelastic cross section in the interval 150-200 MeV/c. We have calculated scattering lengths for the $K^{-}$-${\mathrm{He}}^4$ system using the zero-effective-range approximation. The results for the scattering lengths have been compared with other experiments, including a measurement of $K^{-}$-mesonic x rays in ${\mathrm{He}}^4$. Our results are in disagreement with the latter.