Search

A simple relativistic potential model is constructed in order to describe meson spectroscopy in a unified way, encompassing both light- and heavy-quark systems. A linear potential, transforming as the fourth component of a vector potential, is assumed in accordance with quark-confinement mechanisms appropriate to gauge theories. Scaling arguments are used to derive the Regge behavior of the model. We present evidence that the forces which confine quarks are essentially independent of quark species. The over-all systematics of our predictions for meson energy levels and leptonic decay widths give support to the usefulness of the model. We comment on the application of asymptotic freedom as a means of estimating decay rates which violate the Okubo - Zweig - Iizuka rule.

A measurement of the energy differences of the $n=2$, $^{3}P_{J}$ states of charmonium may give significant information on the dynamics of the spin-dependent quark-antiquark ($q\overline{q}$) forces, since the more rapidly these forces increase with $q\overline{q}$ separation, the larger the ratios ${R}_{1}=\frac{[E(^{3}P_{2})\ensuremath{-}E(^{3}P_{1})]}{[E(^{3}P_{1})\ensuremath{-}E(^{3}P_{0})]} \mathrm{and} {R}_{2}=\frac{[E(^{3}P_{1})\ensuremath{-}E(^{1}P_{1})]}{[E(^{3}P_{1})\ensuremath{-}E(^{3}P_{0})]}$ become if the forces are of vector character.

The large-$N$ limit of $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory with $\mathrm{O}(N)$ symmetry is derived by functional methods. Since the method does not require the cumbersome, detailed analysis of Feynman graphs, it should make possible the study of more complicated theories in the many-field limit.

A $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory of self-interacting boson fields with $\mathrm{O}(N)$ symmetry is considered. A nonperturbative method is developed to compute the effective potential, which becomes exact in the large-$N$ limit. The stability of the vacuum and possible spontaneous symmetry breakdown of the theory are studied in this limit. The existence of a local mimimum requires $0l1+{(N\ensuremath{\lambda})}^{\ensuremath{-}1}96{\ensuremath{\pi}}^{2}$, which may be either a true bound or an indication of the breakdown of the ${N}^{\ensuremath{-}1}$ expansion. The function $\ensuremath{\beta}(\ensuremath{\lambda})$ that appears in the Callan-Symanzik equations is calculated in the large-$N$ limit of the massless theory without reference to a perturbation expansion in $\ensuremath{\lambda}$. It is observed that the presence of a nontrivial zero of $\ensuremath{\beta}(\ensuremath{\lambda})$ is a necessary condition for a spontaneously broken symmetry in the massless theory. On this basis, it is shown that this $\ensuremath{\beta}(\ensuremath{\lambda})$ is unlikely to have a nontrivial zero. The leading-logarithm approximation to the effective potential is discussed within the above framework.

It is argued that the method of Cheng and Dashen gives a calculation of approximately three times the $\ensuremath{\sigma}$ term, rather than the $\ensuremath{\sigma}$ term itself. This conclusion is based on a correction derived from a unitarization of the $t$-channel $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude, consistent with the constraints of current algebra. A detailed analysis of possible model dependence of the result is included.

Several simple models, similar to that of Lee, involving indefinite metric are studied in this paper. In this connection, a dispersion-theoretic treatment is applied to a simple "equal-mass" model. It is shown that, at least for these models, the scattering amplitude is analytic in the upper-half energy plane provided time-reversal invariance holds; the rules of the dispersion-theoretic formulation in the case of an indefinite metric theory are given. The solution is reinterpreted as the exact solution of a slightly different model, which can also be obtained by Hamiltonian techniques; further techniques are generalized to include recoil in a relativistic no-pair model. Certain basic questions of interpretation are discussed in some detail in the concluding section.

We compute the on-mass-shell electromagnetic mass difference ${\ensuremath{\mu}}_{{\ensuremath{\pi}}^{+}}\ensuremath{-}{\ensuremath{\mu}}_{{\ensuremath{\pi}}^{0}}$, in a model consistent with chiral-algebra constraints and partial conservation of axial-vector currents, and find it to be logarithmically divergent. Some implications of this result are discussed.

A model-independert definition for a reduced width in nuclear reactions is proposed. In particular, stripping reactions of the form A(d,p)B are studied. The applicability of polology techniques for the determination of absolute reduced widths is tested by means of a computer experiment. The results indicate that the extrapolation to the pole can give a model-independent value for the reduced width which is accurate to at least an order of magnitude. It is likely that increased accuracy is possible if such model-dependent features as the bound- state wave function are included in the extrapolation formula. (auth)

A method is presented for constructing amplitudes which are the most general solutions to the Ward identities of current algebra. These solutions, expressed in terms of undertermined scalar functions, exhibit a wide class of amplitudes which satisfy the current algebra constraints. Further information is obtained from the model dependent equal-time commulators of the space components of the currents by means of the Bjorken-Johnson-Low limits, which provide boundary conditions for certain asymptotic expansions of the scalar functions. Simple phenomenological models are discussed within this framework, and a connection between the Gilman-Harari model and hard pion models is established.

The singularities nearest to the physical region in the complex cos θ plane (for fixed total energy) are located for stripping reactions using heuristic methods. These analytic properties make it plausible to expect: 1. 1) the exchange stripping contribution not to produce large backward peaks in direct interactions; 2. 2) the Butler pole not to be completely dominant even when it is close to the physical region, because of the nearness of important branch cuts; 3. 3) a properly defined reduced width to be energy independent, in analogy with coupling constants; 4. 4) a characteristics radius to come from form factors (near-by branch cuts) associated with the nucleon-nucleus vertex function; 5. 5) “simple” pology techniques to be inadequate because of near-by branch cuts due to distortions and Coulomb effects.

The technique of Bjorken is combined with the Ward identities for the time-ordered products of currents to obtain a correlation between space-space equal-time commutators and the high-energy behavior of the hadron proper functions. The method is illustrated for the three-point function of conserved isovector currents.

New techniques are developed for treating the $n$-point functions of currents, which are interrelated by means of Ward identities obtained from the equal-time current commutation relations. The $n$-point functions are sorted out so as to define proper vertices which describe the reactions of particles of definite spin. A meson-dominance assumption is made by approximating the proper vertices by simple polynomials in momenta, with the coefficients determined by the Ward identities. The method is discussed in detail for $n=3$ and the currents of chiral $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$, and then applied to the decay processes ${A}_{1}\ensuremath{\rightarrow}\ensuremath{\rho}+\ensuremath{\pi}$ and $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}$.

Several theorems are proved which exhibit the impossibility of constructing nontrivial products of the Lorentz group with internal symmetry groups for some physically interesting examples. It is also shown that if the Lorentz group is replaced by the Galilei group, qualitatively different results are obtained; a "linear" breakdown of internal symmetry is possible.

The application of sum rules derived from current commutation relations to give $\mathrm{SU}(6)$-like results is developed without the assumption of saturation. We first derive a consistency condition by comparing sum rules, and then study the sum rules for $\frac{1}{g_{A}^{}{}_{}{}^{2}}$ and the isovector charge radii, replacing the assumption of saturation with a simple dynamical model for the remaining continuuum contribution. This results in several $\mathrm{SU}(6)$-like predictions which contain important correction factors to the results previously derived assuming saturation. We have also re-examined the experimental data for the sum rule for $\frac{{〈r_{N}^{}{}_{}{}^{2}〉}^{V}}{6}$, and find that it converges below 1 GeV like the sum rule for $\frac{1}{g_{A}^{}{}_{}{}^{2}}$ when account is taken of the contribution to the sum rule of the large resonant ${M}_{1+}$ and nonresonant ${\mathrm{E}}_{0+}$ multipoles in pion photoproduction, and of the ${N}^{**}(1520)$.

We present a technique which allows us to display explicitly all the information which current algebra yields about $n$-point functions of vector and axial-vector currents. We write all three- and four-point functions in terms of a few primitive functions which are not determined by current algebra. Any approximation to these functions (subject to a single constraint) when used in our formulation yields three- and four-point functions guaranteed to satisfy all constraints imposed by current algebra and partially conserved axial-vector current. We give a very simple model where the primitive functions are given by as smooth functions of the momenta as possible and apply this model to the ${\ensuremath{\pi}}^{+}\ensuremath{-}{\ensuremath{\pi}}^{0}$ electromagnetic mass difference, ${A}_{1}$ decay, and $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ and $\ensuremath{\pi}\ensuremath{-}\ensuremath{\rho}$ scattering.

An approximate one-dimensional dispersion relation of the Cini-Fubini type is assumed for the invariant amplitudes of the photoproduction of K mesons on nucleons on the supposition that among the various intermediate states in the three related channels only the single-particle states and the resonant states rnake appreciable contributions. The resonant states that are taken into account are the following: the three-pion-nucleon resonances, the pion-kaon resonance, and the three pion-hyperon resonances. The expressions for the spectral functions are derived by applying the unitarity conditions to each of the three related channels. By the conservation of angular momentum and parity only certain multipole amplitudes determined by the spin and parity of the resonances contribute to the spectral functions, and for the se nonvanishing multipole amplitudes the Breit-Wigner formula is assumed. (auth)

The nonleptonic decay amplitudes of the $K$ mesons are calculated using current algebra and the spectral-function sum rules. The result is convergent in the intermediate-boson model, but not for a direct current-current interaction. Comparison with the observed rate for ${{K}_{1}}^{0}$ decay shows that the intermediate-boson mass is about 8 BeV.

The spectral-function sum rules are given for a general Lie algebra, and then applied to the vector and axial-vector mesons coupled to the currents of SU(3) \ensuremath{\bigotimes} SU(3).

Static charged scalar meson theory with broken charge symmetry is solved in the one-meson approximation. It is argued that spontaneous symmetry breakdown does not appear possible in this model with these approximations. Solutions are given for the scattering amplitude and also the isobar mass difference in terms of the n-p mass difference, without a perturbation expansion in the symmetry-breaking parameters.

A calculation of the inner Coulomb corrections to pionnucleon phase shifts is presented which does not depend on an expansion in powers ofkr0. The estimates obtained depend on the choice of simple, reasonable potential models. It is noted that the same corrections may be useful in analyzing other elementary particle reactions.

The question of the Reggeization of the elementary particles in renormalizable gauge theories with the Higgs-Kibble mechanism is examined. It is concluded that the massive non-Abelian gauge vector mesons and $J=\frac{1}{2}$ fermions of these theoreis lie on Regge trajectories in every case in which a counting criterion due to Mandelstam is satisfied. Several explicit examples are presented which verify that the necessary factorization condition of the Born helicity matrices is satisfied at $J=1$ and $J=\frac{1}{2}$, supporting this conclusion.

Methods are described for determining the leading light-cone singularities of bilocal operators in perturbation theory. Specific results are obtained for the bilocal operator ${T}^{*}{\ensuremath{\psi}(x)\overline{\ensuremath{\psi}}(0)}$ in the scalar-gluon model in lowest nontrivial order, with the finding that the $q$-number part of the bilocal operator is singular in the limit ${x}^{2}\ensuremath{\rightarrow}0$, contrary to usual assumptions. A description of the anomalies of deep-inelastic scattering is given in terms of canonically determined bilocal operators, with due attention to the singular nature of these operators, and with no recourse to new bilocal operators not present at the canonical level. Applications to the Schwinger-term sum rule and electromagnetic mass operator are made, with the same conclusions.

This review is concerned with the application of dispersion relation methods to direct nuclear reactions. We begin with a discussion of the assumptions required to relate analyticity properties to the conventional descriptions of direct-reaction amplitudes. This leads naturally to a description of phenomenological methods such as extrapolation to poles and similar topics. The use of the unitarity condition provides a framework for the discussion of the distorted wave Born approximation from the $S$-matrix point of view Finally we call attention to some dynamical calculations of relevance to our discussion.

The process π− + p → K° + Λ is studied near threshold by means of a resonance model based on the Cini-Fubini approximation using only known resonances. Thus the only term contributing an imaginary part to the amplitude is the N∗(1688) T = 12, F52+ resonance. Reasonable agreement with the total cross-section and angular distribution data is obtained, but the predictions for the polarization are not satisfactory. It is concluded that either several small, complex amplitudes make important contributions to the polarization, or that there are other large partial waves in addition to the F52 resonance. It may be difficult to distinguish between these two possibilities within the framework of this or other conventional resonance models.

A model for the reaction $\ensuremath{\pi}+N\ensuremath{\rightarrow}2\ensuremath{\pi}+N$ at low energies, with $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths as parameters, discussed in a previous analysis is extended by removing the assumption that $\frac{{a}_{0}}{{a}_{2}}=\frac{5}{2}$. The parameters are determined by using the ${\ensuremath{\pi}}^{+}$ angular distribution from ${\ensuremath{\pi}}^{\ensuremath{-}}+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}+{\ensuremath{\pi}}^{+}+n$ at 430 Mev and the total cross section for ${\ensuremath{\pi}}^{+}+p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{+}+n$ at 470 Mev. We find two acceptable sets for the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths: ${a}_{0}=0.50$ ${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$, ${a}_{1}=0.07$ ${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$, ${a}_{2}=0.16$ ${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$ and ${a}_{0}=0.65$ ${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$, ${a}_{1}=0.07$ ${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$, ${a}_{2}=\ensuremath{-}0.14$ ${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$. The total cross sections for the various charge channel and for single partial waves are computed. The predictions of this model are in qualitative agreement with experiment.