Angular Momentum Plane Research Articles

Large-angle elastic scattering of α-particles from 20Ne and 22Ne

Resonances observed in the 177° (lab) excitation function of α-particles scattered elastically from 20Ne are investigated in terms of a single Regge pole. The position of the pole in the complex angular momentum plane is determined from angular distributions measured at incident α-particle energies of 25.8 and 27.0 MeV, at which energy values prominent maxima were observed in the excitation function and the pole contribution to these angular distributions is therefore expected to be a maximum. The complete trajectory of the pole over the energy region investigated (24.6 to 31.7 MeV) is obtained by means of extrapolation. The pole trajectory across the surface absorption region of the target nucleus clearly shows the reason for the occurrence of only three large resonances in this energy region, the remaining resonances being attenuated considerably. An angular distribution obtained for 22Ne(α, α) 22Ne at an incident energy of 27.0 MeV also indicates the presence of the Regge pole found for 20Ne at the same energy, even though its strength is severely reduced due to the isotopic dependence effect.

Rainbow scattering in atomic collisions: A Regge pole analysis

The application of Regge pole techniques to rainbow scattering in atomic collisions is investigated. Semiclassical calculations of Regge pole positions and residues are carried out for Lennard-Jones potentials based on the collision systems CH4-Ar, H+-Ar and He-Ne. The residues of maximum modulus have values in the range 1–1013. The larger values are associated with pronounced rainbow maxima in the scattering. Decaying surface waves, propagating around the repulsive core of the potential, are used for the interpretation of rainbow scattering. The background integral is approximated by the direct scattering from the repulsive core. The dark side of a rainbow can be characterized by the Regge poles lying closest to the real axis in the complex angular momentum plane. In favourable cases, such as the differential cross section possessing widely spaced oscillations, a single Regge pole is sufficient to characterize the scattering on both the bright and dark side of the classical rainbow angle.

An Introduction to Regge Theory and High Energy Physics

Originally published in 1977, this book presents an extended introduction to the theory of hadrons, the elementary particles which occur in the atomic nucleus. The main emphasis is on the theory of the complex angular momentum plane 'Regge theory', which has grown from Regge's demonstration in 1959 that it is useful to regard angular momentum as a complex variable when discussing solutions of the Schrodinger equation for non-relativistic potential scattering. This theory helps to classify the many different particles which have been discovered in recent years, to explain the forces between these particles and to predict the results of high-energy scattering experiments. Regge theory thus serves as a unifying concept drawing together many different features of high-energy physics. This monograph is intended primarily for research students just beginning to concern themselves with particle physics, but more experienced workers will also find much to interest them in this detailed survey of the basic ideas and results of Regge theory.

Regge pole character of resonances in sound scattering from elastic bodies

Previous work [R. D. Doolittle, H. Überall, and Peter Ugincuis, J. Acoust. Soc. Am. 43, 1 (1968)] on the scattering of acoustic waves from submerged elastic cylinders has analyzed the creeping-wave contributions to the scattered field, determined from the residues of poles of the scattered amplitude in the complex angular momentum plane. In a recent publication [L. R. Dragonette, L. Flax, and H. Überall, J. Acoust. Soc. Am. (in press)] we have established the connection between the creeping waves and the resonances corresponding to the eigenvibrations of the elastic body. Further studies are reported here which show the analogy of the resonances with the “Regge Poles” of the nuclear-physics literature. The passage of a given resonance may be followed through a series of individual partial waves in the scattering amplitude, where it is seen to move up in frequency. [Supported by the Office of Naval Research, Code 421 (HÜ).]

Regge trajectories for a velocity dependent potential

We study the distribution of the singularities in the complex angular momentum plane of the S matrix for a velocity-dependent potential, and note some deviations with respect to the general behavior established for static potentials. We analyze the physical implications of our results concerning the existence of bound states and resonances.

O 4 symmetry, light-cone expansion and the electroproduction structure function

The electroproduction structure function is examined by formulating the method of the light-cone expansion via theO4 symmetry advocated by Nachtman. The hadron matrix elements of the expansion operators are expressed in terms of theO4 spherical harmonics. The structure functions, for the elementary and composite particles the wave functions of which are given by the Bethe-Salpeter amplitudes, are studied by taking account of the dynamical effects for the superrenormalizable and renormalizable interacting spinless particles. For the composite particles, by considering the high-rank behaviour of the matrix elements, it is shown that the Drell-Yan-West threshold relation holds for the renormalizable interaction if the anomalous dimension is neglected. In the case of the superrenormalizable theory, the relation holds exactly. There exist possible Regge poles in the complex angular-momentum plane ofO4 symmetry for the matrix elements due to the Regge behaviour in the scattering process for the constituent particles. In the case of the superrenormalizable theory, we get the Regge behaviour for the structure function, but, for the renormalizable theory, the structure function may inevitably have the non-Regge behaviour indicated by De Rujulaet al.

Regge behaviour of the imaginary parts of the np charge exchange amplitudes

The experimental data concerning the polarization in the np charge exchange reaction are analysed. The real parts of the amplitudes, which are dominant, were already determined from the differential cross-section and found to behave quite unconventionally (fixed singularities in the complex angular-momentum plane). By starting from these results, the polarization is shown to be essentially due to the product of φ2−φ4=(g2/4π)B, which is practically real, by Im φ5. The energy dependence of Im φ5 is then found to follow very clearly a Regge-pole pattern (breaking of σ-A2 exchange degeneracy), as expected if the «anomalies» in the energy dependence of dσ/dt are attributed to the presence of fixedJ-plane singularities.

Reggeon field theory in zero transverse dimensions

We study Reggeon field theory in zero transverse dimensions. We calculate the renormalization-group functions in closed form and show that there is an infrared-stable fixed point. We also calculate the critical indices exactly. We find that perturbation theory cannot be used directly to calculate the renormalization-group functions in the neighborhood of the fixed point, but that Pade and Borel-Pade approximants can. We prove that the only singularities of the partial-wave amplitude in the angular momentum plane are isolated poles. The intercept of the leading pole approaches unity only in the limit that the bare intercept goes to infinity. (AIP)

Asymptotic freedom, potential scattering, and high-energy limits

The fixed singularities in the complex angular momentum plane of the amplitude are obtained for potentials which correspond to asymptotically free field theories. For potentials of the form -g$sup 2$(r) r$sup -2$, with g$sup 2$(r) decreasing logarithmically for r $Yields$ 0, one obtains an essential singularity with an accumulation of Regge poles. A somewhat faster decrease with the square of the logarithm gives a special branch point. These singularities replace the fixed cuts seen for g$sup 2$(0) > 0, which corresponds to renormalizable theories with a finite fixed point. The connection of our results with field-theory calculations is discussed. They are also considered from the point of view of the general analytic properties of amplitudes satisfying two- particle and multiparticle t-channel unitarity. In asymptotically free gauge theories, similar singularities may occur at j = 1. (AIP)

Quantum-Number Exchanges at Extremely High Energies

We study systematically the high-energy behavior of backward Compton scattering and pair annihilation in massive quantum electrodynamics. For both of these processes (where a fermion is exchanged), we find, in the complex angular-momentum plane, a fixed cut with a simple pole coming out at a physical value of momentum transfer. These results are used to discuss possible behaviors, at extremely high energies, of hadronic processes with the exchange of quantum numbers.

Mandelstam diagrams are not enough to study the high-energy behavior ofφ3theory

Viewed in the crossed channel, the ${\ensuremath{\varphi}}^{3}$ Mandelstam diagrams for the Reggeon-particle cut represent repeated interaction between two of the three exchanged particles. A much larger class of Feynman diagrams is obtained by allowing all three exchanged particles to participate in the interaction, as required by Bose statistics. It is found that, in the limit of high energies, all diagrams of the same order in this larger class contribute comparably. In particular, the contributions from the Mandelstam and non-Mandelstam diagrams are of the same order of magnitude. This larger class of diagrams is summed in terms of an integral equation. It is found from this integral equation that if the momentum transfer is not too large, then the leading singularity in the angular momentum plane is a pole. As the momentum transfer increases this pole eventually disappears into a cut. These considerations are generalized, in a nontrivial way, to the case of the Reggeon-Reggeon cut. The signature factor plays a most important role, and some of the contributing diagrams show unexpected high-energy behavior. The profound effect of these additional diagrams on the program of Reggeon calculus is discussed.

Unitarity in the complex angular momentum plane and the two-Reggeon cut

The results obtained for a two-Reggeon contribution to a scattering amplitude without continuing in the angular momentum are interpreted from the complex angular momentum plane point of view. It is shown that contributions to continued partial-wave amplitudes from two-Reggeon intermediate states can be derived, and agree with previous calculations; however, this leads to a contribution to the full amplitude which has not previously been taken into account. When this new contribution and the correct threshold in the $s$ channel are included, agreement with the result obtained without continuing in the angular momentum is found.

Fixed Regge singularity and small-distance behavior in renormalizable field theories: A model inφ63

We present a detailed analysis of the skeleton ladder graphs for ${\ensuremath{\varphi}}_{6}^{3}$ theory in six dimensions. The ladder graphs are dominated by an energy-independent branch point in the angular momentum plane reminiscent of similar results in ${\ensuremath{\varphi}}^{4}$ theory. This model then serves as a prototype for models in other exactly renormalizable field theories. These branch points are contained explicitly in the anomalous dimensions of the local operators that occur in the Wilson expansion. We contrast our results with some previous results for all internal particles of zero mass. A general renormalization-group-type argument is advanced to determine the conditions for which the complete theory would contain these fixed cuts.

S-matrix representations for potentials with infinite range

Unitary representations for the partial-waveS-matrix in terms of pole trajectories are derived in the complex angular-momentum plane for potentials with infinite range. Different properties of these representations are discussed; their validity is explicitly verified for the Coulomb potential.

Why we do not see fermion parity doublets

Renormalization group arguments are applied to a Reggeon field theory which couples fermion and pomeron trajectories. The bare (input) fermion propagator has both positive and negative parity poles on the physical angular momentum plane for all values of u. However, after interacting with the Pomeron, the negative parity pole of the renormalized (output) fermion propagator migrates onto an unphysical angular momentum sheet for positive u, and so only the positive parity state appears as a physical particle. This provides a dynamical explanation for the disappearance of the negative parity partner, and is a property which is expected to be possessed by most reasonable Reggeon field theories.

Sommerfeld-watson representation for double-spectral functions

We demonstrate the existence of elastically unitary scattering amplitudes, corresponding to a given Born term, when the latter satisfies certain conditions of Holdercontinuity. The results are directly applicable to nonrelativistic potential scattering; and they will also be useful later in connection with the construction of relativistic, crossing-symmetric amplitudes. In this preliminary study, no singularities are allowed in the right half of the complex angular-momentum plane; but it is intended eventually to introduce such singularities (Regge poles and branch-points).

Semiclassical calculation of Regge poles

We have calculated the locations of the Regge poles for an actual interatomic potential by following the semiclassical formulation. For negative energies, this formulation is equivalent to the Bohr-Sommerfeld quantization condition. For positive energies there are three complex turning points; use of linear and parabolic connection formulas yields a semiclassical quantization condition for the poles. The poles are found to lie symmetrically along lines in the first and third quadrants of the angular-momentum plane. The locations of the poles at a given energy and the motion of these poles as the energy changes are presented. Remler has shown that Regge poles provide a convenient way of parametrizing experimental differential cross sections. We discuss the relation between this parametrization and the present results.

Dual pole eikonal ands-channel singularities

Eikonalization of dual pole amplitudes, such as the Veneziano amplitude, is shown to lead to singularities in the impact parameter plane, which, in the eikonal approximation, may be interpreted as branch points in the direct channel angular momentum plane. This result is discussed in the light of dual absorption models.

Regge behavior and fixed-angle scattering

It is shown that, if one expresses the hadronic scattering amplitude as a sum of contributions from individual quark diagrams, each of these quark diagram contributions will obey a logarithmic scaling law in the high-energy fixed-angle limit. The ingredients of the proof are the linear motion of singularities in the complex angular-momentum plane and certain analyticity assumptions. The logarithmic scaling law emerges from the requirement of consistency of the behaviors of the individual quark diagram amplitude in the fixed-angle high-energy regime and in the Regge regime. Each quark diagram is found to have an intrinsic scale set by the slope of the «dominant» singularity in the complex angular-momentum plane of any one of its channels,i.e. the singularity whose trajectory lies highest for large negative values of that channel's Mandelstam variable. Complex angular-momentum plane singularities that are dominant in different channels of the same quark diagram must therefore have the same slope. As the Pomeranchuk singularity and Regge-Regge cuts are dominant in different channels of the same (twisted loop) quark diagram, it follows that they must have the same slope of ∼0.42 (GeV)−2. If one assumes a small number of quark diagrams to dominate the physical scattering amplitude at present energies and fixed angles, the differential cross-section is found also to obey a logarithmic scaling law which agrees with the wide-angle pp scattering data.

J‐Plane Analysis of the Veneziano Amplitude with Nonlinear Trajectories

AbstractThe structure of singularities of the Veneziano amplitude in the complex angular momentum plane is studied. The introduction of a nonlinear term in the trajectory essentially changes the properties of the partial amplitude as compared to the case of linear trajectories. In the J‐plane, besides simple poles lying on the leading and on the conventional daughter trajectories, additional poles lying on mirage trajectories (situated below the leading trajectory) emerge. In the partial amplitude also non‐reggeised poles in the s‐plane emerge whose position is independent of the angular momentum. Such are, for example, the ancestors.