Regge Residues Research Articles

Asymptotic Behaviour of Total Cross Sections and Relations between Reduced Regge Residues

Asymptotic Behaviour of Total Cross Sections and Relations between Reduced Regge Residues

The two-Reggeon/particle coupling

A study is made of the two-Reggeon/particle coupling function defined through a production amplitude, with particular reference to its dependence on the Toller angle ω. When one of the Reggeon spins takes a physical value, the coupling function reduces to a polynomial in cos ω. This is understood in several ways: (a) as resulting from the vanishing of non-sense amplitudes in a partial wave decomposition of the production amplitude; (b) in terms of a Fourier transform of the production amplitude; (c) in the perturbation theory; (d) as an extrapolation of the properties of the three-particle coupling that may be deduced from a set of interaction Lagrangians. When the spin of one of the Reggeons in physical the non-vanishing sense amplitudes become equal to the Reggeon/two-particle coupling amplitudes. This is verified in the perturbation theory model for the case of zero spin. The results of the investigation are used to suggest a method of parametrising the two-Reggeon/particle coupling function which allows it to be approximated from a knowledge of ordinary Regge residues. A generalisation of the analysis to the case of parity change at the central vertex is discussed. The implications for non-sense dips in differential cross sections is pointed out. The method of treating external particles with spin is indicated.

Closed Forms for the Scattering Amplitudes and Bootstrap Based on Sum Rules

An infinite class of solutions of the superconvergence equations is constructed with no restrictions on masses and Regge residues. Implications for bootstraps based on sum rules are discussed.Received 4 November 1968DOI:https://doi.org/10.1103/PhysRev.178.2165©1969 American Physical Society

Hypothesis ofM1Dominance in32+-Isobar Production Reactions

The hypothesis that the $\ensuremath{\rho}N\ensuremath{\Delta}$ and ${A}_{2}N\ensuremath{\Delta}$ vertices are predominantly of the magnetic-dipole ($M1$) type is investigated in detail with reference to the ${\frac{3}{2}}^{+}$-isobar production reactions at high energies. The incorporation of this hypothesis for the Regge exchanges of $\ensuremath{\rho}$ and ${A}_{2}$ mesons is greatly facilitated by our choice of Reggeizing the invariant amplitudes, a choice which avoids spurious parametrizations of the unknown Regge residues and which automatically satisfies all the constraint conditions at the normal thresholds and pseudothresholds. The simple model which assumes only $M1$ couplings and $\mathrm{SU}(3)$ symmetry at the Regge vertices is in satisfactory agreement with the available high-energy data on the differential cross sections and density matrix elements of the reactions ${\ensuremath{\pi}}^{+}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}(\ensuremath{\eta}){\ensuremath{\Delta}}^{++}$, ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\Delta}}^{+}$, and ${K}^{+}p\ensuremath{\rightarrow}{K}^{0}{\ensuremath{\Delta}}^{++}$. The presence of a small $E2$ coupling at the vector-meson vertex, in addition to the dominant $M1$ coupling, is shown to provide an improved fit with experiment near the forward direction.

Alternative Constructions of Crossing-Symmetric Amplitudes with Regge Behavior

A representation for the scattering amplitude that contains Regge behavior, crossing symmetry, and analyticity is discussed. It is shown that it provides a different ghost-eliminating mechanism (the Mandelstam-Wang one) from that given by Veneziano's proposal. Furthermore, it does not restrict the external masses, but reduces to Veneziano's formula when $\ensuremath{\alpha}(s)+\ensuremath{\alpha}(t)+\ensuremath{\alpha}(u)$ equals a particular integer that depends on the reaction. Several examples are discussed. The behavior of the Regge residue ${\ensuremath{\beta}}_{\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}}(t)$ at ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(t)=0$ is proposed as a test that distinguishes the two representations.

Superconvergent bootstrap, exchange degeneracy and fixed singularities in the complex J-plane

We consider the dynamical scheme based on the generalized superconvergence relations (GSCR) or the finite energy sum rules (FESR). We call this scheme the superconvergent bootstrap, where crossing is imposed by the GSCR (or FESR). The fundamental approximation is that the scattering amplitude can be approximated by a finite number of Regge poles. The bootstrap scheme investigated here is based on the narrow-resonance approximation which allows one to make the simple and systematic parametrization of the Regge parameters based on analiticity and unitarity. The bootstrap program is then to find a set of mutually consistent Regge trajectories and determine their parameters self-consistently. We carry out the superconvergent bootstrap for the reaction P + P → P + V, where P and V stand for the pseudoscalar and vector-meson octet respectively. We assume exact SU(3) symmetry. Solving the bootstrap equations algebraically, we show that the coupled vector- and tensor-octet trajectories can bootstrap themselves quite satisfactorily. We also obtained the interesting mass formula 3 m V 2 = m T 2 + 3 m P 2, where m M denotes the mass of a M-meson octet and T stands for the tensor-meson octet. The formula is well satisfied experimentally. As was shown by Mandelstam and Wang, the singular behavior of the Regge residues when the trajectories pass through wrong-signature nonsense points is critically dependent on whether or not we have fixed singularities in the Froissart-Gribov partial-wave amplitude at nonsential values of J with the wrong signature. or equivalently, whether we neglect or take into account the contribution of the third double-spectral function of the scattering amplitude. From the bootstrap equations we find the close connection between exchange degeneracy and the above mentioned fixed singularities: in the limit of exchange degeneracy any contribution from the wrong-signature singularities of the Regge residues at nonsense points of α vanishes, while conversely, exchange degeneracy comes out when we require the vanishing of the above contribution. This statement is proved quite generally based on the crossing properties of the scattering amplitudes. We also applied the dynamical scheme to the reaction P + P → P + ω, where ω stands for the vector-meson singlet. This reaction is dominated by only the vector-octet trajectory in each channel. Solving the bootstrap equation the single vector-octet trajectory is also shown to bootstrap itself quite satisfactorily.

Universal Coupling of theωMeson and Regge Residues

In this paper we consider some of the experimental data bearing on the hypothesis of universality of $\ensuremath{\omega}$ coupling. The low-energy data on the coupling of the $\ensuremath{\omega}$ meson and the high-energy data on the coupling of the $\ensuremath{\omega}$-type Regge trajectory have been considered. The data seem to support the universality hypothesis. The universal $\ensuremath{\omega}$ coupling constant $\frac{{{g}_{\ensuremath{\omega}}}^{2}}{4\ensuremath{\pi}}$ is estimated to be 4.96\ifmmode\pm\else\textpm\fi{}1.24, which is higher than the universal $\ensuremath{\rho}$ coupling constant $\frac{{{g}_{\ensuremath{\rho}}}^{2}}{4\ensuremath{\pi}}=2.60\ifmmode\pm\else\textpm\fi{}0.68$, where ${g}_{\ensuremath{\rho}}$ and ${g}_{\ensuremath{\omega}}$ are so normalized that ${g}_{\ensuremath{\omega}}={g}_{\ensuremath{\rho}}$ in the $\mathrm{SU}(6)$ limit. To relate the Regge residues of vector trajectories to the universal coupling constant, we factor out the angular momentum dependence of the former and introduce a "reduced" Regge residue and a scale parameter for each trajectory. The scale parameters for the $\ensuremath{\omega}$ and the $\ensuremath{\rho}$ trajectories turn out to be of the order of 290 and 186 MeV, respectively.

Veneziano Formula with Trajectories Spaced by Two Units

By representing a scattering amplitude as a sum of terms of the Veneziano type, we can cancel alternate trajectories and remain with trajectories spaced by two units of angular momentum. This result is obtained without imposing a supplementary condition and without introducing poles in the Regge residues at nonsense wrong-signature integers.

Factorization and Constraints in Regge-Pole Theory

The behavior near zero momentum transfer of the Regge residues for a two-particle reaction is determined by means of analyticity and factorization. It is shown that, at least in all cases of present interest, the solution obtained satisfies the kinematical constraints provided that a classification and certain relations between the trajectories involved (identical to those obtained by group-theoretical methods) are imposed.

Factorisation, kinematic factors of the Regge residue function and conspiracy

When helicity amplitudes are Reggeised, the kinematic singularities and zeros of the helicity amplitudes are not generally compatible with the factorisation theorem for the Regge residue function. We find the simplest (but not unique) class of solutions for the extra zeros which must be added to give factorisation. The residue function for a t-channel helicity amplitude may be written βΛμ( t= = β Λμ(t), where WΛμ( t) is a kinematic function. To satisfy the factorisation theorem we must be able to write WΛμ( t)= VΛ( t) Vμ( t). Our main result is that near t = 0 the vertex function VΛ( t) is proportional to t q ̂ where, for unequal masses, q ̂ = −1 2 α+ 1 2 |M−|Λ|| and, for equal masses (each with spin s), q ̂ = 1 4 [1–(−1) Λ+M] if M ⩾ 2s and q ̂ = 1 4 [1–([t-1) Λ+2s+ 1 2 if M ≤ 2s , where M is a non-negative parameter (integer for boson trajectories, half-integer for fermion trajectories), identified with a quantum number characterising the Toller pole from which the Regge pole at complex angular momentum α arises. The general results for VΛ( t) at thresholds and pseudothresholds, as well as at t = 0, are given. As a subsidiary result we show that the analytic structure of helicity amplitudes at thresholds, and at pseudothresholds for unequal masses, follow from non-relativistic arguments without study of the Trueman and Wick crossing relations. The results are applied, as an example, to the pion conspiracy problem, and some earlier results of Le Bellac are rederived.

Regge-Pole Formalism for Pionic Disintegration of Deuteron

We investigate the reaction ${\ensuremath{\pi}}^{+}+d\ensuremath{\rightarrow}p+p$ within the framework of Regge-pole theory. Reggeization of the amplitudes for this process is carried out in detail, and the asymptotic behavior of the helicity amplitudes at zero momentum transfer is discussed. By assuming that the lowest anomalous threshold singularity in the $t$ channel dominates in the nearby region, we calculate the nucleon-deuteron Regge residue in terms of the pion-nucleon Regge residue. We also discuss the relations to the Regge-pole formalism of various low-energy models used to study this reaction and its inverse.

Families of regge trajectories at nonvanishing energy.—II

A bound-state formalism for a broken trajectory generating algebra is developed. At nonzero energy the «wave function» of a Regge state is represented as a reducible representation of the zero-energy trajectory generating algebra. Perturbation formulas in the total energy for families of Regge trajectories and residues are obtained, their algebraic structure is analyzed. A family of trajectories is phenomenologically parametrized in the Chew-Frautschi approximation by three or four parameters.

Modified Dispersion Relations andππScattering

The $\ensuremath{\pi}\ensuremath{\pi}$ $S$-wave scattering-length predictions of Weinberg have been tested by using dispersion sum rules for the infinite-energy cross section. Reasonable agreement is obtained with the infinite-energy cross section ($\ensuremath{\approx}15$ mb) estimated from the factorization theorem for the Pomeranchon Regge residues. Experimental phase-shift data of Gutay et al., Walker et al., and Baton et al. are used in estimating the dispersion integrals. The analysis seems to rule out an $I=0$ $S$-wave scattering length $\ensuremath{\gtrsim}0.4$${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$.

Phase Contours of Scattering Amplitudes. II. Crossing Symmetry, Resonance Poles, and High-Energy Behavior

Phase contours are used to study consistency conditions for a symmetric scattering amplitude having a given high-energy behavior. The latter is taken to correspond to dominance by Regge poles that have continuously rising trajectories. By means of crossing symmetry we deduce the existence of a sequence of real zeros of the amplitude that lie along lines of symmetry in our model, $s=u$, in $t<0$, for example. The leading zero in this sequence is related to the scattering length. Under quite general conditions these zeros may be related, via complex surfaces of zeros, to the zeros of Regge residues below threshold. By considering complex sections of phase-contour surfaces, we show that these zeros may also be related to a sequence of zeros at complex points that are due to interference between resonance poles on unphysical sheets above threshold.

Kinematic Singularities and Threshold Relations for Helicity Amplitudes

The kinematic singularities of two-body helicity amplitudes at thresholds and the concomitant relations among these amplitudes are discussed in a direct and elementary way, without recourse to the singularity structure of the crossing matrix. The tools are those of nonrelativistic quantum mechanics, as befits a situation where $p\ensuremath{\rightarrow}0$, with spins combined into channel spins S and Russell-Saunders coupling of L+S=J. The kinematic singularities are shown to follow from a mismatch between $J$ and $L$ for each term in the partial-wave series. The method is applicable at pseudothresholds ${({m}_{1}\ensuremath{-}{m}_{2})}^{2}$ as well as normal thresholds ${({m}_{1}+{m}_{2})}^{2}$ with two formal changes involving an intrinsic parity and a helicity-dependent phase. The relations among the different helicity amplitudes at the thresholds are shown to result from the presence at threshold of fewer Russell-Saunders amplitudes than there are independent helicity amplitudes. The use of invariant amplitudes is shown to be an alternative which automatically yields the kinematic singularities and also the threshold relations among the helicity amplitudes. A discussion is given of dynamical exceptions to the threshold constraints, resulting from less singular than standard behavior at a threshold. The threshold relations are important constraints on the amplitudes, and must be satisfied by any realistic model. In the use of $t$-channel amplitudes for peripheral processes in the $s$ channel, the explicit imposition of all the relations at the $t$-channel thresholds is necessary in order to assure a differential cross section without spurious, polelike singularities in $t$ whose variation could in some circumstances completely control the $t$ dependence. The reactions $\ensuremath{\pi}N\ensuremath{\rightarrow}\mathrm{KY}$ and $\ensuremath{\pi}N\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\Delta}$ are used as illustrations. The latter process is especially illuminating because its $t$-channel amplitudes have a pole (rather than a simple inverse-square-root singularity) at the $\overline{N}\ensuremath{\Delta}$ pseudothreshold, $t=0.09 {(\frac{\mathrm{GeV}}{c})}^{2}$. The proximity of this point to the physical region of the $s$ channel means that the threshold relations there are of crucial importance. The consequences of these constraints on the cross-section and decay density matrix of the $\ensuremath{\Delta}$ are discussed within the framework of the Regge-pole model. Comparison with experiment implies that the dynamics make the amplitudes for $\ensuremath{\pi}{\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\overline{N}\ensuremath{\Delta}$ have less than the standard kinematic singularity at $\overline{N}\ensuremath{\Delta}$ pseudothreshold and so avoid almost all the threshold constraints. Examples are cited from the literature where use of Regge-pole formulas possessing the spurious kinematic factors has led to incorrect inferences concerning the dynamic behavior of Regge residues.

Feynman-Diagram Models of Regge and Multi-Regge Couplings

Some Feynman-diagram models are presented which lead to a natural framework for discussions of the kinematic properties of Regge poles. The models do not require an infinite number of recurrences of a trajectory. The basic property of the Feynman prescriptions that is useful here is the fact that the analyticity properties of the amplitude are preserved at all stages of the calculation. The coupling of particles with integer spin to Regge trajectories is discussed in detail. These models allow one to consider various aspects of ghose-killing mechanisms and to clarify the kinematic properties, especially the kinematic singularities, of Regge residues. The extension of the discussion to the coupling of two Regge trajectories to a physical particle is carried out and applied to the multiple-Regge model of production amplitudes. Some experimental consequences of these models are briefly explored for particle production.

Interpretation of Recurring Minima in Elastic Scattering at Large Momentum Transfers

Maxima and minima recently reported in forward elastic scattering at moderately high energy and large momentum transfer are correlated with exceptional points $\ensuremath{\alpha}=0, \ensuremath{-}1, \ensuremath{-}2, \ensuremath{\cdots}$ on the ${P}^{\ensuremath{'}}$ and $\ensuremath{\omega}$ Regge trajectories. An Ansatz for the Regge residue functions is given that reproduces the qualitative features of the new data. This interpretation leads to further predictions that can be experimentally checked.

BackwardπNCharge-Exchange Scattering and Pole Extrapolations of Baryon Regge-Exchange Amplitudes

High-energy ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ backward elastic-scattering data are analyzed in terms of ${N}_{\ensuremath{\alpha}}$ and ${\ensuremath{\Delta}}_{\ensuremath{\delta}}$ Regge exchanges. Using the signs of the Regge residues found from extrapolations to the particle poles (which agree with the signs obtained from interference with direct-channel resonances), we predict the ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}n$ cross section near 180\ifmmode^\circ\else\textdegree\fi{}.

Rising Trajectories, Falling Residues, and Experiment

We attempt here a limited experimental test of a recent speculation that the Regge residue function, $\ensuremath{\beta}(s)$, shows an exponential or faster decrease with $\ensuremath{\alpha}(s)$ as $s\ensuremath{\rightarrow}\ensuremath{\infty}$ to insure polynomial boundeness for the total amplitude.

Strip Approximation and the Shape of the Diffraction Peak

It is pointed out that the sharpness of the diffraction peak is incompatible with the usual justification of the strip approximation to the Mandelstam representation. Discrepancies between the Regge residue functions needed to fit experiment and those obtained from dynamical calculations using the new form of the strip approximation are discussed. It is shown that if the correct shape of the diffraction peak is to be reproduced there are special requirements as to the form of the force---that it oscillate in sign within the strip---which are not satisfied by the Born approximation to that force. We indicate how the correct expression for the force may rectify this, and try to discover what difference having the right sort of residues would make in the slope of the trajectories in $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering, by imposing the experimental shape on the residues artificially. The trajectory slopes are found to be improved, but the drastic nature of the change produced by imposing the experimental form on the output residues makes a bootstrap impossible if only the Born approximation to the force is used.