Right-hand Cut Research Articles

Singularity Structure of the Double-Regge Vertex in the Nonplanar Veneziano Model

Previously studied models of the double-Regge vertex have exhibited only right-hand cuts in the Toller variable. We show that the nonplanar dual model of Mandelstam has left-hand cuts in the Toller variable. We also use this function to calculate all possible total (planar plus nonplanar) Reggeon-Reggeon-particle vertex functions.

Influence of higher-order cut contributions and of the asymptotic behaviour of the ππ scattering phase shift on the decreasing properties of the isovector nucleon form factors

We investigate the strong decrease of the isovector nucleon form factors in the spacelike region—which empirically can be adequately described by the dipole fit—through a dispersion-theoretical description of the ππ →N\(\bar {\mathcal{N}}\) scattering amplitude in the ρ-resonance region. To that extent we consider on the right-hand cut besides the elastic contribution the influence of the uncorrelated 2π exchange and on the left-hand cut besides the nucleon pole and the isobar exchange the contributions of box diagrams in the sense of a strip approximation in the Mandelstam plane. We find that the electric Sachs form factor can be described completely through the nucleon pole and the uncorrelated 2π exchange, whereas for the magnetic form factor one needs in the region below the ρ resonance a contribution which is about 2 1/2 times the magnitude of the nucleon pole. The decrease in the spacelike region is very sensitive to variations of the ππ scattering phase shift above the ρ-resonance.

Unitarized pion-pion Veneziano partial-wave amplitudes

A unitarization method is investigated which is based on dispersion relations for the inverse amplitude; the zeros of the partial-wave amplitudes are taken from the Veneziano model and the discontinuity across the right-hand cut is given by elastic unitarity, whereas crossing symmetry yields the left-hand cut discontinuity (for large negatives-values the latter is assumed to be constant). Two free subtraction constants are fixed by minimizing the violation of crossing symmetry-expressed in terms of a number of exact conditions on the partial-wave amplitudes. It was not possible to satisfy all of the crossing constraints simultaneously, but we arrived at a unique solution of the dispersion relations (with a very broadI = 0S-wave resonance) which is unitary and approximately crossing symmetric.

Dispersion Calculation of Isoscalar3πandKK¯Electromagnetic Form Factors

Using dispersion theory and the assumption that the three-pion cut is well represented by the $\ensuremath{\rho}\ensuremath{\pi}$ cut, the $I=0$, $J=1 \ensuremath{\rho}\ensuremath{\pi}$ and $K\overline{K}$ scattering amplitudes are calculated from a matrix integral equation, using only right-hand cuts, in an effective-range-type approximation. Subtraction constants are determined by the experimental masses and widths of the isoscalar vector resonances. There are no ghosts (spurious first-sheet poles) in the model. The $\ensuremath{\rho}\ensuremath{\pi}$ and $K\overline{K}$ isoscalar form factors are calculated and compared to the available ${e}^{+}{e}^{\ensuremath{-}}$ colliding-beam data near the $\ensuremath{\omega}$ and $\ensuremath{\varphi}$ masses. The isoscalar charge radii are also computed. The agreement of the model with the experimental values is satisfactory. An estimate $\ensuremath{\Gamma}(\ensuremath{\rho}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}\ensuremath{\gamma})\ensuremath{\simeq}40$ keV is obtained (the decay has not yet been observed).

Dispersion Sum Rules for Fixed-uπNScattering Amplitudes

We review the fixed$\ensuremath{-}u$ finite-energy sum rules for pion-nucleon scattering amplitudes. We also derive and test the continuous-dispersion sum rules for these amplitudes. The real part of the amplitudes is restricted to appear on the right-hand cut (${N}_{\ensuremath{\pi}}\ensuremath{\rightarrow}{N}_{\ensuremath{\pi}}$) only, because no information is available for the left-hand cut ($\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}N\overline{N}$). The main result is an evaluation of the $\ensuremath{\sigma}$, $f$, and $g$ coupling constants.

Three-Channel Model forP33πNScattering

A Static three-channel model is presented for the πNI = J = (3/2) p-wave scattering at low and intermediate energies. The (ρN) and (πN*) states are taken into account as inelastic channels. Only the nucleon-exchange poles in each channel are retained as the most effective nearby singularities, but static ND-1 is solved exactly. Some of these poles exist on the right-hand cut and give very important contributions. We find a very small value for the (πN*N*) coupling constant compared with the SU(6) prediction. It is also discussed that this value is quite consistent with small experimental inelasticity of P33 πN collision in the intermediate energy region.

Finite-Energy Sum Rules and Their Application toπNScattering at Fixedu

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 $\ensuremath{\pi}N$ amplitudes with $u$ fixed at ${(1236\phantom{\rule{0ex}{0ex}}\mathrm{M}\mathrm{e}\mathrm{V})}^{2}$, both sides of the resultant four sum rules being obtained from the properties of the low-energy $\ensuremath{\pi}N$ resonances. Results are presented for three choices of end point: ${(1356\phantom{\rule{0ex}{0ex}}\mathrm{M}\mathrm{e}\mathrm{V})}^{2}$, ${(1808\phantom{\rule{0ex}{0ex}}\mathrm{M}\mathrm{e}\mathrm{V})}^{2}$, and ${(2313\phantom{\rule{0ex}{0ex}}\mathrm{M}\mathrm{e}\mathrm{V})}^{2}$. 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 $u={(1236 \mathrm{MeV})}^{2}$, all the $\ensuremath{\pi}N$ 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 $\ensuremath{\pi}N$ Regge trajectories is accompanied by a degenerate partner of opposite parity.

Constraints on moments of S-wave pion-pion amplitudes resulting from crossing symmetry and positivity properties of absorptive parts

New conditions on moments of S-wave pion-pion amplitudes over the interval between the left- and right-hand cuts are derived from crossing symmetry and positivity properties of the absorptive parts. They are applied to the model of Iliopoulos.

Three-Body Calculations with Local Potentials

A two-term separable approximation to the two-particle $t$ matrix is developed which preserves exact behavior at each bound-state or resonance pole. While introducing no additional singularities, the approximation also produces the exact right-hand cut. This separable approximation is applied to the three-particle scattering problem with local potentials, and numerical results are obtained for the three-body binding energy in the case of the Yukawa and exponential potentials. A prescription is given for increasing the accuracy of this approach by systematically increasing the number of separable terms. This procedure is tested numerically, as is a method for estimating the error in such calculations. Binding-energy results are also presented for a modified pole approximation, and a simple explanation is given for the infinite-binding-energy phenomenon observed by Osborn for separable potentials.

Momentum Transfer Dispersion Relations for Three-Particle Potential Scattering Amplitudes

The analytic properties in momentum transfer of a class of three-particle scattering amplitudes are investigated in this paper. The amplitudes considered are those in which there is a two-particle bound state in both the initial and final state. The following results are obtained: (1) The amplitudes are analytic inside a Lehmann ellipse in the scattering angle for all real energies. (2) For real energies below the three-free-particle threshold, the amplitudes are analytic in the momentum transfer plane except for real left- and right-hand cuts.

Shape of theN*(1236)Resonance

Corrections to the Breit-Wigner shape of the ${N}^{*}(1236)$ resonance are calculated using analyticity and inelastic unitarity incorporating the $N\ensuremath{\pi}$ and ${N}^{*}\ensuremath{\pi}$ channels in a propagator formalism. This method, which includes only a right-hand cut and which evaluates the effect of bubble insertions in the propagator, is motivated by the rigorous results which have been proved for the nucleon. It is argued that even though background and left-hand cuts have been neglected, it is the inclusion of inelasticity that enables the ${P}_{33}$ phase-shift data to be reproduced, even at energies well above resonance. The structureless-vertex decay model with a Breit-Wigner propagator gives an ${N}^{*}$ shape too asymmetric, and inclusion of an inelastic channel with an analytic propagator serves to correct this. Assuming ${N}^{*}\ensuremath{\pi}$ as the only inelastic channel, the ${N}^{*++}{N}^{*++}{\ensuremath{\pi}}^{0}$ coupling is estimated as 170\ifmmode\pm\else\textpm\fi{}50, where the unknown behavior of the vertices far off the mass shell causes the uncertainty. This estimate can be compared with about 75 from relativistic $\mathrm{SU}(6)$, and 136 using Adler-Weissberger techniques. The ${P}_{33}$ partial-wave amplitude constructed on this model has a left-hand pole which simulates the effect of the neglected nucleon-exchange short cut, and which tends to lie too far left and with too large a residue. The application of the method to other resonances and bound states is discussed.

Existence of N and D Matrices

This paper establishes the existence of N and D matrices with the property that the partial-wave T matrix has the form T = ND−1. We consider the case of a finite number of two-body channels and prove that, if T is analytic with right- and left-hand cuts and is suitably bounded, then N and D can be constructed with all the usual properties—namely, N and D have the left- and right-hand cuts, respectively, N is finite at the bound-state poles, and D tends to one as the energy goes to infinity.

General Solution of the One-Channel-Scattering Singular Integral Equations

Integral equations for one-channel scattering are written and solved, starting from dispersion relations for the generalized Jost function in the momentum $k$ plane. This method is an alternative to the conventional $\frac{N}{D}$ method, but it allows a simple, physically meaningful generalization to the many-channel case, where dispersion relations and integral equations can be written for a unique generalized Jost function in the complex plane of a suitable variable which uniformizes all the right-hand cuts of the scattering amplitude. Even in the pure elastic-scattering case, a unified treatment is possible, whether the phase shift at infinity is or is not an integral multiple of $\ensuremath{\pi}$. In all cases, our singular integral equations are reduced to a Fredholm-type integral equation with a Hilbert-Schmidt kernel.

The pionic form factor of the first πN resonanceresonance

Off-shell πN-scattering in the 3/2, 3/2 state is treated by the Omnes-Muskhelishvili method. The left-hand cut is represented by nucleon exchange and by approximate double spectral functions coming from iterations of nucleon exchange, and on the right-hand cut some inelasticity is assumed. The off-shell amplitude is roughly determined by the behaviour of the Born term taken at resonance energy, but is not insensitive to the absorption at high energies. If the form factor is approximated by a single pole in momentum transfer, the relevant mass is 1.5 nucleon masses.

TheN/D method and the diffraction picture—II.

Elastic spin 0-spin 0 scattering is considered and it is argued that, in order to have a simple connection between the total amplitude and the partial-wave amplitude in the high-energy region, the latter has to satisfy the boundary condition corresponding to the « diffraction picture ». If the amplitude can be written in the conventionalN l (ν)/D l (ν) form, whereN l (ν) has only a left-hand cut andD l (ν) only a right-hand cut, and where there are no arbitrary subtraction constants, it follows that the imaginary part ofD l (ν) behaves likeν−1/2 forν→∞ (apart from possible logarithmic factors).

TheN/D method and the diffraction picture

Elastic spin 0-spin 0 scattering is considered and it is argued that, in order to have a simple connection between the total amplitude and the partial-wave amplitude in the high-energy region, the latter has to satisfy the boundary condition corresponding to the « diffraction picture ». If the amplitude can be written in the conventionalN l (ν)/D l (ν) form, whereN l (ν) has only a left-hand cut andD l (ν) only a right-hand cut, and where there are no arbitrary subtraction constants, it follows that the imaginary part ofD l (ν) behaves likeν−1/2 forν→∞ (apart from possible logarithmic factors).

Solutions of singularN/D equations

TheN/D equations are derived for a partial-wave amplitude which increases as rapidly as is allowed by unitarity. The resulting marginally singular integral equation is split into a system of two coupled integral equations: one with a singular and one with a Fredholm kernel. The main purpose of the paper is to construct the explicit resolvent kernel of the singular equation. This is done first in the approximation of neglecting the gap between the left- and right-hand cuts, when Mellin transforms yield an explicit resolvent; and then the resolvent is set up as a quadrature in the realistic case, without this approximation. In both cases the analytic properties in an appropriate eigenparameterλ are studied. The asymptotics inλ and in energy are also investigated. The properties of the solution of the complete equation, including the Fredholm part, are then deduced. Finally, explicit expressions for theN andD functions are given in certain simple cases.

Theory of Low-Energy Pion-Pion Scattering. II

We have extended the previously published pion-pion calculations by solving numerically the coupled $s$-and $p$-wave equations based on a method of the inverse amplitude, to see if the lower partial waves near threshold converge to definite values. Elastic unitarity is still used for the right-hand cut. Numerical calculations show that the $s$-and $p$-wave scattering lengths are very much the same as those obtained before by using only the left-hand singularities chosen to have the correct functional forms near the branch point; however, the position and the width of the $p$-wave resonance energy are substantially improved. Solutions exhibit results as a function of $\ensuremath{\lambda}$ similar to those found previously; a $p$-wave attraction corresponds to a strong $s$-wave attraction. Thus we find that the procedure of constructing the amplitude developed in the earlier paper is convincing.

Approximate Formula for a Regge Trajectory with Right-Hand Cut and Its Application

We derive a general ansatz for the imaginary part of a Regge trajectory, with only a right-hand cut, using conformal mapping and threshold behavior. Next we use the dispersion relation and a three-parameter approximation of this ansatz to obtain an analytical formula for the trajectory. If the Pomeranchuk trajectory has no left-hand cut, then application of this formula shows that ${f}^{0}$ does not lie on this trajectory. Further, we find ${\ensuremath{\alpha}}_{P}(\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty})>\ensuremath{-}1 \mathrm{and} \ensuremath{\gtrsim}0$. Applying the formula to ${f}^{0}$ and $\ensuremath{\rho}$, we conclude that, if our approximation is good, then the Regge trajectories corresponding to these resonances must have left-hand cuts.

The one-particle-exchange approximation and Feynman graphs

It is shown, in a scalar model, that the one-particle-exchange approximation with elastic unitarity on the right-hand cut is not equivalent to summing a series of Feynman graphs. The convergence properties of theN/D integrals are studied in the case of vector-meson exchange between two spinless particles: it appears that, although the interaction is not renormalizable, a single subtraction is sufficient to give finite results at any order in the coupling constant.