Partial-wave Projections Research Articles

Partial-wave projection of the one-particle exchange in three-body scattering amplitudes

As the study of three-hadron physics from lattice QCD matures, it is necessary to develop proper analysis tools in order to reliably study a variety of phenomena, including resonance spectroscopy and nuclear structure. Reconstructing the three-particle scattering amplitude requires solving integral equations, which can be written in terms of data-constrained dynamical functions and physical on shell quantities. The driving term in these equations is the so-called one-particle exchange, which leads to a kinematic divergence for particles on mass shell. A vital component in defining three-particle amplitudes with definite parity and total angular momentum, which are used in spectroscopic studies, is to project the one-particle exchange into definite partial waves. We present a general procedure to construct exact analytic partial-wave projections of the one-particle exchange contribution for any system composed of three spinless hadrons. Our result allows one full control over the analytic structure of the projection, which we explore for some low-lying partial waves with applications to three pions. Published by the American Physical Society 2024

Nature of the X(6900) in partial wave decomposition of J/ψJ/ψ scattering

In this letter, we perform partial wave decomposition on coupled-channel scattering amplitudes, $J/\psi J/\psi$-$J/\psi \psi(2S)$-$J/\psi \psi(3770)$, to study the resonance appears in these processes. Effective Lagrangians are used to describe the interactions of four charmed vector mesons, and the scattering amplitudes are calculated up to the next-to-leading order. Partial wave projections are performed, and unitarization is implemented by Pad\'e approximation. Then we fit the amplitudes to the $J/\psi J/\psi$ invariant mass spectra measured by LHCb and determine the unknown couplings. The pole parameters of the $X(6900)$ are extracted as $M=6861.0^{+6.3}_{-8.8}$~MeV and $\Gamma=129.0^{+5.6}_{-3.4}$~MeV. Our analysis implies that its quantum number prefers to be $0^{++}$. The pole counting rule and phase shifts show that it is a normal Breit-Wigner resonance and, hence, should be a compact tetraquark.

Dispersive construction of two-loop P→πππ(P=K,η) amplitudes

We present and develop a general dispersive framework allowing us to construct representations of the amplitudes for the processes $P\pi\to\pi\pi$, $P=K,\eta$, valid at the two-loop level in the low-energy expansion. The construction proceeds through a two-step iteration, starting from the tree-level amplitudes and their S and P partial-wave projections. The one-loop amplitudes are obtained for all possible configurations of pion masses. The second iteration is presented in detail in the cases where either all masses of charged and neutral pions are equal, or for the decay into three neutral pions. Issues related to analyticity properties of the amplitudes and of their lowest partial-wave projections are given particular attention. This study is introduced by a brief survey of the situation, for both experimental and theoretical aspects, of the decay modes into three pions of charged and neutral kaons and of the eta meson.

Momentum-space Argonne V18 interaction

This paper gives a momentum-space representation of the Argonne V18 potential as an expansion in products of spin-isospin operators with scalar coefficient functions of the momentum transfer. Two representations of the scalar coefficient functions for the strong part of the interaction are given. One is as an expansion in an orthonormal basis of rational functions and the other as an expansion in Chebyshev polynomials on different intervals. Both provide practical and efficient representations for computing the momentum-space potential that do not require integration or interpolation. Programs based on both expansions are available as supplementary material. Analytic expressions are given for the scalar coefficient functions of the Fourier transform of the electromagnetic part of the Argonne V18. A simple method for computing the partial-wave projections of these interactions from the operator expressions is also given.

High-energy unitarity of gravitation and strings.

It is known that the behavior of a four-point string amplitude at large center-of-mass energy \ensuremath{\surd}s and fixed momentum transfer q= \ensuremath{\surd}-t is not perturbative. We study this region of phase space by summing multiple Reggeized graviton exchange in the eikonal approximation in D space-time dimensions. It is argued that the eikonal sum is at least representative of the summation of the leading powers of s in a string theory. The masslessness and high spin of the (Reggeized) graviton determine the character of the result. For ${\ensuremath{\kappa}}^{2}$${\mathrm{sq}}^{D\mathrm{\ensuremath{-}}4}$\ensuremath{\lesssim}1 (\ensuremath{\kappa} is the gravitational coupling), the eikonal amplitude is dominated by single Reggeized graviton exchange. The amplitude in the region ${\ensuremath{\kappa}}^{2}$${\mathrm{sq}}^{D\mathrm{\ensuremath{-}}4}$\ensuremath{\gg}1 is quite nonperturbative in character: simple Regge behavior and the Froissart bound are violated, and the amplitude does not satisfy a fixed-momentum-transfer dispersion relation. Although order by order the amplitude exhibits in ${q}^{2}$ the exponential decrease of Regge behavior, the final amplitude has only power-law falloff dependent on the number of space-time dimensions but independent of the Regge slope. The unitarity of the partial-wave projections of the eikonal amplitude is also studied. It is demonstrated that for D\ensuremath{\ge}4 noncompact dimensions, the partial-wave amplitudes are bounded as s\ensuremath{\rightarrow}\ensuremath{\infty} only for large values of angular momentum, l\ensuremath{\gtrsim}${x}_{0}$ \ensuremath{\surd}s , where ${x}_{0}$ is the dominant value of the impact parameter. A heuristic argument is presented that the eikonal approximation is successful in unitarizing Reggeized graviton exchange as t/s\ensuremath{\rightarrow}0 in four dimensions but not in higher dimensions.

Quarkonium energy splittings in perturbative QCD.

A $q\overline{q}$ potential model, derived from perturbative QCD and including one- and two-loop contributions, is applied to calculate energy splittings for heavy-quarkonium states. Calculations are performed in momentum space using analytical partial-wave projections. The two-loop contributions give a significant improvement over the first-order results. The results are consistent with experimental data.

Twenty-four optimal inequalities and several new representations for the Coulomb T matrices in momentum space

We derive new series and integral representations for the Coulomb transition matrix in momentum space, 〈p‖Tc‖p′〉, and for its partial-wave projections, 〈p‖Tcl‖p′〉 (l=0,1,...), to be denoted by Tc and Tcl, respectively. We also consider hypergeometric-function representations for Tc and Tcl and discuss their analytic continuation to the whole complex k plane (k2 is the energy). The new integrals are essentially ∫π0 cosh γt (ρ−cos t)−1 dt for Tc and ∫π0 cosh γt ×Ql(uu′+vv′ cos t)dt for Tcl, where γ is Sommerfeld’s parameter and ρ,u,u′,v, and v′ are variables depending on the energy and the momenta; related integrals follow from these. A well-known and convenient series representation for Tc consists essentially of the sum ∑nyn(n2+γ2)−1, where y depends on the energy and the momenta. We derive its analog for Tcl, the corresponding sum being ∑n(n2+γ2)−1Qnl(u) P−nl(u′), 1<u′<u. This sum is a new member of the family of sums of products of Legendre functions that can be evaluated in a relatively simple closed form; other members of this family have been recently obtained by the author. With the new representations for Tc and Tcl we derive a set of twenty-four optimal inequalities (containing two conjectured inequalities) for these Coulomb T matrices, presumably covering all cases relevant for physics. For the proof of these inequalities several different representations, and in particular the newly derived ones, would appear to be indispensable. Many of the inequalities are new. They are valid for fixed real Coulomb strength and fixed real energy≠0. Because of the complexity of exact closed forms for Tc and Tcl, approximations are needed for numerical calculations; the most natural one consists of replacing the Coulomb T matrix by the Coulomb potential. Our inequalities are useful for estimating the accuracy of this approximation.

The partial-wave projected Coulomb T matrix for all l in closed hypergeometric form

We obtain relatively simple closed expressions for the partial-wave projections of the off-shell Coulomb T matrix in the momentum representation, 〈 p‖Tcl(k2)‖p′〉, for all l=0,1,2,⋅⋅⋅ . These exact analytic expressions consist of three parts: (i) ℱl, simple combinations of the Jacobi polynomial P(iγ,−iγ)l and the hypergeometric function 2F1(1,iγ;1+iγ;⋅), with different arguments [γ is Sommerfeld’s parameter; the Coulomb potential is Vc(r)=2kγ/r]; (ii) a polynomial ℰl; and (iii) ℒl=a polynomial times ln[( p+p′)2/( p−p′)2]. The polynomials under (ii) and (iii) are given in terms of the Jacobi polynomials P(m,−m)l , m=0,1,...,l. We derive interesting relations, especially valuable for the theory of off-shell Coulomb scattering, and we present simple closed expressions for the special cases l=0, 1, and 2; p′→k, p′→p, p′→∞, and γ→ 0.

Off-shell CoulombTmatrix in connection with the exact solution of three-particle equations with Coulomb interaction

We investigate the problems connected with the incorporation of the Coulomb interaction into the equations which describe reactions involving charged particles. In particular, the formalism of Vesselova and of Alt, Sandhas, and Ziegelmann is considered. Here various Coulomb quantities play a role which all have a so-called essential singularity in the zero-energy point. This singularity has to be taken care of when the total three-particle energy is near the breakup threshold. We use formulas, both for the case of a repulsive and an attractive Coulomb potential, by means of which the essential singularity is split off in a satisfactory way. This allows one to make really exact calculations for simple models. It is shown that a comparison of such exact calculations with known approximations is indeed necessary. We present and discuss exact calculations of the three-dimensional Coulomb $T$ matrix and of its partial wave projections for all real energies.NUCLEAR REACTIONS Off-shell Coulomb $T$ matrix, three-particle equations, Coulomb effects when two of the particles charged, discussed exact solution for scattering.

Higher multipole polarizabilities of hadrons from compton scattering amplitudes

Starting with partial wave projections of the Compton scattering invariant amplitudes free of kinematical singularities or zeros, generalized electric and magnetic multipole polarizabilities of spin-0 and spin- 1 2 hadrons are introduced as threshold limits of the non-Born parts of the corresponding electric and magnetic Compton multipoles. These objects, which enter the higher low-energy theorems for elastic γ-hadron scattering, are shown to be acceptable quantum field theoretical generalizations of the usual static multipolar polarizabilities encountered in classical electrodynamics or non-relativistic quantum mechanics. Using the analyticity properties of the amplitudes, sum rules for the generalized electric and magnetic quadrupole polarizabilities of the proton and the pions are written down and evaluated numerically within the two-particle unitarity approximation.

Isobar formalism and one-pion-exchange partial-wave cross sections inπN→ππN

This work is a prelude to a phase-shift analysis of $\ensuremath{\pi}N\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}N$ which we are presently performing. We here review the partial-wave isobar formalism for the above process and introduce the notation being used in our analysis. We compute the partial-wave projections of the one-pion-exchange diagrams because we hope to use the high-partial-wave contributions of these diagrams, which are not modified by the interactions, to remove phase ambiguities in the partial waves being varied. A study of the low partial waves, even though they violate unitarity badly, gives modest insights into the results of previous partial-wave analyses.

Coulomb-scattering states and partial-wave decompositions

It is customary to split the physical Coulomb-scattering wave function ψ o (+) into an incoming part φ i and a scattered part φ g : We derive analytic formulae for the partial-wave projections of φ i and φ g and discuss the asymptotic convergence of the corresponding partial-wave series. It is found that the above splitting is unsatisfactory. We propose a different splitting of ψ o (+) into an incoming part and a scattered part which is in close analogy with the standard separation of the scattering wave function for a short-range potential.

Numerical Evaluation of Partial Wave Contributions to Pion–Nucleon Amplitudes

We present detailed numerical evaluations of the partial wave projections of Feynman diagrams of second- and fourth-order in perturbation for the πN–πN scattering in the [Formula: see text] theory. Perturbative contributions to the S, P, and D waves of isospin 1/2 and 3/2 are given in tables of numerical values. Figures regrouping these results show surprising behavior for the ratios Re(4)/Re(2). These tables and figures allow easy calculations with models using low order perturbation terms such as Padé approximants.

New dispersion relations and their application to partial-wave amplitudes

After reviewing the commonly used dispersion relations, a systematic investigation of more generalized dispersion relations on parametrized curves in the Mandelstam plane fors-u crossing-symmetric amplitudes is made with the aim of obtaining dispersion relations which receive contributions from all three channels, however, in such a way that knowledge of the absorptive parts is only required in regions well inside the various Lehmann ellipses. In addition we require that the dispersion relations receive no contributions from kinematic singularities arising from the parametrization and that they allow partial-wave projections to be made in a relatively simple manner. It is found that dispersion relations on hyperbolic curves in the Mandelstam plane are a natural solution of the problem. The dispersion relations are written in a remarkably simple form similar to the usual fixed-t dispersion relation but with an additionalt-channel contribution. As an interesting application, we derive generalized partial-wave dispersion relations for elastic pion-nucleon scattering, where the left-hand cut contribution is explicitly given by convergent partial-wave series in the crossed channels.

Semi-group approach to multiperipheral dynamics

We consider the multiperipheral integral equation at vanishing four-momentum transfer in the form given by Chew and de Tar and we remark that the angular integrations can be interpreted as a convolution of measures defined in a semi-group S contained in the Lorentz group. We study the geometrical properties and a class Banach-space representations of this semi-group. By projection on these representations, we perform a partial wave analysis of the multiperipheral equation. Under some physically very natural conditions, we prove that the projection integrals converge, the partial wave amplitudes are analytic in a half-plane of the complex angular momentum and the kernel of the partial wave equation represents a bounded operator. We give a preliminary discussion of the inversion problem, i.e., of the construction of the amplitude from its partial wave projections.

Duality and Secondary Trajectories

We show analytically that the slope of the dual or output trajectories is equal to the input slope for the partial-wave projections of the Regge amplitude. This result is independent of the special kinematics, and seems not to require a crossing-symmetric amplitude. We find an infinite set of secondary trajectories, and show that their spacing depends on the slope and approaches $\ensuremath{\Delta}s=\frac{2}{{\ensuremath{\alpha}}^{\ensuremath{'}}} {\mathrm{GeV}}^{2}$ for large $s$. The detailed high-energy structure of some Argand diagrams is shown.

Regge poles, Argand diagrams and inelastic resonances

Partial-wave projections of Regge pole amplitudes are discussed, and it is shown that circles in Argand diagrams are freely produced. It is argued that these should not be interpreted as resonances, and hence that the determination of resonances by phase-shift analyses is questionable.

Contribution ofJ≤32Resonances to theu-Channel Exchange Potential inπNScattering

We calculate the contribution of recently found, highly inelastic $\ensuremath{\pi}N$ resonances in the ${S}_{31}$, ${S}_{11}$, ${P}_{11}$, and ${D}_{13}$ partial waves to the exchange $u$-channel generalized potential for $\ensuremath{\pi}N$ scattering. We use the results of the complex phase-shift analyses for incident pion energies \ensuremath{\le}1 BeV joined to smooth asymptotic functions to obtain the absorptive parts of the amplitudes. Dispersion integrals are performed over these amplitudes in the physical $u$ region and then related to the direct $s$ channel by crossing. Finally, partial-wave projections in the direct $s$ channel give the partial-wave generalized-exchange $u$-channel potential input to the $\frac{N}{D}$ equations, which are solved for the phase shifts in the direct channel. The results are as follows: (i) Compared to the exchange of the nucleon and the ${{N}^{*}}_{33}$ (1238 MeV), the ${S}_{11}$, ${S}_{31}$, and ${P}_{11}$ $u$-channel continua have relatively small contributions to the $s$-channel unphysical cut. (ii) As for the ${P}_{33}$ partial wave, the ${D}_{13}$ contribution to the $u$-channel exchange potential is divergent at large $s$. Thus a cutoff is needed to solve the $\frac{N}{D}$ equations. The effect of including ${D}_{13}$ exchange is to make it harder to obtain agreement with experiment. (iii) The detailed shape of the ${P}_{33}$ resonance reduces its contribution to the exchange potential by a factor of 0.75 as compared with the narrow-width approximation.

Some rigorous inequalities satisfied by double-spectral functions

Combining the elastic unitarity condition in the elastic strips of the ππ scattering amplitude and the positiveness of the partialwave projections of the absorptive parts in the crossed channels we derive some rigorous inequalities on the spectral functions. Some applications are considered.

Analytic Structure of Partial-Wave Amplitudes for Production and Decay Processes

We consider a model theory for studying overlapping final-state interaction effects in 3-body production and decay amplitudes. The model is given in terms of dispersion relations similar to those given by Khuri and Treiman for the process $K\ensuremath{\rightarrow}3\ensuremath{\pi}$. We extend the partial-wave projections into the complex plane, and determine their analytic properties, giving explicitly a set of cuts and discontinuities. These consist of the usual right-hand cut with normal discontinuity, together with a "left-hand" cut for which the discontinuity is expressed as an integral over the projections. The right-hand cut can be factored out in the usual way, and thus one can hope to obtain the solution by iteration for the left-hand cut contribution.