# Construction of Non-Perturbative, Unitary Particle-Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms

## Abstract

Starting from a unitary, Lorentz invariant two-particle scattering amplitude, we show how to use an identification and replacement process to construct a unique, unitary particle-antiparticle amplitude. This process differs from conventional on-shell Mandelstam s,t,u crossing in that the input and constructed amplitudes can be off-diagonal and off-energy shell. Further, amplitudes are constructed using the invariant parameters which are appropriate to use as driving terms in the multi-particle, multichannel nonperturbative, cluster decomposable, relativistic scattering equations of the Faddeev-type integral equations recently presented by Alfred, Kwizera, Lindesay and Noyes. It is therefore anticipated that when so employed, the resulting multi-channel solutions will also be unitary. The process preserves the usual particle-antiparticle symmetries. To illustrate this process, we construct a J=0 scattering length model chosen for simplicity. We also exhibit a class of physical models which contain a finite quantum mass parameter and are Lorentz invariant. These are constructed to reduce in the appropriate limits, and with the proper choice of value and sign of the interaction parameter, to the asymptotic solution of the nonrelativistic Coulomb problem, including the forward scattering singularity , the essential singularity in the phase, and the Bohr bound-state spectrum.

- Authors:

- Publication Date:

- Research Org.:
- Stanford Linear Accelerator Center, Menlo Park, CA (US)

- Sponsoring Org.:
- USDOE Office of Energy Research (ER) (US)

- OSTI Identifier:
- 799052

- Report Number(s):
- SLAC-PUB-9156

TRN: US0204374

- DOE Contract Number:
- AC03-76SF00515

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 12 Mar 2002

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ASYMPTOTIC SOLUTIONS; BOUND STATE; INTEGRAL EQUATIONS; SCATTERING AMPLITUDES; SCATTERING LENGTHS; SINGULARITY; ANTIPARTICLES; PARTICLES

### Citation Formats

```
Lindesay, James V.
```*Construction of Non-Perturbative, Unitary Particle-Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms*. United States: N. p., 2002.
Web. doi:10.2172/799052.

```
Lindesay, James V.
```*Construction of Non-Perturbative, Unitary Particle-Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms*. United States. doi:10.2172/799052.

```
Lindesay, James V. Tue .
"Construction of Non-Perturbative, Unitary Particle-Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms". United States.
doi:10.2172/799052. https://www.osti.gov/servlets/purl/799052.
```

```
@article{osti_799052,
```

title = {Construction of Non-Perturbative, Unitary Particle-Antiparticle Amplitudes for Finite Particle Number Scattering Formalisms},

author = {Lindesay, James V},

abstractNote = {Starting from a unitary, Lorentz invariant two-particle scattering amplitude, we show how to use an identification and replacement process to construct a unique, unitary particle-antiparticle amplitude. This process differs from conventional on-shell Mandelstam s,t,u crossing in that the input and constructed amplitudes can be off-diagonal and off-energy shell. Further, amplitudes are constructed using the invariant parameters which are appropriate to use as driving terms in the multi-particle, multichannel nonperturbative, cluster decomposable, relativistic scattering equations of the Faddeev-type integral equations recently presented by Alfred, Kwizera, Lindesay and Noyes. It is therefore anticipated that when so employed, the resulting multi-channel solutions will also be unitary. The process preserves the usual particle-antiparticle symmetries. To illustrate this process, we construct a J=0 scattering length model chosen for simplicity. We also exhibit a class of physical models which contain a finite quantum mass parameter and are Lorentz invariant. These are constructed to reduce in the appropriate limits, and with the proper choice of value and sign of the interaction parameter, to the asymptotic solution of the nonrelativistic Coulomb problem, including the forward scattering singularity , the essential singularity in the phase, and the Bohr bound-state spectrum.},

doi = {10.2172/799052},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Tue Mar 12 00:00:00 EST 2002},

month = {Tue Mar 12 00:00:00 EST 2002}

}