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Progress of Theoretical Physics

Boson-Triboson Scattering with Yamaguchi Potential. I: Restriction to s-Wave Contribution from the 3+1-Subamplitude

Yoshiko Matsui

Published Sep 1, 1999• 1 citation
Publisher

Abstract

For the investigation of the S-wave boson-triboson phase shift, the Faddeev-Osborn equation is solved numerically by assuming a rank-one separable potential of the Yamaguchi type for the two-particle interaction. The calculation is performed in the exact approach based on rigorous Faddeev theory of the scattering problem. As a first step of the investigation, to gain insight into the origin of a four-body resonance, the contribution from the 3+1-subamplitude is limited to the s-wave only. The calculated phase shift behaves generally like a standard two-body one for a potential that causes one loosely bound state; this phase shift has further characteristic behavior resulting from fine structure consisting of a valley and a peak. The result of the partial cross section indicates the presence of two resonant states in the four-boson system with the Yamaguchi potential.

DOI
10.1143/ptp.102.531

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For the investigation of the S-wave boson-triboson phase shift, the Faddeev-Osborn equation is solved numerically by assuming a rank-one separable potential of the Yamaguchi type for the two-particle interaction. The calculation is performed in the exact approach based on rigorous Faddeev theory of the scattering problem. As a first step of the investigation, to gain insight into the origin of a four-body resonance, the contribution from the 3+1-subamplitude is limited to the s-wave only. The calculated phase shift behaves generally like a standard two-body one for a potential that causes one loosely bound state; this phase shift has further characteristic behavior resulting from fine structure consisting of a valley and a peak. The result of the partial cross section indicates the presence of two resonant states in the four-boson system with the Yamaguchi potential.
We consider the three-boson problem with $\ensuremath{\delta}$-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle off a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired SSS-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum-space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the 1+21+21+2 scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pairwise interactions are short range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.
A numerical study of the Faddeev equation for bosons is made with two-body interactions at or close to the Unitary limit. Separable interactions are obtained from phase-shifts defined by scattering length and effective range. In EFT-language this would correspond to NLO. Both ground and Efimov state energies are calculated. For effective ranges r0>0r_0 > 0r0​>0 and rank-1 potentials the total energy ETE_TET​ is found to converge with momentum cut-off ΛΛΛ for Λ>∼10/r0Λ> \sim 10/r_0Λ>∼10/r0​ . In the Unitary limit (1/a=r0=01/a=r_0= 01/a=r0​=0) the energy does however diverge. It is shown (analytically) that in this case ET=EuΛ2E_T=E_uΛ^2ET​=Eu​Λ2. Calculations give Eu=−0.108E_u=-0.108Eu​=−0.108 for the ground state and Eu=−1.×10−4E_u=-1.\times10^{-4}Eu​=−1.×10−4 for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.
A numerical study of the Faddeev equation for bosons is made with two-body interactions at or close to the Unitary limit. Separable interactions are obtained from phase-shifts defined by scattering length and effective range. In EFT-language this would correspond to NLO. Both ground and Efimov state energies are calculated. For effective ranges r0>0r_0 > 0r0​>0 and rank-1 potentials the total energy ETE_TET​ is found to converge with momentum cut-off Λ\LambdaΛ for Λ>∼10/r0\Lambda > \sim 10/r_0Λ>∼10/r0​ . In the Unitary limit (1/a=r0=01/a=r_0= 01/a=r0​=0) the energy does however diverge. It is shown (analytically) that in this case ET=EuΛ2E_T=E_u\Lambda^2ET​=Eu​Λ2. Calculations give Eu=−0.108E_u=-0.108Eu​=−0.108 for the ground state and Eu=−1.×10−4E_u=-1.\times10^{-4}Eu​=−1.×10−4 for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.
A renormalization method that introduces an auxiliary field to represent a three-body bound state is studied in a three-boson system with a triboson field. A cutoff dependence in the threeboson system emerges as a limit cycle, and the cyclic singularity is renormalized by employing two methods: a standard method and the auxiliary field method. For each method, different sets of diagrams are involved for renormalization; thus, we numerically study three quantities: a counter term for renormalization, the scattering length of s-wave boson-diboson scattering, and the normalized wavefunction. We confirm that the two methods lead to the same result.
Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work will also be discussed.With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The 1S and 1P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the 1S-, 1P-, 1D-, and 1F-waves have been calculated.We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths for the 1,3S and 1,3P partial waves and effective ranges for the 1,3S-wave.
The elastic collision between two ortho-positronium (e.g. S=1) atoms is studied using an ab-initio static-exchange model (SEM) in the center of mass (CM) frame considering the system as a four-body Coulomb problem where all the Coulomb interaction terms in the direct and exchange channels are treated exactly. A coupled-channel methodology in momentum space is used to solve Lippman-Schwinger equation following the integral approach. A new SEM code is developed in which the Born-Oppenheimer (BO) scattering amplitude acts as input to derive the SEM amplitude adapting the partial wave analysis. The s-, p- and d- wave elastic phase shifts and the corresponding partial cross sections for the spin alignment S=0, i.e. singlet (+) and S=2, i.e. triplet (-) states are studied. An augmented-Born approximation is used to include the contribution of higher partial waves more accurately to determine the total/integrated elastic cross section ( ), the quenching cross section ( ) and ortho to para conversion ratio ( ). The effective range theory is used to determine the scattering lengths and effective ranges in the s-wave elastic scattering. The theory includes the non-adiabatic short-range effects due to exchange.