Momentum-space Lippmann-Schwinger-Equation,Fourier-transform with Gauss-Expansion-Method

Momentum-space Lippmann-Schwinger-Equation, Fourier-transform with Gauss-Expansion-Method

Abstract

In these notes we construct the momentum-space potentials from configuration-space using for the Fourier-transformation the Gaussian-Expansion-Method (GEM) Hiy03 (). This has the advantage that the Fourier-Bessel integrals can be performed analytically, avoiding possible problems with oscillations in the Bessel-functions for large r, in particular for . The mass parameters in the exponentials of the Gaussian-base functions are fixed using the geometric progression recipe of Hiyama-Kamimura. The fitting of the expansion coefficients is linearly.

Application for nucleon-nucleon is given in detail for the recent extended soft-core model ESC08c. The NN phase shifts obtained by solving the Lippmann-Schwinger equations agree very well with those obtained in configuration space.

pacs:
13.75.Cs, 12.39.Pn, 21.30.+y

I Introduction

In these notes the soft-core momentum-space potentials are constructed from configuration-space using for the Fourier-transformation the Gaussian-Expansion-Method (GEM) Hiy03 (). With gaussian form factors this gives a most natural practical presentation in momentum space. This can be seen, using the Schwinger representation of the meson propagator, as follows

(1)

where the transition to the approximate form can be realized by using appropriate quadratures. The label represents the set , and the real potentials are sums (integrals) over the set for each type i of potential. The types considered are the central, spin-spin, tensor, spin-orbit, and quadratic-spin-orbit potentials, i.e. .

The necessary Fourier-Bessel integrals can be performed analytically, avoiding possible problems with the oscillations in the Bessel-functions for large r, in particular for .

The mass parameters in the exponentials of the Gaussian-base functions are fixed using the geometric progression recipe of Hiyama-Kamimura. The fitting of the remaining parameters, the expansion coefficients , is linearly, requiring only a couple of steps to reach the optimal parameters. In addition to the potential values at a set of distances , we fit the volume integrals.

The soft-core potentials can be evaluated directly in momentum space using the analytic forms of the potentials, see e.g. RPN02 (). However, in the case of the ESC-model also the momentum space potentials require the execution of various types of numerical integrals for each set , where i and j run over the used mesh-points of the quadrature which is used for solving the Lippmann-Schwinger equation, and is rather time consuming. In the method described here we use the configuration potentials, which for ESC are energy independent, as input and fix the GEM-expansion coefficients for each channel, e.g. in the NN-case for I=0,1, or pp, np and nn. From these the momentum space potentials can be computed very efficiently and fast.

In the case of hyperon-nucleon and hyperon-hypron this method can readily be generalized. Then, the coefficients become matrices in channel space. For example for the coupled channels one has for isospin I=1/2 for each coefficient a 2x2-matrix.

The content of these notes is as follows. In section II and III the Lippmann-Schwinger equation, the relation to the partial wave K-matrix, phase shifts, and the used units are given. In section IV the Fourier transform for a general potential form for the central, tensor, spin-orbit, and quadratic-spin-orbit potential is derived in detail. In section V The form factor is icluded in particular the gaussian form factor which is essential for the Nijmegen soft-core potentials. The Gauss-Bessel radial integrals are evaluated in section VI. In section VII the application to the ESC-potentials is given. Here we introduce the explicit form of the used GEM expansion. In section VIII contains the results for the recent ESC08c potential NRY13a (); NRY14a (). We demonstrate the method by giving the phases obtained by solving the Lippmann-Schwinger equation by either the Kowalski-Noyes Noy65 () or the Haftel-Tabakin Haf70 () method. Both methods give essentially the same results. Finally, in section VIII these notes are closed by a brief discussion and concluding remarks. In Appendix A the general expressions for the Gauss-Bessel integral is checked by a second method of evaluation. In Appendix B the general expressions for the Gauss-Bessel integrals are checked by an explicit evaluation for and .

Ii Lippmann-Schwinger equation, NR-normalization

With the non-relativistic normalization of the two-particle states

(2)

the Lippmann-Schwinger equation LSE) reads

(3)

The partial-wave LSE, restricting ourselves to single channel elastic scattering, in the CM-system reads

(4)

Now, the dimensions are , for units where .

Likewise for the K-matrix the partial wave LSE reads

(5)

Transforming

(6)

leads, in the obvious notation , to the LSE

(7)

Iii K-matrix and Phase-shifts

The differential X-section is given by

(8)

which means , and using gives .
In general units the transformation (6) reads

(9)

Then, in general units The LSE for the K-amplitude is momprog ()

(10)

with

(11)

Comparing (8) and the transformation (9), it follows that for the transformed T-matrix we have that . For the partial waves we have

(12)

which implies for the elastic phase-shift

(13)

Iv Fourier-Transform Configuration vice-versa Momentum Space

For establishing the details this is more complicated for the tensor-, spin-orbit-, and quadratic-spin-orbit-potential than for the central-potential. Therefore we give more explicit details of the derivation of the formulas for these potentials in momentum-space by making a Fourier-transform from configuration space.

iv.1 Configuration- and Momentum-space States

The normalization of the states and wave functions we use GW-Sc ()

(14)
(15)
(16)

For particles with momemtum and orbital angular momentum the state, apart from the spin part, is given by

(17)

here, is an eigenstate of the angular momentum operator . From these introductory definitions and normalizations we obtain the following matrix elements

(18)
(19)

The extension of this last matrix element including spin is straightforward. One has

(20)

where

(21)

Here, denotes a particular spin variable of the BB-system, e.g. the helicity, or the transversal spin component, or the projection along the z-axis. For the latter, which will be used here, .

iv.2 Fourier-Transform Central-Potential

The partial-wave matrix elements in momentum space can be related to those in configuration space as follows

(22)

From Bauer’s formula

(23)

we have that

Substitution into (22) leads finally to the desired formula

(24)

iv.3 Fourier-Transform Tensor-Potential

The partial-wave matrix elements in momentum space can be related, analogous to (22) etc., to those in configuration space as follows

(25)

Here, we used partial integration in order to transfer the derivatives to the basic function , and

(26)
(27)

From Bauer’s formula (23) we have that

Substitution into (25) leads to the desired formula

(28)

which relates the configuration space tensor-potential to the momentum space one. Here,

(29)

with only for total spin non-zero matrix elements. We have two cases:

(i) triplet-uncoupled: : .

(ii) triplet-coupled: and : and are ,

(30)

iv.4 Fourier-Transform Spin-Orbit-Potential

The partial-wave matrix elements in momentum space can be related to those in configuration space as follows

(31)

where

(32)

 

Note: Relation partial wave integrals and . The partial wave projection of the spin-orbit potential is

In passing we note that here for , relevant for the spin-orbit, there is no stock term in the partial integration step. Using the recurrence relation, see AS70 () formula (9.1.27),

we get

(33)

This partial wave integral for the spin-orbit is in accordance with RKS94 (), formula (43)-(44).

 

Now, from it follows that

Then, with application of the Bauer formula (23) etc. one arrives at

(34)

Here, we incorporated the spin S and the total angular momentum J, and