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

 ˜Vi,α(k2) = g2 e−k2/Λ2k2+m2=g2 ∫∞0dα e−α(k2+m2) e−k2/Λ2 = ¯g2 ∫∞1/Λ2dγ e−γm2 e−γ k2,  ¯g2=g2 exp(m2Λ2), Vi,α(r) = ¯g22π√π ∫2Λ0dμ e−μ2r2 ≈∑kw(i)α,k e−μ2kr2, (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

 (p′,s′1;p′2,s′2|p1,s1;p2,s2)=(2π)6δ(p′1−p1)δ(p′2−p2)δs′1,s1δs′2,s2, (2)

 (3,4|T|1,2)=(3,4|V|1,2)+∑n∫d3pn(2πℏ)3(3,4|V|n1,n2)2μn1,n2p2E−p2n+iϵ(n1,n2|T|1,2). (3)

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

 (pf|TJ|pi)=(pf|VJ|pi)+12π2∫∞0dpn p2n (pf|VJ|pn)2μredp2E−p2n+iϵ(pn|TJ|pi) (4)

Now, the dimensions are , for units where .

Likewise for the K-matrix the partial wave LSE reads

 (pf|KJ|pi)=(pf|VJ|pi)+12π2P∫∞0dpn p2n (pf|VJ|pn)2μredp2E−p2n(pn|KJ|pi) (5)

Transforming

 ˜KJ≡14π√2mredKJ √2mred (6)

leads, in the obvious notation , to the LSE

 ˜KJ(pf,pi)=˜VJ(pf,pi)+2πP∫∞0dpn p2n ˜VJ(pf,pn)1p2E−p2n˜KJ(pn,pi) (7)

## Iii K-matrix and Phase-shifts

The differential X-section is given by

 dσdΩ=|F|2,  F=−2μredc24π(ℏc)2 T, (8)

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

 ˜KJ≡14π√2mredc2(ℏc) KJ √2mredc2(ℏc). (9)

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

 ˜KJ(pf,pi) = ˜VJ(pf,pi)+2πP∫∞0d(pnc)(ℏc) ˜VJ(pf,pn)(pnc)2(pEc)2−(pnc)2˜KJ(pn,pi) (10) ≡ ˜VJ(pf,pi)+P∫∞0d(pnc) ˜VJ(pf,pn) ˜gE(pn) ˜KJ(pn,pi),

with

 ˜gE(pn) = +2π(ℏc)−1(pnc)2(p2Ec)−(pnc)2,  tanδ=−pcℏc ˜K(pE,pE). (11)

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

 FJ(p)=−˜TJ=−14π√2mredc2(ℏc) TJ √2mredc2(ℏc), (12)

which implies for the elastic phase-shift

 tanδJ=−pcℏc˜KJ=−2mredc24π(ℏc)2 pcℏc KJ. (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 ()

 ⟨r′|r⟩ = δ3(r′−r), (14) ⟨p′|p⟩ = (2π)3δ3(p′−p), (15) ⟨r|p⟩ = exp(ip⋅r). (16)

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

 |plm⟩=∫dΩ^p Ylm(^p)|p⟩,  |p⟩=∑lm Y∗lm(^p)|plm⟩. (17)

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

 ⟨r|plm⟩ = ∫dΩ^p Ylm(^p)⋅4π∑l′m′il′jl′(kr)Y∗l′m′(^p)Yl′m′(^r) (18) = 4πiljl(kr)Ylm(^r), ⟨p′l′m′|p⟩ = ∫dΩ^p′ Y∗l′m′(^p′)(2π)3δ3(p′−p) (19) = (2π)3δ(p′−p)p2 Y∗l′m′(^p′).

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

 ⟨p′s′|p,LSJM⟩ = (2π)3δ(p′−p)p2 YLSJM(^p′,s′), (20)

where

 YLSJM(^p′,s′) = ∑m,μCJ L SM m μYLm(^p′)χ(S)μ(s′). (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

 ⟨pfLfmf|˜VC(k2)|piLimi⟩=∫d3p′(2π)3∫d3p(2π)3⟨pfLfmf|p′⟩⟨p′|VC|p⟩⟨p|piLimi⟩= ∫dΩ^pf ∫dΩ^pi Y∗Lfmf(^pf)YLimi(^pi)⋅[∫d3r e−ipf⋅r VC(r) e+ipi⋅r]. (22)

From Bauer’s formula

 exp(ip⋅r)=4π∑l,mil jl(pr) Y∗lm(^p)Ylm(^r) (23)

we have that

 ∫dΩ^p YLimi(^p)e+ip⋅r = 4πi−LiYLimi(^r) jLi(pr), ∫dΩ^p′Y∗Lfmf(^p′)e−ip′⋅r = 4πi+LfY∗Lfmf(^r) jLf(pr).

Substitution into (22) leads finally to the desired formula

 ⟨pfLfmf|˜VC(k2)|piLimi⟩=+(4π)2iLf−Li⋅ ×∫d3r [Y∗Lfmf(^r) YLimi(^r)]⋅[jLf(pfr) VC(r) jLi(pir)]⇒ +(4π)2δLf,Liδmf,mi ∫∞0r2dr jLf(pfr)VC(r) jLi(pir). (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

 ⟨pfLfmf|(\boldmathσ1⋅k \boldmathσ2⋅k−13k2% \boldmathσ1⋅\boldmathσ2) ˜V3(k2)|piLimi⟩= ∫d3p′(2π)3d3p(2π)3⟨pfLfmf|p′⟩⟨p′|(\boldmathσ1⋅k \boldmathσ2⋅k−13k2\boldmathσ1⋅\boldmathσ2) ˜V3(k2)|p⟩⟨p|piLimi⟩⇒ −∫dΩ^pf ∫dΩ^pi Y∗Lfmf(^pf)YLimi(^pi)⋅ ×[∫d3r e+ik⋅r (% \boldmathσ1⋅\boldmath∇ \boldmathσ2⋅\boldmath∇−13\boldmathσ1⋅\boldmathσ2 \boldmath∇2) V3(r)]= −∫dΩ^pf ∫dΩ^pi Y∗Lfmf(^pf)YLimi(^pi)⋅ ×[∫d3r e+ik⋅r (% \boldmathσ1⋅^r \boldmathσ2⋅^r−13\boldmathσ1⋅\boldmathσ2) (1rddr−d2dr2)V3(r)]|piLimi⟩≡ −∫dΩ^pf ∫dΩ^pi Y∗Lfmf(^pf)YLimi(^pi)⋅[∫d3r e−ip′⋅r S12(^r) VT(r) e+ip⋅r]|piLimi⟩. (25)

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

 S12(^r) = 3\boldmathσ1⋅^r\boldmathσ2⋅^r−\boldmathσ1⋅\boldmathσ2, (26) VT(r) = −13(1rddr−d2dr2) V3(r). (27)

From Bauer’s formula (23) we have that

 ∫dΩ^p YLimi(^p)e+ip⋅r = 4πi−LiYLimi(^r) jLi(pr), ∫dΩ^p′Y∗Lfmf(^p′)e−ip′⋅r = 4πi+LfY∗Lfmf(^r) jLf(pr).

Substitution into (25) leads to the desired formula

 ⟨pfLfmf|(\boldmathσ1⋅k \boldmathσ2⋅k−13k2% \boldmathσ1⋅\boldmathσ2) ˜V3(k2)|piLimi⟩= −(4π)2iLf−Li∫d3r [Y∗Lfmf(^r) S12(^r) YLimi(^r)]⋅[jLf(pfr) VT(r) jL1(pir)]⇒ −(4π)2iLf−LiδMf,Mi (JMf,SfLf||S12||JMi,LiSi)⋅ ×∫∞0r2dr [jLf(pfr)VT(r) jLi(pir)], (28)

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

 ∫dΩ^r [Y∗J Mf;LfSf(^r) S12(^r) YJ Mi;LiSi(^r)]≡(J,Lf||S12||J,Li)δSf,Si δMf,Mi, (29)

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

(i) triplet-uncoupled: : .

(ii) triplet-coupled: and : and are ,

 ⟨S12⟩=(Lf||S12||Li)=12J+1(−2J+26√J(J+1)6√J(J+1)−2J−4) (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

 ⟨pfLfmf|i2(\boldmathσ1+\boldmathσ2)⋅n ˜V0(k2)|piLimi⟩=∫d3p′(2π)3∫d3p(2π)3⋅ ×⟨pf,Lfmf|p′⟩⟨p′|i2(\boldmathσ1+\boldmathσ2)⋅(q×k) ˜V0(k2)|p⟩⟨p|piLimi⟩⇒ −∫dΩ^p′∫dΩ^p Y∗Lfmf(^p′)YLimi(^p)⋅ ×[∫d3r e+ik⋅r12(\boldmathσ1+\boldmathσ2)⋅(q×\boldmath∇) V0(r)]= −∫dΩ^p′∫dΩ^p Y∗Lfmf(^p′)YLimi(^p)⋅[∫d3r e+ik⋅r S⋅(r×q) VSO(r)] −∫dΩ^p′∫dΩ^p Y∗Lfmf(^p′)YLimi(^p)⋅[∫d3r e+ip′⋅r L⋅S VSO(r) e−ip⋅r], (31)

where

 L=12r×(p′+p),  and  VSO(r)=−1rddr V0(r). (32)

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

 Jn,n = ∫∞0r2dr jn(pfr) VSO(r) jn(pir)=∫∞0r2dr jn(pfr) [1rV0(r)dr] jn(pir) = π2√pfpi∫∞0dr Jn+1/2(pfr) [V0(r)dr] Jn+1/2(pir) = −π2√pfpi∫∞0dr ddr[Jn+1/2(pfr) Jn+1/2(pir)] V0(r) = −π4√pfpi∫∞0dr {AApiJn+1/2(pfr)[AAJn−1/2(pir)−Jn+3/2(pir)] AA+pf[AAJn−1/2(pfr)−Jn+3/2(pfr)] Jn+1/2(pir)} V0(r).

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),

 2νzJν = Jν−1(z)+Jν+1(z),

we get

 Jn,n = −π4√pfpi (2n+1)−1 ∫∞0rdr {AA[AAJn−1/2(pfr)+Jn+3/2(pfr)] [AAJn−1/2(pir)−Jn+3/2(pir)] (33) AA+[AAJn−1/2(pfr)−Jn+3/2(pfr)] [AAJn−1/2(pir)+Jn+3/2(pir)]} = −π2√pfpi (2n+1)−1 ∫∞0rdr {AAJn−1/2(pfr) Jn−1/2(pir)−Jn+3/2(pfr) Jn+3/2(pir)} V0(r) ≡ −(pfpi)⋅(2n+1)−1 [AAIn−1/2,n−1/2−In+3/2,n+3/2].

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

 ⟨pfLfmf|i2(\boldmathσ1+\boldmathσ2)⋅n ˜VSO(k2)|piLimi⟩=−(4π)2iLf−Li ×∫d3r [Y∗Lfmf(^r) L⋅S YLimi(^r)] jLf(pfr)VSO(r)jLi(pir)⇒−(4π)2δLf,Li δMf,Mi⋅ ×(LfSf;JMf||L⋅S||LiSi;JMi) ∫∞0r2dr jLf(pfr)VSO(r) jLi(pir). (34)

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

 ∫dΩ^r [Y∗LfSf,JMf(^r)