Momentum-space Lippmann-Schwinger-Equation,
Fourier-transform with Gauss-Expansion-Method
Th. A. Rijken
arXiv:1409.5593v1 [nucl-th] 19 Sep 2014
Institute for Mathematics, Astrophysics and Particle Physics, University of Nijmegen
(Dated: version of: September 22, 2014)
In these notes we construct the momentum-space potentials from configuration-space using for the Fouriertransformation the Gaussian-Expansion-Method (GEM) [1]. 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 pf 6= pi . 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 numbers: 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 Fouriertransformation the Gaussian-Expansion-Method (GEM) [1]. 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
Z∞
−k2 /Λ2
2
2
2
2
2
2 e
2
e
Vi,α (k ) = g 2
=g
dα e−α(k +m ) e−k /Λ
2
k +m
0
2
Z∞
2
m
2
−γm
−γ k2
2
2
,
= ḡ
dγ e
e
, ḡ = g exp
Λ2
1/Λ2
Z 2Λ
X (i)
2 2
2 2
ḡ2
√
Vi,α (r) =
dµ e−µ r ≈
wα,k e−µk r ,
2π π 0
k
(1.1)
where the transition to the approximate form can be realized by using appropriate quadratures. The label α represents the set
(g, m, Λ), and the real potentials are sums (integrals) over the set {α} for each type i of potential. The types considered are the
central, spin-spin, tensor, spin-orbit, and quadratic-spin-orbit potentials, i.e. i = C, σ, T, SO, Q.
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 pf 6= pi .
The mass parameters in the exponentials of the Gaussian-base functions are fixed using the geometric progression recipe of
(i)
Hiyama-Kamimura. The fitting of the remaining parameters, the expansion coefficients wα,k , is linearly, requiring only a couple
of steps to reach the optimal parameters. In addition to the potential values at a set of distances {rn , n = 1, N}, 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. [2].
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 (pi , pj ), 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 wi become
matrices in channel space. For example for the coupled channels ΛN, ΣN 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 [3, 4]. We demonstrate the method
2
by giving the phases obtained by solving the Lippmann-Schwinger equation by either the Kowalski-Noyes [5] or the HaftelTabakin [6] 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 In,n+2 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 I0,0 and I0,2 .
II. LIPPMANN-SCHWINGER EQUATION, NR-NORMALIZATION
With the non-relativistic normalization of the two-particle states
(p ′ , s1′ ; p2′ , s2′ |p1 , s1 ; p2 , s2 ) = (2π)6 δ(p1′ − p1 )δ(p2′ − p2 )δs1′ ,s1 δs2′ ,s2 ,
(2.1)
the Lippmann-Schwinger equation LSE) reads
(3, 4|T |1, 2) = (3, 4|V|1, 2) +
X Z d3 pn
2µn1 ,n2
(3, 4|V|n1 , n2 ) 2
(n1 , n2 |T |1, 2).
3
2
(2πh̄)
p
E − pn + iǫ
n
The partial-wave LSE, restricting ourselves to single channel elastic scattering, in the CM-system reads
Z
1 ∞
2µred
(pn |T J |pi )
(pf |T J |pi ) = (pf |V J |pi ) + 2
dpn p2n (pf |V J |pn ) 2
2π 0
pE − p2n + iǫ
Now, the dimensions are V J = T J = [MeV]−2 , for units where h̄ = c = 1.
(2.2)
(2.3)
Likewise for the K-matrix the partial wave LSE reads
1
(pf |K |pi ) = (pf |V |pi ) + 2 P
2π
J
J
Z∞
0
dpn p2n (pf |V J |pn )
2µred
(pn |KJ |pi )
p2E − p2n
(2.4)
Transforming
p
p
e J ≡ 1 2mred KJ 2mred
K
4π
e J (pf , pi ) ≡ (pf |K
e J |pi ), to the LSE
leads, in the obvious notation K
Z∞
1
e J (pf , pi ) = V
e J (pf , pi ) + 2 P
e J (pn , pi )
e J (pf , pn )
K
K
dpn p2n V
2
π 0
pE − p2n
(2.5)
(2.6)
III. K-MATRIX AND PHASE-SHIFTS
The differential X-section is given by
dσ
2µred c2
T,
= |F|2 , F = −
dΩ
4π(h̄c)2
which means [F] = [fm], and using [h̄c] = [MeV].[fm] gives [T ] = [V] = [MeV].[fm]3 .
In general units the transformation (2.5) reads
p
p
2
1
2m
c
2mred c2
red
J
J
e ≡
K
K
.
4π
(h̄c)
(h̄c)
Then, in general units The LSE for the K-amplitude is [7]
Z∞
(pn c)2
d(pn c) e J
2
J
J
e J (pn , pi )
e
e
V (pf , pn )
K
K (pf , pi ) = V (pf , pi ) + P
π 0 (h̄c)
(pE c)2 − (pn c)2
Z∞
J
e
e J (pf , pn ) g
e J (pn , pi ),
eE (pn ) K
≡ V (pf , pi ) + P
d(pn c) V
0
(3.1)
(3.2)
(3.3)
3
with
(pn c)2
2
pc e
eE (pn ) = + (h̄c)−1 2
K(pE , pE ).
g
, tan δ = −
2
π
h̄c
(pE c) − (pn c)
(3.4)
Comparing (3.1) and the transformation (3.2), it follows that for the transformed T-matrix we have that Te = −F. For the
partial waves we have
p
p
2
1
2m
c
2mred c2
red
J
J
J
F (p) = −Te = −
T
,
(3.5)
4π
(h̄c)
(h̄c)
which implies for the elastic phase-shift
tan δJ = −
IV.
2mred c2 pc J
pc e J
K =−
K .
h̄c
4π(h̄c)2 h̄c
(3.6)
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 momentumspace by making a Fourier-transform from configuration space.
A. Configuration- and Momentum-space States
The normalization of the states and wave functions we use [8]
hr ′ |ri = δ3 (r ′ − r),
hp ′ |pi = (2π)3 δ3 (p ′ − p),
hr|pi = exp (ip · r) .
For particles with momemtum p and orbital angular momentum l the state, apart from the spin part, is given by
Z
X
∗
|plmi = dΩ^p Ylm (^
p)|pi, |pi =
Ylm
(^
p)|plmi.
(4.1)
(4.2)
(4.3)
(4.4)
lm
here, |plmi is an eigenstate of the angular momentum operator L. From these introductory definitions and normalizations we
obtain the following matrix elements
Z
X ′
hr|plmi = dΩ^p Ylm (^
p) · 4π
p)Yl ′ m ′ (^r)
il jl ′ (kr)Yl∗′ m ′ (^
l ′m ′
l
= 4πi jl (kr)Ylm (^r),
Z
hp ′ l ′ m ′ |pi = dΩ^p ′ Yl∗′ m ′ (^
p ′ )(2π)3 δ3 (p ′ − p)
= (2π)3
δ(p ′ − p) ∗
Yl ′ m ′ (^
p ′ ).
p2
(4.5)
(4.6)
The extension of this last matrix element including spin is straightforward. One has
hp ′ s ′ |p, LSJMi = (2π)3
δ(p ′ − p)
YLSJM (^
p ′ , s ′ ),
p2
(4.7)
where
YLSJM (^
p ′, s ′) =
X
S
′
CJMLm
p ′ )χ(S)
µ (s ).
µ YLm (^
(4.8)
m,µ
Here, s ′ 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, s ′ = µ.
4
B. Fourier-Transform Central-Potential
The partial-wave matrix elements in momentum space can be related to those in configuration space as follows
Z 3 ′ Z 3
d p
eC (k2 )|pi Li mi i = d p
hpf Lf mf |p ′ ihp ′ |VC |pihp|pi Li mi i =
hpf Lf mf |V
(2π)3 (2π)3
Z
Z
Z
dΩ^pf dΩ^pi YL∗f mf (^
pf )YLi mi (^
pi ) · d3 r e−ipf ·r VC (r) e+ipi ·r .
(4.9)
From Bauer’s formula
exp(ip · r) = 4π
X
∗
il jl (pr) Ylm
(^
p)Ylm (^r)
(4.10)
l,m
we have that
Z
Z
dΩ^p YLi mi (^
p)e+ip·r = 4πi−Li YLi mi (^r) jLi (pr),
p ′ )e−ip
dΩ^p ′ YL∗f mf (^
′
·r
= 4πi+Lf YL∗f mf (^r) jLf (pr).
Substitution into (4.9) leads finally to the desired formula
eC (k2 )|pi Li mi i = +(4π)2 iLf −Li ·
hpf Lf mf |V
Z
× d3 r YL∗f mf (^r) YLi mi (^r) · [jLf (pf r) VC (r) jLi (pi r)] ⇒
Z∞
+(4π)2 δLf ,Li δmf ,mi
r2 dr jLf (pf r)VC (r) jLi (pi r).
(4.11)
0
C. Fourier-Transform Tensor-Potential
The partial-wave matrix elements in momentum space can be related, analogous to (4.9) etc., to those in configuration space
as follows
1
e3 (k2 )|pi Li mi i =
hpf Lf mf | σ1 · k σ2 · k − k2 σ1 · σ2 V
3
Z 3 ′ 3
1 2
d p d p
′
′
e3 (k2 )|pihp|pi Li mi i ⇒
hpf Lf mf |p ihp | σ1 · k σ2 · k − k σ1 · σ2 V
(2π)3 (2π)3
3
Z
Z
− dΩ^pf dΩ^pi YL∗f mf (^
pf )YLi mi (^
pi ) ·
Z
1
2
3
+ik·r
V3 (r) =
× d re
σ1 · ∇ σ2 · ∇ − σ1 · σ2 ∇
3
Z
Z
− dΩ^pf dΩ^pi YL∗f mf (^
pf )YLi mi (^
pi ) ·
Z
1
1 d
d2
3
+ik·r
× d re
σ1 · ^r σ2 · ^r − σ1 · σ2
V3 (r) |pi Li mi i ≡
−
3
r dr dr2
Z
Z
Z
∗
3
−ip ′ ·r
+ip·r
− dΩ^pf dΩ^pi YLf mf (^
pf )YLi mi (^
pi ) · d r e
|pi Li mi i.
(4.12)
S12 (^r) VT (r) e
Here, we used partial integration in order to transfer the derivatives to the basic function V0 (r), and
S12 (^r) = 3σ1 · ^rσ2 · ^r − σ1 · σ2 ,
d2
1 1 d
− 2 V3 (r).
VT (r) = −
3 r dr dr
(4.13)
(4.14)
5
From Bauer’s formula (4.10) we have that
Z
dΩ^p YLi mi (^
p)e+ip·r = 4πi−Li YLi mi (^r) jLi (pr),
Z
′
p ′ )e−ip ·r = 4πi+Lf YL∗f mf (^r) jLf (pr).
dΩ^p ′ YL∗f mf (^
Substitution into (4.12) leads to the desired formula
1 2
e3 (k2 )|pi Li mi i =
hpf Lf mf | σ1 · k σ2 · k − k σ1 · σ2 V
3
Z
−(4π)2 iLf −Li d3 r YL∗f mf (^r) S12 (^r) YLi mi (^r) · [jLf (pf r) VT (r) jL1 (pi r)] ⇒
−(4π)2 iLf −Li δMf ,Mi (JMf , Sf Lf ||S12 ||JMi , Li Si ) ·
Z∞
r2 dr [jLf (pf r)VT (r) jLi (pi r)] ,
×
(4.15)
0
which relates the configuration space tensor-potential to the momentum space one. Here,
Z
dΩ^r YJ∗ Mf ;Lf Sf (^r) S12 (^r) YJ Mi ;Li Si (^r) ≡ (J, Lf ||S12 ||J, Li ) δSf ,Si δMf ,Mi ,
(4.16)
with only for total spin Sf = Si = 1 non-zero matrix elements. We have two cases:
(i) triplet-uncoupled: Lf = Li = J: hS12 i = (J||S12 ||J) = 2.
(ii) triplet-coupled: Lf = J ± 1 and Li = J ± 1: Lf and Li are L ± 1,
p
1
−2J + 2 6 J(J + 1)
p
hS12 i = (Lf ||S12 ||Li ) =
2J + 1 6 J(J + 1) −2J − 4
(4.17)
D. Fourier-Transform Spin-Orbit-Potential
The partial-wave matrix elements in momentum space can be related to those in configuration space as follows
Z 3 ′ Z 3
d p
d p
i
2
e
·
hpf Lf mf | (σ1 + σ2 ) · n V0 (k )|pi Li mi i =
3
2
(2π)
(2π)3
i
e0 (k2 )|pihp|pi Li mi i ⇒
×hpf , Lf mf |p ′ ihp ′ | (σ1 + σ2 ) · (q × k) V
2
Z
Z
− dΩ^p ′ dΩ^p YL∗f mf (^
p ′ )YLi mi (^
p) ·
Z
1
× d3 r e+ik·r (σ1 + σ2 ) · (q × ∇) V0 (r) =
2
Z
Z
Z
− dΩ^p ′ dΩ^p YL∗f mf (^
p ′ )YLi mi (^
p) · d3 r e+ik·r S · (r × q) VSO (r)
Z
Z
Z
′
− dΩ^p ′ dΩ^p YL∗f mf (^
p ′ )YLi mi (^
p) · d3 r e+ip ·r L · S VSO (r) e−ip·r ,
(4.18)
where
L=
1
1 d
r × (p ′ + p), and VSO (r) = −
V0 (r).
2
r dr
(4.19)
6
Note: Relation partial wave integrals VSO and V0 . The partial wave projection of the spin-orbit potential is
Z∞
Z∞
1 V0 (r)
2
2
Jn,n =
r dr jn (pf r) VSO (r) jn (pi r) =
r dr jn (pf r)
jn (pi r)
r dr
0
0
Z∞
V0 (r)
π
Jn+1/2 (pi r)
dr Jn+1/2 (pf r)
= √
2 pf pi 0
dr
Z∞
π
d Jn+1/2 (pf r) Jn+1/2 (pi r) V0 (r)
= − √
dr
2 pf pi 0
dr
Z∞
π
= − √
dr pi Jn+1/2 (pf r) Jn−1/2 (pi r) − Jn+3/2 (pi r)
4 pf pi 0
+ pf Jn−1/2 (pf r) − Jn+3/2 (pf r) Jn+1/2 (pi r) V0 (r).
In passing we note that here for n ≥ 1, relevant for the spin-orbit, there is no stock term in the partial integration step. Using the
recurrence relation, see [9] formula (9.1.27),
2ν
Jν = Jν−1 (z) + Jν+1 (z),
z
we get
Z∞
π√
Jn,n = −
pf pi (2n + 1)−1
rdr
Jn−1/2 (pf r) + Jn+3/2 (pf r) Jn−1/2 (pi r) − Jn+3/2 (pi r)
4
0
+ Jn−1/2 (pf r) − Jn+3/2 (pf r) Jn−1/2 (pi r) + Jn+3/2 (pi r)
Z∞
π√
pf pi (2n + 1)−1
= −
rdr Jn−1/2 (pf r) Jn−1/2 (pi r) − Jn+3/2 (pf r) Jn+3/2 (pi r) V0 (r)
2
0
−1
In−1/2,n−1/2 − In+3/2,n+3/2 .
(4.20)
≡ −(pf pi ) · (2n + 1)
This partial wave integral for the spin-orbit is in accordance with [10], formula (43)-(44).
Now, from q = (p ′ + p)/2 it follows that
Z
Z
′
′
d3 r e+ip ·r S · (r × q) VSO (r) e−ip·r =
d3 r e+ip ·r L · S VSO (r) e−ip·r .
Then, with application of the Bauer formula (4.10) etc. one arrives at
i
eSO (k2 )|pi Li mi i = −(4π)2 iLf −Li
hpf Lf mf | (σ1 + σ2 ) · n V
2
Z
× d3 r YL∗f mf (^r) L · S YLi mi (^r) jLf (pf r)VSO (r)jLi (pi r) ⇒ −(4π)2 δLf ,Li δMf ,Mi ·
Z∞
× (Lf Sf ; JMf ||L · S||Li Si ; JMi )
r2 dr jLf (pf r)VSO (r) jLi (pi r).
(4.21)
0
Here, we incorporated the spin S and the total angular momentum J, and
Z
dΩ^r YL∗f Sf ,JMf (^r) (L · S) YLi Si ,JMi (^r) ≡ (Lf Sf , J||L · S||Li Si , J) δMf ,Mi ,
(4.22)
with for total spin S=1 and angular momentum J, the matrix elements are non-zero. We have two cases:
(i) triplet-uncoupled: Lf = Li = J: hL · Si = (J||L · S||J) = −1.
(ii) triplet-coupled: Lf = J ± 1 and Li = J ± 1: Lf and Li are L ± 1, the spin-orbit is diagonal in L:
J−1
0
hL · Si = (Lf Sf J||L · S||Li Si J) =
0 −(J + 2)
(4.23)
7
E. Fourier-Transform Quadratic-Spin-Orbit-Potential
The partial-wave matrix elements in momentum space can be related to those in configuration space as follows
Z 3 ′ Z 3
d p
eQ (k2 )|pi Li mi i = d p
·
hpf Lf mf | σ1 · (q × k) σ2 · (q × k) V
3
(2π)
(2π)3
eQ (k2 )|pihp|pi Li mi i ⇒
×hpf Lf mf |bfp ′ ihp ′ | σ1 · (q × k) σ2 · (q × k) V
Z
Z
− dΩ^p ′ dΩ^p YL∗f mf (^
p ′ )YLi mi (^
p) ·
Z
3
+ik·r
× d re
σ1 · (q × ∇) σ2 · (q × ∇) V0 (r) .
(4.24)
Now, using for F = F(r) the identity
1 ′
1 ′ xl xn
′′
∇l ∇n F(r) = F δln + F − F
,
r
r
r2
one gets
1 ′
σ1 · (q × ∇) σ2 · (q × ∇) = σ1 · σ2 q − σ1 · q σ2 · q
F (r)
r
1
1 ′
1
′′
(σ1 · q × r) (σ2 · q × r) + (σ2 · q × r) (σ1 · q × r) · 2 F − F .
+
2
r
r
2
Neglecting the first non-local term, we arrive at
eQ (k2 )|pi Li mi i
hpf Lf mf | σ1 · (q × k) σ2 · (q × k) V
Z
Z
Z
∗
′
3
−ip ′ ·r
+ip·r
≈ − dΩ^p ′ dΩ^p YLf mf (^
p )YLi mi (^
p) · d r e
.
VQ (r) Q12 (^r)e
(4.25)
Here,
1
(σ1 · L)(σ2 · L) + (σ2 · L)(σ1 · L) ,
Q12 (^r) =
2
2
= 2 (L · S) + L · S − L2 ,
2
3
1
d
1 d
VQ (r) = − 2 VT (r) = − 2
V0 (r).
−
r
r
dr2
r dr
Substitution into (4.25) leads again to the desired formula
eQ (k2 )|pi Li mi i =
hpf Lf mf | σ1 · (q × k) σ2 · (q × k) V
Z
2 Lf −Li
+(4π) i
d3 r YL∗f mf (^r) Q12 (^r) YLi mi (^r) VQ (r) ⇒
Z∞
+(4π)2 iLf −Li δmf ,mi (Lf ||Q12 ||Li )
r2 dr jLf (pf r)VQ (r) jLi (pi r).
(4.26)
(4.27)
(4.28)
0
Here,
Z
dΩ^r YL∗f mf (^r) Q12 (^r) YLi mi (^r) ≡ (Lf ||Q12 ||Li ) δmf ,mi ,
We have three cases:
(i) spin singlet: Lf = Li = J: hQ12 i = (J||Q12 ||J) = −J(J + 1).
(4.29)
8
(ii) spin triplet-uncoupled: Lf = Li = J: hQ12 i = (J||Q12 ||J) = 1 − J(J + 1).
(ii) spin triplet-coupled: Lf = J ± 1 and Li = J ± 1: Lf and Li are L ± 1,
(J − 1)2
0
hQ12 i = (Lf ||Q12 ||Li ) =
0
(J + 2)2
(4.30)
F. Non-local Potential
We note that the q2 -terms contain a non-local and a local part as seen from
1
q2 = q2 + k2 − k2 /4.
4
(4.31)
We note that the second term is contained in the local part of the configuration space potential. Therefore, only the first term
(. . . ) has to be included in addition to the local potentials, which corresponds to the φC and φσ functions.
(i) Central Potential: The non-local factor q2 + k2 /4 = (p2f + p2i )/2 is angle independent and therefore the partial wave
expansion is very similar to that for the local potentials. Analogous to (4.11), we have
enl (k2 )|pi Li mi i = +(4π)2 δL ,L δS ,S δm ,m ·
hpf Lf mf | q2 + k2 /4 V
f
i
f
i
f
i
Z∞
1 2
r2 dr jLf (pf r)Vnl (r) jLi (pi r),
(4.32)
× pf + p2i
2
0
where
Vnl (r) = [φC (r) + (4S − 3) φσ (r)] /(2mred ),
(4.33)
with Sf = Si = S, and 4S − 3 = 2S(S + 1) − 3 for S=0,1.
(ii) Tensor Potential: In ESC-models pseudoscalar exchange include the so-called ’Graz-correction’
σ1 · k σ2 · k
f2NNπ 1
2
2
2
2
exp
−k
/Λ
q
+
k
/4
2
m2π 2MN
k2 + m2
2
2
enl,t (k2 ).
≡ q + k /4 (σ1 · k)(σ2 · k) V
e graz =
∆V
PS
(4.34)
Separation of the spin-spin and the tensor part we get
1. Spin-spin: As in (4.32)
1
enl,t (k2 )|pi Li mi i = +(4π)2 δL ,L δS ,S δm ,m ·
(4S − 3)hpf Lf mf | q2 + k2 /4 k2 V
f
i
f
i
f
i
3
2
Z∞
2
1
2 d
d
× − (4S − 3) p2f + p2i
Vnl,t (r) jLi (pi r).
+
r2 dr jLf (pf r)
2
6
dr
r dr2
0
(4.35)
In the soft-core potentials we have that Vnl,t ∝ φ0C (r), which implies that −∇2 Vnl,t ∝ m2 φ1C (r).
2. Tensor: Now the matrix element is
hpf Lf mf |(q + k /4) σ1 · k σ2 · k −
1 2
2
p + pi hpf Lf mf | σ1 · k σ2 · k −
2 f
2
2
which gives, using the result (4.15),
hpf Lf mf | q2 + k2 /4
1 2
k σ1 · σ2
3
1 2
k σ1 · σ2
3
1
σ1 · k σ2 · k − k2 σ1 · σ2
3
enl,t (k2 )|pi Li mi i =
V
enl,t (k2 )|pi Li mi i,
V
(4.36)
enl,t (k2 )|pi Li mi i =
V
1 2
pf + p2i ·
−(4π)2 iLf −Li δMf ,Mi (JMf , Sf Lf ||S12 ||JMi , Li Si ) ·
2
Z∞
1 d2
1 d
2
Vnl,t (r) jLi (pi r) .
×
r dr jLf (pf r)
−
3 dr2
r dr
0
(4.37)
9
fJ : With an extra (non-local) factor q2 + k2 /4 = (p2 + p2 )/2 the integral-equation does not become
3. Integral equation for K
f
i
a non-fredholm integral equation. The Gaussian form factor gives enough damping for the off-shell potential matrix elements
for guaranteeing Fredholm properties! (See e.g. thesis P. Verhoeven [11]).
V.
YUKAWIAN POTENTIALS WITH CUT-OFF
Gaussian expansion: The Feynman/Yukawa propagator in the Schwinger parametrization
2
2
e−k /Λ
e 2) = g
e2
e2 2
= g
V(k
k + m2
Z∞
2
2
2
2
dα e−α(k +m ) e−k /Λ
0
2 Z∞
2
2
m
2
e exp
= g
dγ e−γm e−γ k .
Λ2
2
1/Λ
In configuration space this gives
VG (r) =
=
2
2
e2
g
√ em /Λ
8π π
2
2
e2
g
√ em /Λ
2π π
(5.1)
Z∞
dγ −r2 /(4γ
with γ = 1/4µ2 →
√ e
1/Λ2 γ γ
Z 2Λ
X
2 2
2 2
dµ e−µ r ≈
wi e−µi r ,
0
(5.2)
i
i.e. a quadrature approximation as a sum of gaussians, with 0 < µi < 2Λ.
It appears from (5.1) and (5.2) that a numerical fit for the ESC potentials with GEM (gaussians) is very natural and superior
over a fit with exponentials!.
VI.
GAUSS-BESSEL RADIAL INTEGRALS
In this section we evaluate the Double-Bessel transform of the potential V(r). These are the radial integrals
Z∞
IL ′ ,L (a, b) =
r2 dr V(r) jL ′ (ar)jL (br),
(6.1)
0
where a = qf and b = qi . Using the standard numerical quadratures gives problems for large momenta, and in particularly for
the far off-energy-shell matrix elements hqf , L ′ |V|qi , Li. Here we describe the method based on the expansion
V(r) =
N
X
k=1
Ak rn exp −µ2k r2 ,
(6.2)
for general L ′ , L, using the closed analytical expression for the integrals (6.1) given in [12]. For the expansion of the potential in
gaussians the partial-wave Namely, the basic integrals are
Z∞
1
β2 + γ 2
βγ
−ρ2 x2
xdx e
Jp (βx) Jp (γx) = 2 exp −
Ip
2ρ
4ρ2
2ρ2
0
1
(6.3)
Re p > −1, |argρ| < , α > 0, β > 0 .
4
10
A. Gaussian Expansion Central Potentials VT (r)
Application of (6.3) gives
In,n (a, b; µ ) =
where fn (X) =
Z∞
2 2
r2 dr e−µ r jn (ar) jn (br)
0
Z∞
2 2
π
rdr e−µ r Jn+ 1 (ar) Jn+ 1 (br)
= √
2
2
2 ab 0
2
2
π
ab
a +b
√ exp −
=
In+1/2
4µ2
2µ2
4µ2 ab
2
√
2
ab
a +b
π
fn
,
exp −
=
3
2
4µ
4µ
2µ2
2
(6.4)
p
π/2X In+1/2 (X), see [9], section (10.2). The recurrence relations read, [9] 10.2.18-10.2.20,
fn−1 (X) − fn+1 (X) = (2n + 1) fn (X)/X,
nfn−1 (X) + (n + 1)fn+1 (X) = (2n + 1) fn′ (X),
(n + 1)fn (X)/X + fn′ (X) = fn−1 (X),
−nfn (X)/X + fn′ (X) = fn+1 (X).
(6.5)
Gaussian Expansion Tensor potentials VT (r) (I)
B.
The tensor-potential in configuration space is given by [13]
Z∞
dk k4 j2 (kr) V3 (k2 ) → r2 for r → 0.
VT (r) = −(1/6π2 )
(6.6)
0
In the case of the soft-core potentials the integral in (6.6) exists for all 0 < r < ∞.
Now,
1 d2
1 d
VT (r) =
V0 (r),
−
3 dr2
r dr
and making the GEM expansion
V0 (r) =
N
X
k=1
ak exp −µ2k r2 ,
we have for the tensor potential the logical expansion
VT (r) =
N
4X
(ak µ2k ) (µk r)2 exp −µ2k r2 .
3
k=1
1. For the diagonal matrix elements hL||VT ||Li we have the following integral (L=n):
2
J(2)
n,n (a, b; µ )
=
Z∞
0
i
h
d
2 −µ2 r2
jn (ar) jn (br) = − 2 In,n (a, b; µ2 )
r dr r e
dµ
2
(6.7)
Performing the derivative gives
√
2
π
a 2 + b2
a + b2
′
3
−
f
(X)
+
2Xf
(X)
exp
−
n
n
8µ5
4µ2
2µ2
2
√
π
a + b2
a 2 + b2
fn (X)
= − 5 exp −
8µ
4µ2
2µ2
2n + 3
2n − 1
−X
fn−1 (X) +
fn+1 (X) ,
2n + 1
2n + 1
2
J(2)
n,n (a, b; µ ) =
(6.8)
11
J
J
where X = (ab)/(2µ2 ). Notice that the expression (6.8) is in complete analogy with the expressions for V1,1
(P) and V3,3
(P),
with n = J − 1 and n = J + 1 respectively, of [10] on p. 13. The connection is given by the substitutions
q
q
a
b
(T )
(6.9)
(q2f + q2i ) sin ψ = qf → √ ,
(q2f + q2i ) cos ψ = qi → √ , VJ → fn (X),
µ 2
µ 2
and
√
2
8π
a + b2
π
→
exp
−
.
3
8µ5
4µ2
2. For the non-diagonal matrix elements hL||VT ||L + 2i we first consider the following integral (L=n):
(2)
Jn,n+2 (a, b; µ2 )
Z∞
i
h
2 2
r2 dr r2 e−µ r jn (ar) jn+2 (br)
0
Z∞
2 2
π
√
=
r3 dr e−µ r Jn+ 1 (ar) Jn+ 5 (br).
2
2
2 ab 0
=
Now,
Jn+5/2 (br) = (2n + 3)
(2n + 1)
1 1 d
− 2
2b2 r2
r b db
Jn+1/2 (br) − Jn+1/2 (br),
(6.10)
(6.11)
which gives in (6.10)
√
Z∞
2 2
2 ab (2)
(2n + 1) 1 d
2
−
J
(a, b; µ ) = (2n + 3)
rdr e−µ r Jn+ 1 (ar) Jn+ 1 (br)
2
2
π n,n+2
2b2
b db 0
Z∞
2 2
d
rdr e−µ r Jn+ 1 (ar) Jn+ 1 (br)
+
2
2
dµ2
0
leading, using
1 1
√
=
b bdb
1
1
+
2b2
bdb
1
√ ,
b
to the expression
(2)
2n
d
1 d
(2n + 3)
+
In,n (a, b; µ2 )
−
2b2 b db
dµ2
√
a 2 + b2
π (2n + 3)
(2)
2
·
exp −
= −Jn,n (a, b, µ ) +
8 (µ3 b2 )
4µ2
b2
ab
× 2n + 2 fn (X) − 2 fn′ (X) .
µ
µ
Jn,n+2 (a, b; µ2 ) =
(6.12)
Now, using the recurrences (6.5),
ab ′
µ2
µ2
′
′
2nfn (X) − 2 fn (X) = 2 2X fn (X) − 2Xfn+1 (X) − 2X fn (X) =
b2
µ
b
2
2
a2 1
2X
µ
′
′
fn (X) − fn+1 (X) = − 2
fn (X) − fn+1 (X) ,
− 2 ·
b n+2
2µ n + 2
which leads to the expression
√
2
a + b2
π
(2)
(2)
Jn,n+2 (a, b; µ2 ) = −Jn,n
(a, b; µ2 ) + 5 (2n + 3) exp −
·
8µ
4µ2
a2 1
′
× fn (X) − 2
fn (X) − fn+1
(X)
2µ n + 2
2
2
√
a + b2
a + b2 2n + 3 a2
π
fn (X)
exp
−
−
=
8µ5
4µ2
2µ2
n + 2 2µ2
ab
2n + 3 a2 ′
− 2 fn+1 (X) +
f
(X) .
µ
n + 2 2µ2 n+1
(6.13)
12
′
where again X = (ab)/(2µ2 ). Now, using fn+1
= fn − (n + 2)fn+1 /X we get for the expression [.....] in (6.13)
ab
a2 1
a 2 + b2
f
−
f
−
(2n
+
3)
fn+1 =
... =
n
n+1
2µ2
µ2
2µ2 X
ab
a2
b2
f
−
f
+
fn+2 ,
n
n+1
2µ2
µ2
2µ2
and
(2)
Jn,n+2 (a, b; µ2 )
2
√
2
a + b2
b
ab
a2
π
=
exp −
fn (X) − 2 fn+1 (X) + 2 fn+2 (X) .
8µ5
4µ2
2µ2
µ
2µ
(6.14)
J
The analog with the expression for V3,1
(P) of [10], eqn. (55), is, using the substitutions (6.9),
√
2
2
a + b2
ab
a2
b
π
(2)
2
Jj−1,j+1 (a, b; µ ) =
exp −
fj (X) −
fj−1 (X) + 2 fj+1 (X) .
8µ5
4µ2
µ2
2µ2
2µ
(6.15)
So, apart from a (-)-sign, which is included elsewhere, (6.14) and (6.15) are in agreement for n = j − 1. compare with [10],
Eqn.’s (38)-(41).
The low momentum behavior of the r.h.s. in (6.14) is given by fn (X) ∼ Xn /(2n + 1)!!, and we get
√
2
π
b2
Xn
a + b2
(2)
Jn,n+2 (a, b; µ2 ) ∼
· 2
exp
−
·
5
2
8µ
4µ
2µ (2n + 5)!!
a4
a2
× (2n + 5)(2n + 3) − (2n + 5) 2 + 4 + ...
µ
4µ
(6.16)
VII. APPLICATION TO ESC-POTENTIALS
A. Set of Gaussians with geometric progression
Analogous to the works of Kamimura, Hiyama, and Kino [1] we expand the potentials Vi (r), with i = C, σ, T, SO, SO2 etc.
Vi (r) =
nX
max
2
G
l −νn r
ai,nl φG
, with
n,l (r), φn,l (r) = Nnl r e
n=1
Nnl =
2l+2 (2νn )l+3/2
√
π(2l + 1)!!
1/2
,
(n = 1 − nmax ),
(7.1)
G
where the constant Nnl normalizes the functions, i.e. hφG
n,l |φn,l i = 1. It is shown by Kamimura, Hiyama, and Kino [1] that an
expansion with high accuracy can be realized using a set of Gaussian range parameters in geometric progression as follows:
νn =
1
, rn = r1 an−1 (n = 1 − nmax ).
r2n
(7.2)
In the application to the ESC-potentials we use√
for the central, spin-spin, spin-orbit l=0. For the tensor potentials we use l=2.
The mass parameters in φG
νn , are displayed in Table I. The maximum mass µmax = 10 h̄c ≈ 2Λ. The
n,l , defined as µn =
radii rn (n = 1, nmax ) in (7.2) are given in Table II.
B. Results for ESC08c-model
(i) For each potential type (i = C, σ, T, SO, ASO, Q) the fitting of the GEM coefficients consists of minimizing the χ2 , which is
defined as
2
N Pnmax
G
X
2
n=1 ai,n φn,l (rk ) − Vi (rk )
,
(7.3)
χ (ai ) =
σ(k)
k=1
13
TABLE I: GEM mass parameters in MeV, with nmax = 30, µmin = h̄c/10, and µmax = 10 h̄c.
19.733
43.653
96.571
213.635
472.607
1045.509
23.129
51.166
113.191
250.402
553.944
1225.444
27.109
59.972
132.671
293.497
649.279
1436.346
31.775
70.293
155.504
344.009
761.021
1683.544
TABLE II: GEM r[fm] parameters in MeV, with nmax = 30, r1 =
10.000
4.520
2.043
0.924
0.418
0.189
8.532
3.857
1.743
0.788
0.356
0.161
7.279
3.290
1.487
0.672
0.304
0.137
6.210
2.807
1.269
0.574
0.259
0.117
37.244
82.391
182.267
403.213
891.995
1973.286
h̄
µmax c
fm, rmax = (h̄c/µmin ) fm.
5.298
2.395
1.083
0.489
0.221
0.100
w.r.t. variations of the coefficients ai,n , where we choose for the errors σ(k) = 1 MeV. Here, the base functions are the
l
2 2
Gaussians φG
n,l (r) = r exp −µn r , where the µn are given in Table I. The radii rk (k = 1, N) in (7.3) are chosen to be an
equidistant set of distances in the interval 0 < rk < 15 fm, where for example N=400. So, they are a different set as those in
(7.2).
(0)
Starting from an initial set parameters ai the χ2 (ai ) is developed up to second order around the initial values and the optimal
2
values are given by minimizing the χ , i.e. the solution of the equation
nX
max
∂χ2
∂ 2 χ2
∂χ2 (ai )
=0=
+
∆ai,m .
∂∆ai,n
∂ai,n
∂ai,n ∂ai,m
(7.4)
m=1
Equation (7.4) is solved for ∆ai,n and via iteration the minimum is approached. Since the χ2 is quadratic in the parameters
ai,n (n = 1, nmax ), equation (7.4) is linear in the parameters. This makes the procedure very fast and in a few steps the
minimum is reached in practice.
(ii) The numerical solution of the partial-wave Lipmann-Schwinger equation is done using either the Kowalski-Noyes [5] or the
Haftel-Tabakin [6] method. The momentum integral over de interval (0, ∞) in the Lippmann-Schwinger equation is transformed
to an integral over interval (−1, +1) in the variable y
Z +1
Z∞
dy g(y)
(7.5)
dp f(p) =
0
−1
by the hyperbolic mapping
p − p0
2p0
y=
, g(y) =
f
p + p0
(1 − y)2
1+y
p0 .
1−y
(7.6)
We use the Gauss quadrature applying it for the interval (−1 < y ≤ 0) and (0 < y ≤ +1). It appeared that 40 points are
adequate with p0 = 1200 MeV. The results are rather insensitive to the precise value of 800 < p0 < 1600 MeV. Also, the
arctangent mapping gives equivalent results.
In Table VI and Table VIII we display the ESC08c phase shifts for the computations in configuration- and momentum-space
respectively. The differences are shown in Fig’s 1 and 2. The solid curves represent the configuration-space and the dashed
curves the momentum-space results. The agreement is satisfactory. The phases for the higher partial waves (L ≥ 3) show for
Tlab ≤ 5 MeV sign changes in the momentum space computations, which seems to indicate that the fitting of the very long
range parts of the potentials should be improved.
14
TABLE III: Meson parameters of the fitted ESC-model. Coupling constants are at k2 = 0. An asterisk denotes that the coupling constant is
not searched, but constrained via SU(3) or simply put to some value used in previous work.
√
√
meson
mass (MeV)
g/ 4π
f/ 4π
Λ (MeV)
π
η
η′
ρ
φ
ω
a0
f0
ε
a1
f1
f ′1
b1
h1
h ′1
Pomeron
Odderon
138.04
547.45
957.75
768.10
1019.41
781.95
982.70
974.10
760.00
1270.00
1420.00
1285.00
1235.00
1380.00
1170.00
223.20
273.91
0.7323
–1.2246∗
3.5574
0.8353
–1.3072
4.3553
–1.1983
0.8153
–0.8672
–0.2039
–0.0622
–0.0344
3.6832
3.7111
0.2689
0.1142∗
0.1264∗
3.7754
2.4639∗
–0.6096
0.9013
–1.8968
1.7293
948.10
,,
942.74
688.75
,,
1124.26
1137.66
,,
1057.64
1203.56
,,
,,
948.10
948.10
948.10
–4.5900
TABLE IV: Pair-meson coupling constants employed in the MPE-potentials. Coupling constants are at k2 = 0. An asterisk denotes that the
coupling constant is set to zero.
JPC
++
0
0++
0++
0++
1−−
1++
1++
1++
1+−
SU(3)-irrep
(αβ)
g/4π
f/4π
{1}
,,
{8}s
(ππ)0
(σσ)
(πη)
(πη ′ )
(ππ)1
(πρ)1
(πσ)
(πP)
(πω)
0*
0*
–0.0701
0*
0.0351
0.9813
–0.0315
0*
–0.0413
–0.3195
{8}a
,,
,,
,,
{8}s
VIII.
DISCUSSION AND CONCLUDING REMARKS
In this paper we presented a very practical and accurate method for obtaining the momentum-space potentials from the
configuration-space ones. The method is demonstrated by the application to the Extended-soft-core (ESC) potentials. The
reproduction of the phase shifts as obtained in the configuration-space computations, e.g. up to ten significant digits, is rather
difficult. But, with a little adjustment by refitting the meson parameters one can produce the same χ2p.d.p for NN with the
momentum space computation. For application to e.g. nuclei this is unnecessary.
The treatment of the tensor potential in this paper enforces the zero at r=0 by using an extra factor r2 in the GEM-expansion. An
alternative is to use instead a factor (1 − exp[−U2 r2 ]), where U is such that only the short-range part of the potential is affected.
The drawback is perhaps that it means the introduction of the new (non-linear) parameter U.
The extension of the method developed in this paper for the computation of the matrix elements of the potentials for spin 1/2spin 1/2 systems to hyperon-nucleon (YN) and hyperon-hyperon (YY) systems is straightforward. The expansion coefficients
become matrices in channel-space, which is obvious.
We have developed an analytical presentation of the ESC-potentials in momentum space for NN [2]. Of course, this also
could be extended to YN and YY. Although by itself this is interesting but it seems that the method proposed in this paper is
15
TABLE V: χ2 and χ2 per datum at the ten energy bins for the Nijmegen93 Partial-Wave-Analysis [14, 15]. Ndata lists the number of data
within each energy bin. The bottom line gives the results for the total 0 − 350 MeV interval. The χ2 -access for the ESC model is denoted by
∆χ2 and ∆^
χ2 , respectively.
Tlab
χ20
♯ data
∆χ2
χ
^ 20
∆^
χ2
0.383
1
5
10
25
50
100
150
215
320
144
68
103
209
352
572
399
676
756
954
137.5549
38.0187
82.2257
257.9946
272.1971
547.6727
382.4493
673.0548
754.5248
945.3772
19.1
58.1
9.8
36.8
45.5
64.0
20.3
99.1
130.5
229.1
0.960
0.560
0.800
1.234
0.773
0.957
0.959
0.996
0.998
0.991
0.132
0.854
0.095
0.127
0.129
0.112
0.051
0.147
0.173
0.240
Total
4233
4091.122
712.2
0.948
0.164
more practical when the Schrödinger and Lippmann-Scwinger equations are employed in the configuration and momentumspace respectively. In the case of e.g. the Kadyshevsky formalism for baryon-baryon, like in pion-nucleon [16], the analytic
presentation in momentum-space is preferable.
Appendix A: Check Jn,n+2 -integral
The purpose of this appendix is to review and check the derivation given in section VI of the integral (6.14):
1. The basic integral (6.4)
In,n (a, b; µ2 ) =
Z∞
r2 dr e−µ
2 2
r
jn (ar) jn (br) =
0
√
2
π
ab
a + b2
f
,
exp
−
n
4µ3
4µ2
2µ2
p
where fn (X) = π/2X In+1/2 (X), see [9], section (10.2).
2. The diagonal tensor-integral, see (6.7) and (6.8),
Z∞
h
i
d
(2)
2
2
2 −µ2 r2
Jn,n (a, b; µ ) =
r dr r e
jn (ar) jn (br) = − 2 In,n (a, b; µ2 )
dµ
0
√
2
2
2
2
π
a +b
a +b
′
=
3−
fn (X) + 2Xfn (X)
exp −
8µ5
4µ2
2µ2
2
√
a 2 + b2
a + b2
π
fn (X)
= − 5 exp −
8µ
4µ2
2µ2
2n + 3
2n − 1
−X
fn−1 (X) +
fn+1 (X) ,
2n + 1
2n + 1
(A1)
(A2)
where X = (ab)/(2µ2 ).
3. The off-diagonal tensor integral (6.10)
(2)
Jn,n+2 (a, b; µ2 ) =
Z∞
0
i
h
2 2
r2 dr r2 e−µ r jn (ar) jn+2 (br)
2
(2)
2
≡ −J(2)
n,n (a, b, µ ) + (2n + 3) Hn (a, b, µ ),
(A3)
16
TABLE VI: ESC08c nuclear-bar pp and np phases in degrees. No φlow constraint, χ2p.d.p. = 1.112
Tlab
♯ data
∆χ2
1
S0 (np)
1
S0
3
S1
ǫ1
3
P0
3
P1
1
P1
3
P2
ǫ2
3
D1
3
D2
1
D2
3
D3
ǫ3
3
F2
3
F3
1
F3
3
F4
ǫ4
3
G3
3
G4
1
G4
3
G5
ǫ5
0.38
144
19
54.56
14.61
159.39
0.03
0.02
–0.01
–0.05
0.00
–0.00
–0.00
0.00
0.00
0.00
0.00
0.00
–0.00
–0.00
0.00
–0.00
–0.00
0.00
0.00
–0.00
0.00
1
68
58
62.01
32.62
147.78
0.11
0.14
–0.08
–0.19
0.01
–0.00
–0.01
0.01
0.00
0.00
0.00
0.00
–0.00
–0.00
0.00
–0.00
–0.00
0.00
0.00
–0.00
0.00
5
103
98
63.49
54.77
118.27
0.68
1.62
–0.90
–1.50
0.22
–0.05
–0.19
0.22
0.04
0.00
0.01
0.00
–0.01
–0.01
0.00
–0.00
–0.00
0.00
0.00
–0.00
0.00
10
290
36
59.75
55.20
102.75
1.18
3.84
–2.05
–3.10
0.67
–0.20
–0.69
0.84
0.16
0.00
0.08
0.01
–0.03
–0.06
0.00
–0.00
–0.00
0.01
0.00
–0.00
0.00
25
352
46
50.52
48.73
80.87
1.82
8.86
–4.89
–6.46
2.51
–0.81
–2.85
3.67
0.68
0.02
0.55
0.11
–0.23
–0.41
0.02
–0.05
–0.05
0.17
0.04
–0.01
0.04
50
572
64
39.85
39.04
63.15
2.13
11.87
–8.27
–9.93
5.79
–1.72
–6.56
9.02
1.69
0.24
1.60
0.34
–0.67
–1.10
0.11
–0.19
–0.26
0.70
0.15
–0.06
0.20
100
399
20
25.41
25.09
43.91
2.43
9.76
–13.24
–14.84
10.82
–2.71
–12.44
17.40
3.77
1.24
3.48
0.81
–1.47
–2.12
0.48
–0.53
–0.97
2.10
0.41
–0.20
0.71
150
676
99
14.94
14.76
31.70
2.81
4.95
–17.36
–18.94
13.94
–2.97
–16.69
22.29
5.69
2.51
4.86
1.13
–2.05
–2.78
1.02
–0.83
–1.75
3.49
0.67
–0.33
1.24
215
756
130
4.19
4.10
20.14
3.43
–1.59
–21.99
–23.41
16.27
–2.79
–20.68
24.97
7.66
3.94
6.04
1.22
–2.61
–3.47
1.89
–1.14
–2.78
5.13
1.06
–0.43
1.88
320
954
229
–9.51
–9.54
6.40
4.59
–11.00
–28.20
–28.68
17.48
–2.10
–25.05
24.71
9.40
5.42
7.07
0.53
–3.40
–4.72
3.14
–1.46
–4.13
7.41
1.72
–0.42
2.72
with, see (6.12),
√
π µ2
8µ5 b2
2n
1 d
=
In,n (a, b; µ2 ) =
−
2b2 b db
2
b2
a + b2
′
2n
+
f
(X)
−
2X
f
(X)
.
exp −
n
n
4µ2
µ2
2
H(2)
n (a, b; µ )
(A4)
4. For checking (6.14) we now consider the combination
′
′
2n fn (X) − 2X fn (X) = 2X fn (X) − fn+1 (X) − 2X fn′ (X) = −2X fn+1 (X),
where we applied the fourth recurrence in (6.5). Next, we apply the first recurrence in (6.5) and obtain
2X2
′
fn (X) − fn+2 (X) .
2n fn (X) − 2X fn (X) = −
(2n + 3)
This gives for (A4) the expression
2
H(2)
n (a, b; µ )
√
2 π
1
a
a 2 + b2
=
fn (X) − fn+2 (X)
fn (X) −
.
exp −
8µ5
4µ2
2n + 3
2µ2
(A5)
17
TABLE VII: ESC08c Low energy parameters: S-wave scattering lengths and effective ranges, deuteron binding energy EB , and electric
quadrupole Qe . ESC08’ is a fit with no φlow constraint. The asterisk denotes that the low-energy parameters were not searched.
1
app ( S0 )
rpp (1 S0 )
experimental data
–7.823 ± 0.010
2.794 ± 0.015
ESC08c
–7.7699
2.7516∗
anp (1 S0 ) –23.715 ± 0.015 –23.7264
2.760 ± 0.030
2.6914∗
rnp (1 S0 )
1
ann ( S0 ) –16.40 ± 0.42
–16.762
2.860 ± 0.15
2.867∗
rnn (1 S0 )
anp (3 S1 ) 5.423 ± 0.005
5.4270∗
3
1.761 ± 0.005
1.7521∗
rnp ( S1 )
EB
–2.224644 ± 0.000046 –2.224621
0.286 ± 0.002
0.2696∗
Qe
ESC08c’
–7.7705
2.7575∗
–23.7178
2.6961∗
–15.7585
2.8723∗
5.4260∗
1.7464∗
–2.224392
0.2601∗
TABLE VIII: ESC08c nuclear-bar pp and np phases in degrees. Computed with LSE via GEM-fit x-space potentials.
Tlab
♯ data
1
S0 (np)
1
S0
3
S1
ǫ1
3
P0
3
P1
1
P1
3
P2
ǫ2
3
D1
3
D2
1
D2
3
D3
ǫ3
3
F2
3
F3
1
F3
3
F4
ǫ4
3
G3
1
G4
3
H4
0.38
144
54.45
14.70
159.38
0.10
0.03
0.02
0.07
0.02
0.00
0.02
0.01
–0.00
–0.00
–0.00
0.00
–0.00
0.00
0.00
0.00
–0.00
–0.00
0.00
1
68
62.03
32.76
147.66
0.24
0.13
0.01
–0.08
0.04
0.01
0.10
0.02
–0.01
–0.02
–0.02
0.02
0.01
0.05
0.01
0.00
–0.00
–0.00
0.00
5
103
63.50
54.87
118.26
0.67
1.63
–0.87
–1.45
0.23
–0.06
–0.04
0.05
0.08
–0.03
–0.01
–0.02
0.03
–0.13
–0.04
0.02
0.09
–0.04
0.12
10
290
59.78
55.30
102.67
1.20
3.82
–2.02
–3.13
0.67
–0.21
–0.62
0.84
0.16
–0.01
0.12
0.01
–0.02
–0.10
–0.01
–0.00
0.36
0.10
–0.03
25
352
50.58
48.84
80.82
1.84
8.84
–4.86
–6.46
2.51
–0.82
–2.80
3.70
0.69
0.02
0.57
0.10
–0.19
–0.44
0.02
–0.05
0.06
0.06
0.02
50
572
39.94
39.19
63.09
2.16
11.85
–8.25
–9.94
5.79
–1.72
–6.50
8.98
1.69
0.26
1.62
0.34
–0.60
–1.12
0.11
–0.20
–0.19
0.16
0.03
100
399
25.56
25.29
43.85
2.49
9.78
–13.26
–14.85
10.80
–2.70
–12.36
17.33
3.75
1.30
3.50
0.81
–1.35
–2.13
0.52
–0.53
–0.89
0.42
0.11
150
676
15.13
15.07
31.65
2.90
5.05
–17.43
–18.92
13.89
–2.96
–16.57
22.20
5.64
2.57
4.85
1.13
–1.90
–2.80
1.07
–0.83
–1.71
0.68
0.20
215
756
4.41
4.41
20.10
3.56
–1.43
–22.17
–23.28
16.22
–2.80
–20.54
24.81
7.59
4.05
6.03
1.23
–2.45
–3.50
1.90
–1.14
–2.73
1.07
0.35
(2)
Collecting terms for Jn,n+2 we have
a2
a2
a 2 + b2
+ (2n + 3) − 2 , fn+2 : + 2 ,
2
2µ
2µ
2µ
2n − 1
2n + 3
X , fn+1 : −
X .
: −
2n + 1
2n + 1
fn :
fn−1
320
954
–9.24
–9.11
6.36
4.76
–10.67
–28.51
–28.24
17.33
–2.15
–24.86
24.45
9.26
5.60
7.06
0.54
–3.22
–4.79
3.17
–1.46
–4.07
1.72
0.56
18
TABLE IX: Born-approximation: ESC08c nuclear-bar pp and np phases in degrees. Computed with LSE via GEM-fit x-space potentials.
Tlab
F3
1
F3
3
F4
ǫ4
3
G3
3
G4
1
G4
3
G5
ǫ5
3
H4
3
H5
1
H5
3
H6
ǫ6
3
0.38
–0.00
0.00
0.00
0.00
–0.00
0.00
–0.00
–0.00
–0.00
0.00
–0.00
0.00
0.00
0.00
1
0.01
0.05
0.01
0.00
–0.00
0.01
–0.00
–0.00
–0.00
0.00
–0.00
0.00
0.00
0.00
5
0.08
–0.13
–0.07
0.01
0.09
0.06
–0.05
–0.12
–0.05
0.12
–0.05
0.24
0.07
0.00
10
–0.01
–0.10
–0.07
–0.01
0.36
–0.31
0.07
0.10
–0.03
–0.03
0.08
–0.17
–0.06
0.01
25
–0.22
–0.44
0.01
–0.06
0.06
0.08
0.07
0.00
0.05
0.01
–0.01
–0.03
0.00
–0.01
50
–0.69
–1.14
0.11
–0.20
–0.19
0.65
0.17
–0.05
0.22
0.02
–0.08
–0.19
–0.00
–0.03
100
–1.50
–2.18
0.48
–0.53
–0.77
2.02
0.42
–0.22
0.72
0.11
–0.29
–0.53
0.04
–0.12
150
–2.09
–2.89
1.00
–0.83
–1.36
3.37
0.69
–0.39
1.25
0.22
–0.52
–0.87
0.10
–0.23
215
–2.69
–3.69
1.73
–1.13
–1.98
4.92
1.06
–0.55
1.87
0.37
–0.78
–1.19
0.23
–0.36
320
–3.60
–5.29
2.84
–1.47
–2.59
6.96
1.69
–0.65
2.68
0.60
–1.07
–1.51
0.51
–0.55
which leads to
(2)
Jn,n+2 (a, b; µ2 )
2
a + b2
π
...
exp −
=
8µ5
4µ2
√
where the expression between the curly brackets, using the previous results, becomes
2
b
... =
+ (2n + 3) fn
2µ2
a2
2n + 3
2n − 1
fn+1 +
fn−1 + 2 fn+2 .
−X
2n + 1
2n + 1
2µ
Now, it is easy to derive that
2n − 1
2n + 3
2n + 3
−X
fn+1 +
fn−1 = −X 2fn+1 +
fn .
2n + 1
2n + 1
X
These results lead to the formula
(2)
Jn,n+2 (a, b; µ2 )
√
2
2
π
a + b2
b
ab
a2
=
exp −
fn (X) − 2 fn+1 (X) + 2 fn+2 (X) ,
8µ5
4µ2
2µ2
µ
2µ
(A6)
which is the same expression as in (6.14).
The low momentum behavior of the r.h.s. in (6.14) is given by fn (X) ∼ Xn /(2n + 1)!!, and we get
2
√
b2
Xn
a + b2
π
(2)
2
·
exp
−
·
Jn,n+2 (a, b; µ ) ∼
8µ5
4µ2
2µ2 (2n + 5)!!
a2
a4
× (2n + 5)(2n + 3) − (2n + 5) 2 + 4 + ...
µ
4µ
(A7)
19
20
1
60
3
S0(pp)
P0
10
40
0
20
-10
0
-20
-20
0
100
200
300
0
0
100
200
300
20
3
P1
-10
15
-20
10
-30
5
3
-40
P2
0
0
100
200
300
1
0
100
200
2
300
3
ε2
F2
0
1
-1
-2
0
-3
-4
-1
0
100
200
300
0
100
200
300
FIG. 1: Solid line: proton-proton I = 1 phase shifts for the ESC08c-model. The dashed line: GEM-method momentum-space computation.
(2)
(2)
Appendix B: Explicit evaluation J0,0 - and J0,2 -integral
1. For the I0,0 -integral we get explicitly
Z∞
Z∞
i sin(ar) sin(br)
h
2 2
2 2
I0,0 ≡
r2 dr e−µ r j0 (ar) j0 (br) =
dr r2 e−µ r ·
·
ar
br
0
0
Z
2 2
1 ∞
= −
dr e−µ r eiar − e−iar eibr − e−ibr
4ab 0
√ h
√
i
2
2
2
2
π
π −(a2 +b2 )/4µ2
ab
= −
e−(a+b) /4µ − e−(a−b) /4µ = +
e
sinh
4abµ
2abµ
2µ2
√
2
2
2
π
= + 3 e−(a +b )/4µ f0 (X).
4µ
(B1)
(2)
2. For the J0,2 -integral we obtain
Z∞
h
i
2 2
(2)
J0,2 ≡
r2 dr r2 e−µ r j0 (ar) j2 (br)
Z0∞
i sin(ar) 1 sin(br)
h
cos(br)
4 −µ2 r2
·
=
dr r e
3
− sin(br) − 3
·
ar
br
(br)2
br
0
Z∞
2 2
sin(br)
3
(2)
− cos(br) .
rdr e−µ r sin(ar)
= −J0,0 +
ab2 0
br
(B2)
20
0
1
60
1
S0(np)
40
P1
-10
20
-20
0
-20
-30
0
100
200
300
0
100
200
300
5
3
160
S1
4
120
3
80
2
40
1
ε1
0
0
0
100
200
300
0
0
100
200
300
30
3
3
D1
-10
20
-20
10
-30
D2
0
0
100
200
300
0
100
200
300
FIG. 2: Solid line: neutron-proton I = 0, and the I=1 1 S0 (NP) phase shifts for the ESC08c-model. The dashed line: GEM-method momentumspace computation.
(2)
3. The J0,0 -integral is defined as
Z∞
Z
h
h
i
i
2 2
1 ∞
(2)
4 −µ2 r2
J0,0 ≡
dr r e
dr r2 e−µ r sin(ar) sin(br)
j0 (ar) j0 (br) =
ab 0
0
Z∞
2 2
1
d
=
dr e−µ r eiar − e−iar eibr − e−ibr .
4ab dµ2
0
(B3)
The explicit expression is derived as
(2)
J0,0
=
=
=
=
=
√ π −(a+b)2 /4µ2
d
1
−(a−b)2 /4µ2
e
−e
+
4ab dµ2 µ
√
2
π d
ab
1 − a24µ+b
2
sinh 2
−
e
2ab dµ2 µ
2µ
√
2
ab
ab
a 2 + b2
ab
π − a24µ+b
2
sinh
+ 2 cosh
e
1−
4abµ3
2µ2
2µ2
µ
2µ2
√
2
2
2
2
a +b
π − a 4µ+b
2
f0 (X) + 2Xf−1 (X)
e
·X 1−
3
4abµ
2µ2
√
π − a2 +b2 a2 + b2
− 5 e 4µ2
f
(X)
−
X
(3f
(X)
−
f
(X))
0
−1
1
8µ
2µ2
Here, in the last step we used the recurrence f0 = X(f−1 − f1 ). We notice that (B4) agrees with (6.8) for n=0.
(B4)
21
(2)
(2)
(2)
4. Writing J0,2 ≡ −J0,0 + H0,2 we have
Z
sin(br)
3 ∞
(2)
−µ2 r2
−
cos(br)
=
rdr
e
sin(ar)
H0,2 =
ab2 0
br
Z
Z
2 2
3 ∞
3 d ∞
−µ2 r2
dr e−µ r cos(ar) cos(br)
dr
e
sin(ar)
sin(br)
+
3
2
ab 0
ab da 0
Here appear two integrals, see [9] formula 7.4.6 for the relevant integral formulas,
Z
Z∞
2 2
2 2
1 ∞
cos(a − b)r − cos(a + b)r
dr e−µ r
J1 =
dr e−µ r sin(ar) sin(br) =
2 0
0
√ 2
(a − b)
(a + b)2
π
exp −
− exp −
=
4µ
4µ2
4µ2
Z∞
Z∞
1
−µ2 r2
−µ2 r2
J2 =
dr e
cos(ar) cos(br) =
cos(a − b)r + cos(a + b)r
dr e
2 0
0
√ (a − b)2
(a + b)2
π
exp −
+
exp
−
.
=
4µ
4µ2
4µ2
(2)
H0,2
(B5)
(B6)
√
2
3
3 π
3 dJ2
a + b2
ab
=
=
sinh
J1 +
exp −
ab3
ab2 da
2µab3
4µ2
2µ2
ab
ab
b
+ (a + b) exp − 2
− 2 (a − b) exp
4µ
2µ2
2µ
√
2
3 π
ab
b2
a + b2
ab
ab
ab
=
sinh
− 2 cosh
+ 2 sinh
exp −
2µab3
4µ2
2µ2
2µ
2µ2
2µ
2µ2
√
2
2
2
2
π
a +b
3µ
b
exp −
=
· 2 f0 (X) − X f−1 (X) + 2 f0 (X) .
5
2
4µ
4µ
b
2µ
(B7)
Useful recurrences are:
Xf−1 = Xf1 + f0 , f2 = f0 − 3f1 /X.
(2)
(2)
5. Collecting the results for J0,0 and H0,2 we finally arrive at
2
√
2
a + b2
π − a24µ+b
µ2
µ2
(2)
2
J0,2 =
e
f0 (X) − 3X f−1 (X) + X f1 (X) + 6 2 f0 − 6 2 X f−1 + 3f0
8µ5
2µ2
b
b
√
2
2
2
2
2
π − a 4µ+b
µ
µ2
a +b
2
e
f
(X)
−
3X
f
(X)
+
X
f
(X)
+
6
f
−
6
X f−1 + 3f0
=
0
−1
1
0
8µ5
2µ2
b2
b2
√
2
a2
3a2
ab
a2
a2
π − a24µ+b
b2
2
e
f
(X)
−
f
(X)
+
f2 (X)
+ 2 f2 − 2 f0 + 2 f1 =
·
0
1
2µ
2µ
2µ X
8µ5
2µ2
µ2
2µ2
µ2
µ2
a2
µ2
−3f0 (X) + 6 2 f0 (X) − 6 2 X f1 (X) − 6 2 f0 (X) + 3f0 (X) + 3 2 f1 (X)
b
b
b
2µ X
√
2
2 +b2
2
a
π − 4µ2
b
ab
a
·
=
e
f0 (X) − 2 f1 (X) + 2 f2 (X) .
8µ5
2µ2
µ
2µ
This result is equal to the expression in (A6) for n=0:
√
2
π
a 2 + b2
ab
a2
b
(2)
2
J0,2 (a, b; µ ) =
exp −
f0 (X) − 2 f1 (X) + 2 f2 (X) .
8µ5
4µ2
2µ2
µ
2µ
[1] E. Hiyama, Y. Kino, and M. Kamimura, Progr. in Part. and Nucl. Phys., 51 (2003) 223-307.
(B8)
(B9)
22
[2] Th.A. Rijken, H. Polinder, and J. Nagata, Phys. Rev. C 66, 044008 (2002); ibid C 66, 044009.
[3] For the demonstration of the method discussed in this paper we use a preliminary version of the final ESC08c-model [4].
[4] M.M. Nagels, Th.A. Rijken, and Y. Yamamoto, Extended-soft-core Baryon-Baryon Model ESC08. I. Nucleon-Nucleon Interactions.,
arXiv:1408.4825v1[nucl-th], 2014.
[5] H.P. Noyes, Phys. Rev. Lett. 15 538 (1965); K.L. Kowalski, Phys. Rev. Lett. 15 798 (1965), erratum 15, 908 (1965).
[6] M.I. Haftel and F. Tabakin, Nucl. Phys. A158 (1970) 1.
[7] In the computer programs pspace.newrep and pspace.newrep12 we solve for the LS-equation for the KJ ′ -matrix, which is related by
fJ and Green function g ′ = g/(4π). These programs are available on request.
KJ ′ = 4π K
[8] For this formalism see e.g.: M.L. Goldberger and K.M. Watson, Collision Theory, New York, Wiley 1964; M.D. Scadron, Advanced
Quantum Theory, New York, Springer-Verlag, 1979.
[9] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc. New York, 1970.
[10] Th.A. Rijken, R.A.M. Klomp, and J.J. de Swart, Soft-Core OBE-Potentials in Momentum Space, THEF-NIJM 91.05, Nijmegen preprint
1994; Published in A Gift of Prophecy, Essays in Celebration of the Life of Robert Eugene Marshak, Editor E.C.G. Sudarshan, World
Scientific, Singapore, 1995.
[11] P.A. Verhoeven, Off-shell Baryon-baryon Scattering, Ph.D.-thesis, University of Nijmegen, 1976 (unpublished).
[12] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Fifth Edition, Academic Press, 1965.
[13] P.M.M. Maessen, Th.A. Rijken, and J.J. de Swart, Phys. Rev. C40 (1989) 2226.
[14] V.G.J. Stoks, R.A.M. Klomp, M.C.M. Rentmeester, and J.J. de Swart, Phys. Rev. C 48 (1993) 792.
[15] R.A.M. Klomp, private communication (unpublished).
[16] J.W. Wagenaar and T.A. Rijken, Phys. Rev. C 80, 055204 (2009); ibid C 80, 055205.