Table Of ContentBoundary Conditions for Three-Body
Scattering in Configuration Space
G. L. Payne
Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242
W. Gl¨ockle
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Institut fu¨r Theoretische Physik II, Ruhr-Universita¨t Bochum, Germany
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2
n
a J. L. Friar
J
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Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
3
v (August 1999)
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Abstract
9
9
/
h
t The asymptotic behavior of three-body scattering wave functions in config-
-
l
c
uration space is studied by considering a model equation that has the same
u
n
: asymptotic form as the Faddeev equations. Boundary conditions for the wave
v
i
X
function are derived, and their validity is verified by numerical calculations.
r
a
Itisshownthattheseboundaryconditionsforthepartialdifferentialequation
can be used to obtain accurate numerical solutions for the wave function.
I. INTRODUCTION
In a previous paper [1] (hereafter referred to as I), we studied the asymptotic form of
a three-body wave function, which results in the propagation of three free particles from
various types of sources. The goal was to determine the boundary conditions appropriate
for the three-body scattering equations in configuration space. In particular, we want to
establish the values of
1
1
3
ρ = (x x )2
i cm
2 i=1 −
r
X
for which the leading asymptotic form of an outgoing wave would be valid and to investi-
gate the correction terms to that form. For a complete discussion of three-body scattering
in configuration space see Ref. [2], which also contains references to earlier work on this
problem.
In I two sources were studied. The first was a localized source corresponding to the
elastic-scattering driving term in the three-body Faddeev equation and is determined by
the overlap of a two-body force in one pair and a two-body bound-state wave function in
another pair. The other was an extended model source that mimics the real source term in
the Faddeev equation including the breakup process in the Faddeev amplitude. The latter
source reaches far out in the distance y between one particle and the center of mass of
the other two particles; it decreases only as O(y−3/2). However, the presence of the pair
interaction limits the extent in the distance x between the other two particles. As expected
in the case of the extended source, the leading form is reached only at a much larger radius
than the localized source, specifically when x is small and y is large.
By inverting the free propagator, one can determine the propagating wave function using
a partialdifferential equation with thegiven source terms. Weestablished suitable boundary
conditions that could be used to solve this problem efficiently. A matching radius of about
100fm was found to be sufficient.
In this article we extend our previous study of the extended source to allow one pair
interaction to be present while the three particles propagate from the given sources, which
is exactly what happens in the Faddeev formulation. We use the same notation as in I. In
Sec. II we evaluate the three-body Green’s function including one pair interaction, apply
it to the extended source, and study the asymptotic behavior in the two- and three-body
fragmentation channels. In Sec. III we solve the related partial differential equation as an
exercise for applying that technique (in a forthcoming article) to the Faddeev equation itself.
Finally, we conclude in Sec. IV.
2
II. THREE-BODY PROPAGATION FROM GIVEN SOURCES
WITH A PAIR INTERACTION
In order toavoid unnecessary complications, we restricted ourconsiderations inIto three
identical bosons interacting by spin-independent s-wave pairwise interactions in a state with
total angular momentum of zero. For this case the Faddeev equation for the channel with
incident wave Φ,
Ψ = Φ+G(E)VPΨ , (1)
where P = P+ +P− is the permutation operator, reads in explicit notation as
∞ ∞
′ ′ ′ ′ ′ ′
ψ(x,y) = φ(x,y)+ dx dy g(x,y;x,y )Q(x,y ) . (2)
Z0 Z0
We have introduced in Eq. (2) thereduced Faddeev amplitude ψ(x,y) andthecorresponding
′ ′
reduced Green’s function g(x,y;x,y ). The coordinates x and y are the standard Jacobi
variables
x x1 = r2 r3 (3)
≡ −
y y1 = r1 1 (r2 +r3) (4)
≡ − 2
expressed in terms of the individual position vectors.
The source term in Eq. (2) is given by
1 xy
Q(x,y) = V(x) dµ ψ(x ,y ) , (5)
2 2
Z−1 x2y2
where x , y , and µ in Eq. (5) result from (3) and (4) by the cyclical permutations, and are
2 2
explicitly given by
x = 1 x2 +y2 +xyµ (6)
2 4
q
y = 9 x2 + 1 y2 3 xyµ , (7)
2 16 4 − 4
q
where µ is the cosine of the angle between x and y. The total wave function is given by
the sum of the three Faddeev amplitudes. In Eq. (1) Ψ is one of the Faddeev amplitudes,
3
the other two are generated by cyclical permutations of the particles, P+Ψ and P−Ψ, and
they appear in the source term as PΨ. The pair interaction is V(x) and this interaction
also occurs in the Green’s function
1
G(E) = .
E (H +V)+iε
0
−
We refer to Ref. [3] for the general background and details on the notation.
The source term has a short-range component arising from the elastic-scattering piece of
the Faddeev amplitude, and a long-range component from the breakup piece of the Faddeev
amplitude. In I we studied the effects of both components; however, for the long-range com-
ponent we used a model source term. Using the asymptotic form of the Faddeev amplitude
derived by the stationary phase approximation, one finds [2] that the asymptotic form of
the source term for three equal mass particles with total energy E is
xei√4/3k0y
Q(x,y) CV(x) , (8)
→ y3/2
where k2 = mE/h¯2 and the constant C is given by the magnitude of the wave function in
0
the asymptotic region. Therefore, to study the effects of this long-range behavior, we used
the model source term
xyei√4/3k0y
Q (x,y) = V(x) , (9)
Model
(y +y )5/2
0
with y = 2fm. This source term has the same asymptotic form as (8), and we have set C
0
to be unity for convenience.
In I we neglected the final-state interaction between one pair in the propagator. As
required by the Faddeev scheme, this will now be included. We will study the second term
in Eq. (2) with the source term replaced by the model source. Thus, we write
∞ ∞
′ ′ ′ ′ ′ ′
F(x,y) = dx dy g(x,y;x,y )Q (x,y ). (10)
Model
Z0 Z0
To simplify the numerical calculations we follow the procedure used in I, and use the
Bargmann two-body potential
4
V e−λx
0
V(x) = ,
−(1+βe−λx)2
where
h¯2λ2
V = 2β .
0
M
(cid:18) (cid:19)
The bound-state wave function for this potential is given by
(1 e−λx)
u (x) = 2κβ − e−κx,
d (1+βe−λx)
q
where κ is the bound-state wave number for a two-body state with the energy ǫ = h¯2κ2/m
−
and
λ+2κ
β = .
λ 2κ
−
For our model calculations we use the values κ = 0.2316fm−1 and λ = 0.7fm−1. We also
need the two-body scattering states u (x) for this potential. They are given by
k
1
u (x) = eiδ(k)eikxh(k,x) e−iδ(k)e−ikxh∗(k,x) ,
k 2i −
n o
with
2k iλ
1+ − βe−λx
2k +iλ
h(k,x) = ,
1+βe−λx
and
2k +iλ k +iκ
eiδ(k) = v v .
u u
u u
u2k iλ uk iκ
u − u −
u u
t t
In addition, we use the arbitrary but fixed laboratory energy of the incident particle of
E = 14MeV and h¯2/m = 41.47MeV fm which corresponds to the case for three nucleons.
lab
·
For this case k = 0.41403fm−1. Henceforth, we set h¯ = 1.
0
For three equal-mass particles with mass m and total energy E, the three-body Green’s
′ ′
function g(x,y;x,y ) for the case with one pair interaction has the well-known form
5
4m sinq y
g(x,y;x′,y′) = u (x) eiq0y> 0 < u (x′)
d d
− 3 q
(cid:18) 0 (cid:19)
+ 2 ∞dku (x) 4m eiqky> sinqky< u (x′), (11)
π Z0 k (cid:18)− 3 qk (cid:19) k
where q = 4m/3(E ǫ), q = 4/3(k2 k2), and k2 = mE. Obviously the propagation
0 − k 0 − 0
q q
from the source can now proceed not only into the unbound states but also into the deuteron
channel. Moreover, the two-body scattering states u (x) include the interaction V(x). It
k
is the purpose of this paper to study the additional effects of V(x) in the propagator, in
contrast to I where only the free propagator was considered. The numerical evaluation of
the second part in (11) containing u (x) requires some explanation. We were not able to
k
find an analytical expression such as we found for g in I, and had to perform the k-integral
0
numerically. Clearly at the upper end of the integral the integrand has rapid oscillations
that make the integral difficult to evaluate numerically. Since u (x) approaches sinkx as
k
′ ′
k , it appears natural to subtract the free propagator g (x,y;x,y ) and add it back in
0
→ ∞
a separate term. Then for k the integrand has a stronger fall off and, moreover, the
→ ∞
path of integration can be rotated into the complex plane, similar to the procedure used in
Ref. [4].
Let us start with the propagation in the deuteron channel:
F (x,y) 4m u (x) ∞dy′ eiq0y> sinq0y< ∞dx′u (x′)Q (x′,y′)
d d d Model
≡ − 3 Z0 q0 Z0
4m ∞ sinq y′ ∞
= u (x)eiq0y dy′ 0 dx′u (x′)Q (x′,y′)
d d Model
− 3 Z0 q0 Z0
4m ∞ sinq (y y′) ∞
u (x) dy′ 0 − dx′u (x′)Q (x′,y′). (12)
d d Model
− 3 Zy q0 Z0
We see that this consists of a flux-conserving term
Fasy(x,y) = u (x)eiq0yf , (13)
d d d
with
4m ∞ sinq y′ ∞
f = dy′ 0 dx′u (x′)Q (x′,y′),
d d Model
− 3 Z0 q0 Z0
6
and a correction term
4m ∞ sinq (y y′) ∞
Fcorr(x,y) u (x) dy′ 0 − dx′u (x′)Q (x′,y′). (14)
d ≡ − 3 d Zy q0 Z0 d Model
′
Inserting Eq. (9) into Eq. (14) and performing one partial integration in the y variable,
one arrives easily at the following asymptotic form
ei√4/3k0y 1 ∞
Fcorr(x,y) u (x) dx′u (x′)x′V(x′). (15)
d y−→→∞ − y3/2 d ǫ Z0 d
Clearlythelong-rangesourcebehaviorcarriesoverintoacorrespondinglong-rangecorrection
term in the deuteron channel. We rewrite Eq. (14) in the form
ei√4/3k0y
Fcorr(x,y) = u (x)fcorr(y), (16)
d y3/2 d d
and show in Fig. 1 the behavior of fcorr(y) as it approaches its asymptotic value given by
d
1 ∞
fcorr(y) dx′u (x′)x′V(x′). (17)
d y−→→∞ − ǫ Z0 d
To illustrate the convergence, we have normalized the plot to the asymptotic value
fcorr( ) = 23.325fm3/2.
d ∞ −
To illustrate the error in the elastic term that results from matching to the asymptotic
boundary conditions at a finite distance, we plot in Fig. 2 the absolute value of f (y)
d
≡
F (x,y)/u (x) and its asymptotic form given by Eq. (13) and Eq. (15). The difference is
d d
less than 2% for y greater than 50fm and less than 1% for y greater than 75fm.
The propagation into the unbound states u (x) is more complicated, since
k
F (x,y) 4m 2 ∞dku (x) ∞dy′eiqky> sinqky< ∞dx′u (x′)Q (x′,y′) (18)
scat ≡ − 3 π Z0 k Z0 qk Z0 k Model
can be written in the form
F (x,y) = 4m 2 k0dku (x)eiqy ∞dy′ sinqky′ ∞dx′u (x′)Q (x′,y′)
scat − 3 π Z0 k Z0 qk Z0 k Model
4m 2 k0dku (x) ∞dy′ sinqk(y −y′) ∞dx′u (x′)Q (x′,y′)
− 3 π Z0 k Zy qk Z0 k Model
4m 2 ∞ y sinh Ky′ ∞
dku (x)e−Ky dy′ dx′u (x′)Q (x′,y′)
− 3 π k K k Model
Zk0 Z0 Z0
4m 2 ∞ sinh Ky ∞ ∞
dku (x) dy′e−Ky′ dx′u (x′)Q (x′,y′) (19)
− 3 π k K k Model
Zk0 Zy Z0
(1) (2) (3) (4)
F (x,y)+F (x,y)+F (x,y)+F (x,y). (20)
scat scat scat scat
≡
7
In the third and fourth terms K 4/3 k2 k2.
≡ − 0
q q
Let us first examine the asymptotic behavior for fixed x and y approaching infinity. One
has
k0
F(1)(x,y) = dku (x)eiqkyT(k) (21)
scat k
Z0
with
4m 2 ∞ sinq y′ ∞
T(k) = dy′ k dx′u (x′)Q (x′,y′). (22)
− 3 π Z0 qk Z0 k Model
There is no saddle-point for this case; thus, the asymptotic form arises from the leading
end-point contribution at k = 0, which is easily evaluated to be
√π ei√4/3k0y
F(1)(x,y) 33/4k3/2eiπ/4 u˜ (x)T˜ , (23)
scat |k≈0 −→ − 4 0 y3/2 0 0
where u˜ (x) u (x)/k and T˜ T(k)/k . We find that
0 ≡ k |k=0 0 ≡ |k=0
4m 2 ∞ sin 4 k y′ ∞
T˜ = dy′ 3 0 dx′u˜ (x′)Q (x′,y′). (24)
0 − 3 π Z0 q43 k0 Z0 0 Model
q
This term has the same dependence on y as the correction term Eq. (15) in the deuteron
channel. Note that T˜ is given by the analytical expression Eq. (22) differentiated with
0
respect to k under the integral. This is obviously true for the localized source; however,
for the extended source, one must rewrite the integral using a contour deformation before
performing the differentiation.
(2)
Let us now consider F (x,y) in Eq. (19) for the model source term given by Eq. (9)
scat
(2) 4m 2 k0 ∞ ′ ′ ′ ′
F (x,y) dku (x) dx u (x)xV(x)
scat −→ − 3 π k k
Z0 Z0
∞ sinq (y y′) ei√4/3k0y′
dy′ k − . (25)
×Zy qk y′3/2
After one partial integration one finds
(2) 2 k0 m ∞ ′ ′ ′ ′ ei√4/3k0y
F (x,y) dku (x) dx u (x)xV(x) . (26)
scat y−→→∞ − π k k2 k y3/2
Z0 Z0
x fixed
8
(3)
The term F (x,y), as it stands, is less obvious in its asymptotic behavior. The contri-
scat
′
butions to the y -integral keep growing towards the upper limit y. Because of the factor
′
exp( Ky), only contributions from the upper end of the y integral have to be considered.
−
Again by partial integration, one easily finds
4m 2 ei√4/3k0y ∞ 1 1
(3)
F (x,y) dku (x)
scat xy−→fi→x∞ed − 3 π y3/2 Zk0 k 2K K +i 4/3k0
∞ q
′ ′ ′ ′
dx u (x)xV(x). (27)
× k
Z0
(4)
Finally, the last piece, F (x,y), can again be handled in a straightforward manner with
scat
the result
4m 2 ei√4/3k0y ∞ 1 1
(4)
F (x,y) dku (x)
scat xy−→fi→x∞ed − 3 π y3/2 Zk0 k 2K K −i 4/3k0
q
∞
′ ′ ′ ′
dx u (x)xV(x). (28)
× k
Z0
Adding equations (26), (27), and (28) we obtain the concise result given in Ref. [5],
2 ei√4/3k0y ∞ m ∞
(2) (3) (4) ′ ′ ′ ′
F +F +F dku (x) dx u (x)xV(x). (29)
scat scat scat −→ π y3/2 k k2 k
Z0 Z0
This can be simplified by using a technique suggested by C. Gignoux [6]. Writing the
two-body Green’s function in the form
1 2 ∞ 1
′ ′ ′
g (x,x;z) u (x) u (x)+ dku (x) u (x), (30)
2 ≡ d z ǫ d π k z k2/m k
− Z0 −
the integral over k occurring in Eq. (29) can be rewritten as
2 ∞ m 1
′ ′ ′
dku (x) u (x) = g (x,x;0)+u (x) u (x). (31)
− π k k2 k 2 d ǫ d
Z0
Thus, we are led to the function
∞
′ ′ ′ ′
u˜(x) dx g (x,x;0)xV(x), (32)
2
≡
Z0
which obeys the inhomogeneous equation
9
1 d2
+V(x) u˜(x) = xV(x). (33)
− m dx2 −
h i
′
Using the explicit form for g (x,x;0), one easily derives
2
∞
′ ′ ′ ′
u˜(x) −→ dx u˜ (x)xV(x) = a, (34)
x→∞ −Z0 0
where a is the scattering length defined by δ(k) approaching ka as k goes to zero. From
−
Eq. (33) and Eq. (34) follows
u˜(x) = u˜ (x) x. (35)
0 −
We now have the concise form
ei√4/3k0y 1 ∞
(2) (3) (4) ′ ′ ′ ′
F +F +F u˜(x)+u (x) dx u (x)xV(x) . (36)
scat scat scat y−→→∞ y3/2 d ǫ d
x fixed h Z0 i
Altogether F (x,y) has the asymptotic form
scat
ei√4/3k0y π
F (x,y) u˜ (x) 33/4k3/2eiπ/4T˜ +u˜ (x) x
scat y−→→∞ y3/2 − 0 4 0 0 0 −
r
x fixed h
1 ∞
′ ′ ′ ′
+u (x) dx u (x)xV(x) . (37)
d d
ǫ
Z0 i
The x-dependence is therefore built up of the zero-energy scattering state, a linear term in
x, and the two-body bound state. The last term cancels exactly against the correction term
Eq. (15) in the deuteron channel and the total amplitude F(x,y) behaves as1
ei√4/3k0y π
F(x,y) u (x)eiq0yf + u˜ (x) 33/4k3/2eiπ/4T˜ +u˜ (x) x . (38)
y−→→∞ d d y3/2 − 0 4 0 0 0 −
(cid:20) r (cid:21)
x fixed
For x outside the range of V(x), the expression in the brackets in Eq. (38) reduces to
√π
(x a) 33/4k3/2eiπ/4T˜ a. (39)
− − 4 0 0 −
1An alternate derivation of this result is given in Section 6.3 of Ref. [2]; however, the solution
g(x) = 1 of Eq. (2.6.19) in Ref. [2] is not given explicitly.
−
10
