Connection between asymptotic normalization coefficients, subthreshold bound states,
and resonances
A. M. Mukhamedzhanov and R. E. Tribble
Cyclotron Institute, Texas A&M University, College Station, Texas 77843
~Received 14 September 1998!
We present here useful relations showing the connection between the asymptotic normalization coefficient
~ANC! and the fitting parameters in K- and R-matrix theory methods which are often used when analyzing low
energy experimental data. It is shown that the ANC of a subthreshold bound state defines the normalization of
both direct radiative capture leading to this state and resonance capture in which the state behaves like a
subthreshold resonance. A determination of the appropriate ANC~s! thus offers an alternative method for
finding the strength of these types of capture reactions, both of which are important in nuclear astrophysics.
@S0556-2813~99!01006-7#
PACS number~s!: 24.30.2v, 25.40.Lw, 26.30.1k
I. INTRODUCTION
Nuclear excited states below the particle emission thresh-
old typically undergo
g
decay to lower lying states. These
decays result in the initial states having their own natural
width. In the case when
g
emission is the only open decay
channel, the natural width G
g
is typically ;eV. If a particle
bound excited state lies very close to the particle threshold,
the natural width can result in the tail of the wave function
extending above the particle threshold. As a result of this
tail, the subthreshold bound state can behave like a reso-
nance state in a capture reaction. Such states are often re-
ferred to as subthreshold resonance states @1# and they can
play an important role in determining reaction rates of inter-
est in nuclear astrophysics.
Consider the capture of particle b by particle a at very low
relative kinetic energy E and assume that there is a sub-
threshold bound state c1 in the system c5 (ab). There are
three possible mechanisms by which the capture can occur
@1#: ~i! direct radiative capture to the ground state c, ~ii!
radiative capture to the ground state through the subthreshold
resonance, and ~iii! direct radiative capture into the sub-
threshold bound state with
g
emission.
Process ~ii! corresponds to nonradiative capture of par-
ticle b into the subthreshold resonance c1. The excited state
then undergoes
g
decay to the ground state c. The energy of
the emitted photon is
E
g
5 E1 «
c
, ~1!
where «
c
is the binding energy of the ground state c5 (ab).
Note that only one gamma is emitted in the process and it
occurs after capture into the c1 state. Process ~iii! results
initially in a photon with energy
E
g
5 E1 «
c1
. ~2!
The subthreshold bound state c1 is then deexcited to the
ground state c by emitting a photon with energy «
c
2 «
c1
.
Note that in mechanisms ~ii! and ~iii! the capture occurs into
the same state, but in ~ii! this state reveals itself as a reso-
nance, while in ~iii! it acts as a real bound state. All three of
these capture processes occur in nature and are important in
determining reaction rates for nuclear astrophysics.
In previous papers @2–5# we have pointed out that the
overall normalization of the cross section for a direct radia-
tive capture reaction at low binding energy is entirely defined
by the asymptotic normalization coefficient ~ANC! of the
final bound state wave function into the two-body channel
corresponding to the colliding particles. Below we show how
to extend this to capture into subthreshold resonance states.
Typically the approaches used to analyze low energy experi-
mental data in order to derive astrophysical factors are the K-
and R-matrix methods. We will present equations relating the
ANC to the residue of the pole corresponding to the sub-
threshold bound state in the K-matrix method and the re-
duced width amplitude in the R-matrix method. In the case of
a Breit-Wigner-type resonance ~above threshold!, the ANC is
related to the resonance width. The equations given here
have direct experimental implications and can be used in the
analysis of experimental data. When analyzing data using the
K-orR-matrix methods, the parameters corresponding to the
subthreshold bound states can be fixed by measuring ANC’s
independently from transfer reactions @4,5#. Also by measur-
ing ANC’s one can simultaneously determine astrophysical
factors both for direct radiative capture to the subthreshold
bound state and for capture to the subthreshold resonance.
The equations presented below are correct for scattering am-
plitudes in K- and R-matrix theory at negative energies, and
so they can be used to find the ANC by extrapolating elastic
scattering data ~phase shifts! to the pole corresponding to the
subthreshold bound state @6#.
In what follows we use the system of units in which \
5 c5 1.
II. ASYMPTOTIC NORMALIZATION COEFFICIENT
We present first some useful equations for the ANC. Let
us consider a virtual decay of nucleus c into two nuclei a and
b. First we introduce the overlap function I of the bound state
wave functions of particles c, a, and b:
PHYSICAL REVIEW C JUNE 1999VOLUME 59, NUMBER 6
PRC 59
0556-2813/99/59~6!/3418~7!/$15.00 3418 ©1999 The American Physical Society
I
ab
c
~
r
!
5
^
w
a
~
z
a
!
w
b
~
z
b
!
u
w
c
~
z
a
,
z
b
;r
!
&
5
(
l
c
m
l
c
j
c
m
j
c
^
J
a
M
a
j
c
m
j
c
u
J
c
M
c
&
3
^
J
b
M
b
l
c
m
l
c
u
j
c
m
j
c
&
i
l
c
Y
l
c
m
l
c
~
r
ˆ
!
I
a,bl
c
j
c
c
~
r
!
, ~3!
where for each nucleus
w
is the bound state wave function,
z
are a set of internal coordinates including spin-isospin vari-
ables, and J and M are the spin and spin projection. Also r is
the relative coordinate of the centers of mass of nuclei a and
b, r
ˆ
5 r/r, j
c
, m
j
c
are the total angular momentum of particle
b and its projection in the nucleus c5 (ab), l
c
, m
l
c
are the
orbital angular momentum of the relative motion of particles
a and b in the bound state c5 (ab) and its projection,
^
j
1
m
1
j
2
m
2
u
j
3
m
3
&
is a Clebsch-Gordan coefficient, Y
l
c
m
c
(r
ˆ
)
is a spherical harmonic, and I
ab,l
c
j
c
c
(r) is the radial overlap
function which includes the antisymmetrization factor due to
identical nucleons. The summation over l
c
and j
c
is carried
out over the values allowed by angular momentum and parity
conservation in the virtual process c→ a1 b. Since the radial
overlap function is not a solution of the Schro
¨
dinger equa-
tion, it is approximated by a model wave function of the
bound state c5 (ab) as follows:
I
ab l
c
j
c
c
~
r
!
5S
abl
c
j
c
1/2
w
n
c
l
c
j
c
~
r
!
. ~4!
Here
w
n
c
l
c
j
c
(r) is the bound state wave function for the rela-
tive motion of a and b which can be calculated, for example,
in the shell model or resonating group method and is normal-
ized by
E
0
`
dr r
2
w
n
c
l
c
j
c
2
~
r
!
51. ~5!
S
abl
c
j
c
is the spectroscopic factor of the configuration (ab)
with quantum numbers l
c
, j
c
in nucleus c. It is defined as the
norm of the radial overlap function @7,4#
S
abl
c
j
c
5
E
0
`
dr r
2
@
I
ab l
c
j
c
c
~
r
!
#
2
. ~6!
The asymptotic normalization coefficient C
abl
c
j
c
c
defining
the amplitude of the tail of the radial overlap function
I
ab l
c
j
c
c
(r) @7,4# is given by
I
ab l
c
j
c
c
~
r
!
→
r.R
N
C
abl
c
j
c
c
W
2
h
c
,l
c
11/2
~
2
k
ab
r
!
r
, ~7!
where R
N
is the nuclear interaction radius between a and b,
W
2
h
c
,l
c
1 1/2
(2
k
c
r) is the Whittaker function describing the
asymptotic behavior of the bound state wave function of two
charged particles,
k
c
5
A
2
m
ab
«
c
is the wave number of the
bound state c5 (ab),
m
ab
is the reduced mass of particles a
and b, and
h
k
c
5 Z
a
Z
b
m
ab
/
k
c
is the Coulomb parameter of
the bound state (ab). The ANC is related to the nuclear
vertex constant G
abl
c
j
c
c
by @7,2#
G
abl
c
j
c
c
52exp
F
i
p
S
l
c
1
h
c
2
D
G
A
p
m
a
C
abl
c
j
c
c
. ~8!
Taking into account the asymptotic behavior of the bound
state wave function
w
n
c
l
c
j
c
~
r
!
→
r. R
N
b
l
c
j
c
W
2
h
c
,l
c11/2
~
2
k
c
r
!
r
, ~9!
where b
l
c
j
c
is the single-particle ANC defining the amplitude
of the tail of the bound state wave function at large r,we
easily derive, from Eqs. ~4!, ~7!, and ~9!,
~
C
abl
c
j
c
c
!
2
5S
abl
c
j
c
b
l
c
j
c
2
. ~10!
The ANC is related to the residue of the elastic scattering
amplitude in the so-called direct pole in the energy plane
corresponding to the bound state. To show this we introduce
the transition matrix T, which is related to the S matrix as
S5 12 T. ~11!
The diagonal partial S-matrix element is given by S
jj
5exp(2i
d
l
), where
d
l
is the full scattering phase shift in the
partial wave l which includes the Coulomb scattering phase
shift
s
l
also. Thus in our approach the S and T matrices
include the Coulomb phase shift if it is nonzero. Note that
usually the Coulomb rescattering is singled out — i.e., only
the Coulomb-modified nuclear phase shift is considered —
but we take into account the total scattering phase shift. Let
us consider now the elastic scattering a1 b→a1 b. Let j
stand for the channel a1 b.Ifaand b can form the bound
state c5 (ab) with binding energy «
c
and relative orbital
angular momentum l ~for simplicity we omit the subscript c
in l), then the elastic scattering amplitude has a pole corre-
sponding to this bound state in the lth partial wave at the
relative kinetic energy of particles a and b, E52«
c
. In the
momentum plane it corresponds to the pole at k5 i
k
c
, where
E5 k
2
/2
m
ab
with k being the relative momentum of particles
a and b. Near this pole the partial elastic transition amplitude
T
jj
in the lth partial wave can be written in the form @8#
T
jj
~
k
!
'
k→i
k
c
~
21
!
l
ie
i
p
h
k
c
u
C
u
2
k2i
k
c
. ~12!
Thus the ANC simultaneously defines the normalization of
the tail of the overlap function and the residue in the pole
corresponding to the bound state of the partial elastic transi-
tion amplitude. This connection follows from the particle
conservation law in nonrelativistic quantum mechanics @8#.
III. K-MATRIX APPROACH AND THE ANC
A. Relating the ANC to the pole residue for the subthreshold
bound state and the resonance width
Consider the radiative capture process
a1 b→ c1
g
, ~13!
where the final nucleus c has an excited bound state which is
very close to the threshold for a1 b. For convenience, we
PRC 59 3419CONNECTION BETWEEN ASYMPTOTIC NORMALIZATION . . .
assume that the constituent particles a and b in channel c are
spinless. We also assume that there are no close resonances
at low relative kinetic energy E between particles a and b.
Then we need to take into account only two channels j and
g
,
which correspond to channels a1 b and c1
g
, respectively.
The transition matrix has two-components, T
jj
which corre-
sponds to the elastic scattering a1 b→ a1 b and T
g
j
which
corresponds to the radiative capture ~13! to the ground state
through the subthreshold resonance. For simplicity we con-
sider only two bound states in the system (ab), the ground
state and the excited subthreshold bound state. Since T
g
j
is
significantly smaller than T
jj
, one can write
T
g
j
5 2ip
g
p
j
K
g
j
11i
m
j
K
jj
, ~14!
T
jj
52ip
j
2
K
jj
11i
m
j
K
jj
. ~15!
The diagonal elements p
j
, p
g
, and
m
j
of the diagonal ma-
trices p and
m
in channels j and
g
are given by
p
j
5 e
2 (
p
h
/2)sgn Re k
G
~
l1 i
h
1 1
!
l!
k
l1 1/2
, ~16!
m
j
5
u
p
j
u
2
, ~17!
p
g
5 k
g
l
g
1 1/2
. ~18!
Here
h
5 Z
a
Z
b
m
ab
/k is the Coulomb parameter, l is the rela-
tive orbital angular momentum of particles a and b in chan-
nel j, k
g
is the momentum of the photon emitted during the
transition from the subthreshold bound state c to the ground
state, and l
g
is its multipolarity. Since we consider only
Re k. 0, even when extrapolating to the bound state pole k
5 lim Re k → 101 i Im k, we can take sgn Re k5 1.
Let us consider the partial element T
jj
in the partial wave
l where particles a and b form the subthreshold bound state
c1. The excited bound state close to threshold has a width
caused by its
g
transition to lower lying bound states. At low
relative energies E in channel j, the subthreshold bound state
can be ‘‘seen’’ by the incident particle b; i.e., it can be cap-
tured into the subthreshold bound state of nucleus c as a
resonance state with subsequent
g
transition to the bound
state. The matrix element describing the capture to the sub-
threshold resonance is given by Eq. ~14!.
For certain classes of local nuclear potentials, the K ma-
trix is a real symmetric matrix. Moreover, the matrix ele-
ments of the K matrix are analytic functions of k
2
at k
2
5 0
with a branch cut on part of the negative real axis and with
isolated poles on the cut in the complex k
2
plane @9,10#.
Since the matrix elements K
jj
and K
g
j
are meromorphic
functions of k
2
except for the cut, we present them in the
Pade
´
form
K
g
j
5
P
N
1
Q
M
, K
jj
5
D
N
2
Q
M
, ~19!
where P
N
, D
N
, Q
N
are polynomials of Nth order in the k
2
plane. Consider first the transition matrix element T
jj
. Tak-
ing into account the Pade
´
parametrization of K
jj
, we get
T
jj
52ip
j
2
D
N
2
Q
M
1i
m
j
D
N
2
. ~20!
As has been indicated, the elastic transition matrix element
T
jj
has a pole at k5 i
k
c1
where
k
c1
5
A
2
m
ab
«
c1
is the wave
number corresponding to the subthreshold bound state c1.
T
jj
also has a pole corresponding to the ground state of c but
we do not consider it as we assume that it is quite far from
the subthreshold bound state. We now show how to relate the
residue of T
jj
in the pole corresponding to the subthreshold
bound state to the ANC. To do this, we must extrapolate T
jj
to the bound state pole located on the physical sheet of the k
plane at k5 i
k
c1
, i.e., to the positive imaginary axis in the
complex k plane or to the negative real axis in the E plane.
Since
m
j
is a modulus of p
j
2
, it is not an analytic function,
and when extrapolated down to negative energies,
m
j
5 0at
E<0. However, Eq. ~16! shows that p
j
2
is an analytic func-
tion in the k plane. If we write
p
j
2
5 e
2i
s
l
m
j
, ~21!
then it becomes clear why the Coulomb scattering, given by
exp(2i
s
l
), was included in T
jj
since without this factor, p
j
2
would not be analytic and its extrapolation to negative ener-
gies would lead to the wrong residue ~see the Appendix!.
Thus at E, 0 we get
T
jj
52ip
j
2
D
N
2
Q
M
. ~22!
Hence the pole of T
jj
at negative energy corresponds to the
zero of Q
M
. It is convenient to represent the ratio D
N
2
/Q
M
as a sum of pole terms plus a background B
j
:
D
N
2
Q
M
5
(
l5 1
M
g
cl
2
k
2
2 k
l
2
1 B
j
. ~23!
Then
T
jj
5
k
2
,0
2ip
j
2
(
l51
M
g
cl
2
k
2
2k
l
2
1B
j
, ~24!
where g
cl
2
is the pole residue. Note that some but not all of
the poles in the expansion ~24! correspond to bound states in
c @9#. Let l5 1 correspond to the subthreshold bound state
c1. Then, at k
2
→k
1
2
52
k
c1
2
,
T
jj
'
k
2
→2
k
c1
2
2ip
j
2
g
c1
2
k
2
1
k
c1
2
. ~25!
Recall that the elastic transition amplitude T
jj
near the pole
corresponding to the bound state was given by Eq. ~12!.
Comparing Eqs. ~25! and ~12! we find the relationship be-
tween g
c1
and the ANC. For k→ i
k
c1
,
p
j
5 e
i
p
h
k
c1
/2
G
~
l1
h
k
c1
1 1
!
l!
~
i
k
c1
!
l1 1/2
. ~26!
Hence
3420 PRC 59A. M. MUKHAMEDZHANOV AND R. E. TRIBBLE
T
jj
'
k
2
→2
k
c1
2
i
~
21
!
l
e
i
p
h
k
c1
F
G
~
l1
h
k
c1
11
!
l!
G
2
k
c1
2l
g
c1
2
k2i
k
c1
,
~27!
with the expression for g
c1
,
g
c1
2
5
1
k
c1
2l
~
l!
!
2
G
2
~
l1 11
h
k
c1
!
u
C
u
2
, ~28!
following from Eqs. ~12! and ~27!. Thus the residue of the
closest pole of K
jj
is proportional to the corresponding ANC.
Although we assumed that particles a and b are spinless, Eq.
~28! is valid also for particles with nonzero spins. Allowing
for spin, the ANC and the residue g
c1
also depend on the
total angular momentum j
c1
of particle b in the bound state
~in jj coupling! or on the spin channel ~in LS coupling!.
Consider now T
g
j
. From ~19! we immediately arrive at
T
g
j
5 2ip
g
p
j
1
Q
M
/P
N
1
1i
m
j
~
D
N
2
/P
N
1
!
. ~29!
Once again we introduce the pole expansion for
K
g
j
5 P
N
1
/Q
M
5
(
l5 1
M
g
g
l
g
cl
k
2
2 k
l
2
1 B
g
. ~30!
At small
k
c1
2
, where k
1
2
52
k
c1
2
, and k
2
→0, we can use the
one pole approximation giving
T
g
j
5
k
2
→0
2ip
j
p
g
g
g
1
g
c1
k
2
1
k
c1
2
1i
m
j
g
c1
2
. ~31!
Comparing this equation with the Breit-Wigner amplitude,
we find the relationship between the partial width G
c1
of the
subthreshold resonance seen by the incident particle b at E
. 0 and the residue g
c1
2
in the K-matrix approach:
G
c1
~
E
!
5
m
j
m
ab
g
c1
2
5
1
m
ab
S
k
k
c1
D
2l
ke
2
p
h
S
u
G
~
l1i
h
11
!
u
G
~
l111
h
k
c1
!
D
2
u
C
u
2
,
~32!
while the
g
width of the subthreshold resonance G
g
and g
g
1
are related by
G
g
~
E
!
5 2k
g
1
2l
g
1 1
g
g
1
2
, ~33!
where k
g
is the momentum of the photon emitted during the
transition from the subthreshold bound state to the ground
state c. The total width of the subthreshold resonance at posi-
tive energies is
G
~
E
!
5 G
c1
~
E
!
1 G
g
~
E
g
!
'G
c1
~
E
!
. ~34!
Thus the total width of the subthreshold resonance at E. 0is
proportional to
u
C
u
2
.
We can now find the behavior of the cross section for
capture to the subthreshold resonance at E→0. The cross
section for this capture is given by
s
l
g
5
~
2l1 1
!
2
p
m
ab
k
2
u
T
g
j
u
2
~35!
5
~
2l1 1
!
p
k
2
G
g
G
c1
~
E1 «
c1
!
2
1 G
c1
2
/4
~36!
5
~
2l1 1
!
p
m
ab
k
S
k
k
c1
D
2l
3e
2
p
h
S
u
G
~
l1i
h
11
!
u
G
~
l111
h
k
c1
!
D
2
3
G
g
u
C
u
2
~
E1«
c1
!
2
1G
c1
2
/4
~37!
'
E→0
~
2l1 1
!
p
2
k
c1
m
ab
2
1
E
e
22
p
h
3
~
h
k
c1
!
2l11
G
2
~
l111
h
k
c1
!
G
g
u
C
u
2
~
E1«
c1
!
2
. ~38!
Hence the astrophysical factor at E→ 0 behaves as
S
~
E
!
5 Ee
2
p
h
s
l
g
'
E→0
~
2l11
!
p
2
k
c1
m
ab
2
3
~
h
k
c1
!
2l11
G
2
~
l111
h
k
c1
!
G
g
u
C
u
2
~
E1«
c1
!
2
. ~39!
Thus we have shown that the ANC of the subthreshold
bound state defines the overall normalization of the cross
section and therefore the astrophysical factor for the capture
into the subthreshold resonance at E→0. Usually when fit-
ting low energy experimental data in the K-matrix approach,
the one pole approximation is not sufficient. Nevertheless,
the main fitting parameter g
c1
can be fixed from an indepen-
dent measurement of the ANC.
B. Subthreshold bound state, ANC, and the scattering length
Consider now the relationship between the ANC and the
scattering length assuming that there is a subthreshold
s-wave (l5 0) bound state c1. The scattering amplitude is
related to T
jj
by
f
jj
52
1
2ik
T
jj
. ~40!
Consider now the behavior of f
jj
at k→ 0:
PRC 59 3421CONNECTION BETWEEN ASYMPTOTIC NORMALIZATION . . .
f
jj
'
k→0
2e
2
p
h
G
2
~
i
h
11
!
K
jj
11
m
j
K
jj
'2e
2
p
h
G
2
~
i
h
11
!
g
c1
2
k
2
1
k
c1
2
1i
m
j
g
c1
2
. ~41!
We used in Eq. ~41! the single-pole approximation for K
jj
K
jj
'
k→0
g
c1
2
k
2
1
k
c1
2
. ~42!
Approximation ~42! is valid at k
2
→0 and small enough
k
c1
2
.
At k→ 0,
f
jj
'
k→0
2e
2i
s
0
C
e
2(
p
h
)
u
G
~
i
h
11
!
u
2
g
c1
2
k
c1
2
. ~43!
The quantity
a
5
g
c1
2
k
c1
2
~44!
is nothing but the scattering length. Taking into account Eq.
~28! we derive the relationship between the scattering length
and the ANC:
a
5
1
k
c1
2
1
G
2
~
11
h
k
c1
!
u
C
u
2
. ~45!
IV. R-MATRIX APPROACH AND THE ANC
Below we present some useful equations relating ANC’s
and paramerters in the R-matrix method. Although the
R-matrix method was developed for analysis of resonance
reactions, the reduced width of the R matrix, which corre-
sponds to the subthreshold resonance, can be related to
ANC’s of the subthreshold bound states. Let us consider the
elastic scattering a1 b→a1 b at k→ 0 assuming the pres-
ence of the subthreshold bound state c1. We note that the
elastic scattering S-matrix element in channel j is given by
@11,12#
S
jj
5e
22i(
f
l
2
s
l
)
1/R
l
2
@
D
l
~
E
!
2 B
l
2 iV
l
~
E
!
#
1/R
l
2
@
D
l
~
E
!
2 B
l
1 iV
l
~
E
!
#
, ~46!
where R
l
is the R matrix for the lth partial wave, D
l
(E) is the
Thomas shift, B
l
is the energy-independent R-matrix bound-
ary condition constant, and V
l
(E) is given by
V
l
~
E
!
5 kr
0
P
l
~
E
!
. ~47!
P
l
(E) is the penetration factor which is given by
P
l
~
E
!
5
1
G
l
2
~
k,r
0
!
1 F
l
2
~
k,r
0
!
, ~48!
where r
0
is the channel radius, and G
l
(k,r
0
) and F
l
(k,r
0
)
are the singular ~at the origin! and regular solutions of the
radial Schro
¨
dinger equation with a pure Coulomb potential at
E. 0, i.e.,
e
2i
f
l
5
G
l
~
k,r
0
!
1 iF
l
~
k,r
0
!
G
l
~
k,r
0
!
2iF
l
~
k,r
0
!
. ~49!
The elastic scattering amplitude is given by the sum of two
terms,
T
jj
512S
jj
5T
jj
(pot)
1 T
jj
, ~50!
where T
jj
(pot)
is the potential scattering amplitude and T
jj
is
the so-called resonance scattering amplitude which is
T
jj
522ie
22i(
f
l
2
s
l
)
V
l
~
E
!
R
l
2
@
D
l
~
E
!
2B
l
1iV
l
~
E
!
#
. ~51!
In the above equation, the R matrix is
R
l
5
(
l
g
cl
2
E
cl
2 E
, ~52!
where E
c
g
are the poles of the R matrix and
g
cl
is the re-
duced width of the lth level. If the energy of the subthresh-
old bound state is very close to threshold and the incident
energy E→0, we can use the one-level R-matrix approxima-
tion which leads to
T
jj
522ie
22i(
f
l
2
s
l
)
V
l
~
E
!
g
c1
2
E
c1
2E2
@
D
l
~
E
!
2B
l
1iV
l
~
E
!
#
g
c1
2
.
~53!
A priori the poles of the R matrix do not coincide with the
poles of the S matrix. However, if we choose the boundary
condition parameter B
l
5 D
l
(2
e
c1
), then the level shift of
the subthreshold bound state disappears, E
c1
52«
c1
@13#,
and Eq. ~53! reduces to
T
jj
522ie
22i(
f
l
2
s
l
)
V
l
~
E
!
g
˜
c1
2
2«
c1
2E2iV
l
~
E
!
g
˜
c1
2
, ~54!
where the effective reduced width of the subthreshold bound
state is
g
˜
c1
2
5
g
c1
2
11
g
c1
2
@
dD
l
~
E
!
/dE
#
u
E52
e
c1
. ~55!
Next we extrapolate Eq. ~54! down to the bound state pole at
E52«
c1
. The factor V
l
(E) can be written as
V
l
~
E
!
5 kr
0
1
u
G
l
~
k,r
0
!
1iF
l
~
k,r
0
!
u
2
. ~56!
This is not an analytic function. As we did when considering
the K matrix, we take V
l
(E)5 0atE,0 in the denominator,
since V
l
(E) is the imaginary part of the logarithmic deriva-
tive of the wave function which is real at negative energies.
However, in the numerator we have
3422 PRC 59A. M. MUKHAMEDZHANOV AND R. E. TRIBBLE