arXiv:0704.1461v4  [nucl-th]  16 Nov 2007
Trajectory of neutron−neutron−18C excited
three-body state
M. T. Yamashita a, T. Frederico b and Lauro Tomio c 1
aUniversidade Estadual Paulista, 18409-010, Itapeva, SP, Brazil.
bDepartamento de F́ısica, ITA, CTA, 12228-900 São José dos Campos, Brazil.
cInstituto de F́ısica Teórica, UNESP, 01405-900 São Paulo, Brazil.
Abstract
The trajectory of the first excited Efimov state is investigated by using a renor-
malized zero-range three-body model for a system with two bound and one virtual
two-body subsystems. The approach is applied to n− n−18C, where the n− n vir-
tual energy and the three-body ground state are kept fixed. It is shown that such
three-body excited state goes from a bound to a virtual state when the n−18C
binding energy is increased. Results obtained for the n−19C elastic cross-section at
low energies also show dominance of an S−matrix pole corresponding to a bound
or virtual Efimov state. It is also presented a brief discussion of these findings in
the context of ultracold atom physics with tunable scattering lengths.
PACS 03.65.Ge, 21.45.+v, 11.80.Jy, 21.10.Dr
Key words: Bound states, scattering theory, Faddeev equation, Few-body
The interest on three-body phenomena occurring for large two-body scattering
lengths have increased in the last years in view of the experimental possibilities
presented in ultracold atomic systems, where the two-body interaction can
be tunned by using Feshbach resonance techniques. Theoretical predictions,
well investigated for three particle systems, such as the increasing number of
three-body bound states when the two-body scattering length goes to infinity -
known as Efimov effect [1] - can actually be checked experimentally in ultracold
atomic laboratories. Indeed, the first indirect evidence of Efimov states came
from recent experiments with ultracold trapped Caesium atoms made by the
Innsbruck group [2].
1 Email: tomio@ift.unesp.br
Preprint submitted to Elsevier Science 24 October 2018
http://arxiv.org/abs/0704.1461v4
In the nuclear context, the investigations of Efimov states are being of renewed
interest with the studies on the properties of exotic nuclei systems with two
halo neutrons (n−n) and a core (c). One of the most promising candidates to
present these states is the 20C [3–5] (c ≡18 C). 20C has a ground state energy
of 3.5 MeV with a sizable error in n−18C two-body energy, 160±110 keV [6].
The proximity of an Efimov state (bound or virtual) to the neutron-core elastic
scattering cut makes the cross-section extremely sensitive to the corresponding
S-matrix pole. However, one should be aware that the analytic properties
of the S-matrix for Borromean systems (where all the two-body subsystems
are unbound, like the Caesium atoms of the Innsbruck experiment [2]) are
expected to be quite different from systems where at least one of the two-
body subsystems are bound, as the present case of n− n−18C.
The trajectory of Efimov states for three particles with equal masses has been
studied in [7,8] using the Amado model [9]. By studying the S−matrix in
the complex plane, varying the two-body binding, it was confirmed previous
analysis [8,10], that Efimov bound states disappear into the unphysical energy
sheet associated to the unitarity cut, becoming virtual states. It was also
verified that, by further increasing the two-body binding, the corresponding
pole trajectories remain in the imaginary axis and never become resonant.
Considering general halo-nuclei systems (n − n − c), in [3] it was mapped a
parametric region defined by the s−wave two-body (bound or virtual) energies,
where the Efimov bound states can exist. By increasing the binding energy of
a two-body subsystem, it was noted that the three-body bound state turns
out to a virtual state, remaining as virtual with further increasing of the two-
body binding, as already verified in [8,11]. In the other side, starting with
zero two-body binding, by increasing the two-body virtual energy we have the
three-body energy going from a bound to a resonant state [12].
Actually, the analysis of the trajectory of Efimov states in the complex plane
can be relevant to study properties of the 20C. In order to study the behavior
of Efimov states, the authors of [4] have recently pointed out the importance of
the analysis of low energy n−19C elastic scattering observables. Their results,
leading to a 20C resonance prediction near the scattering threshold [4], when
the separation energy of the bound halo neutron of 19C is changed, suggest a
different behavior from the one found for three equal-mass particles, where a
bound Efimov state turns into a virtual one as the two-body binding increases.
In view of the importance of the results, not only in the nuclear context, it
is interesting to consider a new independent analysis of three-particle systems
where two subsystems are bound and one is unbound, considering particularly
the case with different masses. As we show in the present letter, the results of
our treatment for three-body system with unequal masses are consistent with
the expectation derived from equal mass systems. Moreover, they are model
independent due to the universal character of the three-body physics at very
low energies. See Ref. [13] for a comment on the results obtained in Ref. [4].
In order to clarify the behavior (in the complex energy plane) of a given Efimov
state for the n− n−18C system, we consider the n− c subsystem bound with
varying energies; the virtual energy of the n − n subsystem and the ground
state energy of 20C are fixed, respectively, at −143 keV [14,15] and −3.5 keV
(these constraints are convenient to follow the trajetory of the excited Efimov
states since their appearance/disappearance depends only on the ratio of the
n− c and n−n− c energies, see pages 327-329 of Ref. [16]). Our present study
of n−n−18C can be easily extended to similar systems such as 12Be, 15B, 23N
and 27F.
In the present case, the bound state equations are extended to the second
Riemann energy sheet through the n−19C elastic scattering cut (see Fig. 1)
using a well-known technique [17]. Consistent with previous results [3,8,11]
for three equal masses particles, by using the present approach that will be
explained in the following, we conclude that (also in this case) the three-
body bound state turns into a virtual (and not a resonance) one when we
increase the n−18C two-body binding energy. The full behavior of the Efimov
virtual states in the unphysical sheet of the complex energy plane, by further
increasing the two-body bound state energy, is still open for investigation.
In our present investigation, we also show that, when the three-body S−matrix
pole of a virtual or excited Efimov state is near the scattering threshold, the
n−19C elastic scattering cross section is dominated by such pole, peaking at
zero relative energy and decreasing monotonically with energy.
Elastic scattering
Dimer breakup
Fig. 1. Analytic structure of the S−matrix. The energies of the neutron-core (bound
system), the three-body bound and the three-body virtual states are, respectively,
given by εnc, E3b, and E3v. The arrow passing through the elastic scattering cut
shows the trajectory of an S−matrix pole to the second Riemann sheet.
Next, we introduce the basic formalism by starting with the coupled spectator
functions for a bound three-body system n − n − c (c ≡ 18C in our specific
case). Our units are such that h̄ = mn = 1, where mn is the mass of the
neutron, with the n− n, n− c and three-body energies respectively given by
Enn = h̄
2εnn/mn, Enc = h̄
2εnc/mn, and E3b = h̄
2E3b/mn, where n − n and
n − c refer to virtual and bound subsystems. After partial wave projection,
the ℓ-wave spectator functions χℓn and χ
c (where the subindex n or c indicates
the spectator particle), for a bound three-body system, are given by [3]
n(q; E3b) = τnc(q; E3b)
2(q, k; E3b)χ
n(k; E3b) +K
1(q, k; E3b)χ
c(k; E3b)
c(q; E3b) = τnn(q; E3b)
1(k, q; E3b)χ
n(k; E3b), (2)
where
Kℓi=1,2(q, k; E3b)≡G
i(q, k; E3b)− δℓ0 G
i(q, k;−µ
2) (3)
Gℓi(q, k; E3b) =
Pℓ(y)
E3b −
A+Ai−1
q2 − A+1
k2 − kqy
, (4)
τnn(q; E3b)≡
|εnn|+
q2 − E3b
, (5)
τnc(q; E3b)≡
|εnc| −
(A+ 2)
2(A+ 1)
q2 − E3b
. (6)
In the above, E3b ≡ εnc −
2(A+1)
κ2b and A is the mass-number of particle c.
The absolute value of the momentum of the spectator particle with respect
to the center-of-mass (CM) of the other two particles is given by q ≡ |~q|;
with k ≡ |~k| being the absolute value of the relative momentum of these two
particles. The n − n virtual state energy is fixed at Enn = −143 keV. In
Eqs. (1) and (2), we have the Kronecker delta δℓ0 (= 1 for ℓ = 0 and =0 for
ℓ 6= 0) in order to renormalize the equations (using a subtraction procedure)
only for the partial wave where such renormalization is necessary, ℓ = 0. In
the cases of ℓ > 0, due to the centrifugal barrier, the Thomas collapse is
absent and such renormalization is not necessary. In this way, with Eqs. (1)
and (2) renormalized, the three-body observables are completely defined by
the two-body energy scales, εnc and εnn. Later on, the definitions (3)-(6) will
be extended also to unbound systems. The regularization scale µ2, used in
the s−wave [see Eq. (3)], is chosen to reproduce the three-body ground-state
energy of 20C, E0 = −3.5 MeV [6]. A limit cycle [16,18] for the scaling function
of s−wave observables is evidenced when µ is let to be infinity. We note that,
a good description of this limit is already reached in the first cycle [11].
The analytic continuation of the bound three-body system given by (1) and
(2) to the second Riemann sheet is performed through the n− (n− c) elastic
scattering cut, following Refs. [11,17,19]. As we are considering n−n−c nuclei
where only the subsystem n − c is bound (“samba-type” nuclei [14,15]), we
have only the n − c cut in the complex energy plane. In order to see how to
perform the analytical continuation from the first to the second sheet of the
complex energy, let us first consider a complex momentum variable ki. As the
energies of the two-body sub-systems are fixed and only the n− c subsystem
is bound, it is convenient to define this momentum as ki =
2(A+1)
(Ei − εnc).
In this case, the bound-state energy (Ei = E3b) is given by ki = iκb, with the
virtual state energy (Ei = E3v) given by ki = −iκv. Next, by removing E3b in
favor of ki in the bound-state equations (1)-(6), with
τ̄nc(q; E) ≡
|εnc|+
(A+ 2)q2
2(A+ 1)
, (7)
and with χc and χn redefined as
χℓc(q; E) ≡ h
c(q; E), χ
n(q; E) ≡ h
n(q; E)/(q
2 − k2i − iǫ), (8)
we observe that the relevant integrals to be considered have the structure
I(ki) =
q2dq [F (q2)/(k2i − q
2 + iǫ)] in the first sheet of the complex energy
plane. To obtain I(ki) in the second sheet we need to change the contour of
integration, as shown in detail in Ref. [7], such that the value of I(ki) in the
second sheet of the complex energy plane is given by I ′(ki) = I(ki)−iπkiF (k
Following this procedure, from Eqs. (1) and (2) we obtain the corresponding
equations in the second sheet of the complex energy plane, where for the
virtual state energy we have ki = −iκv =
2(A+1)
(E3v − εnc):
n(q; E3v) =
2(A+ 1)
τ̄nc(q; E3v)
2(q,−iκv ; E3v)h
n(−iκv ; E3v)+
Kℓ1(q, k; E3v)h
c(k; E3v) +
Kℓ2(q, k; E3v) h
n(k; E3v)
k2 + κ2v
 , (9)
hℓc(q; E3v) =τnn(q; E3v)
1(−iκv , q; E3v)h
n(−iκv ; E3v)+
1(k, q; E3v) h
n(k; E3v)
k2 + κ2v
 . (10)
The above coupled equations, as well as the corresponding coupled equations
for bound state, can be written as single-channel equations for hℓn, by defining
an effective interaction V and considering I = b, v:
hℓn(q; E3I) = 2κvh
n(−iκv; E3v)V
ℓ(q,−iκv ; E3v)δI,v +
dkk2Vℓ(q, k; E3I)
hℓn(k; E3I)
k2 + κ2I
Vℓ(q, k; E3I )≡ π
(A+ 1)
τ̄nc(q; E3I)× (12)
2(q, k; E3I) +
1(q, k
′; E3I)τnn(k
′; E3I)K
1(k, k
′; E3I)
The first term in the right-hand-side (rhs) of (11), which is non-zero only for
the virtual state, corresponds to the contribution of the residue at the pole.
The virtual states are limited by the cut of the elastic scattering amplitude in
the complex plane, corresponding to the second term in the rhs of Eq. (9). In
this case, the cut is given by the zero of the denominator of Gℓ2(q, k; E3v) [See
Eq. (4)], where −1 < y < 1 and q = k = −iκcut. With |E3v| = |εnc|+
2(A+1)
κ2cut,
we obtain the branch points, with the cut given by
2(A+ 1)
A + 2
|εnc| < |E3v| <
2(A+ 1)
|εnc|. (13)
For n − n−18C (A = 18), a virtual state energy can be found in the energy
interval between the threshold of the elastic scattering and the starting of the
above cut (13):
|εnc| < |E3v| < 1.9 |εnc| . (14)
0 100 200 300 400 500 600
90 120 150 180 210
| E19C| (keV)
Fig. 2. Three-body n − n−18C results for the first excited state, with respect to
the threshold (|E20C − E19C|) for varying
19C binding energies. Three-body bound
(virtual) states occur when |E19C| is approximately smaller (larger) than 170 keV.
s−wave results (solid line) are also presented in the inset (with dots). Results for
the p−and d−waves, divided by a factor 10, are shown with dashed and dotted
lines, respectively.
Figure 2 shows how the absolute value of the first excited three-body state
energy with respect to the two-body bound state, |E − Enc|, varies when
increasing the n − c bound-state energy. In this case, with 18C being the
core, we have 19C as the two-body bound subsystem. The solid line of Fig. 2
presents the s−wave results, with a close focus in the region of the threshold
(E = Enc) given by the inset figure. Although the Efimov states do not appear
in higher partial waves due to the existence of the centrifugal barrier, we have
also presented results for the virtual states of p− and d− waves in view of
their possible relevance for the elastic n−19C low-energy cross-section. Such
situation can happen when the s−wave virtual state is far from the elastic
scattering region.
The results shown in Fig. 2, valid for n−n−c system with n−c bound, together
with previous analysis [3,11], are clarifying that the behavior of Efimov states,
when one of the particles have a mass different from the other two, follows
the same pattern as found in the case of three equal-mass particles [8,10]. No
resonances were found for Eqs. (9) and (10) in the complex energy plane.
Next, for consistency, we present results for the s−wave elastic cross-sections,
which are in agreement with the above. The formalism for the partial-wave
elastic n− 19C scattering equations can be obtained from Eqs. (1) and (2), by
first introducing the following boundary condition in the full-wave spectator
function χn(~q):
χn(~q) ≡ (2π)
3δ(~q − ~ki) + 4π
hn(~q; E(ki))
q2 − k2i − iǫ
, (15)
where hn(~q; E(ki)) is the scattering amplitude, and the on-energy-shell in-
coming and final relative momentum are related to the three-body energy
Ei ≡ E(ki) by ki ≡ |~ki| = |~kf | =
[2(A+ 1)/(A+ 2)] (Ei − εnc). With the
same formal expressions (7) and (6) for τ̄nc and τnn, by using the definition
(13), the partial-wave scattering equation can be cast in the following single
channel Lippmann-Schwinger-type equation for hℓn:
n(q; Ei) = V
ℓ(q, ki; Ei) +
ℓ(q, k; Ei) h
n(k; Ei)
k2 − k2i − iǫ
. (16)
Virtual states and resonances correspond to poles of the scattering matrix on
unphysical sheets. So, the most natural method to look for them is to perform
an analytically continuation of the scattering matrix to the unphysical sheet.
For that one needs prior knowledge about the analytic properties of the kernel
of the integral equation, as well as the scattering matrix properties on the
unphysical sheet of the complex energy plane. Although such properties are
easy to derive in simple cases, they can be difficult to obtain in more complex
situations, which can put some restriction on the approach.
For the numerical treatment of Eq. (16), we consider the approach developed
in [8] to find virtual states and resonances on the second energy sheet associ-
ated with the lowest scattering threshold. Such approach does not require prior
knowledge of the analytic properties of the scattering matrix on the unphys-
ical sheet. The solution of the scattering equation is written in a form where
the analytic structure in energy is clearly exhibited. Then, it is analytically
continued to the unphysical sheet. The method relies on the calculation of an
auxiliary (resolvent) function [20], which has an integral structure similar to
the original scattering equation, but with a weaker kernel due to a subtrac-
tion procedure at an arbitrary fixed point k̄i. For real positive energies, this
subtraction point k̄i is identified with ki, such that the corresponding integral
equation does not have the two-body unitarity cut. For non-real or negative en-
ergies, k̄i can be any arbitrary positive real number. For convenience, k̄i = |ki|.
The final solution is obtained by evaluating certain integrals over the auxil-
iary function. In case of scattering solutions, the two-body unitarity cut is
introduced through these integrals. In the present case, we have the following
integral equation for the auxiliary function Γ, and the corresponding solution
for hℓn(q; Ei):
Γℓn(q, ki; Ei) =V
ℓ(q, ki; Ei) +
k2Vℓ(q, k; Ei)− k̄
ℓ(q, ki; Ei)
] Γℓn(k, ki; Ei)
k2 − k2i
hℓn(q; Ei) =Γ
n(q, ki; Ei) + Γ
n(q, k̄i; Ei)
Γℓn(k, ki; Ei)
k2 − k2i − iǫ
Γℓn(k, k̄i; Ei)
k2 − k2i − iǫ
. (17)
For the on-shell scattering amplitude, we have
hℓn(ki; Ei) = [1/Γ
n(ki, ki; Ei)−J ]
−1 (18)
Γℓn(k, ki; Ei)/Γ
n(ki, ki; Ei)− 1
k2 − k2i
+ iki
In order to obtain the bound and virtual energy states, two independent pro-
cedures have been used. The first one, by solving directly the homogeneous
coupled equations (1), (2), (9) and (10), looking for zeros of the corresponding
determinants. The other one, by verifying the position of the poles in the com-
plex energy plane of the scattering amplitude, hℓn(ki; Ei), given by Eq. (18) and
corresponding analytic extension to the second Riemann sheet. In this second
approach, we solve the corresponding inhomogeneous equation with on-energy-
shell momentum ki = +iκb (bound state, Ei = −|E3b|) and ki = −iκv (virtual
state, Ei = −|E3v|)[8].
By comparing the results of both approaches, we checked that they give con-
sistent results. However, for numerical stability and accuracy of the results,
particularly in the case of the search for Efimov states, when the absolute
values of the energies are very close to zero, the second approach is by far
much better.
Results for the total n−19C elastic cross-sections, obtained from dσ/dΩ =
|hn(~kf ; Ei)|
2, refering to bound or virtual states E∗, are presented in Fig. 3 as
functions of the CM kinetic energy,
K(ki) ≡ [(A+ 2) k
i ]/[2(A+ 1)] = E(ki)− Enc. (19)
0 25 50 75 100 125 150
10000
0 30 60 90 120150180
10000
0 100 200 300 400 500
 ] (keV)
 ] (keV)
Fig. 3. n−19C elastic cross sections (in barns) versus the CM kinetic energies
[Eq. (19) with A=18], for different 19C bound energies. In the left-hand-side frame we
show results for two cases that generate three-body energies close to the threshold:
E19C = −150 keV (main figure) and −180 keV (inset), producing respectively three-
-body bound and virtual states. Solid-line are obtained from (16), with dashed-line
from (20). In the rhs we have results for E19C = −500 keV.
Although each higher partial ℓ−wave have a virtual state, below the breakup
the cross-section is completely dominated by the s−wave [19]. In the frame
shown in the left-hand-side, we have two cases of energies close to the scatter-
ing threshold: n − 18C bound at −150 keV, giving an excited Efimov bound
state with E∗ = −150.12 keV; and n − 18C bound at −180 keV, producing
a virtual state with E∗ = −180.12 keV. In both two cases the cross-section
has a huge peak at zero energy due to the presence of the nearby pole. For
comparison, we also show (with dashed-line) the results obtained from the
following effective range expansion (approximately valid for small ki near the
elastic scattering threshold):
σ(ki) =
4πh̄2
1.9mn (Enc − E
∗) + h̄2k2i
2741 keV
E(ki)−E∗
barn, (20)
where we have used A =18, h̄2/mn =414.42 keV barn and Eq. (19). The
case not so close to the threshold (where Eq.(20) fails) is shown in rhs for
Enc = −500 keV, with
20C virtual energy E∗ = −568.73 keV.
The proximity of an Efimov state (bound or virtual) to the neutron and
neutron-core elastic scattering makes the cross-section extremely sensitive to
the corresponding S-matrix pole. We remark that if it will be possible to
dissociate a “samba-type” halo nuclei, like 20C, and measure the correlation
function in the two-body channel corresponding to a neutron and a bound n−c
system for small relative momentum, the information on the final state inter-
action as well as the halo structure will be clearly probed, as the counterpart
seen in the n− n correlation in the breakup of Borromean nuclei [21].
In conclusion, we analyzed the three-body halo system n−n−18C, where two
pairs (n−18C) are bound, and the remaining pair n − n has a virtual-state.
We study the trajectory of three-body Efimov states in the complex energy
plane. As shown, the energy of an excited Efimov state varies from a bound
to a virtual state as the binding energy of the subsystem n−18C is increased,
while keeping fixed the 20C ground-state energy and the virtual energy of
the remaining pair (n − n). In our approach we applied a renormalized zero-
range model, valid in the limit of large scattering lengths. Considering that
low-energy correlations, as the one represented by the Phillips line [22] (cor-
relation between triton and doublet neutron-deuteron scattering length), are
well reproduced by zero-range potentials [23], the present conclusions should
remain valid also for finite two-body interactions when the scattering length
is much larger than the potential range. On the numerical analysis, we should
remark that we have considered two approaches that give consistent results
for bound and virtual state energies. In the case of Efimov physics, where the
poles are very close to zero, the method considered in Ref. [8] was found to
give solutions with much better stability and accuracy in the scattering region
than by using a contour deformation technique.
The present results are extending to n−n−core systems the long ago conclu-
sion reached for three equal-mass particles [8,10]: by increasing the binding
energy of the n−core subsystem, an excited weakly-bound three-body Efimov
state moves to a virtual one and will not become a resonance. From Fig. 3, one
can also observe that the n−19C elastic cross-sections at low energies present a
smooth behavior dominated by the S−matrix pole corresponding to a bound
or virtual three-body state. In contrast with the above conclusion applied to
system where n−c is bound, we should observe that it was also verified that an
excited Efimov state can go from a bound to a resonant state (instead of vir-
tual state) in case of Borromean systems (with all the subsystems unbound),
when the absolute value of a virtual-state energy for the n − c system is in-
creased [12]. Actually, it should be of interest to extend the present analysis
of the trajectory of Efimov states to other possible two-body configurations,
with different mass relations of three-particle systems.
In view of the exciting possibilities of varying the two-body interaction, it can
be of high interest the results of the present study to analyze properties of
three-body systems in ultracold atomic experiments. For negative scattering
lengths the Efimov state goes to a continumm resonance when |a| is decreased,
as observed by the change of resonance peak in the three-body recombination
to deeply bound states towards smaller values of |a| by raising the tempera-
ture [24]. Alternatively, for positive a the recombination rate has a peak when
the Efimov state crosses the threshold and turns into a virtual state when
decreasing a. A dramatic effect will appear in the atom-dimer scattering rate
when the cross-section is dominated by the S-matrix pole near the scattering
threshold. We foreseen that the coupling between atom and molecular con-
densate will respond strongly to the crossing of the triatomic bound state
to a virtual one by changing a. Determined by the dominance of the cou-
pled channel interaction, new condensate phases of the atom-molecule gas are
expected. The proximity of the virtual trimer state to the physical region, im-
plying in a large negative atom-dimer scattering length, will warrant stability
to both condensates, while the positive atom-dimer scattering length, due to a
trimer bound state near threshold, make possible the collapse of the condensed
phases. Indeed, the occurrence of some interesting effects in the condensate
due to Efimov states near the scattering threshold have already been discussed
in Refs. [25]. We hope these exciting new consequences of Efimov physics can
be explored experimentally in the near future.
LT thanks Prof. S.K. Adhikari for helpful suggestions. We also thank Fundação
de Amparo à Pesquisa do Estado de São Paulo and Conselho Nacional de
Desenvolvimento Cient́ıfico e Tecnológico for partial support.
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  The trajectory of the first excited Efimov state is investigated by using a
renormalized zero-range three-body model for a system with two bound and one
virtual two-body subsystems. The approach is applied to $n-n-^{18}$C, where the
$n-n$ virtual energy and the three-body ground state are kept fixed. It is
shown that such three-body excited state goes from a bound to a virtual state
when the $n-^{18}$C binding energy is increased. Results obtained for the
$n-^{19}$C elastic cross-section at low energies also show dominance of an
$S-$matrix pole corresponding to a bound or virtual Efimov state. It is also
presented a brief discussion of these findings in the context of ultracold atom
physics with tunable scattering lengths.

arXiv:0704.1461v4  [nucl-th]  16 Nov 2007
Trajectory of neutron−neutron−18C excited
three-body state
M. T. Yamashita a, T. Frederico b and Lauro Tomio c 1
aUniversidade Estadual Paulista, 18409-010, Itapeva, SP, Brazil.
bDepartamento de F́ısica, ITA, CTA, 12228-900 São José dos Campos, Brazil.
cInstituto de F́ısica Teórica, UNESP, 01405-900 São Paulo, Brazil.
Abstract
The trajectory of the first excited Efimov state is investigated by using a renor-
malized zero-range three-body model for a system with two bound and one virtual
two-body subsystems. The approach is applied to n− n−18C, where the n− n vir-
tual energy and the three-body ground state are kept fixed. It is shown that such
three-body excited state goes from a bound to a virtual state when the n−18C
binding energy is increased. Results obtained for the n−19C elastic cross-section at
low energies also show dominance of an S−matrix pole corresponding to a bound
or virtual Efimov state. It is also presented a brief discussion of these findings in
the context of ultracold atom physics with tunable scattering lengths.
PACS 03.65.Ge, 21.45.+v, 11.80.Jy, 21.10.Dr
Key words: Bound states, scattering theory, Faddeev equation, Few-body
The interest on three-body phenomena occurring for large two-body scattering
lengths have increased in the last years in view of the experimental possibilities
presented in ultracold atomic systems, where the two-body interaction can
be tunned by using Feshbach resonance techniques. Theoretical predictions,
well investigated for three particle systems, such as the increasing number of
three-body bound states when the two-body scattering length goes to infinity -
known as Efimov effect [1] - can actually be checked experimentally in ultracold
atomic laboratories. Indeed, the first indirect evidence of Efimov states came
from recent experiments with ultracold trapped Caesium atoms made by the
Innsbruck group [2].
1 Email: tomio@ift.unesp.br
Preprint submitted to Elsevier Science 24 October 2018
http://arxiv.org/abs/0704.1461v4
In the nuclear context, the investigations of Efimov states are being of renewed
interest with the studies on the properties of exotic nuclei systems with two
halo neutrons (n−n) and a core (c). One of the most promising candidates to
present these states is the 20C [3–5] (c ≡18 C). 20C has a ground state energy
of 3.5 MeV with a sizable error in n−18C two-body energy, 160±110 keV [6].
The proximity of an Efimov state (bound or virtual) to the neutron-core elastic
scattering cut makes the cross-section extremely sensitive to the corresponding
S-matrix pole. However, one should be aware that the analytic properties
of the S-matrix for Borromean systems (where all the two-body subsystems
are unbound, like the Caesium atoms of the Innsbruck experiment [2]) are
expected to be quite different from systems where at least one of the two-
body subsystems are bound, as the present case of n− n−18C.
The trajectory of Efimov states for three particles with equal masses has been
studied in [7,8] using the Amado model [9]. By studying the S−matrix in
the complex plane, varying the two-body binding, it was confirmed previous
analysis [8,10], that Efimov bound states disappear into the unphysical energy
sheet associated to the unitarity cut, becoming virtual states. It was also
verified that, by further increasing the two-body binding, the corresponding
pole trajectories remain in the imaginary axis and never become resonant.
Considering general halo-nuclei systems (n − n − c), in [3] it was mapped a
parametric region defined by the s−wave two-body (bound or virtual) energies,
where the Efimov bound states can exist. By increasing the binding energy of
a two-body subsystem, it was noted that the three-body bound state turns
out to a virtual state, remaining as virtual with further increasing of the two-
body binding, as already verified in [8,11]. In the other side, starting with
zero two-body binding, by increasing the two-body virtual energy we have the
three-body energy going from a bound to a resonant state [12].
Actually, the analysis of the trajectory of Efimov states in the complex plane
can be relevant to study properties of the 20C. In order to study the behavior
of Efimov states, the authors of [4] have recently pointed out the importance of
the analysis of low energy n−19C elastic scattering observables. Their results,
leading to a 20C resonance prediction near the scattering threshold [4], when
the separation energy of the bound halo neutron of 19C is changed, suggest a
different behavior from the one found for three equal-mass particles, where a
bound Efimov state turns into a virtual one as the two-body binding increases.
In view of the importance of the results, not only in the nuclear context, it
is interesting to consider a new independent analysis of three-particle systems
where two subsystems are bound and one is unbound, considering particularly
the case with different masses. As we show in the present letter, the results of
our treatment for three-body system with unequal masses are consistent with
the expectation derived from equal mass systems. Moreover, they are model
independent due to the universal character of the three-body physics at very
low energies. See Ref. [13] for a comment on the results obtained in Ref. [4].
In order to clarify the behavior (in the complex energy plane) of a given Efimov
state for the n− n−18C system, we consider the n− c subsystem bound with
varying energies; the virtual energy of the n − n subsystem and the ground
state energy of 20C are fixed, respectively, at −143 keV [14,15] and −3.5 keV
(these constraints are convenient to follow the trajetory of the excited Efimov
states since their appearance/disappearance depends only on the ratio of the
n− c and n−n− c energies, see pages 327-329 of Ref. [16]). Our present study
of n−n−18C can be easily extended to similar systems such as 12Be, 15B, 23N
and 27F.
In the present case, the bound state equations are extended to the second
Riemann energy sheet through the n−19C elastic scattering cut (see Fig. 1)
using a well-known technique [17]. Consistent with previous results [3,8,11]
for three equal masses particles, by using the present approach that will be
explained in the following, we conclude that (also in this case) the three-
body bound state turns into a virtual (and not a resonance) one when we
increase the n−18C two-body binding energy. The full behavior of the Efimov
virtual states in the unphysical sheet of the complex energy plane, by further
increasing the two-body bound state energy, is still open for investigation.
In our present investigation, we also show that, when the three-body S−matrix
pole of a virtual or excited Efimov state is near the scattering threshold, the
n−19C elastic scattering cross section is dominated by such pole, peaking at
zero relative energy and decreasing monotonically with energy.
Elastic scattering
Dimer breakup
Fig. 1. Analytic structure of the S−matrix. The energies of the neutron-core (bound
system), the three-body bound and the three-body virtual states are, respectively,
given by εnc, E3b, and E3v. The arrow passing through the elastic scattering cut
shows the trajectory of an S−matrix pole to the second Riemann sheet.
Next, we introduce the basic formalism by starting with the coupled spectator
functions for a bound three-body system n − n − c (c ≡ 18C in our specific
case). Our units are such that h̄ = mn = 1, where mn is the mass of the
neutron, with the n− n, n− c and three-body energies respectively given by
Enn = h̄
2εnn/mn, Enc = h̄
2εnc/mn, and E3b = h̄
2E3b/mn, where n − n and
n − c refer to virtual and bound subsystems. After partial wave projection,
the ℓ-wave spectator functions χℓn and χ
c (where the subindex n or c indicates
the spectator particle), for a bound three-body system, are given by [3]
n(q; E3b) = τnc(q; E3b)
2(q, k; E3b)χ
n(k; E3b) +K
1(q, k; E3b)χ
c(k; E3b)
c(q; E3b) = τnn(q; E3b)
1(k, q; E3b)χ
n(k; E3b), (2)
where
Kℓi=1,2(q, k; E3b)≡G
i(q, k; E3b)− δℓ0 G
i(q, k;−µ
2) (3)
Gℓi(q, k; E3b) =
Pℓ(y)
E3b −
A+Ai−1
q2 − A+1
k2 − kqy
, (4)
τnn(q; E3b)≡
|εnn|+
q2 − E3b
, (5)
τnc(q; E3b)≡
|εnc| −
(A+ 2)
2(A+ 1)
q2 − E3b
. (6)
In the above, E3b ≡ εnc −
2(A+1)
κ2b and A is the mass-number of particle c.
The absolute value of the momentum of the spectator particle with respect
to the center-of-mass (CM) of the other two particles is given by q ≡ |~q|;
with k ≡ |~k| being the absolute value of the relative momentum of these two
particles. The n − n virtual state energy is fixed at Enn = −143 keV. In
Eqs. (1) and (2), we have the Kronecker delta δℓ0 (= 1 for ℓ = 0 and =0 for
ℓ 6= 0) in order to renormalize the equations (using a subtraction procedure)
only for the partial wave where such renormalization is necessary, ℓ = 0. In
the cases of ℓ > 0, due to the centrifugal barrier, the Thomas collapse is
absent and such renormalization is not necessary. In this way, with Eqs. (1)
and (2) renormalized, the three-body observables are completely defined by
the two-body energy scales, εnc and εnn. Later on, the definitions (3)-(6) will
be extended also to unbound systems. The regularization scale µ2, used in
the s−wave [see Eq. (3)], is chosen to reproduce the three-body ground-state
energy of 20C, E0 = −3.5 MeV [6]. A limit cycle [16,18] for the scaling function
of s−wave observables is evidenced when µ is let to be infinity. We note that,
a good description of this limit is already reached in the first cycle [11].
The analytic continuation of the bound three-body system given by (1) and
(2) to the second Riemann sheet is performed through the n− (n− c) elastic
scattering cut, following Refs. [11,17,19]. As we are considering n−n−c nuclei
where only the subsystem n − c is bound (“samba-type” nuclei [14,15]), we
have only the n − c cut in the complex energy plane. In order to see how to
perform the analytical continuation from the first to the second sheet of the
complex energy, let us first consider a complex momentum variable ki. As the
energies of the two-body sub-systems are fixed and only the n− c subsystem
is bound, it is convenient to define this momentum as ki =
2(A+1)
(Ei − εnc).
In this case, the bound-state energy (Ei = E3b) is given by ki = iκb, with the
virtual state energy (Ei = E3v) given by ki = −iκv. Next, by removing E3b in
favor of ki in the bound-state equations (1)-(6), with
τ̄nc(q; E) ≡
|εnc|+
(A+ 2)q2
2(A+ 1)
, (7)
and with χc and χn redefined as
χℓc(q; E) ≡ h
c(q; E), χ
n(q; E) ≡ h
n(q; E)/(q
2 − k2i − iǫ), (8)
we observe that the relevant integrals to be considered have the structure
I(ki) =
q2dq [F (q2)/(k2i − q
2 + iǫ)] in the first sheet of the complex energy
plane. To obtain I(ki) in the second sheet we need to change the contour of
integration, as shown in detail in Ref. [7], such that the value of I(ki) in the
second sheet of the complex energy plane is given by I ′(ki) = I(ki)−iπkiF (k
Following this procedure, from Eqs. (1) and (2) we obtain the corresponding
equations in the second sheet of the complex energy plane, where for the
virtual state energy we have ki = −iκv =
2(A+1)
(E3v − εnc):
n(q; E3v) =
2(A+ 1)
τ̄nc(q; E3v)
2(q,−iκv ; E3v)h
n(−iκv ; E3v)+
Kℓ1(q, k; E3v)h
c(k; E3v) +
Kℓ2(q, k; E3v) h
n(k; E3v)
k2 + κ2v
 , (9)
hℓc(q; E3v) =τnn(q; E3v)
1(−iκv , q; E3v)h
n(−iκv ; E3v)+
1(k, q; E3v) h
n(k; E3v)
k2 + κ2v
 . (10)
The above coupled equations, as well as the corresponding coupled equations
for bound state, can be written as single-channel equations for hℓn, by defining
an effective interaction V and considering I = b, v:
hℓn(q; E3I) = 2κvh
n(−iκv; E3v)V
ℓ(q,−iκv ; E3v)δI,v +
dkk2Vℓ(q, k; E3I)
hℓn(k; E3I)
k2 + κ2I
Vℓ(q, k; E3I )≡ π
(A+ 1)
τ̄nc(q; E3I)× (12)
2(q, k; E3I) +
1(q, k
′; E3I)τnn(k
′; E3I)K
1(k, k
′; E3I)
The first term in the right-hand-side (rhs) of (11), which is non-zero only for
the virtual state, corresponds to the contribution of the residue at the pole.
The virtual states are limited by the cut of the elastic scattering amplitude in
the complex plane, corresponding to the second term in the rhs of Eq. (9). In
this case, the cut is given by the zero of the denominator of Gℓ2(q, k; E3v) [See
Eq. (4)], where −1 < y < 1 and q = k = −iκcut. With |E3v| = |εnc|+
2(A+1)
κ2cut,
we obtain the branch points, with the cut given by
2(A+ 1)
A + 2
|εnc| < |E3v| <
2(A+ 1)
|εnc|. (13)
For n − n−18C (A = 18), a virtual state energy can be found in the energy
interval between the threshold of the elastic scattering and the starting of the
above cut (13):
|εnc| < |E3v| < 1.9 |εnc| . (14)
0 100 200 300 400 500 600
90 120 150 180 210
| E19C| (keV)
Fig. 2. Three-body n − n−18C results for the first excited state, with respect to
the threshold (|E20C − E19C|) for varying
19C binding energies. Three-body bound
(virtual) states occur when |E19C| is approximately smaller (larger) than 170 keV.
s−wave results (solid line) are also presented in the inset (with dots). Results for
the p−and d−waves, divided by a factor 10, are shown with dashed and dotted
lines, respectively.
Figure 2 shows how the absolute value of the first excited three-body state
energy with respect to the two-body bound state, |E − Enc|, varies when
increasing the n − c bound-state energy. In this case, with 18C being the
core, we have 19C as the two-body bound subsystem. The solid line of Fig. 2
presents the s−wave results, with a close focus in the region of the threshold
(E = Enc) given by the inset figure. Although the Efimov states do not appear
in higher partial waves due to the existence of the centrifugal barrier, we have
also presented results for the virtual states of p− and d− waves in view of
their possible relevance for the elastic n−19C low-energy cross-section. Such
situation can happen when the s−wave virtual state is far from the elastic
scattering region.
The results shown in Fig. 2, valid for n−n−c system with n−c bound, together
with previous analysis [3,11], are clarifying that the behavior of Efimov states,
when one of the particles have a mass different from the other two, follows
the same pattern as found in the case of three equal-mass particles [8,10]. No
resonances were found for Eqs. (9) and (10) in the complex energy plane.
Next, for consistency, we present results for the s−wave elastic cross-sections,
which are in agreement with the above. The formalism for the partial-wave
elastic n− 19C scattering equations can be obtained from Eqs. (1) and (2), by
first introducing the following boundary condition in the full-wave spectator
function χn(~q):
χn(~q) ≡ (2π)
3δ(~q − ~ki) + 4π
hn(~q; E(ki))
q2 − k2i − iǫ
, (15)
where hn(~q; E(ki)) is the scattering amplitude, and the on-energy-shell in-
coming and final relative momentum are related to the three-body energy
Ei ≡ E(ki) by ki ≡ |~ki| = |~kf | =
[2(A+ 1)/(A+ 2)] (Ei − εnc). With the
same formal expressions (7) and (6) for τ̄nc and τnn, by using the definition
(13), the partial-wave scattering equation can be cast in the following single
channel Lippmann-Schwinger-type equation for hℓn:
n(q; Ei) = V
ℓ(q, ki; Ei) +
ℓ(q, k; Ei) h
n(k; Ei)
k2 − k2i − iǫ
. (16)
Virtual states and resonances correspond to poles of the scattering matrix on
unphysical sheets. So, the most natural method to look for them is to perform
an analytically continuation of the scattering matrix to the unphysical sheet.
For that one needs prior knowledge about the analytic properties of the kernel
of the integral equation, as well as the scattering matrix properties on the
unphysical sheet of the complex energy plane. Although such properties are
easy to derive in simple cases, they can be difficult to obtain in more complex
situations, which can put some restriction on the approach.
For the numerical treatment of Eq. (16), we consider the approach developed
in [8] to find virtual states and resonances on the second energy sheet associ-
ated with the lowest scattering threshold. Such approach does not require prior
knowledge of the analytic properties of the scattering matrix on the unphys-
ical sheet. The solution of the scattering equation is written in a form where
the analytic structure in energy is clearly exhibited. Then, it is analytically
continued to the unphysical sheet. The method relies on the calculation of an
auxiliary (resolvent) function [20], which has an integral structure similar to
the original scattering equation, but with a weaker kernel due to a subtrac-
tion procedure at an arbitrary fixed point k̄i. For real positive energies, this
subtraction point k̄i is identified with ki, such that the corresponding integral
equation does not have the two-body unitarity cut. For non-real or negative en-
ergies, k̄i can be any arbitrary positive real number. For convenience, k̄i = |ki|.
The final solution is obtained by evaluating certain integrals over the auxil-
iary function. In case of scattering solutions, the two-body unitarity cut is
introduced through these integrals. In the present case, we have the following
integral equation for the auxiliary function Γ, and the corresponding solution
for hℓn(q; Ei):
Γℓn(q, ki; Ei) =V
ℓ(q, ki; Ei) +
k2Vℓ(q, k; Ei)− k̄
ℓ(q, ki; Ei)
] Γℓn(k, ki; Ei)
k2 − k2i
hℓn(q; Ei) =Γ
n(q, ki; Ei) + Γ
n(q, k̄i; Ei)
Γℓn(k, ki; Ei)
k2 − k2i − iǫ
Γℓn(k, k̄i; Ei)
k2 − k2i − iǫ
. (17)
For the on-shell scattering amplitude, we have
hℓn(ki; Ei) = [1/Γ
n(ki, ki; Ei)−J ]
−1 (18)
Γℓn(k, ki; Ei)/Γ
n(ki, ki; Ei)− 1
k2 − k2i
+ iki
In order to obtain the bound and virtual energy states, two independent pro-
cedures have been used. The first one, by solving directly the homogeneous
coupled equations (1), (2), (9) and (10), looking for zeros of the corresponding
determinants. The other one, by verifying the position of the poles in the com-
plex energy plane of the scattering amplitude, hℓn(ki; Ei), given by Eq. (18) and
corresponding analytic extension to the second Riemann sheet. In this second
approach, we solve the corresponding inhomogeneous equation with on-energy-
shell momentum ki = +iκb (bound state, Ei = −|E3b|) and ki = −iκv (virtual
state, Ei = −|E3v|)[8].
By comparing the results of both approaches, we checked that they give con-
sistent results. However, for numerical stability and accuracy of the results,
particularly in the case of the search for Efimov states, when the absolute
values of the energies are very close to zero, the second approach is by far
much better.
Results for the total n−19C elastic cross-sections, obtained from dσ/dΩ =
|hn(~kf ; Ei)|
2, refering to bound or virtual states E∗, are presented in Fig. 3 as
functions of the CM kinetic energy,
K(ki) ≡ [(A+ 2) k
i ]/[2(A+ 1)] = E(ki)− Enc. (19)
0 25 50 75 100 125 150
10000
0 30 60 90 120150180
10000
0 100 200 300 400 500
 ] (keV)
 ] (keV)
Fig. 3. n−19C elastic cross sections (in barns) versus the CM kinetic energies
[Eq. (19) with A=18], for different 19C bound energies. In the left-hand-side frame we
show results for two cases that generate three-body energies close to the threshold:
E19C = −150 keV (main figure) and −180 keV (inset), producing respectively three-
-body bound and virtual states. Solid-line are obtained from (16), with dashed-line
from (20). In the rhs we have results for E19C = −500 keV.
Although each higher partial ℓ−wave have a virtual state, below the breakup
the cross-section is completely dominated by the s−wave [19]. In the frame
shown in the left-hand-side, we have two cases of energies close to the scatter-
ing threshold: n − 18C bound at −150 keV, giving an excited Efimov bound
state with E∗ = −150.12 keV; and n − 18C bound at −180 keV, producing
a virtual state with E∗ = −180.12 keV. In both two cases the cross-section
has a huge peak at zero energy due to the presence of the nearby pole. For
comparison, we also show (with dashed-line) the results obtained from the
following effective range expansion (approximately valid for small ki near the
elastic scattering threshold):
σ(ki) =
4πh̄2
1.9mn (Enc − E
∗) + h̄2k2i
2741 keV
E(ki)−E∗
barn, (20)
where we have used A =18, h̄2/mn =414.42 keV barn and Eq. (19). The
case not so close to the threshold (where Eq.(20) fails) is shown in rhs for
Enc = −500 keV, with
20C virtual energy E∗ = −568.73 keV.
The proximity of an Efimov state (bound or virtual) to the neutron and
neutron-core elastic scattering makes the cross-section extremely sensitive to
the corresponding S-matrix pole. We remark that if it will be possible to
dissociate a “samba-type” halo nuclei, like 20C, and measure the correlation
function in the two-body channel corresponding to a neutron and a bound n−c
system for small relative momentum, the information on the final state inter-
action as well as the halo structure will be clearly probed, as the counterpart
seen in the n− n correlation in the breakup of Borromean nuclei [21].
In conclusion, we analyzed the three-body halo system n−n−18C, where two
pairs (n−18C) are bound, and the remaining pair n − n has a virtual-state.
We study the trajectory of three-body Efimov states in the complex energy
plane. As shown, the energy of an excited Efimov state varies from a bound
to a virtual state as the binding energy of the subsystem n−18C is increased,
while keeping fixed the 20C ground-state energy and the virtual energy of
the remaining pair (n − n). In our approach we applied a renormalized zero-
range model, valid in the limit of large scattering lengths. Considering that
low-energy correlations, as the one represented by the Phillips line [22] (cor-
relation between triton and doublet neutron-deuteron scattering length), are
well reproduced by zero-range potentials [23], the present conclusions should
remain valid also for finite two-body interactions when the scattering length
is much larger than the potential range. On the numerical analysis, we should
remark that we have considered two approaches that give consistent results
for bound and virtual state energies. In the case of Efimov physics, where the
poles are very close to zero, the method considered in Ref. [8] was found to
give solutions with much better stability and accuracy in the scattering region
than by using a contour deformation technique.
The present results are extending to n−n−core systems the long ago conclu-
sion reached for three equal-mass particles [8,10]: by increasing the binding
energy of the n−core subsystem, an excited weakly-bound three-body Efimov
state moves to a virtual one and will not become a resonance. From Fig. 3, one
can also observe that the n−19C elastic cross-sections at low energies present a
smooth behavior dominated by the S−matrix pole corresponding to a bound
or virtual three-body state. In contrast with the above conclusion applied to
system where n−c is bound, we should observe that it was also verified that an
excited Efimov state can go from a bound to a resonant state (instead of vir-
tual state) in case of Borromean systems (with all the subsystems unbound),
when the absolute value of a virtual-state energy for the n − c system is in-
creased [12]. Actually, it should be of interest to extend the present analysis
of the trajectory of Efimov states to other possible two-body configurations,
with different mass relations of three-particle systems.
In view of the exciting possibilities of varying the two-body interaction, it can
be of high interest the results of the present study to analyze properties of
three-body systems in ultracold atomic experiments. For negative scattering
lengths the Efimov state goes to a continumm resonance when |a| is decreased,
as observed by the change of resonance peak in the three-body recombination
to deeply bound states towards smaller values of |a| by raising the tempera-
ture [24]. Alternatively, for positive a the recombination rate has a peak when
the Efimov state crosses the threshold and turns into a virtual state when
decreasing a. A dramatic effect will appear in the atom-dimer scattering rate
when the cross-section is dominated by the S-matrix pole near the scattering
threshold. We foreseen that the coupling between atom and molecular con-
densate will respond strongly to the crossing of the triatomic bound state
to a virtual one by changing a. Determined by the dominance of the cou-
pled channel interaction, new condensate phases of the atom-molecule gas are
expected. The proximity of the virtual trimer state to the physical region, im-
plying in a large negative atom-dimer scattering length, will warrant stability
to both condensates, while the positive atom-dimer scattering length, due to a
trimer bound state near threshold, make possible the collapse of the condensed
phases. Indeed, the occurrence of some interesting effects in the condensate
due to Efimov states near the scattering threshold have already been discussed
in Refs. [25]. We hope these exciting new consequences of Efimov physics can
be explored experimentally in the near future.
LT thanks Prof. S.K. Adhikari for helpful suggestions. We also thank Fundação
de Amparo à Pesquisa do Estado de São Paulo and Conselho Nacional de
Desenvolvimento Cient́ıfico e Tecnológico for partial support.
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