PHYSICAL REVIEW A 66, 052702 ~2002!
Scaling limit of virtual states of triatomic systems

M. T. Yamashita,1 T. Frederico,2 A. Delfino,3 and Lauro Tomio4
1Laboratório do Acelerador Linear, Instituto de Fı´sica, Universidade de Sa˜o Paulo, Caixa Postal 66318, CEP 05315-970,

São Paulo, Brazil
2Departamento de Fı´sica, Instituto Tecnolo´gico de Aerona´utica, Centro Te´cnico Aeroespacial, 12228-900 Sa˜o Josédos Campos, Brazil

3Departamento de Fı´sica, Universidade Federal Fluminense, 24210-340 Nitero´i, Rio de Janeiro, Brazil
4Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, 01405-900 Sa˜o Paulo, Brazil

~Received 9 March 2002; revised manuscript received 5 June 2002; published 8 November 2002!

For a system with three identical atoms, the dependence of thes-wave virtual state energy on the weakly
bound dimer and trimer binding energies is calculated in the form of a universal scaling function. The scaling
function is obtained from a renormalizable three-body model with a pairwise Dirac-d interaction. The thresh-
old condition for the appearance of the trimer virtual state was also discussed.

DOI: 10.1103/PhysRevA.66.052702 PACS number~s!: 11.80.Jy, 03.65.Ge, 21.45.1v, 21.10.Dr
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I. INTRODUCTION

Weakly bound three-body zero-angular-momentum sta
appear in a three-boson system, with the number of st
growing to infinity, condensing at zero energy as the p
interactions are just about to bind two particles in thes wave.
These three-body states are known as the Efimov states@1,2#.
Their wave functions, loosely bound, extend far beyo
those of normal states and dominate the low-energy sca
ing phenomena in these systems. The Efimov states h
been studied in a number of model calculations@3–5#, in
atomic and nuclear systems, without yet a clear experime
signature of their occurrence@2,6–10#.

Actually, the search of Efimov states in atomic system
becoming more appealing, due to the experimental real
tion of Bose-Einstein condensation@11#, and due to the pos
sibility of altering the effective scattering length of the low
energy atom-atom interaction in the trap, from large nega
to large positive values crossing the dimer zero-bindi
energy value, by using an external magnetic field@12#. This
possibility of changing the two-body scattering length
large values, as recently shown in Ref.@13#, can alter in an
essential way the balance between the nonlinear first
terms of the mean-field description presented in the eq
tions that model Bose-Einstein condensed gases@14#. This
can certainly open new perspectives for theoretical and
perimental investigations related to the many-body beha
of condensate systems. Even in systems where the oc
rence of an excited bound Efimov state has shown to
doubtful or even not possible, as for example, in the cas
halo nuclei like20C or 18C ~seen as a core with a halo of tw
neutrons! @8#, one can verify the occurrence of three-bo
virtual states. The physics of these three-body system
related to the unusually large size of the wave function co
pared to the range of the potential. Thus, the detailed form
the short-ranged potential is not important for the three-b
observables@15#, which gives to the system universal pro
erties, defined by few physical scales@8#. Strictly speaking,
in the limit of a zero-range interaction, the three-body syst
is parametrized by the physical two- and three-body sca
which are identified with the two-body scattering lengths a
one three-body binding energy@9,16#. The physical reason
1050-2947/2002/66~5!/052702~7!/$20.00 66 0527
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for the sensibility of the three-body binding energy to t
interaction properties comes from the collapse of the sys
in the limit of a zero-range force, which is known as th
Thomas effect@17#.

In the present work, we analyze the possibility that
excited trimer state becomes a virtual state, when the ph
cal scales of the system are changed. This is expecte
occur, for example, near the limit when the two-body sc
tering length goes from large positive to large negative val
the corresponding two-body energy is close to zero and g
from a bound to a virtual state, with the appearance of m
bound and virtual three-body states. The three-body virt
state energy is a pole of theS matrix in the second sheet o
the complex energy plane. In a general case, as the stre
of the two-body potential diminishes, the pole moves
wards the first energy sheet to become a bound state@5#.
More recently, this behavior of the Efimov state going to
virtual state with the increase of the strength of the inter
tion has been confirmed in realistic calculation of the heliu
trimer @18#. Here, we study another aspect of the emerge
of the s-wave virtual state from an Efimov state:it appears
when the ratio between the dimer and trimer binding en
gies grows. This approach goes beyond a previous analy
of excited three-body bound states with short-range inte
tions, that was performed in Ref.@9#. In Ref. @9#, a scaling
function was introduced to analyze the behavior of bou
Efimov states when modifying the triatomic physical scal
Essentially, we are extending to the second sheet of the c
plex energy plane~to include virtual trimer states! a previous
investigation on a universal scaling mechanism that was
plied to two- and three-body bound states@9,10#. The exten-
sion of the scaling function to the second energy shee
performed by following the Efimov states as they move fro
bound to virtual, according to the variation of the ratio of t
dimer to trimer bound-state energies. On the other hand
we present the discussion through a universal scaling me
nism with the results in dimensionless units, all the conc
sions apply equally to any low-energy three-boson syst
For the regularization and renormalization of the zero-ran
model, we compare two different approaches: by using
momentum cutoff parameter@9# and via kernel subtraction
@16,19,20#. As the two-body energy goes to zero~or equiva-
©2002 The American Physical Society02-1



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YAMASHITA et al. PHYSICAL REVIEW A 66, 052702 ~2002!
lently the regularization parameter goes to infinity!, we con-
clude that the results of both methods do not differ.

The paper is organized as follows. In Sec. II, we gene
ize the scaling function defined in Ref.@9# to include virtual
trimer states. In this section, we also revise the connec
between the Thomas and Efimov effects, while introduc
our notation and the homogeneous integral equation for
Faddeev component of the vertex of the wave function
zero-range potential. In Sec. III we present our main num
cal results. In the Sec. III A, we present the subtracted
mogeneous Faddeev equation that we have used for d
mining the trimer bound and virtual states, and we brie
explain how the renormalization method of Refs.@19,20# im-
plies in the subtracted three-body equation first formulate
Ref. @16#. In the Sec. III B, we present our new numeric
results for the virtual-state energies, including the previo
bound-state results and we compare, as well, the results
tained by using the sharp-cutoff and the subtraction schem
Comparison with other calculations are also discussed.
conclusions are summarized in Sec. IV.

II. THOMAS-EFIMOV EFFECT AND THE GENERALIZED
SCALING FUNCTION

In this section, we introduce the generalization of the sc
ing function defined in Ref.@9#, to be used in the secon
energy sheet of the trimer energy. In order clarify this ext
sion, and to define our notation, we begin by revising
main findings of Refs.@9,21#.

The two-boson system in the limit of a zero-range int
action has only one physical scale, which one can choos
the scattering lengtha or the energy of the bound or virtua
state. The two-bodys-wave scattering amplitude in units o
\5m51 is parametrized as a function of the momentumk,
by f (k)5(k cotd02ik)21, where thes-wave phase shiftd0
is given byk cotd052a2111

2r0k
21•••, and r 0 is the effec-

tive range. Fora.0, the two-body system is bound; othe
wise, for a,0, it is virtual. A short-range potential is cha
acterized byr 0uau21!1 and, in this case,f (k)5(2a21

2 ik)21 and a2156AE2 (1 for bound and2 for virtual
state!.

The three-boson system for,50 in three dimensions col
lapses whenr 0→0 with a fixed two-body scale, which i
known as the Thomas effect@17#. Thus, the three-body sys
tem has a characteristic physical scale independent of
two-body ones@16#. In one and two space dimensions, t
collapse is absent@15#. In the limit where the binding energ
of the two-boson system goes to zero, the three-boson
tem has an infinite number of bound Efimov states@1# con-
densing at zero energy. The Thomas and Efimov effects w
shown to be physically equivalent@21#, since in both cases
the ratio between the interaction range and the two-b
scattering length goes to zero.

The integral equation for the Faddeev components,f, of
the three-boson bound-state vertex, for,50, with the zero-
range interaction, needs a momentum cutoffL of the order
of r 0

21, due to the Thomas collapse. According to Ref.@21#,
using units ofL51, we rescale the momentum variables a
the two- and three-body binding energies, respectively, s
05270
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that pW 5LxW , qW 5LyW , E25L2e2, andE35L2e3. In this di-
mensionless variables, after redefiningf as x(xW )
[L3/2f(pW ), we obtain the integral equation@21,7,8#:

x~yW !5
2p22

6Ae22Ae31 3
4 yW 2

E d3x
u~12uxW u!

e31yW 21xW21yW•xW
x~xW !.

~1!

The number of three-body bound states, given by the va
of e3 that satisfies Eq.~1!, grows without limit whene2

decreases to zero;e35e3
(N) (N50,1,2, . . . ), with

e3
(N)/e3

(N11)'500 @1#. They are the energies of the Efimo
states, in units ofL51. But, the limit of e2 going to zero
can be realized either byE2→0 ~with a fixed L) or by L
;r 0

21→` ~with E2 fixed!. In this last case, the range of th
interaction is set to zero and the system collapses;E3

(0)

5e3
(0)L2→`. This is known as the Thomas collapse of t

three-body ground state. Therefore, the Thomas and the
mov states are given by the same limite2→0 of Eq.~1!, and
are related by a scale transformation@21#.

Now, the concept of the scaling function is introduc
according to Ref.@9#. For a nonvanishinge2, the solutions of
Eq. ~1! defines the dimensionless three-body energies
functions of 6Ae2; e3

(N)[e3
(N)(6Ae2). Using theNth en-

ergy to obtainL, thenL25E3
(N)/e3

(N) , and

E3
(N11)5E3

(N)
e3

(N11)~j!

e3
(N)

, ~2!

where j[6Ae256(E2e3
(N)/E3

(N))1/2. In Eq. ~2!, the two-
and three-body physical scales determineE3

(N11) , the next
excited state aboveE3

(N) . In Ref. @9#, E3
(N) was identified

with the three-body scale, as any stateN works equally well
to set the trimer scale. However, we will be interested in
two most excited three-body states that, in practice, we
going to identify with the ground- and the first-excited-sta
in triatomic systems. This identification is unambiguous b
cause, withN and N11 being two consecutive excite
states, the limit

E3
(N11)

E3
(N)

5 lim
N→`

e3
(N11)~j!

e3
(N)

5FS 6A E2

E3
(N)D ~3!

exists and defines the scaling functionF @8,9#. A qualitative
argument to explain the scaling limit has been provided
Ref. @9# based on the notion of the long-range potent
@1,2,22#.

In the following, we provide the generalization of th
scaling function~3!, which is obtained by extending the fo
malism to the second sheet of the three-body complex en
plane. In the present approach, we only consider the t
body subsystem as bound. For this purpose, we define
general scaling functionK, given by
2-2



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SCALING LIMIT OF VIRTUAL STATES OF . . . PHYSICAL REVIEW A 66, 052702 ~2002!
KSA E2

E3
(N)D [6AE3

(N11)2E2

E3
(N)

56Ae3
(N11)2e2

e3
(N)

.

~4!

This defined functionK has its values on the imaginary ax
of a three-body momentum space; a space that is defi
with origin at the point in which the energies of the thre
body system and the bound two-body subsystem are e
(E35E2). In this respect, relative to the bound subsyste
we can define bound and virtual states for the three-b
system.K assumes a negative value for a three-body virt
state and a positive value for a three-body bound state. S
matically, we represent in Fig. 1 the energies of the two- a
three-body system in the complex energy plane. The t
body subsystem is bound and the three-body system ca
bound or virtual, with the energies given, respectively,
e3B and e3V . Through the elastic cut~corresponding to the
atom-dimer elastic scattering!, one defines two sheets; in th
first sheet, we have the three-body bound-state energ
Re(e)52e3B ; in the second sheet, we have the three-bo
virtual-state energy at Re(e)52e3V , as illustrated in Fig. 1.

We would like to add one more comment to this sectio
The existence of a three-body scale implies in the lo
energy universality found in three-body systems, or corre
tions between three-body observables@23,16#. In the scaling
limit, one has

O~E,E3 ,E2!5~E3!hA~AE/E3,AE2 /E3!, ~5!

whereO is a general observable of the three-body system
energyE, with dimension of energy to the powerh. The
scattering amplitude of the elastic processa1bc→a1bc,
f 35AE3

21F(AE/E3,AE2 /E3) for E5E2, implies that the
scattering length is given by a functiona3

5AE3
21F(AE2 /E3). In the three-nucleon system this orig

nates the ‘‘Phillips plot,’’ the correlation between the doub
neutron-deuteron scattering length and the triton energy@24#.
The scaling functions, Eqs.~3! and ~4!, express the correla

FIG. 1. Schematical representation of the complex energy pl
in our dimensionless units.e3B and e3V are, respectively, pictoria
representations of the positions of the three-body bound-
virtual-state energies in the first and second three-body en
sheet. The three-body cut is shown for Re(e).0. The elastic cut
~the narrow one! is shown with the origin at Re(e)52e2, wheree2

is the energy of the two-body bound state.
05270
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tion between the excited- or virtual-state energies of the
mer and its ground-state energy, which can be understoo
particular cases of Eq.~5!.

III. NUMERICAL RESULTS FOR VIRTUAL AND BOUND
TRIMERS

In this section, we present our main results for the trim
bound and virtual states. With the sake to be complete,
first briefly sketch a new derivation of the subtracted eq
tions that were numerically solved.

A. Renormalization and subtracted equations

The homogeneous form of the subtracted Faddeev e
tion @16# for the bound three-boson system with a zero-ran
interaction is given by

x~yW !5
2p22

6Ae22Ae31 3
4 yW 2

E d3xS 1

e31yW 21xW21yW•xW

2
1

11yW 21xW21yW•xW
D x~xW !, ~6!

which is written in units such that the three-body subtract
energy ism (3)

2 51. It has a similar form as that of Eq.~1!
with a different regulator, which expresses the physical c
dition at the subtraction point.

We briefly explain below the main physical steps to der
the three-body renormalized equation@16# used in our nu-
merical calculation of the scaling functions through Eq.~6!
for the bound state and its analytic continuation to the sec
energy sheet for the virtual state. We begin from the gen
Lippman-Schwinger equation expressed in a subtracted f
@19#:

TR~E!5TR~2m2!1TR~2m2!@G0
(1)~E!

2G0~2m2!#TR~E!, ~7!

where TR(2m2) is the T matrix at a given energy scale
2m2 ~negative energy, for convenience!, G0

(1)(E)5@E
2H01 id#21, andH0 is the free Hamiltonian. Equation~7!
defines the renormalizedT matrix in which TR(2m2) is
known and replace the original ill-defined potentialV:

TR~2m2!5@12VG0~2m2!#21V. ~8!

The renormalizedT matrix does not depend on the arbitra
subtraction point2m2 ~oncedV/dm250), which implies in
a Callan-Symanzik-type@19,20# equation forTR(2m2):

d

dm2
TR~2m2!5TR~2m2!@G0~2m2!#2TR~2m2!. ~9!

This expresses the renormalization-group invariance of
subtracted equation.

To solve Eq.~7! for the three-bodyT matrix TR
(3)(E), a

dynamical assumption has to be made at a particular sub

e,

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YAMASHITA et al. PHYSICAL REVIEW A 66, 052702 ~2002!
tion point 2m (3)
2 , where we assume that the three-bodyT

matrix is equal to the driving term, which is given by th
sum of the pairwise two-bodyT matrices. Thus, at the energ
2m (3)

2 , it is assumed that the three-body multiple-scatter
series vanishes beyond the driving term. Observe that th
not true for a regular finite-range potential, only in the lim
of m (3)→`. However, in the scaling limit, in fact, the actu
value ofm (3) tends to infinity such thatE2 /m (3)

2 goes to zero,
as it be will be clear in our numerical calculations.

With our assumption, theT matrix at the subtraction poin
m (3) is given by

TR
(3)~2m (3)

2 !5(
( i j )

TR( i j )
(2) S 2m (3)

2 2
qk

2

2mk( i j )
D , ~10!

where (i , j ,k)5(1,2,3), ~2,3,1!, ~3,1,2!. The summation is
performed over all pairs and the renormalized two-bo
T-matrix elements for the pair (i j ) are given by

^PW 8uTR
(2)(E)uPW &51/@2p2(6AE21 iAE)#. The argument of

the two-bodyT matrix is the center-of-mass pair energ
whereqk is the Jacobi relative momentum canonically co
jugated to the relative coordinate of the particlek to the
center of mass of the pair (i j ), and mk( i j ) is the reduced
mass.

Using Eqs.~7! and ~10! and after some straightforwar
manipulations, the equations for the Faddeev componen
the T matrix at the bound-state pole give Eq.~6!, which has
a natural momentum scale given bym (3)

2 . In principle,m (3)
2

can be varied without changing the content of the theory
long as the three-bodyT matrix at the new subtraction energ
m (3)

2 is found from the solution of Eq.~9! with the boundary
condition Eq.~10!; and consequently, Eq.~6! should be con-
veniently rewritten. In the scaling limit, Eqs.~1! and ~6!
produce the same results~as we are going to illustrate nu
merically!, since they are solved fore2 going to zero, and the
detailed form of the regularization implied in both equatio
is not important anymore. However, Eq.~6! has conceptua
and practical advantages over Eq.~1!, namely, it is explicitly
renormalization-group invariant and also regularized.

To simplify the notation of Eq.~6!, we introduce anothe
definition related to the two-body energy;k2[6Ae2, where
1 refers to bound and2 to virtual two-body-state energies
After partial-wave projection of Eq.~6!, the s-wave integral
equation for the three-boson system is

xs~y!5t~y;e3 ;k2!E
0

`

dx x2G~y,x;e3!xs~x!, ~11!

where

t~y;e3 ;k2!52
2

p FAe31
3

4
y22k2G21

, ~12!
05270
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-

of

s

G~y,x;e3!5~e321!

3E
21

1

dz
1

~e31y21x21yxz!~11y21x21yxz!
.

~13!

For the,th angular-momentum three-body state, the Thom
collapse is forbidden if,.0; consequently, no regulariza
tion is required and the integration over momentum can
extended to infinity even in the limitm (3)→`. For,.0, the
original Skornyakov and Ter-Martirosian equation@25# is
well defined, and the three-body observables are comple
determined by the two-body physical scale corresponding
E2. One finds examples of the disappearance of the dep
dence on the three-body scale inp-wavevirtual states, for the
trineutron system whenn-n is artificially bound@26,27#, and
in three-body halo nuclei~represented as a core with a ha
of two neutrons! @28#.

The analytic continuation to the second energy sheet
the scattering equations for separable potentials, is discu
in detail by Glöckle, in Ref.@26#. The particular case of the
zero-range three-body model@25# is also given in Ref.@29#.
On the second energy sheet, the integral equations are
tained by the analytical continuation through the two-bo
elastic scattering cut corresponding to the atom-dimer s
tering. The elastic scattering cut comes through the pole
the atom-atom elastic scattering amplitude in Eq.~12!. We
perform the analytic continuation of Eq.~11! to the second
energy sheet. By substituting the spectator functionxs(y) by
x̄s(y)[(e3v2e21 3

4 y2)xs(y), wheree3v is the modulus of
the virtual-state energy, the resulting equation in the sec
energy sheet is given by

x̄s~y!5 t̄~y;e3v ;k2!
4pk3v

3
G~y,2 ik3v ;e3v!x̄s~2 ik3v!

1 t̄~y;e3v ;k2!E
0

`

dx x2
G~y,x;e3v!x̄s~x!

e3v2e21
3

4
x2

, ~14!

where the on-energy-shell momentum at the virtual stat

k3v[A 4
3 (e3v2e2), and

t̄~y;e3v ;k2![2
2

p FAe3v1
3

4
y21k2G . ~15!

The cut of the elastic amplitude given by the exchange
one atom between the different possibilities of the bou
dimer subsystems is near the physical region due to the s
value of e2. This cut is given by the values of imaginaryx
between the extreme poles of the free three-body Gre
function G(y,x;e3v), given by Eq.~13!, which appears in
the right-hand side of Eq.~14!,

e3cut1y21x21xyz50, ~16!

with 21,z,1, y5x52 ikcut , and e3cut5
3
4 kcut

2 1e2.
With the above, the cut satisfies
2-4



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SCALING LIMIT OF VIRTUAL STATES OF . . . PHYSICAL REVIEW A 66, 052702 ~2002!
4e2.e3cut.
4
3 e2 . ~17!

The virtual-state energye3v in the second energy sheet
found between the scattering threshold and the cut,e2,e3v
, 4

3 e2.

B. Scaling plots

It is usual to analyze how the Efimov states arise by va
ing the strength of the interaction to change the value of
two-body binding energy. In our case, instead of this pro
dure, we change directly the value of the energies in unit
m51, and by doing this we calculate thes-wave three-body
energy evolution in the complex energy plane, correspond
to the bound and virtual triatomic states from Eqs.~11! and
~14!, respectively. Ase2 goes to zero, a crescent number
weakly bound~in units of m51) Efimov states appear. Th
Thomas-Efimov limit fore2 going to zero is clearly seen i
Fig. 2, where we plote3

(N) as a function ofe2. In this figure
we display only the energies of the first three states. T
main purpose of Fig. 2 is to show the real nature of
energies of the Thomas-Efimov states. The small circles
triangles correspond, respectively, to the first and second
cited virtual-state energies, which begin at the cut from
one-particle exchange mechanism that givese3v5(4/3)e2
~shown in the figure by the dotted line!. The threshold, from
which the virtual three-body states arise, are exhibited
down arrows (↓). When the two-body energy is enough for
trimer bound state to exist, then a decrease ine2 allows the

FIG. 2. Trimer energiese3 as functions of the dimer bound-sta
energye2. The trimer ground-state energy (e3

(0)) is shown by the
curve with crosses; the first excited bound state (e3

(1)) is shown by
the curve with squares; and the second excited bound state (e3

(2)) by
the curve with diamonds. The behavior of two trimer virtual-sta
energies,e3v

(1) ~small circles! and e3v
(2) ~small triangles!, are also

shown as functions of the two-body energy, varying from t
thresholde35e2 ~solid line! to the threshold for the one-particle
exchange cute35

4
3 e2 ~dotted line!. All the energies are given in

arbitrary units.
05270
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e
-

of

g

f

e
e
nd
x-
e

y

virtual state to appear from the one-particle-exchange
Further decrease ine2 favors the appearance of the excite
state, which emerges from the second energy sheet to the
one at the threshold valuee35e2 ~solid line!, indicated by
the up arrow (↑). The critical value ofe2 is given by the
ratio (e2 /e3

(N))1/250.38 where the excited state is labeled
N11, and in the figure is indicated by the up arrow. Th
figure also strongly suggests that the Thomas-Efimov st
cannot be completely understood only through the abso
value ofE2 itself, because the critical value for the appe
ance of the (N11) excited state depends only on the ra
E2 /E3

(N)5e2 /e3
(N) , which is independent of the absolu

scale. Therefore, to show that this argument is universal,
study the functionE3

(N11)/E25e3
(N11)/e2 as a function of

E2 /E3
(N)5e2 /e3

(N) , where the (N11) state can be virtual o
bound. This study is presented in Fig. 3.

The plot of Fig. 3 is constructed with the results for th
first and second Thomas-Efimov states. This plot practic
coincides with the corresponding one obtained from the s
ond and third states~not shown!. Figure 3 shows a universa
route for the energy of the (N11) trimer state in the com-
plex energy plane, from the second energy sheet to the
one, as the ratioE2 /E3

(N)5e2 /e3
(N) decreases. The three

body virtual-state energy reaches 4E2/3 at E2 /E3
(N)50.71.

Also realistic calculations for the helium trimer are availab
and are displayed in this figure. The agreement between
calculations and the realistic ones, shows the significanc
our scaling picture. Unfortunately, there is not yet, to o
knowledge, realistic calculations of the virtual state in h
lium trimer or even in any other weakly bound three-bos

FIG. 3. Ratio of the trimer excited or virtual (N11)th state
energy as a function of the ratio of the dimer energy and trimerNth
bound-state energy. The results for the trimer excited bound-s
energies are shown by the solid curve, and the virtual-state ene
are shown by the dotted curves. Our calculations show that
results forN50 andN51 practically coincide. The symbols rep
resent results from other calculations: empty squares (s wave! and
empty circles (s1d waves! are from Ref.@30# ~for N50); crossed
squares are from Ref.@31#; the crossed circle is from Ref.@32#; the
triangle is from Ref.@33#; and the lozenge is from Ref.@34#.
2-5



siz
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i

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otal
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ity,
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om
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dy
ion

an
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tate
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ide
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YAMASHITA et al. PHYSICAL REVIEW A 66, 052702 ~2002!
system, in which our route should also apply. We empha
that although we have presented results only for the sec
and third Thomas-Efimov states, the scaling limit is prac
cally approached as we see in Fig. 3. We expect that go
further in diminishing the absolute value ofE2, the new
excited states will also follow the same route. The claim
of course, that the route is universal for all states in
scaling limit.

The results for the energy of the excited Efimov state
4He3 molecule given byK(z) (z5@E2 /E3

(N)#1/2), obtained
by solving Eqs.~1!, ~11!, and ~14! in the scaling limit, are
compared to the realistic model calculations also prese
in Fig. 4. The homogeneous integral equation with the sh
cutoff momentum regulator, which generalizes Eq.~1! for the
virtual trimer state, is not written explicitly in the text as
can be easily derived. We observe the ratio@(E3

(N11)

2E2)/E3
(N)#1/2 depends onz for realistic models as well. In

this plot we only show results for a bound dimer andN
50. The extreme limit ofz allowing the excited state ar
given by K(z)50, which givesz50.38. The solution of
Eqs.~1! and~11! in the scaling limit qualitatively reproduce
the results for several interatomic potentials. A deviation
seen forz'0.4, which is due to corrections from the fini
range of the potential. The excited (N11) three-body state
becomes virtual forE2 /E3

(N).0.145~as seen in Fig. 3!, im-
plying that E3

(N),6.9\2/(ma2) in this case. This threshold
value agrees with the value previously found in Refs.@8,9#,
recently confirmed in Ref.@35#, for the condition of the dis-
appearance of the excited trimer state in the limit of a ze
range interaction. Let us stress that the regulariza
schemes used in Eqs.~1! and~11! are consistent not only fo
the calculation of the bound excited trimer energies but a
for the virtual trimer energies, as shown in Fig. 4. The sm
difference between the two regularization schemes tend
vanish fast for higher values ofN.

FIG. 4. Results for the trimer bound and virtual excited (N
11)th state energies, scaled by theNth bound-state energy. A com
parison between calculations performed with cutoff and subtrac
methods for the regularizations is given forN50. We also presen
results from other calculations, as described in Fig. 3.
05270
e
nd
-
g

,
e

n

ed
rp

s

-
n

o
ll
to

IV. CONCLUSIONS

Natural scales determine the physics of quantum fe
body systems with short-range interactions. The phys
scales of three interacting particles, in the state of zero t
angular momentum, are identified with the bound or virtu
subsystem energy and the ground-state three-body bin
energy. The scaling limit is found when the ratio between
scattering length and the interaction range tends to infin
while the ratio between the physical scales are kept fix
This defines a scaling function for a given observable. Fr
the formal point of view, we showed the relation of the sc
ing limit and the renormalization aspects of a few-bo
model with a zero-range interaction, through the derivat
of subtracted three-bodyT-matrix equations that are
renormalization-group invariant.

In the present work, we investigate the behavior of
excited Thomas-Efimov state as the binding energy of
subsystem increases with respect to the energy of the
lower bound three-body state. As shown, by allowing t
two-body binding energy to increase in respect to the thr
particle ground-state energy, the excited three-body state
appears, and a corresponding three-body virtual s
emerges. The threshold for the three-body virtual state
found to be at the energy of the weakly bound trimer equa
6.9\2/(ma2) for large positive scattering lengthsa. The de-
pendence of thes-wave virtual-state three-body energy o
the two- and three-atom ground-state binding energies is
culated in the limit of a zero-range potential in a form of
universal scaling function. The scaling plots are an use
tool to classify observables and provide first guess to gu
realistic calculations, as well as for planning experimen
with the aim of looking for weakly bound excited state
triatomic molecules.

The results of the present study can also be particul
relevant to the interpretation of experiments in atomic co
densation, in which the effective atom-atom scattering len
can be altered from negative to positive, in a wide range
values crossing zero-energy bound dimer@12#. For large
positive scattering lengths, our estimate gives the thresh
for the zero-binding trimer state, which allows to settle t
experimental conditions for an investigation of the Efim
effect, and search for their influence on the observables
condensed systems. On the other hand, large negative
body scattering lengths have been recently investigate
Ref. @13#. There is the possibility that the observed discre
ancy related with previous theoretical predictions can h
their explanations in three-body effects as well, because la
two-body scattering lengths give the conditions where thr
body ~bound or virtual! Efimov states are likely to occur.

ACKNOWLEDGMENTS

We would like to thank Fundac¸ão de Amparo a` Pesquisa
do Estado de Sa˜o Paulo~FAPESP! and Conselho Naciona
de Desenvolvimento Cientı´fico e Tecnolo´gico ~CNPq! for
partial support.

n

2-6



ys

A

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.

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M

A.

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ys

ys.

v.

.

. A

r.

s.

s.

nt

SCALING LIMIT OF VIRTUAL STATES OF . . . PHYSICAL REVIEW A 66, 052702 ~2002!
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