Charged three-body system with arbitrary masses near conformal invariance
A. Delfino,1T. Frederico,2and Lauro Tomio3,4 1
Instituto de Física, Universidade Federal Fluminense, 24210-900 Niterói, RJ, Brazil 2
Departamento de Física, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil 3
Instituto de Física Teórica, Universidade Estadual Paulista (UNESP), 01140-070 São Paulo, SP, Brazil 4
Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150, 22290-180 Rio de Janeiro, RJ, Brazil
共Received 10 June 2009; published 30 November 2009兲
Within an adiabatic approximation to the three-body Coulomb system, we study the strength of the leading-order conformaly invariant attractive dipole interaction produced when a slow charged particle q3
共with massm3兲is captured by the first-excited state of a dimer关with individual masses and charges共m1,q1兲
and 共m2,q2= −q1兲兴. The approach leads to a universal mass-charge critical condition for the existence of three-body level condensation,共m1−1+m2−1兲/关共m1+m2兲−1+m
3 −1兴⬎兩
q1/共24q3兲兩, as well as the ratio between the geometrically scaled energy levels. The resulting expressions can be relevant in the analysis of recent experi-mental setups with charged three-body systems, such as the interactions of excitons, or other matter-antimatter dimers, with a slow charged particle.
DOI:10.1103/PhysRevA.80.052509 PACS number共s兲: 36.10.⫺k, 03.65.Ge
I. INTRODUCTION
In view of the actual experimental possibilities, we recall some general characteristics of three-body charged systems with arbitrary masses and two different charges, in order to derive the charge-mass dependence of the leading-order strength of the attractive dipole interaction produced by a bound two-body subsystem 共with individual chargesq1 and
q2= −q1兲 in the third particle with charge q3. We start our investigation by considering the old and well-known case of the interaction of a positronium 共Ps, a bound state of an electrone−⬅−eand a positrone+⬅+e, whereeis the abso-lute value of the electron charge兲with a spectator electron. This is the ionized negative positronium共Ps−兲. Early calcu-lations, by Wheeler in 1946 关1兴, have already predicted a bound state for such system, confirmed by Mills in 1981关2兴. Further investigations on the properties ofe−e+e−system 关3–6兴, as well as on other Coulombic three-body systems, since 1960s up to recent years关7–11兴, have also been moti-vated by the increasing interest in matter-antimatter interac-tions 关12兴. Up to 1995, the theoretical and experimental ad-vances in understanding Coulombic three-body systems and matter-antimatter interaction can be found, respectively, in two reviews关10,12兴. As emphasized in关12兴, small number of leptons, electrons, muons and their antiparticles, are impor-tant to test fundamental theories of quantum electrodynam-ics; and systems with small number of protons and antipro-tons can also provide relevant tests of the strong interaction. The actual interest on the properties of few-body charged systems is evidenced by the recent report on the production of a molecular dipositronium Ps2 关13兴. For a recent review, particularly concerned on the stability of quantum charged few-body systems, see 关11兴. In Ref. 关4兴, Rost and Wintgen explored and classified the Ps−dynamics considering the ob-servation that it has a molecular structure similar to the ion-ized hydrogen moleculeH2+关共pe−兲p兴. They have also reported the existence of a 1S resonance pattern unknown in three-body Coulomb systems. Such results, combined with results obtained for the hydrogen ionH−关共pe−兲e−兴, lead them to
sug-gest the existence of a similar resonance spectrum for all
ABACoulomb systems with charge and mass ratios such that 兩qA/qB兩= 1 and mA/mBⱖ1, respectively. More recently, by considering a molecular adiabatic 共MA兲 treatment for the Ps−, it was also reported an accumulation of three-body reso-nancesabovethe two-body threshold关5兴.
Surprisingly, besides the number of recent studies con-cerned with few-charged quantum systems关11兴, and the rec-ognized relevance of a mass-charge universal relation in view of actual experimental facilities 关4兴, a particular straightforward condition relating charges and masses for a general three-body system 共with arbitrary masses兲 is still missing in the literature. In the case of exotic molecular three-body systems, where the subsystem is hydrogenlike 共withe−replaced by−or −兲, it was shown in关14兴that a relation for the structure of the spectrum can be obtained in the frame of Born-Oppenheimer approximation, following spectral properties of long-ranged 1/R2 interactions 关15兴, which are known to be conformally invariant关16,17兴.
The study of resonance patterns, which can occur in few-body interactions, became very relevant in trapped ultracold atom experiments, as the presence of several resonances at experimentally accessible magnetic fields can allow precise tuning of atom-ion interaction 关18兴. Theoretical predictions such as the increasing number of three-body bound states when the two-body scattering length goes to infinity, known as Efimov effect 关19兴, can actually be checked experimen-tally in ultracold atomic laboratories 关20兴. The spectrum of Efimov states, exhibiting a geometrical scaling, is generated by an attractive effective potential proportional to 1/R2, whereRis the distance of one of the particles to the center of mass of the remaining pair, considering short-range two-body interactions 关19,21兴 共on the scaling mechanism and conformal invariance behind this effect, see关16,17,22兴兲. As it will be shown, the long-ranged Coulombic interactions, for certain configurations of three-charged particles, can exhibit the same kind of effective interaction.
For recent relevant applications in ultracold laboratories, of a study with three-charged systems, we can mention the possibility of exciton 关23,24兴 or positronium 关25,26兴
densed gas interacting with a charged particle. In case of excitons 共electron-hole bound pair in a semiconductor兲, the electron and hole in the interacting pair can acquire effective individual masses distinct from the free-electron mass 关24兴. When interacting with a slow charged particle, a charge-mass dependent resonance pattern should emerge. So, well-based simple charge-mass conditions can be very helpful to analyze the relation between effective masses and the observed spec-trum.
Motivated by the above discussion we show examples in atomic physics of charged three-body systems in which ap-pears an effective long-range 1/R2 potential, where the strength is modulated by the arbitrary individual masses and charges, restricted to the condition that the bound subsystem is neutral. A robust mass-charge critical condition is derived for the weakly bound three-body spectrum below the state
n= 2 of the subsystem, in case of arbitrary masses. Here, we should observe that the degeneracy between the 2S and 2P
levels of the chargeless two-body subsystem is broken by the dipole potential. When it is attractive it rises to the spectrum
below关4兴that we consider; the repulsive part is responsible for a set of resonancesabove the 2S− 2P state 关5兴. The ap-proach is strictly valid for the cases where the dimer is made up withpointlike charged particles. However, it can also be taken as a first approach when the charged particles are more complex objects.
The basic equations of our formalism, in the adiabatic approximation, are given in the next section, which lead to the effective 1/R2 long-range interaction and a spectrum geometrically scaled. In Sec.III, after analyzing the range of values for the strength of the effective interaction in terms of the particle masses, we illustrate the main results with ex-amples. A general discussion on the applicability of the present approach and a summary of our conclusions are pro-vided in Sec.IV.
II. FORMALISM
In this section, we present the basic formalism for a gen-eral charged three-body system, where the masses 共m1,m2,m3兲 can be arbitrary, and the charges are such that we have a bound chargeless two-body subsystem 共q2= −q1兲 interacting with a slow charged particleq3共the charges are in units of the absolute value of the electron charge 兩e兩兲. The system Hamiltonian is given by
H= − ប 2
2R ⵜ
R 2
− ប 2
2r ⵜ
r 2
+V共rជ,Rជ兲, 共1兲
V共rជ,Rជ兲 ⬅q1q2
r +
q2q3
冏
rm2
r ជ+Rជ
冏
+ q3q1
冏
rm1
r ជ−Rជ
冏
, 共2兲
where rជ is the distance vector between the charges q1
andq2andRជ the distance vector from the center of mass of the subsystem to the third共spectator兲 particle. Here, we de-fine r as the reduced mass of the subsystem 共q1q2兲, r⬅12⬅共m1
−1
+m2−1兲−1; and
R the corresponding reduced mass of the subsystem 共q1q2兲 interacting with the third particle:R⬅共12兲3⬅关共m1+m2兲−1+m3
−1 兴−1
.
A. Adiabatic approximation for three charges
The adiabatic approximation implies in solving a coupled equation for the total wave function ⌿共Rជ,rជ兲=共Rជ,rជ兲⌽共Rជ兲, such that we first solve the Schrödinger equation in the vari-ablerជfor the wave function共Rជ,rជ兲, usingRជ as a parameter, with energy solution U共Rជ兲,
冉
− ប 22r ⵜ
r 2
+V共rជ,Rជ兲
冊
共Rជ,ជr兲=U共Rជ兲共Rជ,rជ兲. 共3兲Next, the total energyE is given by
冉
− ប 22R
ⵜR2+U共Rជ兲
冊
⌽共Rជ兲=E⌽共Rជ兲. 共4兲The separation of the Schrödinger equation by means of the adiabatic approximation is justified in case the energy of the relative motion of the center of mass of the two-body sub-system, with respect to the third particle, is small in compari-son with the subsystem binding energy.
Here, the main interest is the behavior of U⬅U共Rជ兲as R
goes to infinity. The case where the third particle interacts with the共q1q2兲subsystem. Using the expansion for the Cou-lomb potential, with兩Rជ兩Ⰷ兩rជ兩andPlthe usual Legendre poly-nomials of order l, we have
U共Rជ,rជ兲=
冋
− ប 22r ⵜ
r 2
+q1q2
r +
q3
R
兺
l=0再
q1冉
−rr
m1R
冊
l
+q2
冉
rr m2R冊
l
冎
Pl共rˆ.Rˆ兲册
共Rជ,rជ兲. 共5兲 Assuming a neutral subsystem 共q1= −q2兲, the leading-orderterm 共conformally invariant兲 in the asymptotic expansion
R→⬁is given by
U共Rជ,rជ兲=
冋
− ប 22r ⵜr2−q1
2
r − q1q3r
R2 P1共cos兲
册
共Rជ,rជ兲,共6兲
whereis the angle between the vectorsrជandRជ. The next-to-leading-order contribution to the adiabatic potential is O共R−4兲P3共cos兲 if m1=m2, otherwise a term like
O共R−3兲P2共cos兲appears, which however does not contribute due to parity conservation. Note that the higher-order terms in Eq.共5兲break the conformal invariance.
The interaction between the third particle and the sub-system 共q1q2兲 at large distances is dominated by a dipole potential, which breaks the degenerated character of the op-posite parity states. According to Ref. 关3兴, there is an accu-mulation of resonances below the threshold of the n= 2 ex-cited state of the positronium negative ion, due to an attractive dipole interaction, which appears as the leading asymptotic term of the potential at large hyperspherical ra-dius. Therefore, it is reasonable to assume that for large val-ues of R the first low lying levels of the共q1q2兲-subsystem, i.e., the first-excitedS andPstates are weakly perturbed by the dipole interaction and the degeneracy is broken. For
by 共Rជ,rជ兲=a共R兲共20兲共r兲Y00共rˆ兲+b共R兲共21兲共r兲Y10共rˆ兲, where
Ylm共rˆ兲 共with lm= 00,10兲 are the usual spherical harmonic function and nl 共with nl= 20,21兲 are the two-body hydro-genic like wave functions for the chargesq1and −q1. Insert-ing共Rជ,rជ兲 in Eq.共6兲, we obtain
U共R兲= −E1⫾
q1q3
R2 兩K兩, 共7兲
K⬅
冕
d3r20ⴱ共r兲Y 00ⴱ 共rˆ兲rP1共cos兲21共r兲Y10共rˆ兲
= − 3共ប2/共 rq1
2
兲兲, 共8兲
whereE1 is the energy of the first-excited state of the sub-system共q1q2兲. The two possible signs in Eq.共7兲come from the diagonalization of the eigenvalue Eq. 共6兲 in the 2S-2P
subspace.
From Eqs.共4兲,共7兲, and共8兲, we obtain the corresponding equation for the third particle motion related to the center of mass of the subsystem,
冉
−ប 2ⵜR 2
2R ⫾3ប
2q 3 rq1
1
R2
冊
⌽=共E+E1兲⌽⬅ប2 2R
E⌽. 共9兲
Here we note how the degeneracy between the 2S and 2P
levels of the chargeless two-body subsystem is broken by the dipole potential. The motion of the wave function ⌽⬅⌽共R兲is given by an attractive type potential. The effect discussed in Ref.关4兴of level condensation near theNth state of positronium is analogous to this situation, as the long-range attractive potential 1/R2 is responsible for forming such resonances. The other solution to U共R兲 gives a long-range repulsion which means that, asymptotically, no bound states can be formed. This repulsive 1/R2 interaction is re-sponsible for the set of resonances above the 2S-2P states 关5兴. Focusing our approach in the weakly bound states of Eq. 共9兲, we end up with an attractive dipole interaction, /R2, generated by the particles with massesm1andm2, where the dimensionless is given by
= 6
冏
q3q1
冏
Rr = 6
冏
q3q1
冏
共m1−1 +m2
−1 兲 共m1+m2兲−1
+m3−1. 共10兲
Now, to solve Eq. 共9兲, we employ a well-known result de-rived for attractive dipole potential共see关15兴兲, which implies in a spectrum of infinite weakly bound levels E
below
E= −E1, if ⱖ1/4; otherwise, no other bound state. So, the following condition emerges for the existence of such spec-trum:
共m1 −1
+m2 −1兲
共m1+m2兲−1+m3 −1ⱖ
1 24
冏
q1
q3
冏
. 共11兲
In order to examine the condition suggested in 关4兴 for the occurrence of level condensation, we restrict the above to an
ABA system with兩qB兩=兩qA兩: Ifm3=m1=mA andm2=mB, we can easily verify that in all the cases we will have ⱖ1/4, implying in a level spectrum below the state n= 2 of the subsystem 共AB兲. The mass ratio mA/mB in this case can only control the level spacing of the spectrum. However, if
the bound subsystem is of a particle and antiparticle 共mA⬅m1=m2兲, with the captured particle having mass mB, the condition ⱖ1/4 will give usmA/mBⱕ47.5. This can be exemplified with the possible configurations of three-body systems with protons, electrons and their antiparticles, such as共e−e+兲p,共e−e+兲¯p, and共¯ pp 兲e⫾, where level spectrum is ex-pected to occur only for共e−e+兲p and共e−e+兲¯p.
B. Geometrical scaling
Equation 共11兲gives us a quite general relation which al-lows infinite number of energy levels that scale geometri-cally in most of three-body systems involving the usual stable charged particles and their antiparticles. The number of states are very dense when Ⰷ1/4, as the ratio between the levels is shown to be given by 关14,15兴 E−1/E= exp共2/
冑
− 1/4兲, implying in the geometrically scaled energy levelsE=E0exp共− 2/
冑
− 1/4兲, 共12兲 where E0 is a reference energy, solution of Eq. 共9兲, deter-mined by nonadiabatic effects. Observe that Eq. 共9兲 is not well defined for small values ofR共of about few Bohr radii兲, leading to the collapse of the reference ground-state energy关15兴. Therefore, the dipole potential must be modified at
some short distance, which in our case corresponds to a re-gion where nonadiabatic effects start to be relevant.
Two other important remarks related to Eqs. 共10兲 and 共11兲:共i兲the strength of the dipole interaction, as well as the corresponding level ratio E−1/E 共when the occurrence of infinite number of levels is possible兲, depend only on the ratio of reduced masses 共12兲3/12 and ratio of the charges 兩q3/q1兩, implying that systems such as 共e−e+兲e⫾,共K−K+兲K⫾, or 共pp¯兲p have the same and energy ratios E−1/E. 共ii兲 Another relevant characteristic of Eq.共10兲is its dependence on the specific three-body configuration and identification of the spectator particle: Let us consider two possible configu-rations for the same three-body system, where m3Ⰶm1 and
m3Ⰶm2andq3=q2= −q1. With⬅共ij兲kfor the configuration 共mimj兲mk, we obtain共12兲3共13兲2⬇36. An obvious example is that of 共pe−兲¯p or 共¯ep +兲p
, where⬇3共mp/me兲Ⰷ1/4 should lead to a dense level spectrum below the n= 2 state of the subsystem. For the counterpart configuration, 共pp¯兲e⫾,
= 12共me/mp兲Ⰶ1/4, a similar spectrum is not expected to occur.
III. RESULTS AND EXAMPLES
Figure1illustrates our conclusions for the values ofas a function of the mass ratios m2/m1 and m3/m1, taking
m2ⱖm1and兩q3兩=兩q1兩. The critical value of= 1/4 is shown by the dashed straight line. As observed, only in the cases that 24⬍m1/m3⬍47.5 it is possible to reach the value = 1/4 for some ratios ofm2/m1. There is no infinite number of levels in case thatm1/m3⬎47.5, independently ofm2/m1. Whenm1/m3= 47.5, the only solution ism2=m1; and, when
m1/m3ⱕ24, ⬎1/4 for all choices ofm2 andm1.
in Table I. As verified, level condensation below the n= 2 state of the subsystem 共shown in the first two lines of the table兲is not possible when the mass of the captured particle is much smaller than the lighter particle of the subsystem. In this table we also present a few results motivated by the actual interest in matter-antimatter interaction, and also mo-tivated by experiments with exotic mesonic atoms 关where protons 共p兲, deuterons 共d兲 or tritons 共t兲 are combined with
muons共兲, kaons共K兲or pions共兲兴. Our共p−兲p
and共d−兲d results correspond to the ones given in 关14兴. Obviously, the same results apply, respectively, for 共p−兲¯p and 共d−兲¯d. TableIdisplays the striking difference of such results when compared with the ones for 共pp¯兲−and 共dd¯兲−. Related to Ps−we should note that our approach gives= 8, to be com-pared with = 7.06 obtained by Botero 关3兴 in a more in-volved calculation.
IV. DISCUSSION AND SUMMARY
Before our conclusions, it is relevant a discussion on the applicability of the Born-Oppenheimer approximation to the particular configuration of excited states in three-charged systems. The separation of variables approximation, ex-pressed in the assumption of a product form for the wave function of the three-body system, considered in Eq. 共4兲, breaks down when the third particle gets close to the neutral subsystem, with R of the order of the corresponding Bohr radius. For some three-body systems, as for example systems with proton and antiproton, nuclear force effects can also be relevant to be considered in addition to Coulomb effects. While in the asymptotic region the excited-state three-charged wave function can be described well by the Born-Oppenheimer approximation, as the size of the system gets smaller the magnitude of nonadiabatic effects is enhanced. In this case, in order to go beyond the adiabatic approximation, the appropriate description of the wave function will consist of an expansion with several higher-order terms, such that one cannot disregard the coupling between them. However, once one large three-charged state is formed with a size much larger than the dipole radius, i.e., with the wave func-tion having major contribufunc-tion from the asymptotic region, the nonadiabatic effects can be translated to a boundary con-dition on the wave function, at a radius characteristic of the region where nonadiabatic effects start to be important, which can be roughly estimated to be of the order of few Bohr radii. Within our present approximation we cannot ob-tain the first large bound excited state that spreads out in the asymptotic region of the dipole potential, as its energy is determined by nonadiabatic contributions. But, once this en-ergy is known, the energies of the other large excited states follow the expression derived from the dipole interaction, emerging the geometrically scaled pattern given in Eq.共12兲. In summary, considering a charged particle captured by a dimer 共bound two-body subsystem兲, within an adiabatic ap-proximation to the Coulombic three-body system with arbi-trary masses, we derived the strength of the attractive dipole interaction with the critical condition for level condensation below the first-excited state of the dimer, as given in Eqs. 共10兲 and 共11兲. The adiabatic approach is justified when the energy of the relative motion of the dimer center of mass, with respect to the third particle, is small compared to the subsystem binding energy; and physically reasonable for a weak dipole potential interacting with the spectator particle. The particular straightforward charge-mass expression ob-tained for the leading term of the dipole interaction, as well as the condition to occur the geometrically scaled energy spectrum, can be very useful as an insight into the analysis of
1 5 9 13 17 21
m 2/ m1
0.1 0.2 0.3 0.4
m
1/m3= 24, 25, 27, 30, 35, 40, 47.5
24
47.5 40 35 30 27 25
FIG. 1. 共Color online兲 Behavior of in terms of the ratio m2/m1. Each plot refers to a different ratiom1/m3, as given inside the figure. ⬎1/4 is satisfied for all values of m1⬍24m3; and
⬍1/4 for all values ofm1⬎47.5m3.
TABLE I. Strength of the attractive dipole interaction共in di-mensionless units兲generated by the chargesq1andq2= −q1, when capturing a charge q3, for兩q3兩=兩q1兩 and different mass configura-tions. In the examples, the estimated value foris the same for the cases with the corresponding antiparticle configurations. Estima-tions of level ratiosE−1/Eare given when ⬎1/4. The massesm andM are for arbitrary defined particles, with conditionMⰇm.
System E−1/E System E−1/E
共pp¯兲e⫾ 0.0065 共+−兲e⫾ 0.058
共K+K−兲e⫾ 0.012 共p−兲e⫾ 0.025
共m−m+兲m⫾ 8 9.55 共M⫾m⫿兲m⫿ 6 13.74
共e−e+兲p 24 3.63 共e−e+兲⫾ 23.75 3.66
共p−兲⫾ 6.06 13.55 共pK−兲K⫾ 6.81 11.63
共pe−兲p 5508 1.09 共pe−兲⫾ 1428 1.18
共pp¯兲⫾ 1.66 198.62 共pp¯兲⫾ 1.28 488.34
共pp¯兲K⫾ 5 17.87 共pp¯兲d 12 6.25
共dd¯兲⫾ 0.66 1.8⫻104 共tt¯兲⫾ 0.44 1.8⫻106
共p−兲p 24.77 3.56 共p−兲p 31.22 3.09
共d−兲d 44.89 2.56 共d−兲d 57.82 2.29
共t−兲t 65.04 2.18 共t−兲t 84.45 1.98
共t−兲p 32.13 3.04 共t−兲p 41.84 2.65
three-body charged systems with different mass relations. Of particular interest is the use of such approach to study inter-action properties of matter and antimatter. Another relevant application can be found in the interaction of excitons with electrons in semiconductors关23兴. As the electron and hole in the interacting pair of excitons can acquire effective indi-vidual masses distinct from the free-electron mass关24兴, gen-eral charge-mass conditions can be very helpful to analyze the relation between effective masses and the observed spec-trum. We finally note that the present approach can be very useful to study capture reactions of a charged particle by a neutral two-body system, in view of possible three-body con-figurations. From a configuration without spectrum below the state n= 2, one can generate a system with very dense
spectrum by exchanging the spectator particle with the same charge particle of the neutral system. An exchange mecha-nism can change drastically the observed spectrum: for 共e−e+兲pwe haveE
−1/E= 3.63共levels below the first-excited
state of Ps兲, whereas for 共e−p兲e+ we have E−1/E= 13.74 共levels below the first-excited state of the hydrogen atom兲.
ACKNOWLEDGMENTS
L.T. thanks Professor S. B. Duarte and Professor R. Shel-lard for the hospitality in CBPF. We acknowledge financial support from Fundação de Amparo à Pesquisa do Estado de São Paulo and Conselho Nacional de Desenvolvimento Científico e Tecnológico.
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