Anomaly in Conformal Quantum Mechanics: from Molecular Physics to Black Holes Horacio E

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12-30-2003 Anomaly in Conformal Quantum Mechanics: From Molecular Physics to Black Holes Horacio E. Camblong University of San Francisco, [email protected]

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Recommended Citation Camblong, Horacio E., "Anomaly in Conformal Quantum Mechanics: From Molecular Physics to Black Holes" (2003). Physics and Astronomy. Paper 6. http://repository.usfca.edu/phys/6

This Article is brought to you for free and open access by the College of Arts and Sciences at USF Scholarship: a digital repository @ Gleeson Library | Geschke Center. It has been accepted for inclusion in Physics and Astronomy by an authorized administrator of USF Scholarship: a digital repository @ Gleeson Library | Geschke Center. For more information, please contact [email protected]. PHYSICAL REVIEW D 68, 125013 ͑2003͒

Anomaly in conformal quantum mechanics: From molecular physics to black holes

Horacio E. Camblong Department of Physics, University of San Francisco, San Francisco, California 94117-1080, USA

Carlos R. Ordo´˜nez Department of Physics, University of Houston, Houston, Texas 77204-5506, USA and World Laboratory Center for Pan-American Collaboration in Science and Technology, University of Houston Center, Houston, Texas 77204-5506, USA ͑Received 19 March 2003; revised manuscript received 18 August 2003; published 30 December 2003͒ A number of physical systems exhibit a particular form of asymptotic conformal invariance: within a particular domain of distances, they are characterized by a long-range conformal interaction ͑inverse square potential͒, the apparent absence of dimensional scales, and an SO͑2,1͒ symmetry algebra. Examples from molecular physics to black holes are provided and discussed within a uniﬁed treatment. When such systems are physically realized in the appropriate strong-coupling regime, the occurrence of quantum symmetry breaking is possible. This anomaly is revealed by the failure of the symmetry generators to close the algebra in a manner shown to be independent of the renormalization procedure.

DOI: 10.1103/PhysRevD.68.125013 PACS number͑s͒: 11.10.Gh, 03.65.Fd, 11.25.Hf, 11.30.Qc

I. INTRODUCTION problems, from molecular physics to black holes, and ͑ii͒ to show the details of the breakdown of the commutator algebra An anomaly is defined as the symmetry breaking of a ͑1͒ for the long-range conformal interaction. In Sec. II we classical invariance at the quantum level. This intriguing introduce a number of examples that can be regarded as phenomenon has played a crucial role in theoretical physics physical realizations of conformal quantum mechanics. In since its discovery in the 1960s ͓1͔. In addition to its use in Sec. III we show that the origin of the anomaly can be traced particle phenomenology of the standard model ͓2͔ and its to the short-distance singular behavior of the conformal in- extensions, it has been a fruitful tool for the study of confor- teraction. In Sec. IV we introduce a generic class of real- mal invariance in string theory ͓3͔. space regulators, within the philosophy of the effective-field Surprisingly, the presence of an infinite number of degrees theory program. In Sec. V we compute the anomaly for the of freedom does not appear to be a prerequisite for the emer- regularized theory and show that it is independent of the gence of anomalies. This fact was first recognized within a details of the ultraviolet physics, and in Sec. VI we comment model with conformal invariance: the two-dimensional con- on various renormalization frameworks. After the conclu- tact interaction in quantum mechanics ͓4͔. In conformal sions in Sec. VII, we summarize a number of technical re- quantum mechanics, a physical system is classically invari- sults: a derivation of the anisotropic generalization of the ant under the most general combination of the following time conformal long-range interaction ͑Appendix A͒; a study of reparametrizations: time translations, generated by the interdimensional dependence ͑Appendix B͒; a derivation of Hamiltonian H; scale transformations, generated by the dila- the near-horizon properties of black holes ͑Appendix C͒; and ϵ Ϫ ϩ a derivation of useful integral identities ͑Appendix D͒. tion operator D tH (p•r r•p)/4; and translations of re- ciprocal time, generated by the special conformal operator ϵ Ϫ 2 ϩ 2 K 2tD t H mr /2. These generators define the noncom- II. RELEVANT PHYSICAL APPLICATIONS pact SO(2,1)ϷSL(2,R) Lie algebra ͓5͔ In recent years, diverse examples of systems have been ͑ ͒ ͓D,H͔ϭϪiបH, ͓K,H͔ϭϪ2iបD, ͓D,K͔ϭiបK. studied from the viewpoint of the conformal algebra 1 , as- ͑1͒ sumed to be a representation of an approximate symmetry within specific scale domains. In the applicable conformally This symmetry algebra has also been recognized in the free invariant domain, the relevant physics is described by a nonrelativistic particle ͓6͔, the inverse square potential ͓7,8͔, d-dimensional effective Hamiltonian the magnetic monopole ͓9͔, the magnetic vortex ͓10͔, and various nonrelativistic quantum field theories ͓6,11,12͔. Fur- p2 g Hϭ Ϫ , ͑2͒ thermore, conformal quantum mechanics has been fertile 2m r2 ground for the study of singular potentials and renormaliza- ͓ ͔ tion, using Hamiltonian 13–15 as well as path integral which involves a long-range conformal interaction; or, alter- ͓ ͔ methods 16 . Most importantly, a variety of physical real- natively, by its anisotropic counterpart izations of conformal quantum mechanics have been recently identified, as discussed in the next section. 2 ͑ ͒ p g The main goals of this paper are i to illustrate the rel- Hϭ Ϫ F͑⍀͒, ͑3͒ evance of conformal quantum mechanics for several physical 2m r2

⍀ ⍀ ␭( )ϭ␯2 ␥ where stands for the angular variables and F( )isa where eff* eff for each eigenvalue of the angular equa- generic anisotropy factor that accounts for the angular depen- tion. When this outline is implemented, according to the pro- dence. Equation ͑3͒ is discussed in Appendix A. cedure of Ref. ͓17͔ or its generalization of Appendix A, the In the problems considered below, ␭ϭ2mg/ប2 is the di- existence of a critical dipole moment p(*) for binding is mensionless form of the coupling constant and ␯ϭ(d predicted; the order of magnitude of its ‘‘conformal value,’’ Ϫ2)/2; furthermore, the choice បϭ1ϭ2m will be made for ␭(*)Ϸ1.279, or p(*)Ϸ1.625 D, has been verified in numer- the problem involving black holes. In all cases, the strong- ous experiments ͓18,19͔. In particular, when binding occurs, coupling regime ͓14͔ is defined by the condition gуg(*), extended states known as dipole-bound anions are formed. ␭( )ϵ␭( )ϭ ϩ␯ 2 with a critical dimensionless coupling * l * (l ) These conclusions have also been confirmed by detailed ab ͑for angular momentum l), when the Hamiltonian model ͑2͒ initio calculations ͓18,19͔ and by studies that incorporate the is adopted ͓14͔. In addition, in the strong-coupling regime, as effects of rotational degrees of freedom ͓20͔, which also deduced in Sec. III, an uncontrolled oscillatory behavior of modify slightly the value of p(*). the Bessel functions of imaginary order i⌰ makes the con- In short, the central issue in this analysis—also shared by formal system singular and regularization is called for. The the other physical realizations discussed in this paper—is the characteristic parameter ⌰ϭͱ␭Ϫ(lϩ␯)2 strictly corre- existence of a conformally invariant domain whose ultravio- sponds to the Hamiltonian ͑2͒; in physical applications, such let boundary leads to the anomalous emergence of bound as those of Secs. II A, II B, and II C, we will define states via renormalization. As a result, these states break the original conformal symmetry of the model and modify the ⌰ϵ⌰ ϭͱ␭ Ϫ␭( ) ͑ ͒ commutators ͑1͒, as we will show in the next few sections. eff eff eff* , 4 This simple fact alone captures the essence of the observed which will turn out to be crucial in parametrizing the anoma- critical dipole moment in polar molecules and leads to an lous physics of the conformal system in the presence of sym- analytical prediction for the energies of the conformal states, metry breaking. In discussing these realizations, we will ex- as discussed in Sec. VI and Appendix A. plicitly use a subscript to emphasize the effective nature of the parameter of Eq. ͑4͒—as arising from a reduction frame- B. Near-horizon black hole physics work. The same notational convention will apply to the di- A generic class of applications of conformal quantum me- mensionality (deff). As shown in Appendix B, even when chanics arises from the near-horizon conformal invariance of interdimensional equivalences are introduced, the value of black holes, its impact on their thermodynamics ͓21͔, and its ͑ ͒ the parameter 4 is a dimensional invariant. extension to superconformal quantum mechanics ͓22͔.In particular, analyses based on the Hamiltonian ͑2͒ have been A. Dipole-bound anions and anisotropic conformal interaction used to explore horizon states ͓23,24͔ and to shed light on ͓ ͔ The three-dimensional (d ϭ3or␯ ϭ1/2) interaction black hole thermodynamics 24 . Another class of current eff eff ͓ ͔ between an electron ͑charge QϭϪe) and a polar molecule applications 25 involves a many-body generalization of Eq. ͑ ͑2͒: the Calogero model, which has also been directly linked dipole moment p) was the first physical application to be ͓ ͔ recognized as a realization of this anomaly ͓17͔. When the to black holes 26 . These remarkable connections seem to confirm the conjecture that it is the horizon itself that en- molecule is modeled as a point dipole, this interaction can be ͓ ͔ effectively described with an anisotropic long-range confor- codes the quantum properties of a black hole 27 . mal interaction of the form ͑3͒: V(r)ϭϪg cos ␪/r2, in which In this context, we consider the spherically symmetric ¨ the polar angle ␪ is subtended from the direction of the di- Reissner-Nordstrom geometry in D spacetime dimensions, pole moment. For this potential, the dimensionless coupling whose metric ␭ϭϪ ប2ϭ Ϫ is 2mKepQ/ p/p0, with m being the reduced mass 2ϭϪ ͒ 2ϩ ͒ 1 2ϩ 2 ⍀ ͑ ͒ ds f ͑r dt ͓ f ͑r ͔ dr r d DϪ2 6 of the system and Ke the electrostatic constant. Thus, the relevant scale for phenomenological analyses is provided by is minimally coupled to a scalar field ⌽(x) with action (c Ϸ ͑ ͒ p0 1.271 D where D stands for the Debye . ϭ1 and បϭ1) As shown in Appendix A, in some sense, the generic an- ͑ ͒ 1 isotropic conformal interaction 3 —of which the electron- D ␮␯ 2 2 ␯⌽ϩm ⌽ ͔. ͑7͒ץ␮⌽ץ SϭϪ ͵ d xͱϪg͓g molecule interaction is a particular case—can be reduced to 2 an effective isotropic conformal interaction for the zero ͓ ͑ ͔͒ ͑ ͒ ⍀ angular-momentum channel see Eq. A7 ; this corresponds In Eq. 6 , d DϪ2 stands for the metric on the unit (D ͑ ͒ Ϫ ϭ Ϫ DϪ3ϩ 2(DϪ3) to an effective Hamiltonian of the type 2 , with an appro- 2)-sphere, f (r) 1 2(aM /r) (bQ /r) , and ␭ priate effective coupling eff . More precisely, this equiva- the lengths aM and bQ are determined from the mass M and lence is achieved, after separation of variables in spherical charge Q of the black hole respectively ͓28͔. In this ap- coordinates, at the level of the radial equation. In addition, proach, the conformal structure is revealed by a two-step ␭ the corresponding value of eff is identical to the eigenvalue procedure discussed in Appendix C and consisting of: ͑a͒ a ␥ of the angular equation, which is a function of the dipole reduction to an effective Schro¨dinger-like equation, to be coupling ␭. The effective conformal parameter ͑4͒ becomes analyzed in its frequency (␻) components; ͑b͒ the introduction of a near-horizon expansion in the variable xϭrϪrϩ ⌰ ϭͱ␥Ϫ␯2 ͑ ͒ eff eff, 5 ͓with rϭrϮ being the roots of f (r)ϭ0, and rϩуrϪ]. Two

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ϭ distinct scenarios emerge from this reduction: the extremal deff 2d for the internal dynamics of the three-body system and nonextremal cases, when rϩϭrϪ and rϩÞrϪ , respec- ͑as the total number of coordinates is 3d, but d of them are tively. We will omit any discussion of the extremal case, eliminated in favor of the center-of-mass coordinates͒. Con- which is known to pose a number of conceptual difficulties sequently, when a hyperspherical adiabatic expansion ͓32͔ is and is otherwise beyond the scope of the framework pre- combined with a Faddeev decomposition of the wave func- ͓ ͔ ϭ sented in this paper. As for the nonextremal case, the follow- tion 33 , a reduction to a deff 2d realization of our confor- ing facts arise from this reduction: mal model ͑2͒ is obtained. These conclusions can be gleaned ͑i͒ The ensuing effective problem is described by an in- from the conformal nature of the effective adiabatic poten- ͓ ͔ teraction tials Veff(r) arising from this reduction framework 30 , ͑near horizon͒ Ϫ2 g V͑x͒ ϰ Ϫx , ͑8͒ ͑ ͒ϭϪ eff ␭ ϭ͑ Ϫ ͒2ϩ⌰2 ϭ Veff r , eff d 1 eff, deff 2d, r2 which is conformally invariant with respect to the near- ͑10͒ horizon coordinate x. ͑ ͒ ␭ ͑ ͒ ii The effective Hamiltonian, still being a d-dimensional where geff and eff are related as described after Eq. 3 . realization of the conformal interaction, does not have the Incidentally, due to the interdimensional equivalence of Ap- usual form corresponding to the radial part of a multidimen- pendix B, this result is often quoted in its one-dimensional ͓ ͑ ͔͒ ␭ ϭ ϭ␭ Ϫ Ϫ 2 sional Schro¨dinger problem. In particular, the angular mo- reduced form from Eq. B4 , (d 1) eff (d 1) ϩ ϭ⌰2 ϩ mentum variables appear at a higher order in the near- 1/4 eff 1/4. For example, for the all-important case of ϭ ␭ ϭ horizon expansion. ordinary three-dimensional space, deff 6 and eff 4 ͑iii͒ The coupling constant ␭ is supercritical for all non- ϩ⌰2 ͑ ͒ eff eff . Furthermore, the coupling constant in Eq. 10 de- zero frequencies. This can be seen from Eq. ͑C11͒, which pends upon the physical parameters defining the system: implies that when the scattering lengths are large, it is function of the three ratios of particle masses. In particular, for the lowest ␻ ⌰ ϭ ͑ ͒ angular eigenvalue of a three-body three-dimensional system eff . 9 ͉f Ј͑rϩ͉͒ of identical bosons with zero-range two-particle interactions, the characteristic conformal parameter ͑4͒ is approximately The conclusion from this procedure is that the relevant phys- given by the solution of the transcendental equation ͓30͔ ics occurs in the strong-coupling regime, in which the frame- ␲⌰ ␲⌰ work discussed in this paper can be applied. eff eff 8 sinhͩ ͪ ϭͱ3⌰ coshͩ ͪ , ͑11͒ 6 eff 2 C. Other applications ⌰ Ϸ so that eff 1.006, which corresponds to the strong- While Secs. II A and II B conform to the title of this pa- coupling regime. per, applications in other areas of physics are also likely. In short, the essential feature shared by the problems dis- ͓ ͔ Among these, the Efimov effect 29,30 stands out. This ef- cussed above is the existence of an effective description in fect is expected to arise in a three-body system with short- terms of SO͑2,1͒ conformal invariance, which results from a range interactions, in which at least two of the two-body prescribed reduction framework. We now turn our attention subsystems have virtual or bound s-states near zero energy. to this generic effective problem, characterized by the As in the case of the dipole-bound anions of Sec. II A, these Hamiltonian of Eq. ͑2͒. As different dimensionalities are re- are spatially extended and weakly bound states. Unfortu- quired for the applications to which Eq. ͑2͒ refers, we will nately, the combination of phenomenological parameters analyze this problem for the arbitrary d-dimensional case. needed to form these states, together with their weakly Our goal is to investigate and characterize the possible real- bound nature, has defied experimental detection to date. ization of a conformal anomaly within this scope. Nonetheless, this effect is regarded as relevant in the description of the three-body nucleon interaction ͓31͔. The most III. CONFORMAL ANOMALY AND SHORT-DISTANCE outstanding feature of these three-body interactions in three PHYSICS spatial dimensions is the fact that these problems are reduced to an effective equation with a long-range conformal interac- Conformal symmetry is guaranteed at the quantum level tion in the strong-coupling regime. In terms of possible ex- when the naive scaling of operators, described by the algebra perimental detection, this effect is currently being studied for ͑1͒, is maintained. A measure of the deviation from this scal- the description of various systems, including helium trimers ing is afforded by the ‘‘anomaly’’ ͓34͔ and nuclear three-body halos ͓30͔. The conformal nature of the effective interaction, for the 1 1 A͑r͒ϵ ͓D,H͔ϩHϭͫ ϩ E ͬV͑r͒ ͑12͒ three-body systems described above, can be deduced as fol- iប 1 2 r lows. Typically, one starts by introducing hyperspherical coordinates with hyperradius ␳ϵr,inad -dimensional con- eff rdϪ2 rV͑r͒ figuration space for the internal degrees of freedom; if the ϭ ͫ ͬ ͑13͒ one-particle dynamics occurs in a d-dimensional space, then 2 “• rdϪ2

125013-3 H. E. CAMBLONG AND C. R. ORDO´ N˜ EZ PHYSICAL REVIEW D 68, 125013 ͑2003͒

͑valid for arbitrary d spatial dimensions͒, in which 1 is the IV. REGULARIZATION AND RENORMALIZATION: E ϭ THE EFFECTIVE-FIELD THEORY PROGRAM identity operator and r r•“. At first sight, the right-hand side of Eq. ͑12͒ appears to be zero for any scale-invariant The Hamiltonian ͑2͒, in the strong-coupling regime, de- potential; however, upon closer examination, this apparent scribes an effective system with singular behavior for short- cancellation may break down at rϭ0, where the interaction distance scales. This interpretation, in which regularization is singular. Equations ͑12͒ and ͑13͒ can be directly applied to and renormalization are mandatory, is inspired by the any of the interactions within the conformal quantum me- effective-field theory program ͓36͔. The required regulariza- chanics class, and reduce to the familiar results known for tion procedure is implemented in real space, where the ultra- the two-dimensional contact interaction ͓34,35͔. However, violet physics is replaced over length scales rՇa. The effec- the most interesting case is provided by the Hamiltonian ͑2͒, tive theory that comes out of this renormalization is expected whose symmetry breaking can be made apparent by means to be applicable within a domain of energies of magnitude ͓ ˆ dϪ1͔ of the formal d-dimensional identity “• r/r ͉E͉ӶE ϵប2/2ma2. The scale E defines an approximate ϭ⍀ ␦(d) ⍀ a a dϪ1 (r), in which dϪ1 is the surface area of the unit limit of the conformal regime from the ultraviolet side; ef- Ϫ dϪ1 (d 1)-sphere S ; then, fectively, this limit prevents the singular interaction from yielding unphysical divergent results for supercritical cou- ⍀ dϪ1 A͑r͒ϭϪg rdϪ2␦(d)͑r͒. ͑14͒ pling. 2 Specifically, we consider a generic class of regularization schemes that explicitly modify the ultraviolet physics; each Despite its misleading appearance, this term is not identically scheme is described by a potential V(Ͻ)(r), for rՇa, where equal to zero, due to the singular nature of the interaction at a is a small real-space regulator. An appropriate procedure rϭ0. The recognition of this remarkable singular term, as for the selection of solutions of this singular conformal in- well as of its regularized and renormalized counterparts, teraction was proposed in Ref. ͓37͔, using a constant poten- leads to the central result of our paper: the proof of the ex- tial for rՇa. Our approach is based on a generalization of istence of a conformal anomaly. this method, in which a core interaction V(Ͻ)(r) is intro- However, two important points should be clarified. First, duced. Eq. ͑14͒ is merely a formal identity, whose physical meaning Incidentally, in this section, we consider a core V(Ͻ)(r) can only be manifested through appropriate integral expres- ϵV(Ͻ)(r) with central symmetry V(Ͻ)(r). Even though this sions. Second, the coordinate singularity highlights the need condition is not strictly necessary, it leads to a tractable deri- to determine the behavior of the wave function near rϭ0. vation. Moreover, it is also consistent with the original rota- Therefore, nontrivial consequences of Eq. ͑14͒ can only be tional invariance of the isotropic singular interaction and displayed by the expectation value with a normalized state captures the singular behavior of the potential, which origi- ͉⌿͘, nates from its radial dependence ͑even in the anisotropic case͒. The generalization for an anisotropic conformal inter- d ⍀ Ϫ d 1 d (d) ␯ 2 action is nontrivial, but when this interaction is reduced to an ͗D͘⌿ϭ͗A͑r͒͘⌿ϭϪg ͵ d r␦ ͑r͉͒r ⌿͑r͉͒ . dt 2 effective radial problem, the procedure developed in this sec- ͑15͒ tion can be applied. The core interaction is subject to the conditions of finite- A similar analysis applies to the anisotropic interaction of ness Eq. ͑3͒; in this case, ប2 Ꭽ ϪϱϽ ϵ (Ͻ) ͒ ϵϪ ͑ ͒ V0 min͓V ͑r ͔ 17 d 2m a2 ͗D͘⌿ϭ͗A͑r͒͘⌿ dt and continuous matching with the external inverse square ⍀ dϪ1 ␯ potential at rϭa, ϭϪg ͵ ddr␦(d)͑r͉͒r ⌿͑r͉͒2F͑⍀͒. 2 V(Ͻ)͑a͒ϭV(Ͼ)͑a͒ϭϪg/a2. ͑18͒ ͑16͒ Ͻ It should be noticed that these restrictions imply that V0 0 It should be noticed that the intermediate steps leading to or ᎭϾ0, and that Ꭽϭ␭ϩ␵, where ␵Ͼ0 is the dimensionless Eqs. ͑15͒ and ͑16͒ are based on formal identities involving energy difference between the minimum V0 and the match- the d-dimensional ␦ function. For the unregularized inverse ing value ͑18͒. In addition, in this approach, the energies for square potential, the integrals in Eqs. ͑15͒ and ͑16͒ select the the interior problem will be conveniently redefined from the limit r→0 of the product r␯⌿(r), which is known to be minimum value V0; specifically, proportional to a Bessel function of order i⌰, with ⌰ defined in Eq. ͑4͒. This limit is ill defined in the strong- ͒ϵ (Ͻ) ͒Ϫ ˜ ϭ Ϫ ͑ ͒ U͑r V ͑r V0 , E E V0 . 19 coupling regime, due to the uncontrolled oscillatory behavior of the Bessel functions of imaginary order. Consequently, a For the spherically symmetric long-range conformal inter- regularization procedure is called for; inter alia, this proce- action of Eq. ͑2͒, central symmetry leads to the separable dure will assign a meaningful value to Eqs. ͑15͒ and ͑16͒. solution

125013-4 ANOMALY IN CONFORMAL QUANTUM MECHANICS:... PHYSICAL REVIEW D 68, 125013 ͑2003͒

Yˇ ͑⍀͒v͑r͒ where we have conveniently redeﬁned the logarithmic de- ␰ץ ץ⌿͑ ͒ϭ lm ͵ ⍀ ͉ ˇ ͑⍀͉͒2ϭ⍀ L (Ͼ) ␰ ϵE ͓ ␰ ͔ E ϭ␰ r ␯ , d dϪ1 Y lm dϪ1 , rivatives from i⌰ ( ) ␰ ln Ki⌰( ) , with ␰ / , and r (Ͻ) similarly for L (˜␰;˜k) in terms of w ϩ␯(˜␰;˜k). Explicitly, ͑20͒ lϩ␯ l Eq. ͑28͒ takes the form ˇ ⍀ in which Y lm( ) stands for the ultraspherical harmonics on 2 Ϫ2 2 Ϫ2 Ϫ ⍀ Ϫ ͓B ˜k J ϩ␯͑˜␰;˜k͒ϩA ␬ K ⌰͑␰͔͒ϭ1, ͑29͒ Sd 1 ͓38͔, which have been conveniently redeﬁned with a d 1 l,␯ l l,␯ i ⍀ normalization integral equal to the solid angle dϪ1. Then, in which the normalization constants can be chosen to be ¨ the corresponding effective radial Schrodinger equation for real, and where v(r) becomes ϱ 2 2 K ͑␰͒ϵ ͵ ͓ ͑ ͔͒2 ͑ ͒ d 1 d ͑lϩ␯͒ i⌰ dzz Ki⌰ z 30 ͭ ϩ ϩͫ k2Ϫ ϪV͑r͒ͬͮ v͑r͒ϭ0, ͑21͒ ␰ dr2 r dr r2 and where V(r)ϭ2mV(r)/ប2 and k2ϭ2mE/ប2. In particular, ϭ ␬ ͑ ͒ ˜␰ for bound states, k i and Eq. 21 provides solutions of J ͑˜␰ ˜ ͒ϵ ͵ ͓ ͑ ˜ ͔͒2 ͑ ͒ lϩ␯ ;k dzz wlϩ␯ z;k . 31 the form 0

(Ͻ) ͒ϭ ˜ ˜ ͒ р ͑ ͒ ͑ ͒ v ͑r Bl,␯wlϩ␯͑kr;k for r a, Equations 26 and 29 then provide the values of the con- v ͑r͒ϭͭ ͑22͒ a (Ͼ) ͒ϭ ␬ ͒ у stants Al,␯ and Bl,␯ ; for example, v ͑r Al,␯Ki⌰͑ r for r a, Ϫ ␬ ␰2 ͑˜␰ ˜ ͒ 2 1/2 ͓ ͔ wlϩ␯ ;k in which Ki⌰(z) is the Macdonald function 39 , and where ϭ J ͑˜␰ ˜ ͒ϩͫ ͬ K ͑␰͒ B ␯ ͭ ϩ␯ ;k ⌰ ͮ . l, l i ˜ ˜ ϭប2˜ 2 ˜ϭͱϪ␬2ϪV ͱ⍀ ˜␰2 K ⌰͑␰͒ k is deﬁned from E k /2m, so that k 0, with dϪ1 i V ϭ ប2Ͻ ͑ ͒ 0 2mV0 / 0. The regularizing core is arbitrary and 32 ˜ ˜ wlϩ␯(kr;k) is a particular real solution in that region, For reasons that will become clear in the next section, it is convenient to rewrite Eqs. ͑30͒ and ͑31͒ in an alternative d2 1 d ͑lϩ␯͒2 2 way, using the generalized Lommel integrals of Appendix D. ͭ ϩ ϩͫ˜k Ϫ ϪU͑r͒ͬͮ w ϩ␯͑˜kr;˜k͒ϭ0, l dr2 r dr r2 First, the integral deﬁned by Eq. ͑30͒, which applies to the ͑23͒ external domain (rуa), can be expressed as

U ϭV ϪV ˜ ˜ 1 where (r) (r) 0; as an example, wlϩ␯(kr;k)isa 2 (Ͼ) K ⌰͑␰͒ϭ ͓K ⌰͑␰͔͒ M ͑␰͒, ͑33͒ Bessel function of order lϩ␯ when the potential V(Ͻ)(r)isa i 2 i i⌰ constant. The solution ͑22͒ can be completely determined by en- where forcing the following three additional physical conditions: M (Ͼ)͑␰͒ϵ͓L (Ͼ)͑␰͔͒2ϩ⌰2Ϫ␰2 ͑ ͒ ͑a͒ continuity at rϭa of the radial wave function; ͑b͒ conti- i⌰ i⌰ . 34 ϭ nuity at r a of the logarithmic derivative of the radial wave ͑ ͒ function; and ͑c͒ normalization of the wave function. In what Similarly, the integral deﬁned by Eq. 31 , which applies to р follows, these conditions will be stated using the auxiliary the internal domain (r a), takes the form parameters 1 J ͑˜␰ ˜ ͒ϭ ͓ ͑˜␰ ˜ ͔͒2M (Ͻ) ͑˜␰ ˜ ͒ϩU ͑˜␰ ˜ ͒ lϩ␯ ;k wlϩ␯ ;k lϩ␯ ;k lϩ␯ ;k , ␰ϭ␬a, ˜␰ϭ˜ka, ͑24͒ 2 ͑35͒ which satisfy Eq. ͑19͒, i.e., where ˜␰2ϩ␰2ϭᎭ ͑ ͒ . 25 M (Ͻ) ͑˜␰ ˜ ͒ϵ͓L (Ͻ) ͑˜␰ ˜ ͔͒2ϩ͓˜␰2Ϫ͑ ϩ␯͒2Ϫ˜␰2 ˇ ͑˜␰ ˜ ͔͒ lϩ␯ ;k lϩ␯ ;k l U ;k ͑36͒ Consequently, these conditions ͑a͒–͑c͒ become, respectively, and ˜␰ ˜ ͒ϭ ␰͒ ͑ ͒ Bl,␯wlϩ␯͑ ;k Al,␯Ki⌰͑ , 26 ˜␰ 1 U ͑˜␰ ˜ ͒ϵ ͵ ͓ ͑ ˜ ͔͒2ͫͩ ϩ E ͪ ˇ ͑ ˜ ͒ͬ L(Ͻ) ͑˜␰ ˜ ͒ϭL (Ͼ)͑␰͒ ͑ ͒ lϩ␯ ;k dzz wlϩ␯ z;k 1 z U z;k , lϩ␯ ;k i⌰ , 27 0 2 ͑37͒ and ͓cf. Eq. ͑20͔͒ ץ ץ ϵ ˜ E ϭ ˇ with U U/E and z z / z. The Lommel integral relation ϱ ͑ ͒ ͑ ͒ ͵ d ͉⌿͑ ͉͒2ϭ⍀ ͵ ͉ ͑ ͉͒2ϭ ͑ ͒ 33 appears to be simpler than Eq. 35 because of the ab- d r r dϪ1 drr v r 1, 28 0 sence of an extra core Uˇ (z;˜k) in the external domain.

125013-5 H. E. CAMBLONG AND C. R. ORDO´ N˜ EZ PHYSICAL REVIEW D 68, 125013 ͑2003͒

In addition, the continuity conditions of the potential, Eq. cludes the origin, when applied to any homogeneous poten- ͑18͒, and of the logarithmic derivatives, Eq. ͑27͒, imply the tial of degree Ϫ2 ͑a defining characteristic of the external equality of the ‘‘matching functions’’ ͑34͒ and ͑36͒, i.e., conformal interaction͒. Once Eq. ͑42͒ is established, the anomaly can be com- M (Ͻ) ͑˜␰ ˜ ͒ϭM (Ͼ)͑␰͒ ͑ ͒ lϩ␯ ;k i⌰ . 38 puted from the contribution arising from the ultraviolet domain rрa, As a corollary, a combined Lommel relation can be obtained by elimination of the matching functions from Eqs. ͑33͒ d 1 ͗ ͘ ϭ͗A ͑ ͒͘(Ͻ)ϭ ͵ d ͫͩ ϩ E ͪ ͑ ͉͒ͬ⌿ ͑ ͉͒2 and ͑35͒, D ⌿ a r ⌿ d r 1 r V r a r . dt a rрa 2 ͑43͒ ˜␰ ˜ ͒ 2 wlϩ␯͑ ;k J ͑˜␰ ˜ ͒ϪU ͑˜␰ ˜ ͒ϭͫ ͬ K ͑␰͒ ͑ ͒ Ͻ lϩ␯ ;k lϩ␯ ;k i⌰ . 39 ͑ ͒ ϵ ( ) ͑ ͒ K ⌰͑␰͒ In Eq. 43 , V(r) V (r) can be replaced using Eq. 19 , i ⌿ ͑ ͒ ͑ ͒ and a(r) using Eqs. 20 and 22 ; when these substitutions Even though the implementation of a renormalization pro- are made and the dimensionless variable ˜␰ in Eq. ͑24͒ is cedure is a necessary condition for the emergence of the introduced, Eq. ͑43͒ becomes conformal anomaly, the actual details of this procedure are not explicitly required. It suffices to know that these details 2 d ⍀ Ϫ B ␯ ˜␰ d 1 l, 2 are to be consistently derived from Eqs. ͑24͒–͑39͒, which ͗D͘⌿ϭ ͵ dzz͓w ϩ␯͑z;˜k͔͒ 2 l permit the exact evaluation of all relevant expectation values, dt ˜k 0 and by enforcing the finiteness of a particular bound state 1 z energy. ϫͭ ͩ ϩ E ͪͫ V ϩUͩ ͪͬͮ . ͑44͒ 1 z 0 2 ˜k V. COMPUTATION OF THE CONFORMAL ANOMALY Despite its cumbersome appearance, the integral in Eq. ͑44͒ The value of the anomalous part of the commutator can be easily evaluated once the definitions ͑31͒ and ͑37͒ are ͓D,H͔ is given as the ‘‘anomaly’’ A(r) in Eq. ͑12͒. In Sec. introduced, so that III, this quantity was computed for the unregularized inverse square potential in terms of the formal identity ͑14͒; this ⍀ 2 1 d 1 dϪ1Bl,␯ expression, in turn, led to an ill-defined expectation value ͗D͘⌿ϭ ͓V J ϩ␯͑˜␰;˜k͒ϩ˜EU ϩ␯͑˜␰;˜k͔͒ 2 0 l l ͑15͒. This difficulty can be overcome when the singular con- E dt E ˜k formal interaction is regularized according to the generic ͑45͒ scheme introduced in Sec. IV. Then, Eq. ͑12͒ will in prin- р ⍀ 2 ␰2 ciple yield two different contributions: one for r a and one dϪ1Bl,␯ у ͑ ͒ ϭ ͭ J ϩ␯͑˜␰;˜k͒ for r a, with the latter being of the form 14 ; thus, ␬2 ˜␰2 l 1 A ͑ ͒ϭͫͩ ϩ E ͪ (Ͻ)͑ ͒ͬ␪͑ Ϫ ͒ a r 1 r V r a r ϩ͓J ϩ␯͑˜␰;˜k͒ϪU ϩ␯͑˜␰;˜k͔͒ͮ , ͑46͒ 2 l l ⍀ dϪ1 Ϫ Ϫg rd 2␦(d)͑r͒␪͑rϪa͒, ͑40͒ ͑ ͒ ͑ ͒ 2 where V0 was replaced through the relation 19 or 25 , and EϭϪប2␬2/2m. Furthermore, in Eq. ͑46͒, the difference ␪ J ˜␰ ˜ ϪU ˜␰ ˜ ͑ ͒ where (z) stands for the Heaviside function, is the regular- lϩ␯( ;k) lϩ␯( ;k) can be evaluated employing Eq. 39 , ized counterpart of Eq. ͑14͒. Explicitly, this leads to an ex- so that pectation value ⍀ 2 ␰2 1 d dϪ1Bl,␯ d ͗D͘⌿ϭ ͭ J ϩ␯͑˜␰;˜k͒ ϭ A ͒ (Ͻ)ϩ A ͒ (Ͼ) ͑ ͒ 2 2 l ͗D͘⌿ ͓͗ a͑r ͘⌿ ͗ a͑r ͘⌿ ͔, 41 E dt ␬ ˜␰ dt a a 2 w ϩ␯͑˜␰;˜k͒ where the integration range is split into the two regions: 0 ϩͫ l ͬ K ͑␰͒ ͑ ͒ ⌰ ͮ . 47 ͑␰͒ i рrрa and rуa. Moreover, the identically vanishing sec- Ki⌰ ond term ͑ ͒ Finally, the coefficient Bl,␯ can be eliminated using Eq. 32 , (Ͼ) ͗A ͑r͒͘⌿ ϭ0 ͑42͒ which shows that the right-hand side of Eq. ͑47͒ is identi- a a cally equal to one for any bound state. This remarkable sim- in Eq. ͑40͒ shows that the source of the conformal anomaly plification concludes the proof that the anomaly defined in is confined to an arbitrarily small region about the origin. Eq. ͑12͒ is indeed given by This result can be confirmed from a straightforward replace- ment of Eq. ͑12͒ by A(r)ϭϪ 1 (dϪ2)V(r)ϩ 1 ͕rV(r)͖, d 2 2 “• ͗D͘⌿ϭE, ͑48͒ which is identically equal to zero for any domain that ex- dt

125013-6 ANOMALY IN CONFORMAL QUANTUM MECHANICS:... PHYSICAL REVIEW D 68, 125013 ͑2003͒

where E is the energy of the corresponding stationary nor- portantly, this condition is systematically applied to derive malized state. physical predictions in a direct manner, within the prescrip- In short, we have validated the relation ͑48͒—which tions of Sec. IV. As a result, Eq. ͑49͒ leads to the bound-state agrees with the formal prediction from properties of expec- energy levels ͓41͔ tation values ͓34͔. This validation has been established using 2␲n a generic regularization procedure. Therefore, regardless of ϭ ͩ Ϫ ͪ ͑ ͒ En E0exp ⌰ , 51 the renormalization framework used, an anomaly is generated. The generality of Eq. ͑48͒ makes it available for a Ͻ in which E0 0 is an arbitrary proportionality constant. This variety of physical applications, and is a necessary condition derivation also shows that, as ultraviolet physics sets in for when the theory is renormalized. ͉ ͉տ ͑ ␬ տ E Ea that is, for a 1), no claim can be made as to the nature of the states on these deeper scales. VI. RENORMALIZATION FRAMEWORKS A few comments are in order regarding Eq. ͑51͒. First, it In the previous section we showed that the property ͑48͒ explicitly displays a breakdown of the conformal symmetry, ͉ ͉ and related symmetry-breaking results are independent of the by the introduction of a scale E0 and an associated se- ͉ ͉ details of the regularization procedure. Because of the gen- quence of bound states. Second, the scale E0 arises from erality of the real-space regularization approach presented in the renormalization procedure. Third, as a renormalization ͉ ͉ this paper, these results extend the two-dimensional analysis scale, E0 cannot be predicted by the conformal model and of Ref. ͓40͔ in a number of nontrivial ways: it is to be adjusted experimentally. Fourth, once the experi- ͑i͒ For arbitrary renormalization frameworks, other than mental determination is carried out, an unambiguous predic- ͓ ͑ ͒ ͑ ͔͒ the ‘‘intrinsic’’ one of Ref. ͓40͔͑see below͒. tion from Eqs. 48 and 51 follows, ͑ ͒ ii For any dimensionality d. Again, the two-dimensional d͗D͘⌿ case of Ref. ͓40͔ has unique features that considerably sim- nϩ1 ␲ plify the derivation within the intrinsic framework. This is Enϩ1 dt 2 ϭ ϭexpͩ Ϫ ͪ , ͑52͒ particularly relevant because the physical applications that E d͗D͘⌿ ⌰ n n appear to be most interesting are d-dimensional realizations ϵ Þ dt of this phenomenon, with d deff 2. ͑iii͒ For any bound state and angular momentum channel within the range of applicability, ␬aӶ1. This is in agree- ͑and not just for the lϭ0 channel associated with the ground ment with the conclusions of phenomenological analyses of state considered in Refs. ͓34,35,40͔͒. the Efimov effect ͓30͔. In this section we highlight the relevance of these results The alternative intrinsic and core frameworks are charac- with an overview of the real-space ‘‘effective,’’ ‘‘intrinsic,’’ terized by the fact that the limit ␰ϭ␬a→0isstrictly applied and ‘‘core’’ renormalization frameworks ͑according to the before drawing any conclusions about the physics. There- presentation of Ref. ͓41͔͒, and discuss their relationship to fore, in order to keep the bound-state energies and ␬ values the present anomaly calculation. Despite their apparent dif- finite, a running coupling parameter is explicitly introduced, ferences, these frameworks share the basic physical require- so that Eq. ͑49͒ is still maintained in this limit. The running ment that the system is renormalized under the assumption parameter is either the conformal coupling g, in the intrinsic that the ultraviolet physics dictates the possible existence of framework, or the strength Ꭽ of the regularizing core inter- bound states of finite energy; the corresponding energies E action, in the core framework. and values of ␬ϰͱ͉E͉ are then required to remain finite. In the case of the intrinsic framework, the dependence g In order to facilitate the implementation of this renormal- ϭg(a), equivalent to ⌰ϭ⌰(a), is enforced. This leads to ization program, it is convenient to display the specific lim- the asymptotic running behavior ⌰ϳ0, which ensures that iting form that Eq. ͑27͒ takes when a→0; more precisely, the left-hand side of Eq. ͑49͒ remains well defined. This

͑␬ Ӷ ͒ limiting procedure leads to the renormalization framework of a 1 1 ͓ ͔ ͑ ͓͒ ⌰ cot͓␣,͑⌰,␬a͔͒ ϳ L (Ͻ)͑Ꭽ͒, ͑49͒ Refs. 13,14 ; in particular, Eq. 52 with the condition ⌰ ϳ0] implies the existence of a single bound state. In its

␬ Ӷ original form, the renormalization framework of Refs. ( a 1) ͓ ͔ where L (Ͻ)(˜ka;˜k) ϳ L (Ͻ)(Ꭽ) and 13,14 was based upon a Dirichlet boundary condition, which we now reinterpret as an effective Dirichlet boundary → ␬a (a 0) ␣͑⌰ ␬ ͒ϵ⌰ͫ ͩ ͪ ϩ␥ ͬ ͑ ͒ condition ͓16͔ u(rϭa) ϳ 0, for the reduced radial wave , a ln ⌰ , 50 2 function u(r)ϭͱrv(r). This result is guaranteed by the prefactor ͱr, regardless of the behavior of v(r). As for ␥ ϭϪ͕ ͓⌫ ϩ ⌰ ͔͖ ⌰͑ with ⌰ phase (1 i ) / which reduces to the v(r), two distinct cases should be considered: ͑i͒ the special ␥ ͓ ͔ ⌰→ Euler-Mascheroni constant 42 in the limit 0). case characterized by the simultaneous assignments dϭ2, l In the effective renormalization framework, the system is ϭ0, and constant V(Ͻ)(r) ͓or, to be more precise, with ␵ ͉ ͉Ӷ ϵប2 2 regularized maintaining ﬁnite values of E Ea /2ma . ϭ͉V ͉ 2Ϫ␭ϭ 2 ˜␰ϭ˜ ϭ ⌰ 0 a o(a )], for which ka O( ) and This condition deﬁnes an asymptotic conformally invariant (a→0) domain; within that domain, the condition ␬aӶ1 limits the cos ␣(⌰,␬a) ϳ 0; ͑ii͒ the generic case, characterized by d ultraviolet applicability of this effective scheme. Most im- Þ2, or lÞ0, or V(Ͻ)(r) not being constant, for which the

125013-7 H. E. CAMBLONG AND C. R. ORDO´ N˜ EZ PHYSICAL REVIEW D 68, 125013 ͑2003͒ variable ˜␰ϭ˜ka acquires a nonvanishing limit value ͱᎭ ACKNOWLEDGMENTS ϭͱ(lϩ␯)2ϩ␵͓as either (lϩ␯)Þ0or␵Þ0], and This research was supported in part by NSF under Grant (a→0) ͑ ͒ ␣ ⌰ ␬ ϳ No. PHY-0308300 H.E.C. and C.R.O. and by the Univer- sin ( , a) 0. sity of San Francisco Faculty Development Fund ͑H.E.C.͒. The smallness of the variable ˜␰ for case ͑i͒ above is the We also thank Dean Stanley Nel for generous travel support main reason for the simplicity of the derivation of Ref. ͓40͔. and Professors Cliff Burgess, Luis N. Epele, Huner Fan- In effect, in this case, Eq. ͑43͒ can be approximated using the chiotti, and Carlos A. Garcıá Canal for early discussions of small-argument behavior of Bessel functions without explic- this work. itly computing full-fledged Lommel integrals. Thus, (a→0) J ˜ ϳ ⌰2 ͉ ͉ APPENDIX A: ANISOTROPIC LONG-RANGE lϩ␯ϭ0(ka) /2 and Blϭ0,␯ϭ0 (a→0) CONFORMAL INTERACTION AND CONFORMAL ϭ͉ ␬ ˜ ͉ ϳ ␬ ͱ␲⌰ Alϭ0,␯ϭ0Ki⌰( a)/J0(ka) /( ), leading to BEHAVIOR OF DIPOLE-BOUND ANIONS (a→0) (a→0) ͗ ͘ ϳ ␲ 2 J ˜ ˜ 2 ϳ d D ⌿ /dt 2 B0,0 0(ka)V0 /k E, as discussed in In this appendix we show the mathematical procedure that Ref. ͓40͔. By contrast, for the generic case ͑ii͒, the analysis reduces the anisotropic inverse square potential to an effec- presented in this paper, based on the theory developed in tive isotropic interaction. Sec. IV and Appendix D, is inescapable. The Schro¨dinger equation for the Hamiltonian ͑3͒ can be Finally, in the core renormalization framework, the separated in spherical coordinates by means of strength of the core interaction becomes a running coupling ᎭϭᎭ ⌶͑⍀͒u͑r͒ parameter: (a), but the conformal coupling g remains ⌿͑r͒ϭ , ͑A1͒ constant ͓15,43͔. As a result, Eq. ͑49͒ provides the limit- r␯ϩ1/2 cycle running that has been used in renormalization analyses of the three-body problem ͓15,31,43͔. with normalization Incidentally, the ‘‘effective’’ renormalization framework discussed in Ref. ͓41͔͑and summarized in this section͒ leads 2 ͵ d⍀ Ϫ ͉⌶͑⍀͉͒ ϭ1. ͑A2͒ directly to a characterization of the thermodynamics of black d 1 holes. In essence, this amounts to a reinterpretation of ’t ⌶ ⍀ Hooft’s brick wall method ͓27͔, in which ultraviolet ‘‘new’’ As a result, the angular part ( ) of the wave function is physics sets in within a distance of the order of the Planck no longer a solution to Laplace’s equation on the unit (d Ϫ dϪ1 scale from the horizon. The computation of Appendix C 1)-sphere S ; instead, it satisfies the modified equation shows that the leading behavior near the horizon is confor- Aˆ ⌶͑⍀͒ϭ␥⌶͑⍀͒, ͑A3͒ mal and nontrivial, in that the effective system is placed in the supercritical regime. This asymptotic leading contribu- where tion, governed by the effective conformal interaction, re- quires renormalization and provides the correct thermody- Aˆ ϭϪ⌳2ϩ␭F͑⍀͒ ͑A4͒ namics ͓44͔. It should be noticed that there is an alternative treatment, based upon the method of self-adjoint extensions, and ⌳2ϭL2/ប2 is the dimensionless squared angular mo- which has been recently discussed in Refs. ͓23,24͔. mentum. The corresponding radial equation

d2u͑r͒ ␥Ϫ␯2ϩ1/4 ϩͩ k2ϩ ͪ u͑r͒ϭ0 ͑A5͒ VII. CONCLUSIONS dr2 r2 Realizations of the conformal anomaly involve a break- is coupled to Eq. ͑A3͒ through the separation constant ␥. down of the associated SO͑2,1͒ algebra. In this paper we Equation ͑A5͒ can be compared against the radial equation have shown that the actual emergence and value of the con- of an isotropic inverse square potential, which is obtained by formal anomaly rely upon the application of a renormaliza- another Liouville transformation ͓45͔ of the form ͑A1͒, but tion procedure, but are otherwise independent of the details with ultraspherical harmonics instead of ⌶(⍀) and for of the ultraviolet physics. In this sense, the results derived Ϫ V (r)ϰr 2 without angular dependence; the effective equa- herein are robust and totally general. As such, they are in- eff tion tended to shed light on the physics of any system with a conformally invariant domain for which the short-distance 2 ͒ ␭ Ϫ ϩ␯͒2ϩ d u͑r eff ͑l 1/4 physics dictates the existence of bound states. ϩͫ k2ϩ ͬu͑r͒ϭ0 ͑A6͒ In particular, the dipole-bound anions of molecular phys- dr2 r2 ics and the Eﬁmov effect are physical realizations of this ͑ ͒ unusual anomaly. In addition, the intriguing near-horizon is identical to Eq. A5 when the following identiﬁcations are physics of black holes appears to suggest yet another ex- made: ample of this ubiquitous phenomenon; the details of the ther- g ប2 modynamics arising from this conformal description will be ͒ϭϪ eff ϭ ␭ ␭ ͉ ϭ␥ ͑ ͒ Veff͑r , geff eff , eff lϭ0 . A7 reported elsewhere. r2 2m

125013-8 ANOMALY IN CONFORMAL QUANTUM MECHANICS:... PHYSICAL REVIEW D 68, 125013 ͑2003͒

Consequently, Eq. ͑A5͒ can be thought of as the radial part where of a d-dimensional effective isotropic conformal interaction ϭ for l 0. ϭ ͵ ⍀ ͑⍀͒ ͑⍀͒ ͑⍀͒ Ilm,lЈmЈ;lЉmЉ d dϪ1Y* Y lЉmЉ Y lЈmЈ . Furthermore, the values ␥ are quantized from the angular lm equation ͑A3͒ and depend upon the coupling ␭ of the aniso- ͑A12͒ tropic potential. This relationship can be made more explicit Finally, the components ⌶ of the angular wave function by expanding, in the ultraspherical-harmonic basis Y (⍀), lm lm can be formally obtained for every eigenvalue ␥ in the usual the anisotropy factor way, and satisfy ͓from Eq. ͑A2͔͒

͑⍀͒ϭ ͑⍀͒ ͑ ͒ F ͚ FlmY lm A8 ͉⌶ ͉2ϭ ͑ ͒ l,m ͚ lm 1. A13 l,m

and the angular wave function As an example of this general theory, one can consider the particular three-dimensional case (␯ϭ1/2) of the electron- ⌶ ⍀͒ϭ ⌶ ⍀͒ ͑ ͒ ͑ ͚ lmY lm͑ . A9 polar molecule interaction described in Sec. II A. In this l,m case, the matrix elements ͑A11͒ become This decomposition yields the matrix counterpart of Eq. ͗lm͉M͑␥,␭͉͒lЈmЈ͘ ͑A3͒, whence the anticipated relationship can be formally ϭ ϩ ͒ϩ␥ ␦ ␦ displayed by the inﬁnite secular determinant ͓l͑l 1 ͔ llЈ mmЈ ͑␥ ␭͒ϵ ͑␥ ␭͒ ͑␥ ␭͒ϭϪ ͑␭͒ϩ␥ ͑lϩm͒͑lϪm͒ D , det M , , M , A 1, Ϫ␭ͭͱ ␦ ␦ ͑A10͒ ͑2lϪ1͒͑2lϩ1͒ lЈ,lϪ1 mmЈ

in which 1 is the identity matrix; the matrix elements in Eq. ͑lϩmϩ1͒͑lϪmϩ1͒ ͑ ͒ ϩͱ ␦ ␦ ͮ ͑ ͒ A10 are ϩ , A14 ͑2lϩ1͒͑2lϩ3͒ lЈ,l 1 mmЈ ͗lm͉M͑␥,␭͉͒lЈmЈ͘ϭ͓l͑lϩ2␯͒ϩ␥͔␦ ␦ llЈ mmЈ which correspond to a matrix of block-diagonal form with respect to m and tridiagonal in l. Then, the secular determi- Ϫ␭ ͚ ͑ ͒ ␥ ␭ ϭ⌸ ␥ ␭ Ilm,lЈmЈ;lЉmЉFlЉmЉ , nant A10 factors out in the form D( , ) mDm( , ), lЉ,mЉ ␥ ␭ with the reduced determinant Dm( , ) in the m sector; thus, ͑A11͒ for given m, the equation det M(␥,␭)ϭ0 implies that

␭ ␥ Ϫ ͱ1Ϫm2 0 ͱ3 ••• ␭ ␭ Ϫ ͱ1Ϫm2 ͑2ϩ␥͒ Ϫ ͱ4Ϫm2 ͑␥ ␭͒ϭ ͱ ͱ ••• ϭ ͑ ͒ Dm , ͯ 3 15 ͯ 0. A15 ␭ 0 Ϫ ͱ4Ϫm2 ͑6ϩ␥͒ ͱ15 •••

••• ••• ••• •••

Equation ͑A15͒ has been used for the determination of the ␭Ͻ␭(*), no such values produce binding; a first ‘‘binding ␭( )Ϸ ͓ ͔ ␥ϭ␥( ) ␥ ␭у␭( ) critical dipole moment * 1.279 17 when * eigenvalue’’ 0,0 is obtained when * , for the first root ϭ1/4. When the determinant is expanded ͑to high orders͒, with mϭ0; as the strength ␭ of the interaction increases, a additional roots appear for the critical condition ␥(*)ϭ1/4 ␥ second binding eigenvalue 0,1 is produced for the first root and for different values of m. This pattern also illustrates with mϭ1, when ␭Ϸ7.58 or pϷ9.63 D; the next eigenvalue how one would completely solve the generic anisotropic ␥ ϭ ͑ ͒ ͑ ͒ 1,0 arises from the second root with m 0; etc. For each problem: Eq. A15 or its generalization A10 can be used ␥ϭ␥ to obtain the eigenvalues ␥ that correspond to a given cou- one of these values of j,m , an energy spectrum of con- ͑ ͒ ⌰ pling ␭; these eigenvalues replace the usual angular momen- formal states is governed by Eq. 51 , with eff given by Eq. tum numbers. In the molecular physics case described above, ͑5͒. These bound states have been observed experimentally ͓ ͔ ␥ the values of ␥ can be easily evaluated numerically. When 18,19 for the case when 0,0 is the only binding eigenvalue,

125013-9 H. E. CAMBLONG AND C. R. ORDO´ N˜ EZ PHYSICAL REVIEW D 68, 125013 ͑2003͒ a condition that corresponds to typical molecular dipole mo- 1 ͑dϪ2͒2 ␭͑dЈϭ1;lЈϭ0͒ϭ␭͑d;lϭ0͒ϩ Ϫ . ͑B4͒ ments. 4 4 Most importantly, this analysis confirms that the conformal anisotropic problem can be reduced to the isotropic one, Even in the special case of the equivalence described by and the same symmetry-breaking considerations apply. Eq. ͑B4͒, the full-fledged wave functions still retain a trace of the ‘‘physical dimensionality’’ d, because ͑with an obvious ͒ ϵ⌿͉ ϭ (dϪ1)/2⌿͉ APPENDIX B: DIMENSIONALITIES notation u(r) dϭ1(r) r d(r); for example, in AND INTERDIMENSIONAL DEPENDENCE the case of the three-dimensional Efimov effect, the full- fledged wave functions are of the form ⌿(r)ϰrϪ5/2u(r), The spatial dimensionality d of a physical realization of ϭ eff reflecting the fact that deff 6. conformal quantum mechanics is best characterized or de- The example of the near-horizon conformal behavior of fined as the dimension of the configuration space needed for black holes presents a number of peculiar features that de- a complete description of the dynamics within the conformal serve a separate treatment in Appendix C. approximation. Typically, this quantity can be directly identified from the nature of the radial variable used in the de- APPENDIX C: NEAR-HORIZON CONFORMAL scription of scale and conformal symmetries. BEHAVIOR OF BLACK HOLES For instance, with this convention, molecular anions can be naturally seen as a three-dimensional realization (deff In this appendix we present an algebraic derivation of the ϭ3); the Efimov effect, in a d-dimensional one-particle conformal invariance exhibited near the horizon of a black ϭ space, as a (2d)-dimensional realization (deff 2d); and the hole. near-horizon conformal physics of black holes, in Dϭdϩ1 From Eqs. ͑6͒ and ͑7͒, it follows that the equation of spacetime dimensions, as a d-dimensional realization (deff motion satisfied by the scalar field in the black-hole gravita- ϭd). tional background is Of course, there is a certain degree of arbitrariness in the selection of d , due to the existence of a formal relationship 1 1 eff 2 ͱ ␮␯ 2 ¨ ␯⌽͒Ϫm ⌽ϭϪ ⌽ϩ f ⌽Љץ ␮͑ Ϫggץ connecting problems of different dimensionalities. This can ͑ᮀϪm ͒⌽ϵ ͱϪg f be seen from the reduced Schro¨dinger-like radial equation of ͑ ͒ a conformal problem 2 , ͑DϪ2͒f 1 ϩͩ Јϩ ͪ ⌽Јϩ ᭝ ⌽Ϫ 2⌽ f Ϫ m r 2 D 2 d2u͑r͒ ␭Ϫ͑lϩ␯͒2ϩ1/4 r ϩͫ k2ϩ ͬu͑r͒ϭ0. ͑B1͒ dr2 r2 ϭ0, ͑C1͒ where the dots stand for time derivatives and the primes for Equation ͑B1͒ depends on the number of spatial dimensions radial derivatives in the chosen coordinate description of the only through the combination lϩ␯, a property known as background, while ᭝ Ϫ is the Laplacian on the unit (D interdimensional dependence ͓46͔. As a consequence, the ra- D 2 Ϫ2)-sphere. In addition, by separation of the time and an- dial part of the solutions for any two conformal problems are gular variables, identical when their coupling constants are related by ⌽ ⍀͒ϭ Ϫi␻t␸ ͒ ⍀͒ ͑ ͒ ͑t,r, e lm͑r Y lm͑ , C2 ␭͑dЈ;lЈ͒ϭ␭͑d;l͒ϩ͑lЈϪ1ϩdЈ/2͒2Ϫ͑lϪ1ϩd/2͒2. ͑B2͒ Eq. ͑C1͒ turns into

Moreover, f Ј ͑DϪ2͒ ␻2 m2 ␣ ␸Љ͑r͒ϩͩ ϩ ͪ ␸Ј͑r͒ϩͩ Ϫ Ϫ ͪ ␸͑r͒ϭ0, f r f 2 f r2 f ⌰͑dЈ͒ϭ⌰͑d͒ ͑B3͒ ͑C3͒ is a dimensional invariant of these formal transformations. with ␣ϭl(lϩDϪ3) being the eigenvalue of the operator Correspondingly, the conformal physics is totally determined Ϫ᭝ ͑ ͒ DϪ2. Equation C3 can be further reduced, by means of by the invariant value of this parameter. a Liouville transformation ͓45͔ However, the interdimensional equivalence of Eq. ͑B2͒ is severely limited by the fact that the full-ﬂedged solutions ␸͑r͒ϭg͑r͒u͑r͒, ͑wave functions͒ are not identical, because the angular mo- menta are different in different dimensionalities. The only 1 f Ј ͑DϪ2͒ ϭ g͑r͒ϭexpͭ Ϫ ͵ ͫ ϩ ͬdrͮ ϭ f Ϫ1/2rϪ(DϪ2)/2, exception to this is the formal equivalence among the l 0 2 f r angular momentum channels of problems with arbitrary di- ͑C4͒ mensionalities ͑as these channels do not involve additional dimension-dependent angular variables͒; in particular, an ef- to its normal or canonical form fective one-dimensional coupling can always be introduced for a d-dimensional problem with lϭ0: uЉ͑r͒ϩI͑r͒u͑r͒ϭ0, ͑C5͒

125013-10 ANOMALY IN CONFORMAL QUANTUM MECHANICS:... PHYSICAL REVIEW D 68, 125013 ͑2003͒

with normal invariant ͑DϪ2͒͑DϪ4͒ ␣ 1 Ϫͫ ϩ ͬ 2 ␻2 m2 ͑DϪ2͒͑DϪ4͒ ␣ 1 4 f r I͑r͒ϭ Ϫ Ϫͫ ϩ ͬ f 2 f 4 f r2 ͑lϩ␯͒2 1 1 1 1 ϭϪͫ Ϫ ϩ␯2ͩ 1Ϫ ͪͬ ϭOͩ ͪ ͑C12͒ 2 1 f Љ 1 f Ј2 ͑DϪ2͒f Ј f 4 f r x Ϫ ϩ Ϫ . ͑C6͒ 2 f 4 f 2 2rf ͓with ␯ϭ(dϪ2)/2ϭ(DϪ3)/2]. Thus, the angular The conformal behavior of the Schro¨dinger-like equation momentum—together with its associated dimensionality ͑C5͒ near the horizon can be studied by means of an expan- variable—decouples from the conformal interaction ͑C11͒ in sion in the variable the near-horizon limit. It should be noticed that we had to rewrite Eq. ͑C10͒ in the lϭ0, d-dimensional format in order ϭ Ϫ ͑ ͒ x r rϩ , C7 to present this problem within our uniﬁed conformal model ͑2͒. Alternatively, one could write Eq. ͑C11͒ in a simpler with rϭrϩ being the largest root of f (r)ϭ0. The nonextre- one-dimensional reduced form ͓from Eq. ͑B4͔͒, ␭(dϭ1) mal case is characterized by the condition ϭ␭ Ϫ␯2ϩ ϭ⌰2 ϩ eff 1/4 eff 1/4, with the same value for the dimensional invariant ⌰ . f ϩЈ ϵ f Ј͑rϩ͒Þ0, ͑C8͒ eff

equivalent to rϩÞrϪ . Then, APPENDIX D: GENERALIZED LOMMEL INTEGRALS ͑ ͒ϭ Ј ͓ ϩ ͑ ͔͒ f r f ϩx 1 O x , In this appendix we derive a generalization of the Lom- mel integrals ͓47͔ for an arbitrary Sturm-Liouville problem f Ј͑r͒ϭ f ϩЈ ͓1ϩO͑x͔͒,

͒ϭ Љ ϩ ͒ ͑ ͒ ˆ ͒ϭ␮ ͒ ͒ ͑ ͒ f Љ͑r f ϩ͓1 O͑x ͔, C9 Lxv͑x %͑x v͑x , D1

where f ϩЉ ϵ f Љ(rϩ). Thus, with corrective multiplicative factors of the order ͓1ϩO(x)͔, it follows that f Љ/ f d d ˆ ϭϪͭ ͫ ͑ ͒ ͬϩ ͑ ͒ͮ ͑ ͒ Lx p x q x , D2 ϳ f ϩЉ /(f ϩЈ x) and f Ј/ f ϳ1/x, while rϳrϩ , so that the only dx dx 2 2 2 2 leading terms in Eq. ͑C6͒ are ␻ / f ϳ␻ /(f ϩЈ x) and f Ј2/(4f 2)ϳ1/(4x2). As a result, Eq. ͑C5͒ is asymptotically ͑ ͒ reduced to the conformally invariant form and apply it to the reduced radial Schro¨dinger equation 21 . These generalized integrals are needed for the exact evalua- 1 ␻2 tion of expectation values in the anomaly calculation. uЉ͑x͒ϩͫ ϩ ͬxϪ2͓1ϩO͑x͔͒u͑x͒ϭ0, ͑C10͒ In what follows, we rewrite the differential equation ͑D1͒ 4 Ј 2 ͑ f ϩ͒ in the form

where, by abuse of notation, we have replaced u(r)byu(x). Equation ͑C10͒ indicates the existence of an asymptotic con- d ͓p͑x͒vЈ͑x͔͒ϭϪ͓␣2%͑x͒ϩq͑x͔͒v͑x͒, ͑D3͒ formal symmetry driven by the effective interaction dx ␭ ͑ ͒ϭϪ eff Veff x , with an eigenvalue ␮ϭ␣2 and where the prime stands for a x2 derivative with respect to x; moreover, v(x) can be chosen to 2 be a real function. Next, after conveniently multiplying both ␻ Ј ␭ ϭ␯2ϩ⌰2 ⌰2 ϭͫ ͬ ͑ ͒ sides by 2p(x)v (x), and integrating them with respect to x, eff eff, eff , C11 ͑ ͒ f Ј͑rϩ͒ Eq. D3 turns into

as follows by rewriting Eq. ͑C10͒ in the d-dimensional for- ͑ ͒ ͓p͑x͒vЈ͑x͔͒2͉x2 mat of Eq. B1 . This proves the claims made in Sec. II B x1 and, in particular, Eqs. ͑8͒ and ͑9͒. A ﬁnal remark is in order. The effective Hamiltonian x2 d ϭϪ͵ dxp͑x͓͒␣2 ͑x͒ϩq͑x͔͒ ͓v͑x͔͒2, ͑D4͒ ͑C10͒ did not fall ‘‘automatically’’ within the d-dimensional % x1 dx format of Eq. ͑B1͒. The extra terms Ϫ͓(lϩ␯)2Ϫ1/4)]/r2, usually obtained by reduction of a multidimensional Schro¨- dinger equation in ﬂat space, are still present, but at higher in which both the lower (x1) and upper limits (x2) are com- orders in the expansion with respect to the near-horizon co- pletely arbitrary. Finally, after integration by parts and rear- ordinate x;inEq.͑C6͒, they correspond to rangement of terms, Eq. ͑D4͒ leads to

125013-11 H. E. CAMBLONG AND C. R. ORDO´ N˜ EZ PHYSICAL REVIEW D 68, 125013 ͑2003͒

x2 x2 ␣2 ͵ dx͓p͑x͒%͑x͔͒Ј͓v͑x͔͒2 ␣2 ͵ dxx͓v͑x͔͒2 x1 x1

͑ ͒ 2 x2 vЈ x 1 x ϭ͓v͑x͔͒2ͭͫ p͑x͒ ͬ ϩp͑x͓͒␣2%͑x͒ϩq͑x͔͒ͮͯ ϭ ͓v͑x͔͒2͕͓L͑x͔͒2ϩ͓͑␣x͒2Ϫx2W͑x͔͖͉͒ 2 v͑x͒ 2 x1 x1 x2 1 x2 ϩ ͵ dxx͓v͑x͔͒2ͩ ϩ E ͪ W͑x͒, ͑D8͒ Ϫ ͵ dx͓p͑x͒q͑x͔͒Ј͓v͑x͔͒2, ͑D5͒ 1 x x1 2 x1 where L(x)ϭxvЈ(x)/v(x) and both limits are still arbitrary. which generalizes the well-known second Lommel integral Equation ͑D8͒ is the desired generalization that can be di- ͓47͔ of the theory of Bessel functions. A similar procedure rectly applied to the reduced Schro¨dinger equations ͑21͒ and could be applied for a generalization of the first Lommel ͑23͒ to derive Eqs. ͑33͒ and ͑35͒, as we will show next. integral, but this is not needed for the present purposes. First, for the interior problem (rрa), Eq. ͑D8͒ turns into The integral relation ͑D5͒ can be rewritten in a convenient Eq. ͑35͒, by means of the substitutions form for the reduced radial Schro¨dinger equation ͑21͒, which ϭ ␣ϭ˜ 2W͑ ͒ϭ͑ ϩ␯͒2ϩ 2U͑ ͒ is of the generalized Bessel form x r, k, x x l r r , ͒ϭ ˜ ˜ ͒ ϭ˜ ͑ ͒ v͑x wlϩ␯͑kr;k , z kr, D9 d2 1 d ϩ ϩ ␣2ϪW ϭ ͑ ͒ ͭ ͓ ͑x͔͒ͮ v͑x͒ 0. D6 ෈͓ ˜␰͔ ˜␰ϭ˜ dx2 x dx and with integration interval z 0, , where ka. For this case, when r2U(r)→0, that is, for regular core potentials, This is a particular case of the Sturm-Liouville equation the behavior of the differential equation at the origin implies that the contribution from the first term on the right-hand ͑D3͒, with density function %(x)ϭx, p(x)ϭx, and q(x) side of Eq. ͑D8͒ is zero for rϭ0. ϭϪxW(x); however, it is also true that a straightforward set ͓ ͔ ͑ ͒ Second, in a similar manner, for the exterior problem (r of two Liouville transformations 45 makes Eqs. D1 and у ͑ ͒ ͑ ͒ ͑D6͒ formally equivalent to each other. For Eq. ͑D6͒, a), Eq. D8 turns into Eq. 33 , by means of the substi- ͓p(x)q(x)͔ЈϭϪ͓x2W(x)͔Ј, and the final term in Eq. ͑D5͒ tutions can be evaluated with the help of xϭr, ␣ϭkϭi␬, x2W͑x͒ϭ͑lϩ␯͒2Ϫ␭ϭϪ⌰2, ͑ ͒ϭ ͑␬ ͒ ϭ␬ ͑ ͒ d 1 v x Ki⌰ r , z r, D10 ͓W͑ ͒ 2͔ϭ ͩ ϩ E ͪ W͑ ͒ ͑ ͒ x x 2x 1 x x , D7 dx 2 and with integration interval z෈͓␰,ϱ͔, with ␰ϭ␬a. Here, the behavior of the differential equation at infinity implies ץ ץ E ϭ where 1 is the identity operator and x x / x, as in Sec. IV. that the contribution from the first term on the right-hand As a consequence, Eq. ͑D5͒ becomes side of Eq. ͑D8͒ is also zero at that point.

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