Effective Field Theory Program for Conformal Quantum Anomalies Horacio E

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9-19-2005 Effective Field Theory Program for Conformal Quantum Anomalies Horacio E. Camblong University of San Francisco, [email protected]

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Recommended Citation Camblong, Horacio E., "Effective Field Theory Program for Conformal Quantum Anomalies" (2005). Physics and Astronomy. Paper 14. http://repository.usfca.edu/phys/14

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 A 72, 032107 ͑2005͒

Effective ﬁeld theory program for conformal quantum anomalies

Horacio E. Camblong,1 Luis N. Epele,2 Huner Fanchiotti,2 Carlos A. García Canal,2 and Carlos R. Ordóñez3 1Department of Physics, University of San Francisco, San Francisco, California 94117-1080, USA 2Laboratorio de Física Teórica, Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 67–1900 La Plata, Argentina 3Department of Physics, University of Houston, Houston, Texas 77204-5506, USA ͑Received 20 April 2005; published 19 September 2005͒

The emergence of conformal states is established for any problem involving a domain of scales where the long-range SO͑2,1͒ conformally invariant interaction is applicable. Whenever a clear-cut separation of ultraviolet and infrared cutoffs is in place, this renormalization mechanism is capable of producing binding in the strong-coupling regime. A realization of this phenomenon, in the form of dipole-bound anions, is discussed.

DOI: 10.1103/PhysRevA.72.032107 PACS number͑s͒: 11.30.Er, 11.10.Gh, 11.10.St, 31.15.Ϫp

I. INTRODUCTION theory ͓5͔. As we will see in the next section, the effective The renormalization program ͓1͔ provides an insightful field approach also provides the natural connection between framework for the description of physical scales within a this work and the standard results of rotationally adiabatic given problem. This assumes that the characteristic dimen- theory ͓8–10͔. sional scales are sufficiently separated, as required by effective field theory ͓1,2͔. Moreover, symmetry considerations A. Conformal physics of dipole-bound states usually furnish further analytical control over what contrib- The dominant part of the electron-molecule interaction uting factors might be relevant for the hierarchy of scales. In can be described with a point dipole—the electron does not addition to the well-known examples in high-energy physics and condensed matter physics, an effective renormalization significantly probe radial scales smaller than the size a of the of a system in molecular physics was introduced in Ref. ͓3͔, molecule. Then, in three spatial dimensions, the associated where a symmetry-centered approach was developed for the anisotropic Hamiltonian reads formation of dipole-bound anions by electron capture. In the p2 g ␪ ͑ ͒ relevant domain of scales, the dominant physics—governed H = − 2 cos , 1 by an inverse square potential ͓4͔—takes a scale-invariant 2me r form known as conformal quantum mechanics. in which the coupling g can be recast into a dimensionless The central purpose of this paper is to develop an effec- ␭ ប2 form =2meg/ . Under time reparametrizations, this sys- tive field theory program for the quantum anomaly of Ref. tem displays an SO͑2,1͒ conformal symmetry, whose break- ͓3͔. Specifically, we address the role played by additional ing at the quantum-mechanical level can be interpreted as a degrees of freedom—for example, the rotational ones in the quantum anomaly. As a first step, introducing separation of molecular case. In this manner, we extensively use recent ⌿͑ ⍀͒ ͑ ͒⌶͑⍀͒ work on the renormalization and anomalous symmetry variables: r, =u r /r in spherical coordinates. breaking of conformal quantum mechanics ͓5͔. As a conse- This leads to a scale-invariant radial equation quence, we will establish the following results. d2u͑r͒ ␥͑␭͒ ͑ ͒ ͫ 2 ͬ ͑ ͒ ͑ ͒ i The conformal analysis is robust and fairly insensitive 2 + k + 2 u r =0 2 to the ultraviolet and infrared physics. dr r ͑ ͒ ii The effective field approach—centered on renormal- coupled to a scale-independent angular operator equation ization techniques—sheds light, e.g., on the properties of dipole-bound anions; this is in sharp contrast with the state- Aˆ ͑␭͒⌶͑⍀͒ = ␥͑␭͒⌶͑⍀͒, ͑3͒ ments of Ref. ͓6͔. ͑iii͒ The origin of a critical dipole moment for binding where the eigenvalue ␥ϵ␥͑␭͒ plays the role of a separation can be directly traced to the conformal interaction. constant and In short, the predictions of the conformal framework of Ref. ͓3͔ are not significantly altered by the inclusion of ad- Aˆ ͑␭͒ =−l2 + ␭ cos␪, ͑4͒ ditional degrees of freedom. Moreover, a similar analysis can with l being the relative orbital angular momentum of the be applied to other problems for which the conformal quan- electron about the molecule. The problem defined by the tum anomaly is relevant, for example, for the Efimov effect equations above is completely characterized by the solutions ͓7͔. of conformal quantum mechanics. II. CONFORMAL QUANTUM MECHANICS AND DIPOLE-BOUND STATES B. Radial conformal quantum mechanics In this section, we start by summarizing the results of Ref. Conformal quantum mechanics applies to the description ͓3͔ for dipole-bound states in the language of effective field of the radial problem. All the properties and conclusions dis-

1050-2947/2005/72͑3͒/032107͑6͒/$23.00032107-1 ©2005 The American Physical Society CAMBLONG et al. PHYSICAL REVIEW A 72, 032107 ͑2005͒ cussed herein rely on the existence of a domain of scales in C. Angular eigenvalue equation which the dominant physics is scale invariant. A symmetry- The angular problem for an anisotropic conformal inter- centered analysis in the relevant conformally invariant do- action is given by Eq. ͑3͒, whose secular-determinant form ͑ ͒ main shows that the theory retains the SO 2,1 symmetry at D͑␥,␭͒ϵdet M͑␥,␭͒=0 involves the infinite matrix the quantum level when ␥Ͻ1/4, with ␥=1/4 being a critical M͑␥,␭͒=−A͑␭͒+␥1, with 1 being the identity matrix. In par- point of the conformal framework. The existence of a con- ticular, in the angular momentum basis ͉l,m͘, the matrix el- formal critical point ͗ ͉ ͑␥ ␭͉͒ ͘ ␦ ͑␥ ␭ ͒ ements lm M , lЈmЈ = mmЈMllЈ , ;m are diagonal ͑ ͒ ͑ ͒ with respect to m, with tridiagonal blocks ␥ * ϵ ␥͑␭ * ͒ = 1/4 ͑5͒ ͑␥ ␭ ͒ ͓ ͑ ͒ ␥͔␦ ␭͓ ͑ ͒␦ ͑ ↔ ͔͒ MllЈ , ;m = l l +1 + llЈ − Nl m l,lЈ−1 + l lЈ , is the crucial ingredient that explains the experimental fact ͑ ͒ that electron binding by molecular anions only occurs for 9 ͑*͒ ͓ ͔ dipole moments greater than a critical value p 3 . where N ͑m͒=ͱ͓͑l+1͒2 −m2͔/͓͑2l+1͒͑2l+3͔͒. As a result, ␥ജ l Conformal quantum mechanics is singular for 1/4, the secular determinant takes the factorized form D͑␥,␭͒ but can be rescued by the use of renormalization, which ⌸ ͑␥ ␭͒ ͑ = mDm , and the eigenvalues are given by the roots of yields conformal bound states with energies En =E0 exp ͑␥ ␭͒ϵ ͓ ͑␥ ␭ ͔͒ the reduced determinants Dm , det MllЈ , ;m =0, −2␲n/⌰͒, where n is a positive integer, E is the arbitrary 0 for all integer values of m. At this purely conformal level, for ground-state energy, and the conformal parameter ⌰ is de- ␥ every m, the roots h,m can be arranged in a decreasing se- rived from the coupling according to the rule ͓5͔ ␥ ജ␥ ജ␥ ജ … quence: 0,m 1,m 2,m ¯, with h=0, 1, , and compared against the condition for conformal criticality: ␥ 1 ␥͑*͒ ͑ ͒ ␥ ⌰ = ͱ␥ − . ͑6͒ = =1/4. Equation 9 implies the following trends: is a 4 monotonic function with respect to both ␭ and m, increasing with ␭ and decreasing with m. In particular, for any finite The specific value of the characteristic scale E0 defined by dipole moment p ͑i.e., finite ␭͒, there exist only a finite num- the conformal tower of states is sensitive to the ultraviolet ber of supercritical values of ␥; in turn, for each ␥, there is physics and cannot be predicted by a renormalization ap- an infinite tower of conformal states—possibly limited by proach alone. However, the scale E0 is not relevant in the the onset of nonconformal physics for long-distance scales. determination of the relative values of bound-state energies, Hence the conformal bound states are completely character- as exhibited by the geometric scaling ized by the set of quantum numbers ͑n,h,m͒, in which the ͑ ͒ ␥ subset h,m determines h,m, while n labels the ordering of EnЈ 2␲͑nЈ − n͒ the conformal tower or geometric scaling. The existence of = expͫ− ͬ, ͑7͒ ⌰ these states in the “supercritical regime” yields anomalous En breaking of the SO͑2,1͒ commutator algebra ͓5͔. ␥ which is a remnant of the original scale invariance. In par- An important related question is: for the largest root 0,0, ͑ ͒ ticular, the geometric ratio e−2␲/⌰ of adjacent energy levels what is the value ␭ * that generates a conformal critical ␥ ␥͑*͒ has a universal form that is independent of the cutoff and point? By setting 0,0= =1/4, the “principal conformal ␭͑*͒ Ϸ impervious to the ultraviolet physics . Finally, the conformal critical coupling” becomes conf 1.279 whence the required ͑*͒ ␭͑*͒ Ϸ ͓ ͔ states are characterized by normalized radial wave functions critical dipole moment is p =p0 1.625 D 3,12,13 . ␥ of the form Likewise, for each of the other roots h,m, the criticality con- ␥ ␥͑*͒ dition h,m = =1/4 defines additional, increasingly larger ͑ ͒ ͱ2 sinh͑␲⌰͒ͱ values ␭ * of the critical dipole moment. Each of these u͑r͒ = ␬ rK ⌰͑␬r͒, ͑8͒ h,m ␲⌰ i represents the onset of a new tower of conformal states of the form ͑7͒. The sequence of critical values of the dipole mo- ͑*͒ ͑*͒ K ⌰͑z͒ ␭ Ϸ ␭ Ϸ … where i is the Macdonald function of imaginary index ment includes 0,0 1.279; 0,1 7.58; . However, the ex- ͓11͔. This is the function whose properties guarantee the uni- perimentally observed bound states ͓14,15͔ appear to be lim- ͑ ͒ ␥ versal geometric scaling 7 . In addition, the same function ited to the highest root 0,0 because of the characteristic leads to an estimate of the characteristic radial size of the order of magnitude of the molecular dipole moments realized electron probability distribution, given by ␬−1, with relative in nature. ratios ␬ /␬ =e␲͑nЈ−n͒/⌰ exhibiting a similar kind of univer- n nЈ III. ROTATIONAL DEGREES OF FREEDOM sal geometric scaling. OF DIPOLE-BOUND ANIONS In short, the generic properties of conformal quantum mechanics determine the nature of the bound states of molecular We now turn, through an appropriate length-scale hierar- anions and are parametrized by the possible values of the chy, to a derivation of the connection between the approach conformal parameter ⌰. In turn, ⌰ is described, from Eq. of Refs. ͓6,8–10͔ and the conformal treatment of Ref. ͓3͔. ͑6͒, in terms of the effective coupling ␥=⌰2 +1/4, which is completely determined by the angular dependence of the in- A. Rotationally adiabatic theory teraction, through the eigenvalue equation ͑3͒. This is the In the rotationally adiabatic theory ͓9͔, the pseudopoten- problem to which we now turn. tial

032107-2 EFFECTIVE FIELD THEORY PROGRAM FOR … PHYSICAL REVIEW A 72, 032107 ͑2005͒

ប2 ⌫ ␭;F͑r͒ sions from the conformal framework are not substantially V͑r͒ =− „ …G͑r͒͑10͒ 2m r2 altered. The fundamental concept that underlies this surpris- e ing result—and which makes our construction successful—is for the radial electron wave function is an eigenvalue of the the clear-cut separation of scales. This is the essential as- reduced Hamiltonian sumption that underlies renormalization theory ͓1͔,asde- scribed in the effective field theory language ͓2͔. Specifi- 2 Aˆ ␭ ͑ ͒ ប „ ;F r … cally, the two characteristic length scales for the molecular Hˆ =− G͑r͒, ͑11͒ 2m r2 anions are ͑i͒ a scale of the order of the molecular size a; and e ͑ ͒ ͑ ͒ ii the rotational scale rB of Eq. 13 , whose size can be and the radial function G͑r͒ can be selected by comparison gleaned from IϳMa2, with M being the mass of the mol- with different expressions used in the literature ͓6,8–10͔.In ecule. Then, the scale hierarchy particular, the lowest eigenvalue gives the standard adiabatic ⑀ ͑ ͒ϵV͑ ͒ M potential: adiab r r . In addition, the nontrivial part of ϳ ͱ ӷ ͑ ͒ ͑ ͒ rB a a 15 the effective Hamiltonian of Eq. 11 arises from the adia- me batic approximation for the rotational motion of the mol- ϳ ϳ ecule, which provides the operator ͓6,9,10͔ shows that LUV a, and LIR rB play the role of “ultraviolet” and “infrared” scales, respectively. Moreover, Eq. ͑15͒ pro- Aˆ ␭ ͑ ͒ ͑ ͒ 2 ␭ ␪ ͑ ͒ vides a justification for the adiabatic approximation used in „ ;F r … =−F r l + cos , 12 Refs. ͓6,8,9͔; remarkably, this approximation is just a state- ͑ ͒ ͑ ͒ ͑ ͒2 where the function F r has the form F r =1+ r/rB ,in ment about length scales within an effective-field-theory de- which the length scale scription of molecular physics ͓16͔. Thus the conformal treatment constitutes a satisfactory framework for the phys- ប2 ͱ ͑ ͒ ics of dipole-bound molecular anions. This description can rB = 13 2meB be further justified by introducing a systematic reduction procedure. First, the dependence of V͑r͒ for rӷr plays a is associated with the rotator constant B=ប2 /2I ͑with I being B the moment of inertia͒. Simple inspection shows that secondary role for the problem of criticality. This can be rigorously established by an asymptotic analysis of the de- Aˆ ␭ ͑ ͒ ˆ ͑␭͒ ( ;F r ) is a generalization of A , in which the replace- terminant ͑14͒. Most importantly, the existence of a critical 2 → ͑ ͒ 2 ment l F r l is made; therefore their angular operator value and the ensuing bound states follow from the relevant structures are identical. Using again the orbital angular mo- Շ scales r rB: criticality does not originate in the infrared mentum basis ͉l,m͘ of the electron, the eigenvalue ⌫ sector. Second, the critical dipole moment arises from the ϵ⌫͑␭;F͒ of Aˆ ͑␭;F͒ can be found from the secular equation ultraviolet boundary and can be established by a renormalization framework. Therefore the dominant physics can be D ⌫ ␭ ͑ ͒ ϵ ͓M ⌫ ␭ ͑ ͒ ͔ ͑ ͒ m„ , ;F r … det llЈ„ , ;m;F r … =0, 14 extracted by considering the intermediate scales, with aՇr Ӷr . In that range, F͑r͒Ϸ1 and ⌫͑␭;F͒ in Eq. ͑14͒ can be where M(⌫,␭;F͑r͒)=−A(␭;F͑r͒)+⌫1, so that B M ⌫ ␭ ͑ ͒ ͑ ͒ replaced by a constant ␥͑␭͒ϵ⌫͑␭;1͒. Thus, in this “scale llЈ( , ;m;F r ) is obtained from Eq. 9 by the replace- ments l͑l+1͒→l͑l+1͒F͑r͒ and ␥→⌫ in the diagonal terms. window,” the adiabatic potential approximately reduces to a V͑ ͒ ប2␥ ͑ 2͒ Therefore the eigenvalues arising from Eq. ͑14͒ can be la- long-range conformal potential r =− / 2mer . Retrac- beled just as those derived from the conformal secular ing the previous steps, this reduction establishes the Hamil- ⌫ ⌫ tonian ͑1͒, whose conformal symmetry is reminiscent of the determinant: h,m. In particular, the largest one, 0,0, ⑀ ͑ ͒ corresponding description in high-energy physics ͓17͔:at leads to the standard adiabatic potential adiab r =−ប2⌫ (␭;F͑r͒)G͑r͒/͑2m r2͒ in Eq. ͑10͒. sufficiently small distances the problem becomes scale in- 0,0 e variant. Finally, when a length scale of the order a is reached, “new physics” emerges and a more detailed treat- B. Separation of scales: Renormalization theory ment is in order—for which a specific form of the factor G͑r͒ The current reformulation of the rotationally adiabatic would be needed. theory permits a direct comparison with the results of the conformal framework, to which it reduces by the use of ef- IV. GENERALIZED CONFORMAL FRAMEWORK: fective field theory arguments. The reason for this lies in PREDICTIONS AND NATURE OF THE CORRECTIONS that, in a renormalization treatment, the phenomenological factor G͑r͒ merely amounts to an ultraviolet regulator—only The length-scale analysis leads to a noteworthy adjust- needed for distances rՇa, where a is the size of the mol- ment to the previous results: the restriction of the conformal ecule. In other words, the details of the position dependence tower of bound states to the relevant range of scales. This is of G͑r͒ are of secondary importance because G͑r͒Ϸ1 for r due to the fact that the dominant physics is described by a տ a and the conformal potential effectively dominates the “conformal window” limited by the characteristic scales LUV ͓ ͔ relevant physics. Consequently, the only significant addition and LIR, which act as ultraviolet and infrared cutoffs 5 . The to the conformal framework appears to be the inclusion of existence of an ultraviolet boundary is directly involved in rotational degrees of freedom via the function F͑r͒. How- the renormalization process and drives the fundamental prop- ever, a careful analysis of Eq. ͑14͒ shows that the conclu- erties of the renormalized conformal framework. By contrast,

032107-3 CAMBLONG et al. PHYSICAL REVIEW A 72, 032107 ͑2005͒ as shown in Ref. ͓5͔, the infrared boundary only restricts the ␭͑*͒ ⑀ ϵ −1 ͑18͒ range of the dominant physics. ␭͑*͒ Most importantly, there are a number of predictions aris- conf ing from this generalized conformal framework, which— can be computed from the secular equation ͑14͒, by means of with appropriate refinements—could be tested experimen- Eq. ͑6͒, in which ␥=1/4 for the purely conformal theory, ␥ ⌰2 tally and compared against results from alternative while ˜ = gs+1/4 for the theory with an infrared cutoff, so approaches. We will illustrate these results by considering that the dominant sector of the theory in the subspace S ͑l m=0 1 r 2 −2 =0,1͒ of quantum numbers l=0 and l=1 for the secular ⌬␥ = ˜␥ − = ⌰2 =4␲2͑1−␦͒2ͫlnͩ B ͪ ͬ . ͑19͒ ͑ ͒ ⌫ ͱ 2 ␭2 4 gs a determinant 14 with m=0, in which 0,0=−F+ F + /3 ͓ ͔ 9 . In particular, in the restriction of the theory to the dominant The first prediction arises directly from the existence of a subspace S ͑l=0,1͒, the quantity ⑀ in Eq. ͑18͒ becomes conformal domain, which implies that the number of confor- m=0 mal bound states undergoes a cutoff process leading to a 20 ⑀ = ͱ͓1+4͑˜␥ − ␥͔͓͒1+ 4 ͑˜␥ − ␥͔͒ −1Ϸ ͑˜␥ − ␥͒, finite value Nconf. It turns out that the approximate number 9 9 ͑ ͒ ⌰ L 20 N ϳ lnͩ IR ͪ, ͑16͒ conf ␲ LUV where the approximate equality arises from the relatively small values of ͑˜␥−␥͒, which are due to the separation of which is predicted from renormalization, is also in good scales. Consequently, Eqs. ͑19͒ and ͑20͒ imply that agreement with known bound-state estimates ͓18,19͔. For 20 80␲2 r 2 −2 typical values of the parameters involved, the logarithmic ⑀ Ϸ ⌰2 Ϸ ͑ ␦͒2ͫ ͩ B ͪ ͬ ͑ ͒ gs 1− ln . 21 nature of Nconf yields the generally accepted result that 9 9 a dipole-bound molecular anions sustain only one or two bound states. Therefore, in contrast with the claims of Ref. As expected, this correction becomes more prominent for ͓6͔, our approach shows that the presence of a conformal decreasing values of I and increases the critical dipole from domain is the actual cause for the existence of bound states its ideal conformal value. In addition, the fractional state and of the critical dipole moment. contribution ␦ in the compensatory factor ͑1−␦͒ can be de- The second important prediction of the generalized renor- termined using standard estimates for the number of bound malization framework consists of corrections to the critical states ͓19͔. With these building blocks, Eq. ͑21͒ gives the ͑ ͒ value ␭͑*͒. Within the effective-field reduction, as a zeroth- leading dependence of the critical value ␭ * with respect to ␳ ␳ϵ ͑ 2͒ 2 2 order approximation, Eq. ͑14͓͒with F͑r͒Ϸ1͔ provides the the infrared scale through ln , with I/ mea =rB /a be- ␭͑*͒ ing the dimensionless molecular moment of inertia. The required critical dimensionless dipole moment conf, which is purely conformal in nature. Broadly speaking, when a dipole logarithmic dependence ln ␳ is the trademark of the underly- moment is sufficiently different from the critical value, the ing renormalization-induced physics and explains the slow ␭͑*͒ ␭͑*͒ predictions of the conformal framework are remarkably ac- convergence of towards conf. This analysis ultimately curate. However, very near criticality, ⌰ϳ0 and ␬ϳ0; this shows that, even when rotational degrees of freedom are in- is due to the fact that the condition of criticality amounts to cluded in the description of this problem, renormalization is ͑ ͒ the emergence of a ground state from the continuum. The still responsible for the predicted values of p * , including: corresponding enlarged characteristic size of the ground-state ͑i͒ the existence of a critical value whose order of mag- conformal wave function links the relevant scales and cor- nitude is given by the conformal critical point ͑5͒; and rections are unavoidable in the presence of an infrared cutoff. ͑ii͒ the underlying physics of the logarithmic correction One possible way of dealing with this is through a perturba- ͑21͒. tive evaluation of ␭͑*͒ at the level of Eq. ͑14͒; nevertheless, Most importantly, the results ͑16͒–͑21͒ are universal, i.e., because of the extremely long range of the wave function model-independent, within the conformal framework. ͑8͒, one would have to consider all orders of perturbation In addition, we acknowledge the existence of model- theory and carry out infinite resummations. An alternative, dependent corrections to this framework. For molecular di- more direct estimate can be established from the emergence pole anions, these effects can be represented by means of a of the first bound state, pseudopotential comprised of electrostatic terms—described by the multipole expansion—combined with many-body ␦ ͑ ͒ N = Nconf + =1, 17 contributions of two kinds: a polarization part and an exchange part due to the Pauli exclusion principle ␦ ␦ ␦ ͓ ͔ where = IR+ UV is the partial contribution of the infrared 10,15,20–22 . The long-distance electrostatic and polariza- and ultraviolet sectors to the number of states. The criticality tion terms do not substantially affect the rotational infrared condition ͑17͒, combined with Eq. ͑16͒, can then be used to corrections to the purely conformal problem because their ⌰ 2 ͑ evaluate the conformal parameter gs of the critical ground- coupling constants are proportional to a with the relevant ⌰ 2 state wave function; the fact that gs is small but finite is due rotational degrees of freedom being proportional to rB, and ӷ ͓͒ ͔ to the self-consistent restriction of the theory in the infrared. rB a 10,21 . The short-distance behavior, which contrib- Thus the fractional correction to the critical dipole value utes to the ultraviolet physics with a scale of the order of

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ϳ LUV a, involves electrostatic and exchange many-body ef- malization for a system with a conformally invariant domain fects ͓10,21͔. In the case of the exchange effects, the char- whose ultraviolet boundary dictates binding. The ensuing acteristic scale is determined by the overlap of orbitals asso- quantum symmetry breaking within this framework captures ciated with tightly bound electrons, and the corresponding the essence of the observed critical dipole moment for the repulsive core is highly dependent on the nature of the mo- formation of dipole-bound anions. ͓ ͔ ␦ Ͻ lecular species 23 , with UV 0. This negative value par- Moreover, the tools developed in this paper, as exempli- ␦ ͑ ͒ ͑ ͒ tially compensates the positive term IR and favors the agree- fied by Eqs. 16 – 22 , show that this conformal framework: ment with the observed critical dipole moment in complex ͑1͒ permits the extraction of universal properties for molecular species. Consequently, the scale analysis confirms physical problems with a conformally invariant domain; and the remarkable fact that the dipole-bound anionic state exists ͑2͒ provides a description of dipole-bound anions in primarily due to the conformal interaction ͓24͔. One of the which model-dependent and model-independent contribu- simplest characterizations of these model-dependent corrections can be conveniently organized. tions is afforded by the dominant limiting infrared behavior In principle, this generalized conformal framework could of the rotationally adiabatic theory of Ref. ͓9͔, which yields be used as the starting point of a systematic approximation ␦Ϸ␦ Ϸͱ ␭͑*͒ ͑ ⑀͒ ␲ scheme for the description of dipole-bound molecular an- IR 6 conf 1+ /3 . With these assignments, intro- ͓͑ͱ ␭͑*͒ ␲͒−1 ͔−1Ϸ ions. The estimate ͑21͒ is a typical illustration of this: its ducing the parameters c= 6 conf/3 −1 0.498, A =80␲2L−2/͓9͑c+1͒2͔, and L=ln␳, the fractional correction numerical coefficients could be further refined by an im- to the dipole moment becomes ⑀Ϸ͕͓1+1/͑2cA͔͒ proved matching of the conformal domain with the infrared −ͱ͓1+1/͑2cA͔͒2 −1͖/c; for example, for various values of and ultraviolet sectors, as well as by considering higher orders ͑with respect to l͒. Thus our problem is similar to that the dimensionless molecular moment of inertia: ␳=2ϫ108, encountered in many other areas of physics, in which a ze- ␳=2ϫ106, and ␳=4ϫ104, the corresponding fractional cor- roth order approximation captures the essential ingredients, rections are, respectively, ⑀Ϸ0.11, ⑀Ϸ0.16, and ⑀Ϸ0.26 which are to be subsequently improved upon by the use of ͓25͔. miscellaneous approximation techniques. Finally, let us consider another universal prediction for an Most intriguingly, our approach exhibits many similarities experimental realization with at least two conformal bound with the recently developed chiral-Lagrangian program for states ͓26͔. For such a system, Eq. ͑7͒ yields the ratio nuclear physics ͓27,28͔, in which the underlying chiral sym- E /E =exp͑−2␲/⌰͒ from which the relative value of the 1 0 metry from QCD provides a guiding principle within a dipole moment, compared to the critical dipole, is power-counting scheme that selects the terms in the Lagrang- ␭ 20 80 ␲2 E −2 ian for nucleons and pions—with the first terms capturing the −1Ϸ ⌰2 = ͫlnͩ 1 ͪͬ , ͑22͒ ␭͑*͒ 9 9 E dominant, model-independent contributions. Likewise, our 0 conformal framework, based on the SO͑2,1͒ invariance and S ͑ ͒ which can be derived with the restriction to m=0 l=0,1 , the use of effective-field theory concepts, is a discriminating and supplemented by critical-diople corrections just as in Eq. scheme to elucidate the dominant model-independent fea- ͑21͒. This “inversion” makes a simple prediction solely tures of the molecular anions and similar systems with a based on conformal quantum mechanics and which can be conformally invariant domain; in this context, it would be explicitly compared against the improved critical value ͑21͒, interesting to develop the analog of the chiral power- using the known dipole moment ␭ for the given polar mol- counting scheme. ecule. In essence, this is a test of the residual scale invariance of the geometric scaling ͑7͒ of the conformal tower of states. ACKNOWLEDGMENTS

V. CONCLUSIONS This work was supported by CONICET, ANPCyT, the University of San Francisco Faculty Development Fund, and In conclusion, the central concept put forward in this pa- by the National Science Foundation under Grants No. per is the anomalous emergence of bound states via renor- 0308300 and No. 0308435.

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