The quantum-mechanical D-dimensional inverse square potential is analyzed
using field-theoretic renormalization techniques. A solution is presented for
both the bound-state and scattering sectors of the theory using cutoff and
dimensional regularization. In the renormalized version of the theory, there is
a strong-coupling regime where quantum-mechanical breaking of scale symmetry
takes place through dimensional transmutation, with the creation of a single
bound state and of an energy-dependent s-wave scattering matrix element.Comment: 5 page

B. Simon

C. Desfrançois

C. F. de Araujo, Jr.

C. G. Bollini

C. G. Simander

C. Manuel

C. Quigg

C. Radin

C. Schwartz

C. Thorn

Carlos A. García Canal

E. Marinari

G. 't Hooft

G. Parisi

H. Narnhofer

H. van Haeringen

Horacio E. Camblong

Huner Fanchiotti

I. S. Gradshteyn

J. D. Louck

J. F. Ashmore

J.-M. Lévy-Leblond

K. Huang

K. M. Case

K. S. Gupta

L. D. Landau

L. R. Mead

Luis N. Epele

O. H. Crawford

P. Gosdzinsky

P. M. Morse

R. Jackiw

R. O. Mastalir

S. Coleman

S. K. Adhikari

S. P. Alliluev

W. M. Frank

English

arXiv

Crossref

Physical Review Letters

Renormalization of the Inverse Square Potential

The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wave scattering matrix element.Departamento de Físic

Camblong, Horacio E.

Epele, Luis Nicolás

Fanchiotti, Huner

García Canal, Carlos Alberto

Servicio de Difusión de la Creación Intelectual

VOLUME 85, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 AUGUST 2000Renormalization of the Inverse Square PotentialHoracio E. Camblong,1 Luis N. Epele,2 Huner Fanchiotti,2 and Carlos A. García Canal21Department of Physics, University of San Francisco, San Francisco, California 94117-10802Laboratorio de Física Teórica, Departamento de Física, Universidad Nacional de La Plata, C.C. 67 – 1900 La Plata, Argentina(Received 2 March 2000)The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoreticrenormalization techniques. A solution is presented for both the bound-state and scattering sectors ofthe theory using cutoff and dimensional regularization. In the renormalized version of the theory, thereis a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place throughdimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wavescattering matrix element.PACS numbers: 03.65.Ge, 03.65.Nk, 11.10.Gh, 31.15.–pThe quantum-mechanical inverse square potential isa singular problem that has generated controversy fordecades. For instance, the solution proposed in Ref. [1]failed to give a Hamiltonian bounded from below, and thisled to a number of alternative regularization techniques[2–4] based on appropriate parametrizations of thepotential—including the replacement [5] of self-adjointness by an interpretation of the “fall of the particleto the center” [6]. However, it is generally recognizedthat the singular nature of this problem lies in that itsHamiltonian, being symmetric but not self-adjoint, admitsself-adjoint extensions [7]. Recently, a renormalizedsolution was presented using field-theoretic techniques[8], but it was just limited to the one-dimensional caseand cutoff renormalization.In this Letter (i) we generalize the results of Ref. [8] toD dimensions (including the all-important D ­ 3 case) us-ing cutoff regularization in configuration space; (ii) presenta complete picture of the renormalized theory; and (iii)confirm the same conclusions using dimensional regular-ization [9]. This problem is crucial for the analysis andinterpretation of the point dipole interaction of molecularphysics [10,11], and may be relevant in polymer physics[12]. In addition (i) it displays remarkable similarities withthe two-dimensional d-function potential [13–15]; (ii) itprovides another example of dimensional transmutation[16] in a system with a finite number of degrees of free-dom; and (iii) it illustrates the relevance of field-theoreticconcepts in quantum mechanics [13–15,17].This problem is ideally suited for implementation inconfiguration space [18], where the radial Schrödingerequation for a particle subject to the r22 potential in Ddimensions [19] reads (with h̄ ­ 1 and 2m ­ 1)∑1rD21ddrµrD21ddr∂1 E2lsl 1 D 2 2d 2 lr2∏Rlsrd ­ 0 , (1)which is explicitly scale invariant because l is dimension-less [20]. In Eq. (1), l is the angular momentum quantum1590 0031-9007y00y85(8)y1590(4)$15.00number and l . 0 corresponds to an attractive potential;with the transformation Rlsrd ­ r2sD21dy2ulsrd, Eq. (1) isrecognized to have solutions Rlsrd ­ r2sDy221dZsl spE rd,where Zsl szd represents an appropriate linear combinationof Bessel functions of order sl ­ flspdl 2 lg1y2, withlspdl ­ sl 1 Dy2 2 1d2. (2)If l were allowed to vary, one would see that the natureof the solutions changes around the critical value lspdl , foreach angular momentum state. For l , lspdl (including re-pulsive potentials), the order sl of the Bessel functions isreal, so that the solution regular at the origin is propor-tional to the Bessel function of the first kind Jsl spE rd.However, the same solution fails to satisfy the requiredbehavior at infinity for bound states (E , 0); in otherwords, in the weak-coupling regime, the potential cannotsustain bound states. Moreover, the scattering solutionsare scale invariant [20], with D-dimensional phase shiftsdsDdl ­ hflspdl g1y2 2 flspdl 2 lg1y2jpy2. Nothing is sur-prising here: the potential r22 is explicitly scale invariantand no additional scale arises at the level of the solutions,which are well-behaved—one could say that the potentiallooks like a regular “repulsive” one. However, this picturechanges dramatically for l . lspdl : all the Bessel functionsacquire an uncontrollable oscillatory character through theimaginary order sl ­ iQl , where Ql ­ fl 2 lspdl g1y2, aswe shall see next.For the remainder of this Letter, we will mainly analyzethe strong-coupling regime l . lspdl . First, for the bound-state sector, from Eq. (1), ulsrd ~pr KiQl spjEjrd, withKsl szd being the modified Bessel function of the secondkind [21], whose behavior near the origin is of the formKiQl szdsz!0d, 2rpQl sinhspQldsin∑Ql lnµz2∂2 dQl∏3 f1 1 Osz2dg , (3)where dQl is the phase of Gs1 1 iQld. In Eq. (3), thewave function oscillates with a monotonically increasing© 2000 The American Physical SocietyVOLUME 85, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 AUGUST 2000frequency as r ! 0. As a result, there is no criterion forthe selection of a particular subset of states and the bound-state spectrum is continuous and not bounded from below.Clearly, the problem should be renormalized in such a waythat the Hamiltonian recovers its self-adjoint character [7].A first attempt [1,22] is to use Eq. (3) and recognizethat the orthogonality condition for the eigenstates restoresthe discrete nature of the spectrum; unfortunately, in thisapproach, the Hamiltonian is not bounded from below.However, as was proposed in Ref. [8] for the particularsimple case D ­ 1, Eq. (3) can be regularized by intro-ducing a short-distance cutoff a, with a ø jEj21y2, sothat the regular boundary condition ulsad ­ 0 is imple-mented in lieu of the undefined behavior at r ­ 0. Then,Eq. (3) gives the zeros of the modified Bessel function ofthe second kind with imaginary order, zn ­ 2esdQl 2npdyQl[up to a correction factor 1 1 Osz2nyQld], where n isan integer; moreover, the assumption that zn ø 1, withQl $ 0, implies that s2nd , 0, with the conclusion thatn ­ 1, 2, 3, . . . . Parenthetically, zn ø 1 only if Ql ø 1,so that dQl ­ 2gQl 1 OsQ2l d (with g being the Euler-Mascheroni constant) and the energy levels becomeEnr l ­ 2µ2 e2ga∂2expµ22pnrQl∂, (4)where n ­ nr stands for the radial quantum number.Equation (4) should now be renormalized by re-quiring that Ql ­ Qlsad in the limit a ! 0. Moreprecisely, in order for the ground state [characterizedby the quantum numbers sgsd ; snr ­ 1, l ­ 0d] to“survive” the renormalization prescription with a fi-nite energy, it is required that Qsgsdsadsa!0d! 01. Thiscondition amounts to a “critically strong” coupling,lsadsa!0d! lspdsgsd 1 01 (where the notation Q0 ­ Qsgsdand lspd0 ­ lspdsgsd is understood for the ground state). Inparticular, with this ground-state renormalization, therequired relation between Qsgsdsad and a, for a small, is2gs0d ­2 pQsgsdsad1 2 lnµma2∂1 2g , (5)where m is an arbitrary renormalization scale with dimen-sions of inverse length and gs0d is an arbitrary finite partassociated with the coupling, such thatEsgsd ­ 2m2 expfgs0dg \ 2 m2. (6)In Eq. (6), it is understood that, due to the arbitrariness ofboth gs0d and m, the simple choice gs0d ­ 0 can be made.Finally, the ground-state wave function is obtained in thelimit Qsgsdsadsa!0d! 01, which yields [23]Csgsdsrd ­sGµD2∂ µm2p∂Dy2 K0smrdsmrdDy221, (7)whose functional form, up to a factor r2sDy221d, is dimen-sionally invariant [24].The existence of a ground state with a dimensional scalem ~ jEsgsdj1y2 violates the manifest scale invariance of thetheory defined by Eq. (1), but its magnitude is totally ar-bitrary and spontaneously generated by renormalization.Here we recognize the fingerprints of dimensional trans-mutation [16].The next question refers to the possible existence of ex-cited states in the renormalized theory. For any hypotheti-cal state with angular momentum quantum number l . 0,this question can be straightforwardly answered from theground-state renormalization condition Qsgsdsadsa!0d! 01,which, together with Eq. (2), provides the inequality l ­lspdsgsd ­ sDy2 2 1d2 , lspdl . Then, if such a state existed,it would automatically be pushed into the weak-couplingregime, with the implication that it could not survive therenormalization process. This means that there are no ex-cited states with l . 0. Next, the question arises as to thepossible existence of bound states with l ­ 0 and nr fi 0.The fact that these hypothetical bound states also cease toexist in the renormalized theory follows from the exponen-tial suppressionÇEnr0EsgsdÇ­ exp∑22psnr 2 1dQsgsd∏sQsgsd!0,nr .1d! 0 . (8)Moreover, it is easy to see that, for these hypotheticalstates, the corresponding limit of the wave function be-comes ill defined, so that they effectively vanish. Inconclusion, the renormalization process annihilates all can-didates for a renormalized bound state, with the only ex-ception of the ground state of the regularized theory, whichacquires the finite energy value (6) and the normalizedwave function (7).Similarly, the scattering solutions can be studied bygoing back to Eq. (1), which implies that ulsrdypr isa linear combination of the Hankel functions Hs1,2diQl skrd[21], whose asymptotic behavior (r ! `), combined withthe regularized boundary condition ulsad ­ 0, providesthe scattering matrix elements SsDdl sk; ad and phase shiftsdsDdl sk; ad. For example, the phase function fsDdl sk; ad ­dsDdl sk; ad 2 sl 1 Dy2 2 1dpy2 is given bytan sfsDdl sk; add ­ tanhµpQl2∂1 2 Tlsk; ad%lTlsk; ad 1 %l, (9)where Tlsk; ad ­ tanfQl lnskay2dg and %l ­ y2,lyiy1,l ,with y6,l ­ Gs1 2 iQld 6 Gs1 1 iQld. Equation (9)is ill defined in the limit a ! 0; in effect, the variableTlsk; ad oscillates wildly between 2` and `, unlessQl ! 0, just as for the bound-state sector. From Eqs. (5)and (6), for l ­ 0, in the limit a ! 0, the renormalizeds-wave phase shift becomes1591VOLUME 85, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 AUGUST 2000tan sdsD0d0 skd 2 sDy2 2 1d py2d ­plnsk2yjEsgsdjd.(10)Equation (10) explicitly displays the scattering behavior ofs states, as well as its relation with the bound-state sectorof the theory. Both the functional form of Eq. (10) andthe existence of a unique bound state in the renormalizedtheory are properties shared by the two-dimensional d-function potential [13–15].The analysis leading to Eq. (10) refers to l ­ 0. For allother angular momenta, l . 0, the coupling will be weak,so that the phase shifts will be given by their unregularizedvalues, with the condition that l ­ lspdsgsd ­ sDy2 2 1d2;then,dsDdl jlfi0 ­ fsl 1 Dy2 2 1d 2plsl 1 D 2 2dgp2,(11)which is a scale-invariant expression.We now turn to an outline of a similar analysis usingdimensional renormalization [9]. In particular, we will fo-cus on the bound-state sector of the theory, to illustrateand emphasize the fact that proper renormalization usingdifferent regularizations yields the same physics. In this al-ternative regularization scheme, we define the dimension-ally regularized potential in D0 dimensions in terms of itsmomentum-space expression, according to [23]V sD0dsr 0d ­ 2lBZ dD0k0s2pdD0eik0?r0∑ZdDr e2ik?r1r2∏k­k0­ 2lBpey2Gs1 2 ey2dysr 0d22e , (12)where e ­ D 2 D0 and lB is the dimensional bare cou-pling, which will be rewritten as mB ­ lme , with flg ­ 1and m being the floating renormalization scale. The cor-responding D0-dimensional Schrödinger equation for thereduced radial wave function ulsrd ­ r sD021dy2Rlsrd canbe converted, by means of a duality transformation [4,25]8<: jEj1y2r ­ z2ye ,jEj2D0y4 ulsrd ­ wl,eszdz1ye21y2,(13)intoΩd2dz21 eh 2 fVeszd 2 p2 2 1y4z2æwl,eszd ­ 0 ,(14)where fVeszd ­ 24 sgnsEdz4ye22ye2. In Eq. (14), thenew parameters areeh ­ 4lpey2 Gs1 2 ey2de2µjEjm2∂2ey2, (15)and p ­ 2sl 1 D0y2 2 1dye. The key to solvingEq. (14) is that (i) the parameter p is asymptotically1592infinite; and (ii) the term fVeszd in Eq. (14) behaves as aninfinite hyperspherical potential well in the limit e ! 0.Then, for bound states, as a first approximation, theparticle is trapped in a well with a smooth left boundaryproportional to 1yz2 and an infinite-well boundary atz2 ø 1; as the left turning point is z1 ø pyeh1y2, theWKB quantization condition—which we expect to beasymptotically correct for p ! `—becomesZ 1pyeh1y2seh 2 p2 2 1y4z2dz øµnr 214∂p , (16)so that eh1y2 ­ p 1 Cnr p1y3, where Cnr ­ f3psnr 21y4dg2y3y2. Therefore, from Eq. (15), it follows that theregularized energies arejEnr lj ­ m2"llspdl#2yeexpfGnr lsedg , (17)whereGnrlsed ­ 2 24y3Cnr slspdl d21y3e21y31 flnp 1 g 1 2slspdl d21y2g . (18)Equation (17) can be renormalized by demanding that itbe finite for the ground state and by letting l ­ lsed;explicitly,lsed ­ lspdsgsdΩ1 1e2fgs0d 2 Gsgsdsedgæ1 osed , (19)with an arbitrary finite part gs0d. In particular,lsedse!0d! lspdsgsd 1 01, i.e., upon renormalization,the coupling becomes critically strong with respect to sstates. Just as for cutoff regularization, it follows thatonly bound states with l ­ 0 survive the renormalizationprocess. As for the excited states with l ­ 0 in Eq. (17),they are exponentially suppressed according toÇEnr0EsgsdÇ­ expf224y3sCnr 2 C1d slspdsgsdd21y3 e21y3gse!0,nr .1d! 0 . (20)Parenthetically, the regularized energies of Eqs. (4) and(17), for finite a and e, are noticeably different; nonethe-less, as expected, their renormalized counterparts have ex-actly the same informational content.In short, we have analyzed the inverse square potentialand found that (i) a critical coupling divides the pos-sible behaviors into two regimes; (ii) in the strong-couplingregime, the theory is ill defined and requires renormaliza-tion; and (iii) upon renormalization of the strong-couplingregime, only one bound state survives and s-wavescattering breaks scale invariance with a characteristiclogarithmic dependence. The existence and order ofmagnitude of a critical coupling lspdsgsd ­ 1y4 for D ­ 3VOLUME 85, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 21 AUGUST 2000are in agreement with recent experimental results [10,11]for a wide range of polar molecules [26].A final remark is in order. Strictly, even though a morecareful treatment with dimensional regularization changesEq. (17), the difference appears only at the level of thefinite parts (linear in e) and is immaterial to the argumentspresented here. These corrections, as well as a detailedtreatment of the scattering sector of the theory, will bepresented elsewhere.This research was supported in part by CONICET andANPCyT, Argentina (L. N. E., H. F., and C. A. G. C.) andby the University of San Francisco Faculty DevelopmentFund (H. E. C.). The hospitality of the University ofHouston and instructive discussions with Professor CarlosR. Ordóñez are gratefully acknowledged by H. E. C.[1] K. M. Case, Phys. Rev. 80, 797 (1950).[2] W. M. Frank, D. J. Land, and R. M. Spector, Rev. Mod.Phys. 43, 36 (1971), and references therein.[3] G. Parisi and F. Zirilli, J. Math. Phys. 14, 243 (1973);C. Radin, J. Math. Phys. 16, 544 (1975); R. O. Mastalir,J. Math. Phys. 16, 743 (1975); 16, 749 (1975); 16, 752(1975); H. van Haeringen, J. Math. Phys. 19, 2171 (1978).[4] C. Schwartz, J. Math. Phys. 17, 863 (1976).[5] S. P. Alliluev, Sov. Phys. JETP 34, 8 (1972).[6] L. D. Landau and E. M. Lifshitz, Quantum Mechanics,Course of Theoretical Physics Vol. 3 (Pergamon Press, Ox-ford, 1977), pp. 114–117, 3rd ed.[7] B. Simon, Arch. Ration. Mech. Anal. 52, 44 (1974); C. G.Simander, Math. Z. 138, 53 (1974); H. Narnhofer, ActaPhys. Austriaca 40, 306 (1974), and references therein.[8] K. S. Gupta and S. G. Rajeev, Phys. Rev. D 48, 5940(1993).[9] C. G. Bollini and J. J. Giambiagi, Nuovo Cimento Soc. Ital.Fis. 12B, 20 (1972); G. ’t Hooft and M. Veltman, Nucl.Phys. B44, 189 (1972); J. F. Ashmore, Nuovo CimentoLett. 4, 289 (1972).[10] J.-M. Lévy-Leblond, Phys. Rev. 153, 1 (1967); O. H. Craw-ford, Proc. Phys. Soc. London 91, 279 (1967), and refer-ences therein.[11] C. Desfrançois, H. Abdoul-Carime, N. Khelifa, and J. P.Schermann, Phys. Rev. Lett. 73, 2436 (1994).[12] E. Marinari and G. Parisi, Europhys. Lett. 15, 721 (1991).[13] Momentum-space analyses of the two-dimensional d-function potential include C. Thorn, Phys. Rev. D 19, 639(1979); K. Huang, Quarks, Leptons, and Gauge Fields(World Scientific, Singapore, 1982), Secs. 10.7 and 10.8;R. Jackiw, in M. A. B. Bég Memorial Volume, edited byA. Ali and P. Hoodbhoy (World Scientific, Singapore,1991).[14] A momentum-space renormalization-group analysis ofthe two-dimensional d-function potential is developed inS. K. Adhikari and T. Frederico, Phys. Rev. Lett. 74, 4572(1995).[15] Configuration-space analyses of the two-dimensional d-function potential include C. Manuel and R. Tarrach, Phys.Lett. B 328, 113 (1994); P. Gosdzinsky and R. Tarrach,Am. J. Phys. 59, 70 (1991); L. R. Mead and J. Godiness,Am. J. Phys. 59, 935 (1991).[16] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888 (1973).[17] See also S. K. Adhikari and A. Ghosh, J. Phys. A 30, 6553(1997); S. K. Adhikari, T. Frederico, and I. D. Goldman,Phys. Rev. Lett. 74, 487 (1995).[18] As an alternative, the momentum-space scheme ofRef. [14] is applied numerically to a related problem(inverse square potential with exponential screening)in C. F. de Araujo, Jr., L. Tomio, S. K. Adhikari, andT. Frederico, J. Phys. A 30, 4687 (1997).[19] J. D. Louck and W. H. Shaffer, J. Mol. Spectrosc. 4, 285(1960); J. D. Louck, J. Mol. Spectrosc. 4, 298 (1960); 4,334 (1960).[20] R. Jackiw, Phys. Today 25, No. 1, 23 (1972).[21] Handbook of Mathematical Functions, edited byM. Abramowitz and I. A. Stegun (Dover Publications,New York, 1972).[22] P. M. Morse and H. Feshbach, Methods of Theoreti-cal Physics (McGraw-Hill, New York, 1953), Vol. 2,pp. 1665–1667.[23] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Se-ries, and Products (Academic Press, New York, 1980).[24] Incidentally, our theory in D dimensions has a two-dimensional appearance, due to dimensional transmuta-tion. This is in sharp contrast with the behavior of thesolutions of the wave equation and of the free-particleSchrödinger equation. See also Eq. (10).[25] C. Quigg and J. L. Rosner, Phys. Rep. 56, 167 (1979).[26] The present solution of the inverse square potential (withl ­ 0) applies to the radial part of the wave function undera dipole potential; the angular part can be dealt with byusing the method of Ref. [10].1593

Renormalization of the inverse square potential

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Renormalization of the Inverse Square Potential

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