Baryons from quarks in curved space and deconfinement
aa r X i v : . [ h e p - ph ] S e p Baryons from quarks in curved space and deconfinement
M. Kirchbach and C. B. Compean,Instituto de Fis´ıca, Universidad Autonoma de San Luis Potos´ı,Av. Manuel Nava 6, Zona Universitaria,S.L.P. 78290, M´exico
Abstract:
Detailed account is given of the fact that the Cornell potential predicted by LatticeQCD and its exactly solvable trigonometric extension recently reported by us can be viewed asthe respective approximate and exact counterparts on a curved space to an 1 /r flat spacepotential. The “curved” potential describes a confinement phenomenon as it is of infinite depthand has only bound states. It furthermore has the remarkable property of preserving both the SO (4) and SO (2 ,
1) symmetries characterizing the ordinary 1 /r potential. We first make thecase that this particular geometric vision on confinement provides a remarkably adequatedescription of both nucleon and ∆ spectra and the proton mean square charge radius as well,and suggests an intriguing venue toward quark deconfinement as a shut-down of the curvatureconsidered as temperature dependent. Next we observe that the SO (2 ,
1) symmetry of the“curved” potential allows to place it within the context of
AdS /CF T correspondence and toestablish in this manner the algebraic link of the latter to QCD potentiology.“...there will be no contradiction in our mind if we assume that some natural forces are governedby one special geometry, while other forces by another.”N. I. Lobachevsky One of the major achievements of contemporary physics concerns the insight on the non-trivialgeometry of the Universe. According to general theory of relativity, the space is curved by thepresence of mass, a well established concept which was successful in explaining the precession ofthe orbit of Mercury and the bending of light in the vicinity of the Sun. Various geometries havebeen under consideration ever since in gravity, the closed Einstein’s universe of constant positiveand the open one of Lobachevsky of constant negative curvature being among the most prominentversions (see [1],[2] for contemporary treatises). On long term, the ideas of general relativityexercised a profound impact on the development of quantum physics. On the one hand, they ledto the geometric interpretation of gauge theories (see [3] for a pedagogic presentation) and onthe other, they triggered progress in merging external space-time with internal gauge degrees offreedom which culminated in superstring and supergravity theories.Perhaps nothing expresses relevance of the above movements better but the surprisingly mod-ernly sounding statement made by Lobachevsky approximately 200 years ago (taken from ref. [4])which we used as motto to the present article. Indeed, the geometric view on both space timeand gauge theories suggests that some of the fundamental physics phenomena might be related tocurvature, an obvious candidate being the color confinement phenomenon.The idea that quark confinement might reflect some kind of curvature has been pioneered bySalam and Strathdee [5] who found a black-hole solution of Einstein’s equation approximate to theanti-de Sitter
AdS geometry which describes strongly interacting tensor fields confined in a micro-universe of a radius fixed by the negative cosmological constant and which they interpreted as a1ort of hadron bag. In this manner, particles have been described within gravitational context asstrongly curved universes, an idea pursued by several authors within various contexts [6]. Amongthe achievements of this idea we count (i) the explanation of the flavor independent level spacingsof the radial excitation spectra of mesons following from the local isomorphism between the anti-de Sitter group SO (3 ,
2) and Sp (4 , R ), the symmetry group of the two-dimensional harmonicoscillator, (ii) the understanding of quark confinement as divergence of the bag radius due to itsthermal dependence leading to space flattening [7]. Some of the constituent quark models, such asthe MIT bag, the “Chashire cat” or the Skyrme models have been interpreted (roughly speaking)as black holes of anti-de Sitter geometry.From the outgoing 90ies onward, the idea of the geometric confinement and the study of AdSspace-time manifolds experienced a strong push, now from the new perspective of the AdS /CF T correspondence according to which a maximal supersymmetric Yang-Mills conformal field theory(CFT) in four dimensional Minkowski space is equivalent to a type IIB closed superstring theory inten dimensions described by the product manifold AdS × S [8]. More recently, the correspondencebetween string theory in ten dimensional anti-de Sitter space and SO (4 ,
2) invariant conformalYang-Mills theories has been adapted in [9] to the description of hadron properties.Parallel to the above achievements, significant progress in understanding hadron propertieshas been reached independently through the elaboration of the connection between the QCDLagrangian and the potential models as deduced within the framework of effective field theoriesand especially through the non-perturbative methods such as lattice simulations [10],[11], the mostprominent outcome being the linear plus Coulomb confinement potential [12], [13]. The potentialsderived from the QCD Lagrangian have been most successful in the description of heavy quarkoniaand heavy baryon properties [14]. Although the Cornell potential has found applications also innucleon and ∆ quark models [15], the provided level of quality in the description in the non-strangesector stays below the one reached for the heavy flavor sector. This behavior reflects insufficiencyof the one gluon exchange (giving rise to the Coulomb-like term) and of the flux-tube interaction(associated with the linear part) to account for the complexity of the dynamics of three lightquarks. Various improvements have been under consideration in the literature such as screeningeffects in combination with spin-spin forces (see [16] and reference’s therein).Very recently, the Cornell potential has been updated through its extension toward an exactlysolvable (in the sense of the Schr¨odinger equation, or, the Klein-Gordon equation with equal scalarand vector potentials) trigonometric quark confinement potential [17]. The latter potential, whichhas the property of interpolating between a Coulomb-like potential and the infinite well whilepassing through a region of linear growth, was shown to provide a remarkably good descriptionof the spectra of the non-strange baryons, the nucleon and the ∆, considered as quark-diquarksystems, and of the proton charge radius as well. Especially the observed hydrogen-like degeneracypatterns in the above spectra in the non-trivial combination with the non-hydrogen like (becauseincreasing instead of decreasing) level spacings found a stringent explanation in terms of the SO (4)symmetry of the trigonometric potential.The main goal of the present work is to draw attention to the fact that similarlyto the Coulomb-potential, the trigonometrically extended Cornell (TEC) potential,possesses next to SO (4) also SO (2 ,
1) symmetry as manifest through the possibilityto cast its spectrum as well in terms of SO (4) as in terms of SO (2 ,
1) Casimirs. The SO (2 ,
1) versus SO (4) symmetry correspondence allows to place the TEC potentialwithin the context of AdS versus CF T correspondence and in this manner to link thealgebraic aspects of
AdS /CF T to QCD potentiology [10]. Such is possible becausewhile SO (2 ,
1) appears in one of the possible
AdS reduction chains [18], namely, SO (3 , ⊂ SO (2 , ⊂ SO (2 , ⊂ SO (2) , (1) SO (4) appears within the reduction chain associated with CF T , SO (4 , ⊂ SO (4 , ⊂ SO (4) ⊂ SO (3) ⊂ SO (2) . (2)2t is important to be aware of the fact that the algebraic AdS /CF T criteria alone are notsufficient to fix uniquely the potential. One has to complement them by the requirement oncompatibility with the QCD Lagrangian too, a condition which imposes severe restrictions on theallowed potential shapes.Our next point is that algebraically the AdS /CF T correspondence translates into SO (2 , /SO (4) symmetry correspondence of the potential and that it is the trigono-metrically extended Cornell potential, treated as quark-diquark potential, the one thatmeets best both the AdS /CF T and QCD criteria and provides the link between them.Predicted degeneracy patterns and level splitting are such that none of the observed N states drops out of the corresponding systematics which also applies equally well tothe ∆ spectra (except accommodation of the hybrid ∆(1600)). The scenario providesa remarkable description of the proton charge electric form-factor too and moreoverimplies a deconfinement mechanism as a shut-down of the curvature considered astemperature dependent.The group symmetries under discussion appear within the context of a Schr¨odinger equationwritten in different variables. Specifically for the case of the hydrogen atom these different vari-ables have been extensively studied and are well known. The SO (4) appears as symmetry of thestandard radial part, R ( r ), of the Schr¨odinger wave function, with r standing as usual for theradial distance, while SO (2 ,
1) is the symmetry of same equation when transformed to r = y and R ( r ) = y − Y ( y ) variables [19],[20]. Obviously, in the specific case under consideration, bothsymmetries are physically indistinguishable because they both lead to same physical observablessuch as spectrum and transition probabilities. Preferring the one over the other as mathematicaltool in hydrogen description is a pure matter of convenience and giving preference to SO (4) isonly more popular. The Coulomb potential is an example that matches algebraically AdS /CF T correspondence but is at odds with the non-perturbative aspects of QCD dynamics.The TEC case of major interest in this work presents itself bit more involved. While theSchr¨odinger equation giving rise to the SO (4) symmetric spectrum is well studied and well under-stood in terms of a potential satisfying the Laplace-Beltrami equation on the three dimensional(3D) hypersphere, S R , of constant radius, R , i.e. on a curved space of a constant positive cur-vature, knowledge on the differential realization of the SO (2 ,
1) symmetry on a hyperbolic spaceis quite scarce indeed (see next section). Filling this technical gap should certainly be beyondthe scope of the present study which main focus are spectroscopic observables. However, a strongthough indirect hint on the relevance of SO (2 ,
1) for the TEC problem is provided by the pos-sibility to recast its spectrum in terms of SO (2 ,
1) Casimir eigenvalues (admittedly, for limitedvalues of the strength parameter [21]). In other words, at least for some particular values of thestrength parameters a manifest coordinate transformation of the Schr¨odinger equation with theTEC potential from a four-dimensional Euclidean to a three dimensional hyperbolic geometry islikely to exist. Of course, unless such a transformation has not been explicitly constructed, no al-gebraic formalization of a possible indistinguishability between the respective SO (4) and SO (2 , SO (4) as part of CF T , it can also be embedded within an (admittedly,Euclidean) anti-de Sitter space [22], − x + x + x + x + x = − l , (3)where the x i with i = 1 , , x and x are the two time-like dimensions, from which the second has been “Wick rotated”, and − /l is the negative curvature. This additional insight provides, in our view, a further legitimizationfor accepting, without too much a loss of generality, the hypersphere x + x + x + x = R asa mathematical tool in the description of quark confinement as infinite potential barrier, a venuethat takes directly to the trigonometric extension of the Cornell potential. In taking this path,however, one should treat the positive curvature, κ = 1 /R , introduced in that manner, with3ome care and detain from equipping it with too deep a physical meaning. Rather it should beviewed as a second phenomenological parameter next to the potential strength which so far staysuncorrelated to the physical cosmological constant, λ = − /l . We shall show that while spectraand charge form factors remain by and large insensitive to this parameter, it provides a valuablephenomenological tool for deconfinement description in the spirit of ref. [7].The outline of the paper is as follows. The next section is a historic survey on the quantumKepler problem in a space of constant positive curvature, the 3D hypersphere, S R , a surveythat begins with early work by Schr¨odinger [23]. A detailed account is given of the fact that theharmonic potential on S R , i.e. the one that satisfies the four-dimensional angular Laplace-Beltramiequation, takes the form of the trigonometric confinement potential − b cot χ + l ( l + 1) csc χ (with χ standing for the second polar angle). Depending on the χ parametrization in terms ofcoordinates in ordinary position space, a variety of potentials can be created. Examples are thethe exactly solvable trigonometric extension of the Cornell quark confinement potential, aroundwhich the present work is centered, and which corresponds to χ = r √ κπ where r is the absolutevalue of the radius vector, and κ is the curvature of the hypersphere. Another version would bea gradient dependent confinement potential for particles with position and curvature dependentmass as needed for the purposes of quantum dots. We attend also this more subtle version becauseof its possible relevance for the description of the evolution of finite valence to vanishing partonquark masses. These two examples are presented in section 3. Section 4 focuses on the applicationof the trigonometric extension of the Cornell potential to the spectra of the nucleon and the ∆considered as quark–diquark systems. It further contains the description of same spectra withinthe SO (2 ,
1) symmetric version of same potential thus revealing its link to the algebraic aspects ofthe
AdS /CF T scenario. The section ends with a calculation of the mean square proton chargeradius. Section 5 presents the property of energy spectrum and wave functions of the TEC problemto collapse upon curvature shut-down (i.e. in the large R limit) to the bound and scattering statesof ordinary flat-space 1 /r potential, a peculiarity that we employ (in parallel to Takagi’s work [7]mentioned above) to interpret deconfinement as flattening of space due to a thermal dependenceof the curvature parameter. The paper closes with brief summary and outlooks. S R :The survey The first to have considered the Coulomb potential on a curved space has been Schr¨odinger whosolved in [23] the quantum mechanical Coulomb problem in the cosmological context of Einstein’suniverse, i.e. on the three dimensional (3D) hypersphere, S R , of a constant radius R . Schr¨odinger’sprime result, the presence of curvature provokes that the orbiting particle appears confined withina trigonometric potential of infinite depth and the hydrogen spectrum shows only bound states.An especially interesting observation was that the O (4) degeneracy of the levels observed in theflat space H atom spectrum was preserved by the curved space spectrum too in the sense thatalso there the levels could be labeled by the standard atomic indices n , l , and m , and the energydepended on n alone. However, contrary to flat space, no explicit form of O (4) generators couldbe immediately exhibited. Although Higgs [24] and Leemon [25] succeeded in constructing on S R the respective analogue to the Runge-Lenz vector in flat space, no way was found to incorporateit into the O (4) group algebra. Instead, Barut and collaborators [26] designed a version of thepotential as a differential su (1 ,
1) Casimir operator in allowing one of the potential parametersto depend on the principal quantum number in a very particular way. However, this version,strictly speaking, does not share the original SO (4) degeneracy patterns and is not of interest tothe present study.Perhaps because Schr¨odinger used the curved space Coulomb problem as an example for exactsolubility of his celebrated equation by means of the factorization technique, it became more popu-lar in that very context than in any other giving rise to the field of physics known as supersymmetricquantum mechancs (SUSYQM), a historical development mainly triggered by subsequent extensivework by Infeld and collaborators [27] and later on by Witten [28].Nonetheless, also the geometric aspect of Schr¨odinger’s work was independently picked up by4everal researchers and placed within various contexts. The idea of using such “curved” potentialsgradually breached into several areas of quantum physics from atomic [29] to the utmost modernnano-tubes physics [30], and sophisticated non-linear W algebra symmetries [31] viable in stringtheories. Bessis et al. [29] applied it to fine structure analysis of atomic spectra, Ballesterosand Herraz [32] considered it within the context of quantum algebras, while Roy and Roychoud-huri formulated SUSYQM in a 3D curved space [33]. Also the Russian school provided notablecontributions especially regarding the mathematical aspects of the solutions [34].It is worth noticing that also the harmonic oscillator (HO) potential has been considered on S R in [24], [36]. Finally, both the Coulomb and the HO problems have been also solved in spaceswith a negative constant curvature (Lobachevsky geometry) in the second reference [27], and in[37], [38], to mention only few representative examples of such studies (see also ref. [2] for a recentup-date).The current section is a historic survey on the quantum Kepler problem in a space of constantpositive curvature, the 3D hypersphere, S R . It contains a detailed account of the fact that theharmonic potential on S R , i.e. the one that satisfies the four-dimensional Laplace-Beltrami equa-tion, takes the form of the trigonometric confinement potential − b cot χ + l ( l + 1) csc χ (with χ standing for the second polar angle). S R parametrization and curved space Coulomb-like potential From now onward the usual three dimensional flat Euclidean space, E , will be embedded in thefour dimensional Euclidean space, E . A set of generalized Cartesian coordinates, { x , x , x , x } ,in E chosen to parametrize a three-dimensional spherical surface there, has to satisfy the condi-tion, s = x + x + x + x , κ = 1 s , < s < ∞ , (4)where s is the hyper-radius, and κ the corresponding curvature. A Coulomb-like potential in any E n space is defined from the requirement to be harmonic, i.e. to obey the respective n -dimensionalLaplace-Beltrami equation in charge free spaces. Specifically in E such a potential, call it v (¯ x ),where ¯ x denotes the radius vector of a generic point on the hypersphere, is most easily found inCartesian coordinates (cid:3) v (¯ x ) = i =4 X i =1 ∂ ∂x i v (¯ x ) = 0 , (5)and reads v (¯ x ) = c x ¯ r , ¯ r = q x + x + x . (6)Here, ¯ r is the length of the radius vector in the E subspace of E , c is a constant, and usehas been made of the fact that the 1 / ¯ r potential is harmonic in E as it satisfies there thethree-dimensional Laplace equation, ~ ∇ (1 / ¯ r ) = 0. Changing now to hyperspherical coordinates,Ω = { χ, θ, ϕ } , results in x = ¯ r sin θ cos ϕ, x = ¯ r sin θ sin ϕ,x = ¯ r cos θ, x = s cos χ, ¯ r = s sin χ, ≤ χ ≤ π, ≤ θ ≤ π, ≤ ϕ ≤ π. (7)In terms of χ , the “curved” 1 / ¯ r potential is read off from eq. (7) as the following trigonometricpotential, v (cid:16) x ¯ r (cid:17) ≡ v ( χ ) = c cot χ . (8)This is a very interesting situation in so far as in E the r´ole of the radial coordinate of infiniterange in ordinary flat space, 0 < ¯ r < ∞ , has been taken by the angular variable, χ , of finiterange. In other words, while the harmonic potential in E is a central one, in E it is non-central.Moreover, 5he inverse distance potential of finite depth in E is converted to an infinite barrierand therefore to a confinement potential in the higher dimensional E space, a propertyof fundamental importance throughout the paper. This section contains a detailed account of Schr¨odinger’s treatment [23] of the quantum mechanicalCoulomb problem within Einstein’s cosmological concept, i.e. on the three dimensional sphere ofconstant radius, S R . For this purpose, Schr¨odinger had to solve his celebrated equation in E . Theequation is not only especially simple to solve on the hypersphere of constant radius, s = R =const,where it is purely angular, but it is also there where it acquires a special physical importance, tobe revealed below.When written in the hyper-spherical coordinates, it takes the following form, (cid:18) − κ ~ µ b (cid:3) + c cot χ (cid:19) Ψ( χ, θ, ϕ, κ ) = E ( κ )Ψ( χ, θ, ϕ, κ ) , κ = 1 R = const , (9)where b (cid:3) denotes the angular (hyperspherical) part of the E Laplace-Beltrami operator. Here,and without loss of generality, the potential strength has been kept unspecified so far and denotedby the constant c . Using the well known representation of b (cid:3) , b (cid:3) = (cid:20) χ ∂∂χ sin χ ∂∂χ − L sin χ (cid:21) ,L = − (cid:20) θ ∂∂θ sin θ ∂∂θ + 1sin θ ∂ ∂ϕ (cid:21) , (10)where L is the standard three dimensional orbital angular momentum operator in E , and sep-arating variables in the solution as Ψ( χ, θ, ϕ ) = ψ ( χ, κ ) Y ml ( θ, ϕ ), the following equation in the χ variable (hyper-angular equation) emerges, h − κ ~ µ χ ∂∂χ (cid:18) sin χ ∂∂χ (cid:19) + V ( χ, κ ) − E ( κ ) i ψ ( χ, κ ) = 0 , V ( χ, κ ) = κ ~ µ l ( l + 1)sin χ + c cot χ . (11)Multiplying eq. (11) by (cid:0) − sin χ (cid:1) and changing variable to X ( χ, κ ) = sin χψ ( χ, κ ) , (12)allows to cast it into the form of the following one-dimensional Schr¨odinger equation in the angularvariable χ , (cid:20) − κ ~ µ d d χ + V ( χ, κ ) (cid:21) X ( χ, κ ) = (cid:18) E ( κ ) + κ ~ µ (cid:19) X ( χ, κ ) . (13)This is precisely the equation first obtained by Schr¨odinger [23]. Before proceeding further, it isquite instructive to first take a close look on the free particle motion on S R , i.e. c = 0, a subjecttreated in the next section. S R For a free particle motion on S R equation (13) reduces to (cid:20) − κ ~ µ d d χ + κ ~ µ l ( l + 1)sin χ (cid:21) S ( χ, κ ) = E ( c =0) ( κ ) S ( χ, κ ) , (14)6ith S ( χ, κ ) denoting the free-particle solution. The second term on the l.h.s. of this equationdescribes the centrifugal energy, U l ( χ, κ ), of a particle of a non-zero orbital angular momentumon S R , i.e., U l ( χ, κ ) = κ ~ µ l ( l + 1)sin χ . (15)This term provides an infinite barrier and thereby a confinement, an observation to acquire pro-found importance in what follows.The energy spectrum of eq. (14) is easily found from the observation on its inherent O (4)symmetry. Indeed, the angular part, b (cid:3) , of the four-dimensional Laplacian, (cid:3) , represents theoperator of the four-dimensional angular momentum, here denoted by K , according to, (cid:3) = − R ˆ (cid:3) = − R K , whose action on the states is given by [39] K | K, l, m i = K ( K + 2) | K, l, m i . (16)Here, the O (4) states have been equipped by the quantum numbers, K , l , and m defining theeigenvalues of the respective four–, three– and two–dimensional angular momentum operatorsupon same states. These quantum numbers correspond to the O (4) /O (3) /O (2) reduction chainand satisfy the branching rules, l = 0 , , , ..K , and m = − l, ..., + l .Therefore, the corresponding energy spectrum has to be E ( c =0) K ( κ ) = κ ~ µ K ( K + 2) . (17)When cast in terms of n = K + 1, the latter spectrum takes the form E ( c =0) n ( κ ) = κ ~ µ (cid:0) n − (cid:1) , (18)which coincides (up to an additive constant) with the spectrum of a particle confined withinan infinitely deep spherical quantum-box well. Then n acquires meaning of principal quantumnumber.The solutions of eq. (14) are text-book knowledge [39] ,[40] and rely in the following way uponthe Gegenbauer polynomials, C αm , the O (4) orthogonal polynomials, S Kl ( χ, κ ) = √ κ l +1 l ! s κ ( K + 1)( K − l )!2 π ( K + l + 1)! sin l χC l +1 K − l (cos χ ) . (19)The complete solutions to eq. (14) and on the unit hypersphere are the well known hyper-sphericalharmonics given by | Klm > = Z Klm ( χ, θ, ϕ ) = S Kl ( χ, κ = 1) Y ml ( θ, ϕ ) , (20)where Y ml ( θ, ϕ ) are the standard spherical harmonics in ordinary three space. cot χ barrier Various potentials in conventional flat E space appear as images to the cot χ potential in eq. (8).Their explicit forms are determined by the choice of coordinates on S R which shape the lineelement, d s . The general expression of the line element in the space under consideration and inhyper-spherical coordinates, Ω = { χ, θ, ϕ } , readsd s = 1 κ [d χ + sin χ (d θ + sin θ d ϕ )] . (21) The analogue on the two-dimensional sphere of a constant radius r = a is the well known relation ~ ∇ = − a L . χ = f ( r ), restricted to 0 ≤ f ( r ) ≤ π , eq. (21) takes the formd s = 1 κ [( f ′ ( r )) d r + sin f ( r )(d θ + sin θ d ϕ )] , ≡ D ( r, κ ) d r r + R ( r, κ )(d θ + sin θ d ϕ ) , (22) D ( r, κ ) ≡ r √ κ f ′ ( r ) , R ( r, κ ) ≡ sin f ( r ) √ κ , (23)where D ( r, κ ), and R ( r, κ ) are usually referred to as “gauge metric tensor” and “scale factor”,respectively [41].Changing variable in eq. (13) correspondingly is standard and various choices for f ( r ) giverise to a variety of radial equations in ordinary flat space with effective potentials which arenot even necessarily central. All these equations, no matter how different that may look, are ofcourse equivalent, they have same spectra, and the transition probabilities between the levels areindependent on the choice for f ( r ). Nonetheless, some of the scenarios provided by the differentchoices for f ( r ) can be more efficient in the description of particular phenomena than others.Precisely here lies the power of the curvature concept as the common prototype ofconfinement phenomena of different disguises. In the following we shall present twotypical examples for f ( r ). D ( r, κ ) = r √ κ r κ gauge and a gradient dependent confinement potential witha position dependent reduced mass A prominent choice for the transformation of the angular χ variable to the r variable has beenmade in ref. [42] for the purpose of quantum dots physics. This gauge is of general interest inso far as in flat E space it describes a particle with position and curvature dependent massmoving within a confinement potential whose infinite barrier is generated by gradient terms. Thetransformation under consideration reads χ = tan − r √ κ, ≤ r √ κ < ∞ , (24)and corresponds to a parametrization of the “upper” hemisphere in terms of tangential projectivecoordinates with respect to the “North” pole. The line element in this gauge becomesd s = 1(1 + r κ ) r d r + 1 κ sin (cid:0) tan − r √ κ (cid:1) (d θ + sin θ d ϕ ) . (25)The intriguing aspect of this gauge is that in the r variable the cot χ potential on S R is portrayedby a gradient dependent potential with a position and curvature dependent reduced mass. In-deed, changing the χ variable in the principal curved space Schr¨odinger wave equation in (13) inaccordance with eq. (24) and upon the substitution , ψ (tan − r √ κ ) = (1 + κr )Φ( r, κ ) , (26)amounts after some straightforward algebra to − ~ µ (cid:0) κr (cid:1) h (1 + κr ) ∂ ∂r + 2 r (1 + 3 κr ) ∂∂r + 6 κ − l ( l + 1) r i Φ( r, κ )+ α cot(tan − r √ κ )Φ( r, κ ) = E ( κ )Φ( r, κ ) , α = e Zǫ . (27)Now introducing the position and curvature dependent mass as µ ∗ ( r, κ ) = µ κr , (28) This substitution ensures that the Φ( r, κ )’s are normalized as wave functions in E . (cid:0) tan − r √ κ (cid:1) = 1 / ( r √ κ ), allows to casteq. (27) into the following symmetrized form of the kinetic terms, h (cid:18) µ ∗ ( r, κ ) ∆ r + ∆ r µ ∗ ( r, κ ) (cid:19) + v (cid:18) r, ∂∂r , κ (cid:19) i Φ( r, κ ) = E ( κ )Φ( r, κ ) . (29)The explicit expression for the gradient potential reads: v (cid:18) r, ∂∂r , κ (cid:19) = αr √ κ − ~ κ µ ∗ "(cid:18) r ∂∂r (cid:19) + 3 (cid:18) r ∂∂r (cid:19) + 3 − ~ κ µ r (cid:18) r ∂∂r + 1 (cid:19) , ∆ r = ∂ ∂r + 2 r ∂∂r − l ( l + 1) r . (30)Equation (29) describes particles with position and curvature dependent masses confined within agradient potential, a scenario suited for the case of electrons confined in semi-conductor quantumdots. This confinement phenomenon, that occurs due to the electron-crystal interaction, maynot restrict to quantum dots alone. Also quarks with position dependent masses dueto quark-sea interaction may be of interest too [43]. D ( r, κ ) = πr gauge and the central trigonometric Rosen-Morse potential An especially simple and convenient parametrization of the χ variable in terms of r , also used bySchr¨odinger [23] and corresponding to the D ( r ) = πr gauge is χ = rR π ≡ rd , d = Rπ , rR ∈ [0 , , κ → e κ = 1 d , (31)in which case the line element takes the formd s = π e κ (cid:18) d (cid:16) r √ e κ (cid:17) + sin (cid:16) r √ e κ (cid:17) (d θ + sin θ d ϕ ) (cid:19) . (32)Here, the length parameter d assumes the r´ole of rescaled hyper-radius. Correspondingly, in thisparticular gauge, the place of the genuine curvature, κ = 1 /R , is taken by the rescaled one, e κ = 1 /d . Setting now c = − G √ e κ , eq. (11) takes the form of a radial Schr¨odinger equation witha particular central potential of infinite depth (confinement potential) known in SUSYQM underthe name of the trigonometric Rosen-Morse potential (or, Rosen-Morse I) [44]. This equationreads h − e κ ~ µ d d (cid:16) r √ e κ (cid:17) + V (cid:16) r √ e κ, e κ (cid:17) i X (cid:16) r √ e κ, e κ (cid:17) = (cid:18) E ( e κ ) + ~ µ e κ (cid:19) X (cid:16) r √ e κ, e κ (cid:17) , V (cid:16) r √ e κ, e κ (cid:17) = e κ ~ µ l ( l + 1)sin (cid:16) r √ e κ (cid:17) − G √ e κ cot (cid:16) r √ e κ (cid:17) . (33)Important to note, SUSYQM suppresses the curvature dependence of V by absorbingit into the constants and the variable through the replacements, G √ e κ → b , and e κl ( l +1) → a ( a + 1), and refers to r √ e κ as to a dimensionless position variable, r ∈ [0 , π ].Besides Schr¨odinger, eq. (33) has been solved by various authors using different schemes. Thesolutions obtained in [45] are built on top of Jacobi polynomials of imaginary arguments andparameters that are complex conjugate to each other, while ref. [34] expands the wave functions ofthe interacting case in the free particle basis. The most recent construction in our previous work946] instead relies upon real Romanovski polynomials. In the χ variable and according to eq. (12)(versus ψ ( r ) = r R ( r ) in E ) our solutions take the form, X ( Kl ) ( χ, e κ ) = N ( Kl ) sin K +1 χe − bχK +1 R ( bK +1 , − ( K +1)) K − l (cot χ ) , b = 2 µG √ e κ ~ .K = 0 , , , ..., l = 0 , , ..., K, (34)where N ( Kl ) is a normalization constant. The R ( α,β ) n (cot χ ) functions are the non-classical Ro-manovski polynomials [47, 48] which are defined by the following Rodrigues formula, R ( α,β ) n ( x ) = e α cot − x (1 + x ) − β +1 × d n d x n e − α cot − x (1 + x ) β − n , (35)where x = cot r √ e κ (see ref. [49] for a recent review).The energy spectrum of V (cid:16) r √ e κ, e κ (cid:17) is given by E K ( e κ ) = − G ~ µ K + 1) + e κ ~ µ (( K + 1) − , l = 0 , , , ..., K. (36)Giving ( K + 1) the interpretation of a principal quantum number n = 0 , , , ... (as in the H atom), one easily recognizes that the energy in eq. (36) is defined by the Balmer term and itsinverse of opposite sign, thus revealing O (4) as dynamical symmetry of the problem. Stateddifferently, particular levels bound within different potentials (distinct by the values of l ) carrysame energies and align to levels (multiplets) characterized, similarly to the free case in eq. (16),by the four dimensional angular momentum, K . The K -levels belong to the irreducible O (4)representations of the type (cid:0) K , K (cid:1) . When the confined particle carries spin-1/2, as is the case ofelectrons in quantum dots, or quarks in baryons, one has to couple the spin, i.e. the (cid:0) , (cid:1) ⊕ (cid:0) , (cid:1) representation, to the previous multiplet, ending up with the (reducible) O (4) representation | K, l, m, s = 12 i = (cid:18) K , K (cid:19) ⊗ (cid:20)(cid:18) , (cid:19) ⊕ (cid:18) , (cid:19)(cid:21) . (37)This representation contains K parity dyads and a state of maximal spin, J max = K + , withoutparity companion and of either positive ( π = +) or, negative ( π = − ) parity,12 ± , ..., (cid:18) K − (cid:19) ± , (cid:18) K + 12 (cid:19) π ∈ | K, l, m, s = 12 i . (38)As we shall see below this scenario turns to be the one adequate for the description of non-strangebaryon structure.The above examples are illustrative of the power of curved space potentials as sourcesfor a variety of effective confinement potentials in ordinary flat E space.However, much care is in demand when working with such potentials. One should be aware of thefact that physical observables do not depend on the gauge chosen and performing in the r variablehas to be consistent with performing in the χ variable. We shall come back to this point in thenext section. S R potentiology:The baryons SO (4) symmetry scheme The spectrum of the nucleon continues being enigmatic despite the long history of the respectivestudies (see refs. [50], [51] for recent reviews). Unprejudiced inspection of the data reported by10he Particle Data Group [52] reveals a systematic degeneracy of the excited states of the baryonsof the best coverage, the nucleon ( N ) and the ∆(1232). Our case is that • levels and level splittings of the nucleon and ∆ spectra match the spectrum of in eq. (36), • the curvature induced confinement potential in eq. (33) is the exactly solvable extension ofto the Cornell potential predicted by Lattice QCD.1. The N and ∆ spectra: Afreque ll the observed nucleon resonances with masses below 2 . K = 1 , , F and H , statesstill “missing”, an observation due to refs. [53]. Moreover, the level splittings follow with anamazing accuracy eq. (36). This is in fact a result already reported in our previous work inref. [17] where we assumed dominance of a quark-diquark configuration in nucleon structure,and fitted the nucleon spectrum in Fig. 2 to the spectrum of the trigonometric Rosen-Morsepotential (c.f. eq. (33)) by the following set of parameters, µ = 1 .
06 fm − , G = 237 .
55 MeV · fm , d = 2 .
31 fm . (39)However, in ref. [17] the curvature concept has not been taken into consideration and be-cause of that the d quantity did not have any deeper meaning but the one of some lengthmatching parameter. This contrasts the present work, which in being entirely focused onthe geometric aspect of confinement, places d on the firmer ground of a parameter encodinga space curvature.Almost same set of parameters, up to a modification of d to d = 3 fm, fits the ∆(1232)spectrum, which exhibits exactly same degeneracy patterns, and from which only the three P , P , and D states from the K = 5 level are “missing”. Remarkably, none of thereported states, with exception of the ∆(1600) resonance, presumably a hybrid, drops fromthe systematics. The unnatural parity of the K = 3 , − internal excitation of the diquark which, when coupledto its maximal spin 1 + , can produce a pseudoscalar in one of the possibilities. The change ofparity from natural to unnatural can be given the interpretation of a chiral phase transition inbaryon spectra. Levels with K = 2 , N and ∆ spectra. To them, natural parities have been assigned on the basis ofa detailed analysis of the 1 p − h Hilbert space of three quarks and its decomposition in the | K, l, m, s = i basis [53]. The ∆(1232) spectrum obtained in this way is shown in Fig. 3.We predict a total of 33 unobserved resonances of a dominant quark-diquark configurationsin the N and ∆(1232) spectra with masses below ∼ central two-parameter potential in E and without reference to S R , areason for which the values of the parameter accompanying the csc term had to be takenas integer ad hoc and for the only sake of a better fit to the spectra, i.e., without any deeperjustification. Instead, in the present work,we fully recognize that the higher dimensional potential V ( χ, κ ) in eq. (13), whichacts as the prototype of Rosen-Morse I, is a non-central one-parameter potentialin which the strength of the csc term, the centrifugal barrier on S R , is uniquelyfixed by the eigenvalues of underlying three-dimensional angular momentum.2. Relationship to QCD dynamics: The convenience of the scenario under consideration is ad-ditionally backed by the fact that the Cornell quark confinement potential [13] predictedby Lattice QCD [12] is no more but the small-angle approximation to cot r √ e κ in eq. (8).Indeed, the first terms of the series expansion are − G √ e κ cot r √ e κ + e κ ~ µ l ( l + 1)sin (cid:16) r √ e κ (cid:17) ≈ − Gr + 2 G e κ r + ~ µ l ( l + 1) r , (40)11ith e κ = d = π R .Therefore, V ( r √ e κ, e κ ), is the exactly solvable trigonometric extension to the Cornell potential, a reason for which we shall frequently refer to V ( r √ e κ, e κ ) as TEC po-tential.Finally, the “curved” 1 /r potential has been completely independently used in [41] within thecontext of charmonium physics. However, in refs. [41] at the end the degeneracy of the states hasbeen removed through an ad hoc extension of the Hamiltonian to include an additional L /r -term of sign opposite to four-dimensional centrifugal barrier, L / sin χ , so that all states with l = 1 , , ..., K could be pushed below the S state belonging to a given K . In this way a betterdescription of the charmonium states has been achieved indeed but on the cost of compromisingconsistency of the geometric S R concept. In contrast to the charmonium, in N and ∆(1232)baryon spectra the S R degeneracy patterns are pretty well pronounced especially in the mostreliable region below 2000 MeV where our predicted K = 1 , S R geometric concept intact and the exactwave functions unaltered, so far. In subsection C below these wave functions will be put at workin the description of the mean square proton charge radius. SO (2 , symmetry scheme The energy spectrum in eq. (36) can equivalently be cast in terms of the eigenvalues of the SO (2 , J = − J ± J ∓ + J ± J [19],[20]. Here, the op-erators J ± and J satisfy the group algebra [ J + , J − ] = − J , and [ J , J ± ] = ± J ± . The eigenstatesare labeled by the quantum numbers j , and m ′ which define the respective eigenvalues of J and J according to J | j, m ′ i = m ′ | j, m ′ i , and J | j, m ′ i = (cid:16)(cid:0) j − (cid:1) − (cid:17) | j, m ′ i , respectively.The group SO (2 , D ± ( m ′ ) j where the positive, and negative upper signs refer in their turn to m ′ values limited from either below, m ′ = j + n with n non-negative integer, or above, m ′ = j − n where we used a nomenclature of positive integer or half-integer j (also known as Bargmann index).Back to eq. (36), it is not obvious how to re-express the general two-term energy formulacontaining both the quadratic and inverse quadratic eigenvalues of the SO (4) Casimir in terms of SO (2 ,
1) quantum numbers. The most obvious option consists in nullifying the potential strength,i.e. setting G = 0, which takes one back to the free particle on the hypersphere. In this case onlythe quadratic terms survives which is easily equivalently rewritten to E j ( e κ ) = e κ ~ µ (cid:16) ( m ′ ) − (cid:17) , j = l + 1 , m ′ = j + n, (41)where l is the ordinary angular momentum label, while n is the radial quantum number (it equalsthe order of the polynomial shaping the wave function labeled by K in eq. (34). The m ′ labelis limited from below and the whole spectrum can be associated with the basis of the infiniteunitary SO (2 ,
1) representation, D +( m ′ ) j . It is obvious that the degeneracy patterns in the SO (2 , SO (4) ones.Perhaps nothing expresses the SO (2 , /SO (4) symmetry correspondence better but this ex-treme case in which the manifestly SO (4) symmetric centrifugal energy on the (3 D ) hypersphereis cast in terms of SO (2 ,
1) pseudo-angular momentum values.Although the bare l ( l +1) csc potential is algebraically in line with AdS /CF T correspondence,it completely misses the perturbative aspect of QCD dynamics. The better option for getting ridof the inverse-quadratic term in eq. (36) is to permit K dependence of the potential strengthand choose G = g ( K + 1) with g being a new free parameter. Such a choice (up to notationaldifferences) has been made in [21]. If so, then the energy takes the form E j ( e κ ) = − g ~ µ + e κ ~ µ (cid:16) ( m ′ ) − (cid:17) , j = 1 , , , .... (42)12he above manipulation does not affect the degeneracy patterns as it only provokes a shift in thespectrum by a constant. Compared to eq. (41) the new choice allows the former inverse quadraticterm to still keep presence as a contribution to the energy depending on a free constant parameter, g . In this manner, the SO (2 ,
1) energy spectrum continues being described by a two-term formula,a circumstance that allows for a best fit to the SO (4) description.Once having ensured that the SO (2 ,
1) and SO (4) spectra share same degeneracy patterns,one is only left with the task to check consistency of the level splittings predicted by the twoschemes. Comparison of eqs. (36) and (42) shows that for the high-lying levels where the inversequadratic term becomes negligible, both formulas can be made to coincide to high accuracy by aproper choice for g . That very g parameter can be used once again to fit the low lying levels tothe SO (4) description, now by a value possibly different from the previous one.This strategy allows to make the SO (2 ,
1) and SO (4) descriptions of non-strange baryon spectrasufficiently close and establish the symmetry correspondence. In that manner we confirm ourstatement quoted in the introduction that the TEC potential is in line with both the algebraicaspects of AdS /CF T and QCD dynamics and provides a bridge between them. In this section we shall test the potential parameters in eq. (39) and the wave function ineqs. (12), (34) in the calculation of the proton electric form-factor, the touch stone of any spectro-scopic model. As everywhere through the paper, the internal nucleon structure is approximatedby a quark-diquark configuration. In conventional three-dimensional flat space the electric formfactor is defined in the standard way [54] as the matrix element of the charge component, J ( r ),of the proton electric current between the states of the incoming, p i , and outgoing, p f , electronsin the dispersion process, G pE ( | q | ) = < p f | J ( r ) | p i >, q = p i − p f . (43)The mean square charge radius is then defined in terms of the slope of the electric charge formfactor at origin and reads, h r i = − ∂G pE ( | q | ) ∂ | q | (cid:12)(cid:12)(cid:12) | q | =0 . (44)On S R , the three-dimensional radius vector, r , has to be replaced by, ¯r with | ¯r | = R sin χ =sin χ/ √ κ in accordance with eqs. (7). The evaluation of eq. (43) as four-dimensional Fouriertransform requires the four-dimensional plane wave, e iq · ¯ x = e i | q || ¯r | cos θ = e i | q | sin χ √ κ cos θ , | ¯r | = R sin χ = sin χ √ κ . (45)The latter refers to a z axis chosen along the momentum vector (a choice justified in elasticscattering ), and a position vector of the confined quark having in general a non-zero projectionon the extra dimension axis in E .The integration volume on S R is given by sin χ sin θ d χ d θ d ϕ . The explicit form of the nucleonground state wave function obtained from eq. (34) in the χ variable reads X (00) ( χ, e κ ) = N (00) e − bχ sin χ,N (00) = 4 b ( b + 1)1 − e − πb , b = 2 µG √ e κ ~ . (46)With that, the charge-density takes the form, J ( χ, e κ ) = e p | ψ gst ( χ, e κ ) | , e p = 1. In effect, eq. (43)amounts to the calculation of the following integral, G pE ( | q | , e κ ) = √ κ Z π d χ (cid:0) X (00) ( χ, e κ ) (cid:1) sin( | q | sin χ √ κ ) | q | sin χ , (47) A consistent definition of the four-dimensional plane wave in E would require an Euclidean q vector. However,for elastic scattering processes, of zero energy transfer, where q = 0, the q vector can be chosen to lie entirely in E , and be identified with the physical space-like momentum transfer. S R curvature.hyper-radius R κ = R rescaled hyper-radius d = Rπ e κ = d N spectrum 7.26 fm 0.019 fm − − ∆ spectrum 9.42 fm 0.011 fm − − form factor 10.46 fm 0.009 fm − − where the dependence of the form factor on the curvature has been indicated explicitly.The integral is taken numerically and the resulting charge form factor of the proton is displayedin Fig. 4 together with data. The best fit values for R (equivalently, d ), and the related curvaturesare given in Table 1, for illustrative purpose. The best fit value of the mean square charge radiusis found as h ¯r i = 0 .
87 fm , (48)and reproduces well the corresponding experimental value of h r i exp = 0 . [52]. We fur-ther observe that our best fit is of the quality of the calculation of same observable within theframework of the Bethe-Salpeter equation based upon an instanton induced two-body potential[55].A comment is in order on how the present resut compares to our previous work [17] where sameobservable has been calculated without reference to the curvature concept. There, the corre-sponding Schr¨odinger equation (with essentially same potential) has been written in terms of thethree-dimensional Laplacian versus four-dimensional in the present work. Using eq. (33) it can beshown that the E form factor in ref. [17] represents the small χ limit, sin χ ≈ χ with χ = r √ e κ ,of the S R case, an occasion that allowed to take it in closed form. Comparing the E to the S R calculation reveals the insignificant differences shown in Figs. 5, and 5. The coincidence is due tothe rapid exponential fall of the ground state wave function in eq. (46) which strongly damps thelarge χ angle contributions to the integral in eq. (47) (c.f. Fig. 6).The result shows that specifically in the D ( r, κ ) = πr gauge, • performing in E is consistent with performing on S R in the small χ angle limit, in whichsin χ ≈ χ with χ = r √ e κ , • the proton charge electric form-factor is not sufficiently a sensitive observable toward thecurvature parameter.This contrasts excited states whose wave functions for l > χ angledependences (c.f. Fig. 6) in which case the Fourier transforms on S R will become distinguishablefrom those in E . This is visualized in Fig. 7 by the electric charge form-factor for an l = 2 statefrom the second observed level with K = 3 which corresponds to the first F resonance. The presence of the curvature parameter in the trigonometrically extended Cornell confinementpotential opens an intriguing venue toward deconfinement as a S R curvature shut-down. It canbe shown that 14igh-lying bound states from the trigonometrically extended Cornell confinement po-tential approach scattering states of the Coulomb-like potential in ordinary flat space.Stated differently, the TEC confinement gradually fades away with vanishing curvatureand allows for deconfinement.The latter is most easily demonstrated for the case of a TEC potential with a nullified G parameterand reduced to the csc term, the S R centrifugal barrier. Indeed, for small curvatures such that K √ κ ∼ k, with “k” a constant, eq. (18) goes into E ( c =0) K ( κ ) κ → −→ ~ µ k , (49)and describes a continuous energy spectrum. Moreover, in parallel with the asymptotic behaviorof the energy spectrum in eq. (49), also the wave functions from the confinement phase, S Kl ( χ, κ ),approach in same limit the wave functions relevant for the deconfinement phase which are thescattering states of the inverse distance potential in flat E space. In order to see this it is usefulto recall the following differential recursive relation satisfied by the S Kl functions [34], S Kl ( χ, κ ) = sin l χ p ( n − ... ( n − l ) d l (d cos χ ) l S K ( χ, κ ) , S K ( χ, κ ) = r κπ sin( K + 1) χ sin χ . (50)In the limits when κ →
0, and χ → χ/ √ κ stays finite and approaches χ/ √ κ → r (here, the factor π has been absorbed by r for simplicity), while K √ κ → k with aconstant “k”, i.e., lim κ → ( K + 1) √ κ → k , lim κ → χ √ κ → r, (51)one also finds sin( K + 1) χ −→ sin( K + 1) √ κr −→ sin k r, sin χ → √ κr, d cos χ = − sin χ d χ −→ − κr d r. (52)Accounting for the latter relations, eq. (50) takes the form of the Reyleigh formula [35] for thespherical Bessel functions,lim κ → S Kl ( χ, κ ) → s π ( − l (k r ) l (cid:18) r dd(k r ) (cid:19) l sin k r k r = s π j l (k r ) . (53)The latter wave functions are precisely the ones that describe scattering states in ordinary flat E space, i.e., they are the radial functions in the Helmholtz equation describing free motion in E .This example, though a very simplistic one, is already illustrative of the effect that the curvatureshut-down can have as deconfinement mechanism.In the presence of the cot χ barrier the spectrum is shaped after eq. (36). In the unconditional κ → H atom-like bound states. In the conditional √ κK → k limit from above, where “k” is a constant, theterm in question approaches the scattering continuum. In effect, the V ( χ, κ ) spectrum collapsesdown to the regular Coulomb-like potential, E K ( κ ) κ → −→ − G ~ µ n + ~ µ k , l = 0 , , , ..., n − . (54)The rigorous proof that also the wave functions the complete TEC potential collapse to those ofthe corresponding Coulomb-like problem for vanishing curvature is a bit more involved and canbe found in [26], [34]. 15n other words, as curvature goes down as it can happen because of its thermal depen-dence, confinement fades away, an observation that is suggestive of a deconfinementscenario controlled by the curvature parameter of the TEC potential.Deconfinement as gradual flattening of space has earlier been considered by Takagi [7]. Comparedto [7], our scheme brings the advantage that the flattening of space is paralleled by a temperatureevolution of the curved TEC– to a flat Coulomb-like potential, and correspondingly, by the temper-ature evolution of the TEC wave functions from the confined to the Coloumb-like wave-functionsfrom the deconfined phases, in accordance with eqs. (53), and (54). We emphasized importance of designing confinement phenomena in terms of infinite potentialbarriers emerging on curved spaces. Especially, quark confinement and QCD dynamics have beenmodeled in terms of a trigonometric potential that emerges as harmonic potential on the three-dimensional hypersphere of constant curvature, i.e., a potential that satisfies the Laplace-Beltramiequation there. The potential under consideration interpolates between the 1 /r – and infinite wellpotentials while passing through a region of linear growth. This trigonometric confinement poten-tial is exactly solvable at the level of the Schr¨odinger equation and moreover, contains the Cornellpotential predicted by Lattice QCD and topological field theory [56],[57], [58] as leading terms ofits Taylor decomposition. When employed as a quark-diquark potential, it led to a remarkably ad-equate description of the N and ∆ spectra in explaining their O (4) /SO (2 ,
1) degeneracy patterns,level splittings, number of states, and proton electric charge-form factor. Moreover, the trigono-metrically extended Cornell (TEC) potential, in carrying simultaneously the SO (4) and SO (2 , H atom!), matches the algebraic aspects of AdS /CF T correspondence andestablishes its link to QCD potentiology. A further advantage of the TEC potential is the pos-sibility to employ its curvature parameter, considered as temperature dependent, as a driver ofthe confinement-deconfinement transition in which case the wave functions of the confined phaseapproach bound and scattering states of ordinary flat space 1 /r potential.All in all, we view the concept of curved spaces as a promising one especially within the contextof quark-gluon dynamics. Acknowledgments
One of us (M.K) acknowledges hospitality by the Argonne National Laboratory in April 2008 andstimulating interest and discussions with T.S.H. Lee. We thank M. Krivoruchenko for providingassess to ref. [41] and Nora Breton for assistance in clarifying some aspects of
AdS gravity. Worksupported by CONACyT-M´exico under grant number CB-2006-01/61286.16 eferences [1] A. Gersten, Euclidean special relativity,
Found. Phys. , 1237-1252 (2003).[2] J. F. Cari˜nena, M. F. Ra˜nada, M. Santander, Superintegrability on curved spaces, orbits andmomentum hodographs: revisiting a classical result by Hamilton,
J. Phys. A:Math.Theor. ,13645-13666 (2007).[3] J. M. Nester, Gravity, Torsion, and Gauge Theories, in “Chalk River 1983”, Proceedings “AnIntroduction to Kaluza-Klein Theories”, pp. 83-115 (1983).[4] T. G. Vosmishcheva, A. A. Oshemkov,
Topological analysis of the two-center problem on thetwo-dimensional sphere,
Sbornik:Mathematics , 1103-1138 (2002).[5] Abdus Salam, J. Strathdee,
Confinement through tensor gauge fields,
Phys. Rev. , 4596-4609 (1978).[6] C. Dullemond, T. A. Rijken, E. van Beveren, Quark-gluon model with conformal symmetry,
Il Nuovo Cimento A , 401-428 (1984);I. Bediaga, M. Gaseprini, E. Predazzi, Thermal expansion and critical temperature in a geo-metric representation of quark deconfinement,
Phys. Rev. D , 1626-1627 (1988);M. G¨urses, New class of f − g fields relevant to quark confinement, Phys. Rev. D , 1019-1021 (1979);Saulo Carneiro, The large numbers hypothesis and quantum mechanics,
Found.Phys.Lett. ,95-103 (1998).[7] F. Takagi, Quark deconfinement at finite temperature in the bag model,
Phys. Rev. D , 2226–2229 (1987).[8] J. Maldacena, The large N limit of superconformal field theory and supergravity, Adv. Theor.Math. Phys. , 231-252 (1998); Int. J. Theor. Phys. Hadronic spectrum of holographic dual of QCD,
Phys.Rev. Lett. , 201601-1–4 (2005).[10] G. S. Bali, QCD Potentiology, *Wien 2000, Quark confinement and the hadron spectrum*, Effective field theory Lagrangians for baryons with twoand three heavy quarks,
Phys.Rev.D , 034021 (2005);N. Brambilla, Effective Field Theories for Q Q Q and Q Q q baryons,
AIP Conf.Proc. ,366-368 (2005).[12] T. T. Takahashi, H. Suganuma, Y. Nemoto, H. Matsufuru,
Detailed analysis of the three-quarkpotential in SU(3) lattice QCD,
Phys. Rev. D , 114509-1–19 (2002).[13] E. Eichten, H. Gottfried, T. Kinoshita, K. D. Lane, T.M. Yan, Charmonium:The model,
Phys. Rev. D , 3090-3117 (1978);E. Eichten, H. Gottfried, T. Kinoshita, K. D. Lane, T.M. Yan, Charmonium:Comparison withexperiment,
Phys. Rev. D , 203-233 (1980).[14] Quarkonium Working Group (N. Brambilla et al.), Heavy quarkonium physics,
CERN YellowReport, CERN-2005-005, Geneva: CERN, (2005); e-Print: hep-ph/0412158[15] P. Gonzalez, J. Vijande, A. Valcarce, H. Garcilazo,
Spectral patterns in the nonstrange baryonspectrum,
Eur.Phys.J. A , 235–244 (2006).[16] H. Garcilazo, Momentum-space Faddeev calculations for confining potentials,
Phys. Rev. C , 055203-1–9 (2003). 1717] C. B. Compean, M. Kirchbach, Trigonometric quark confinement potential of QCD traits,
Eur. Phys. J. A , 1-4 (2007).[18] J. Fischer, J. Niederle, R. Raczka, Generalized Spherical Functions for the Noncompact Ro-tation Groups,
J. Math. Phys. , 816-821 (1966)[19] Brian G. Wybourne, Classical groups for physicists (Wiley-Inter-science, N.Y., 1974).[20] M. Novaes,
Some basics of su (1 , , Revista Brasileira de Ensino de Fisica , 351-357 (2004).[21] R. Koc, M. Koca, A systematic study on the exact solution of the position dependent massSchr¨odinger equation,
J. Phys. A:Math.Gen. , 8105-8112 (2003).[22] M. Ba˜ n ados, A. Gomberoff, C. Martinez, Anti-de Sitter space and black holes,
Class. QuantumGravity , 3575-3598 (1998).[23] E. Schr¨odinger, A method of determining quantum mechanical eigenvalues and eigenfunction,
Proc. Roy. Irish Acad. A , 9-16 (1940).[24] P. W. Higgs, Dynamical symmetries in spherical geometry,
J. Phys. A:Math.Gen. , 309-323(1979).[25] H. I. Leemon, Dynamical symmetries in a spherical geometry II,
J. Phys. A:Math.Gen. ,489-501 (1979).[26] A. O. Barut, Taj Wilson, On the dynamical group of the Kepler problem in a curved space ofconstant curvature,
Phys. Lett. A , 351-354 (1985);A. O. Barut, A. Inomata, G. Junker,
Path integral treatment of the hydrogen atom in a curvedspace of constant curvature,
J. Phys. A:Math.Gen. , 6271-6280 (1987).[27] L. Infeld, On a new treatment of some eigenvalue problems,
Phys. Rev. , 737-747 (1941);L. Infeld, A. Schild, A note on the Kepler problem in a space of constant negative curvature,
Phys. Rev. , 121-122 (1945);L. Infeld, T. E. Hull, The factorization method,
Rev. Mod. Phys. , 21-68 (1951).[28] E. Witten, Dynamical breaking of supersymmetry,
Nucl. Phys. B , 513-590 (1981).[29] N. Bessis, G. Bessis,
Electronic wave functions in a space of constant curvature,
J. Phys.A:Math.Gen. (11), 1991-1997 (1979);N. Bessis, G. Bessis, D. Roux, Space-curvature effects in the interaction between atoms andexternal fields: Zeeman and Stark effects in a space of constant positive curvature,
Phys. Rev.A , 324-336 (1986).[30] Yu. Kurochkin, Dz. Shoukavy, The tunnel-effect in the Lobachevsky space,
Acta PhysicaPolonica B , 2423-2431 (2006);V. V. Gritsev, Ya. A. Kurochkin, V. S. Otchik, Nonlinear symmetry algebra of the MIC-Keplerproblem on the sphere S , J. Phys. A:Math.Gen. A , 4903-4910 (2000).[31] J. Boer, F. Harmsze, T. Tijn, Non-linear finite W symmetries and applications in elementarysystems, Phys. Rep. , 139- 214 (1996).[32] A. Ballesteros, F. J. Herranz,
The Kepler problem on 3D spaces of variable and constantcurvature from quantum algebras, “Workshop in honor of Prof. Jos´e F. Cari˜nena, “Groups,Geometry and Physics”, December 9-10, 2005, Zaragoza (Spain); math-ph/0604009 (2006).[33] Pinaki Roy, Rajkumar Roychoudhury,
Supersymmetry in curved space,
Phys. Rev. D ,1787-1790 (1986). 1834] S. I. Vinitsky, L. G. Mardoian, G. S. Pogosyan, A. N. Sissakian, T. A. Strizh, A hydrogenatom in curved space.Expansion over free solutions on the three dimensional sphere,
Phys.Atom. Nucl. , 321-327 (1993);V. N. Pervushin, G. S. Pogosyan, A. N. Sissakian, S. I. Vinitsky, Equation for quasi-radialfunctions in momentum representation on a three-dimensional sphere,
Phys. Atom. Nucl. ,1027-1043 (1993).[35] G. B. Arfken, H.-J. Weber, Mathematical Methods for Physicists,
Fifth-Edition (AcademicPress, N.Y., 2001).[36] D. Bonatsos, C. Daskaloyannis, K. Kokkotas,
Deformed oscillator algebras for two-dimensional quantum super-integrable systems,
Phys. Rev. A , 3700-3709 (1994).[37] A. Del Sol Mesa, C. Quesne, Connection between type A and E factorizations and constructionof satellite algebras,
J. Phys. A:Math.Gen. , 4059-4071 (2000).[38] C. C. Barros Jr., Space-time and hadrons,
Braz. J. Phys. , 17-19 (2007).[39] Y. S. Kim, M. E. Noz, Theory and application of the Poincar´e group (D. Reidel, Dordrecht,1986).[40] Taco Nieuwenhuis, J. A. Tjon,
O(4) Expansion of the ladder Bethe-Salpeter equation,
FewBody Syst. , 167-185 (1996).[41] A. A. Ismest’ev, Exactly solvable potential model for quarkonia,
Sov. J. Nucl. Phys. , 1066-1076 (1990).[42] V. V. Gritsev, Yu. A. Kurochkin, Model of excitations in quantum dots based on quantummechanics in spaces of constant curvature,
Phys. Rev. B , 035308-1–9 (2001).[43] Ravit Efraty, Self-intersection, axial anomaly and the string picture of QCD,
Phys. Lett. B , 184-190 (1994).[44] R. De, R. Dutt, U. Sukhatme,
Mapping of shape invariant potentials under point canonicaltransformations,
J. Phys. A:Math.Gen. , L843-L850 (1992).[45] A. F. Stevenson, Note on the “Kepler Problem” in a spherical space, and the factorizationmethod of solving eigenvalue problems,
Phys. Rev. , 842-843 (1941).[46] C. B. Compean, M. Kirchbach, The trigonometric Rosen-Morse potential in the supersym-metric quantum mechanics and its exact solutions,
J. Phys. A:Math.Gen. , 547-557 (2006).[47] E. J. Routh, On some properties of certain solutions of a differential equation of second order,
Proc. London Math. Soc. , 245 (1884).[48] V. Romanovski, Sur quelques classes nouvelles de polynomes orthogonaux,
C. R. Acad. Sci.Paris, , 1023-1025 (1929).[49] A. Raposo, H. J. Weber, D. E. Alvarez-Castillo, M. Kirchbach,
Romanovski polynomials inselected physics problems,
C. Eur. J. Phys. , 253-284 (2007).[50] V. D. Burkert, T. S. H. Lee, Electromagnetic meson production in the nucleon resonanceregion,
Int. J. Mod. Phys. E , 1035-1112 (2004).[51] S. S. Afonin, Parity doublets in particle physics,
Int. J. Mod. Phys. A , 4537-4586 (2007).[52] S. Eidelman et al., Review of particle physics,
Phys. Lett. B , 1-1109 (2004).1953] M. Kirchbach,
On the parity degeneracy of baryons,
Mod. Phys. Lett. A , 2373-2386 (1997);M. Kirchbach, Classifying reported and “missing” resonances according to their P and C properties, Int. J. Mod. Phys. A , 1435-1451 (2000);M. Kirchbach, M. Moshinsky, Yu. F. Smirnov, Baryons in O(4) and vibron model,
Phys. Rev.D , 114005-1–11 (2001).[54] J. Arrington, C. D. Roberts, J. M. Zanotti, Nucleon electromagnetic form factors,
J. Phys.G:Nucl.Part. ∼ metsch/jlab2004[56] G. t’Hooft, On the phase transition towards permanent quark confinement,
Nucl. Phys. B , 1-25 (1979).[57] E. Witten,
Quantum field theory and the Jones polynomial,
Comm. Math. Phys. , 351(1989); IASSNS-HEP-88/33.[58] H. Reinhardt, C. Feuchtler,
Yang-Mills wave functional in Coulomb gauge,
Phys. Rev. D ,105002-1–6 (2005).[59] K. Holland, P. Minkowski, M. Pape, U.-J. Wiese, Exceptional confinement in G(2) theory,
Nucl. Phys. B , 207-236 (2003). 20igure 1: Position and curvature dependence of the reduced mass µ ∗ ( r, κ ). Besides effectiveelectron masses in quantum dots, one may entertain applicability of this scenario to evolutionof finite valence to vanishing parton quark masses as effect of transversal (relative to the extradimension) displacement on the three-dimensional “plane” tangential to the “North” pole of thehemisphere. 21igure 2: Assignments of the reported N excitations to the K levels of the S R potential, V ( r √ e κ, e κ ),in eq. (33), taken as the quark-diquark confinement potential. The potential parameters are thosefrom eq. (39). Double bars represent parity dyads, single bars the unpaired states of maximumspin. The notion L I, J ( −− ) has been used for resonances “missing” from a level. The modelpredicts two more levels of maximal spins J π = 5 / + , and J π = 9 / + , respectively, which arecompletely “missing”. In order not to overload the figure with notations, the names of the reso-nances belonging to them have been suppressed. The predicted energy at rest (equal to the mass)of each level is given to its most right.Figure 3: Assignment of the reported ∆(1232) excitations to the K levels of the S R potential V ( r √ e κ, e κ ) in eq. (33), taken as the quark-diquark confinement potential. The potential parametersare those from eq. (39) with exception of the d value taken here as d = 3 fm. Other notationssame as in Fig. 2. 22igure 4: The electric charge form factor of the proton calculated for various curvature parameters.The upper curve corresponds to the curvature as fitted to the nucleon spectrum, the curvatureleading to the middle curve has been fitted to the experimental value of the mean square of thecharge radius (see Table 1). The lowest curve follows from a Bethe-Salpeter calculation based uponan instanton induced two-body potential and has been presented in ref. [55]. Data compilationtaken from [55].Figure 5: Difference between the electric charge form factor, G pE ( | q | , e κ ), calculated in E , with e κ =1 /d , d = 2 .
31 fm, and G pE ( | e q | ) with | e q | ≡ | q | d calculated in [17] in ordinary three space in closedform for the same d value. On the figure the form factors have been plotted as functions of Q = − q = | q | . The insignificance of this difference illustrates consistency of the three-dimensionalFourier-transform with the small χ angle approximation to the four-dimensional Fourier transformin the D ( r, κ ) = rπ gauge. 23igure 6: Charge density profiles in the ground state (solid line), and the K = 3 , l = 2 excitation(dashed line), corresponding to the first F , resonance. The wave function have been takenunnormalized. It is visible that while the former damps the small χ angle contributions from thefour dimensional plane wave, the latter captures a significant amount of them which enables it todistinguish flat from curved spaces.Figure 7: Electric charge form factor of the K = 3, l = 2 resonance from plane space (dashedline) versus the S R one (solid line). It is visible that in the vicinity of Q = 1 GeV2