Contents
###### Abstract

We construct an effective QCD light-front Hamiltonian for both mesons and baryons in the chiral limit based on the generalized supercharges of a superconformal graded algebra. The superconformal construction is shown to be equivalent to a semi-classical approximation to light-front QCD and its embedding in AdS space. The specific breaking of conformal invariance inside the graded algebra uniquely determines the effective confinement potential. The generalized supercharges connect the baryon and meson spectra to each other in a remarkable manner. In particular, the Regge trajectory is identified as the superpartner of the nucleon trajectory. However, the lowest-lying state on this trajectory, the -meson is massless in the chiral limit and has no supersymmetric partner.

SLAC–PUB–16184

Superconformal Baryon-Meson Symmetry

and Light-Front Holographic QCD

Hans Günter Dosch

Institut für Theoretische Physik, Philosophenweg 16, D-6900 Heidelberg, Germany

Guy F. de Téramond

Universidad de Costa Rica, San José, Costa Rica

Stanley J. Brodsky

SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA

## 1 Introduction

Light front holographic QCD (LFHQCD) has brought important insights into hadron dynamics, especially to the color confinement problem. In Refs. [1, 2] a remarkable equivalence between the bound-state equations of the light-front Hamiltonian in 3+1 physical space-time 444For a review of light-front physics see [3]. and those obtained in five-dimensional anti-de Sitter space (AdS) has been observed: The holographic coordinate in AdS space can be identified with the boost-invariant light front (LF) separation between constituents 555We therefore will use in the following the variable both as light-front variable in LFHQCD and as the holographic (fifth-dimensional) coordinate in AdS.. This holographic equivalence allows one to relate the effective light-front potential for bosons to the symmetry-breaking factor introduced in AdS. In the case of integer spin fields, the breaking of the conformal isometries of AdS space can be done by introducing an additional model-dependent factor into the AdS action – a dilaton term . The specific form of the symmetry-breaking factor, however, is not fixed a priori, but it can be deduced from the comparison with the experimentally observed spectra. Linear Regge trajectories demand for the form [4, 5]. The resulting LF effective potential is harmonic and confining, and it also includes a -dependent constant term. This extra term is a consequence of the separation between kinematical and dynamical quantities for arbitrary spin [6], prescribed by the light-front mapping of AdS bound-state equations. The extra constant term has important phenomenological consequences; in particular, it leads in the chiral limit to a massless pion.

A large step forward in understanding why the effective potential must have the form of a confining harmonic potential was made by applying a method developed in conformal quantum mechanics by de Alfaro, Fubini and Furlan [7] (dAFF) to the light-front bound-state equations [8]. Starting from a conformally invariant action, a new Hamiltonian can be constructed as a superposition of the generators of the conformal algebra. Remarkably, the action remains conformally invariant, and the form of the resulting confining potential is uniquely fixed [8]. It has the form of a harmonic oscillator and corresponds to the quadratic dilaton term previously introduced before by purely phenomenological arguments [4, 5]. However, the -dependent constant term, referred to above, cannot be derived from the DAFF procedure. Furthermore, for half-integer spin a dilaton term in the AdS action does not lead to confinement [9], and therefore an additional Yukawa-like interaction term has to be added to the fermionic action. This Yukawa interaction term again has to be determined phenomenologically – one finds that the linear baryon Regge trajectories, with equal spacing in the orbital and radial excitations, as observed phenomenologically, requires the form  [10, 11].

Recently, we have shown [12] that a comparison of the half-integer LF bound-state equations with the Hamiltonian equations of superconformal quantum mechanics fixes the form of the LF potential in full agreement with the phenomenologically deduced form . This procedure, originally developed by Fubini and Rabinovici (FR) [13], is the superconformal extension of the procedure applied by dAFF [7]. In brief: a new evolution Hamiltonian can be constructed using a generalized supercharge which is a superposition of the original supercharge together with a spinor operator which occurs only in the superconformal algebra. The resulting superconformal quantum mechanics applied to the fermionic LF bound-state equations is completely dual to AdS; this is in contrast to conformal quantum mechanics without supersymmetry, which is dual to the bosonic sector of AdS only up to a constant term, which in turn is fixed only by embedding the LF wave equations for arbitrary integer spin into AdS.

As we shall discuss in this paper, superconformal quantum mechanics applied to LF Holographic QCD also implies striking similarities between the meson and baryon spectra. In fact, as we shall show, the holographic QCD light-front Hamiltonians for the states on the pion and proton trajectories are identical if one shifts the internal angular momenta of the meson () and baryon () by one unit: . The baryon and meson trajectories are actually observed to be linear in the squared masses , as predicted by holographic QCD, a feature not obvious for states satisfying effective bound-state equations (Dirac or generalized Rarita-Schwinger). The slope of the trajectories in the principal quantum number and the orbital angular momentum are also very similar. In fact, the best fits to the numerical values for the Regge slopes agree within for all hadrons, mesons and baryons; this leads to a near-degeneracy of meson and baryon levels in the model.

The idea to apply supersymmetry to hadron physics is certainly not new [14, 15, 16]. In [14] mesons and baryons are grouped together in a big supermultiplet, a representation of . In [15] the supersymmetry results of Miyazawa [14] are recovered in a QCD framework, provided that a diquark configuration emerges through an effective string interaction. This approach relies heavily on the fact that in a diquark can be in the same color representation as an antiquark, namely a . A meson is formed by a quark-antiquark pair and a baryon by a quark and a diquark, which remains color singlet. It is plausible to assume that the color force between a quark and a diquark is approximately equal to that between a quark and an antiquark; and from this, an effective supersymmetry between mesons and baryons follows. An apparent difficulty in this approach is that the pion and the nucleon would have the same mass and thus, supersymmetry would be badly broken [16]. In fact, in the chiral limit – the limit of massless quarks – the pion is massless, and this state has no obvious supersymmetric partner: there is no (nearly) massless baryonic state. In the direct diquark approach [14, 15, 16] there is no natural way to take into account the special role of the pion.

In certain aspects, our approach is similar to the diquark picture described above. The light-front clustering decomposition used here divides the baryon constituents into a special constituent, the active quark, and the rest, the spectator cluster, which could be identified with a diquark. However, in contrast to the direct diquark picture, the problem of a baryonic partner of the pion does not occur in our approach. It yields a massless pion, but the supercharge, which transforms meson into baryon wave functions, annihilates the pion wave function and therefore it has no baryonic partner. The details of this mechanism, which only occurs for a massless pion, are explained in Sec. 4.1.

The approach described here, in contrast to the direct diquark picture of Refs. [14, 15, 16], is by no means restricted to a special number of colors. Indeed, in this effective theory the color quantum number does not appear explicitly. However, since it is an offspring of the Maldacena AdS/CFT correspondence [17], it is reminiscent of an theory. This interpretation is also in accordance with the zero width of all states, including the excited ones. It is interesting to note that there exists a genuine supersymmetric approach to the meson-baryon relation relying on the limit. Armoni and Patella [18] consider supersymmetric ); in their approach, the meson is formed by a bosonic string from a quark to an anti-quark, whereas the baryon is formed by a fermionic string between two quarks. In the large limit the string tension for both objects become equal: “Supersymmetric relics” [19] from the supersymmetric theory lead to equal string tension for mesons and baryons in ).

We emphasize that the supersymmetric relations between the observed baryons and mesons, which we derive here, are not a consequence of supersymmetric QCD with scalar quarks and gluinos. Since no supersymmetric partners of the fundamental QCD fields have been observed, such a theory is evidently broken below the TeV scale. The relations derived here are relations between the wave functions of hadrons, not field operators. The relations obtained in the framework of supersymmetric quantum mechanics reflect properties of the confining mechanism in an effective semiclassical theory. One thus expects deviations from experiment which are of the same order as in light-front holographic QCD.

This article is organized as follows: After briefly reviewing some important results of light-front holographic QCD in Sec. 2, we discuss in Sec. 3 the construction of the bound-state Hamiltonian within the superconformal algebra and the breaking of dilation invariance following [13]. The search for the supersymmetric partners of the baryon trajectories is discussed in Sec. 4. A summary of the main results and our conclusions are presented in Sec. 5. Some useful formulae for the derivations presented in this article are given in the appendices.

## 2 Light Front Holographic QCD

We first briefly review some principal results of light-front holographic QCD 666For an extended review see [20].. In holographic QCD an integer-spin field in AdS, with a free hadronic field at the four-dimensional border , is split into a component , describing the behavior in the bulk, and a plane wave with an integer -spinor describing the Minkowski space-time behavior (See [20], Sec. 5.1.1):

 Φν1⋯νJ(P,ζ)=ΦJ(ζ)eiP⋅xϵν1⋯νJ(P). (1)

The four-momentum squared is the mass squared of the hadron represented by the free field, .

A Schrödinger-like wave equation [2, 6] follows from the AdS action for arbitrary integer spin- modified by a dilaton term :

 (−d2dζ2+4L2−14ζ2+U(ζ,J))ϕJ(ζ)=0, (2)

where we have factored out the scale and dilaton factors from the AdS field by writing . Equation (2) has exactly the form of a LF wave equation for massless quarks with a LF effective potential and LF angular momentum . The latter is related to the total spin and the product of the AdS mass with the AdS radius by

 (μR)2=L2−(J−2)2. (3)

The potential is related to the dilaton profile by [5, 6]

 U(ζ,J)=12φ′′(ζ)+14φ′(ζ)2+2J−32ζφ′(ζ). (4)

The holographic variable is identified with the LF invariant invariant transverse separation:  [1, 2], where is the transverse separation of the constituents and is the longitudinal light-front momentum fraction.

In the case of the quadratic dilaton profile , the LF effective potential is , and the holographic bound-state wave equation (2) can be written as

 (−d2dζ2+λ2Mζ2+2λM(J−1)+4ν2−14ζ2)ϕJ=M2ϕJ, (5)

for a meson with total spin . Near the regular solution behaves as , corresponding to twist . In LFHQCD one thus has , where is the LF angular momentum of the meson, . The eigenvalues of (5) predict the meson spectrum

 M2n,L,J=4(n+J+LM2)λM, (6)

for , where indicates the radial excitation quantum number: the number of notes in the wavefunction.

Similarly, the AdS field for arbitrary half-integer spin- can be factorized into a bulk wave function and a plane wave with a Rarita-Schwinger or Dirac spinor with momentum and mass , representing a freely propagating baryon at the AdS border (See [20], Sec. (5.2))

 Ψ±ν1⋯νJ−1/2(P,ζ)=Ψ±J(ζ)eiPxu±ν1⋯νJ−1/2(P), (7)

where the chiral spinors satisfy the equations

 γ⋅Pu±ν1⋯νJ−1/2(P)=Mu∓ν1⋯νJ−1/2(P);γν1u±ν1⋯νJ−1/2(P)=0. (8)

The spinors have positive and negative chirality, respectively.

The bound-state wave equations for the AdS bulk wave functions can be derived from the action for arbitrary half-integer spin- if one includes the effective interaction . The result is [6]

 (−d2dζ2+λ2Bζ2+2λB(ν+12)+λB+4ν2−14ζ2)ψ+J = M2ψ+J, (9) (−d2dζ2+λ2Bζ2+2λB(ν+12)−λB+4(ν+1)2−14ζ2)ψ−J = M2ψ−J, (10)

where we have factored out the scale by writing , and is related to the product of the AdS fermionic mass and the AdS radius by

 ν=μR−12. (11)

The baryon spectrum which follows from (9,10) is

 M2n,ν=4(n+ν+1)λB, (12)

for . The eigenvalues given by (12) do not depend explicitly on , an important result also found in Ref. [21].

## 3 Superconformal Algebra and Breaking of Dilatation Symmetry

We will now show how the preceding results can be systematically derived using superconformal algebra, but with important new consequences. One starts with the simplest graded algebra of two fermionic operators, the supercharges and , and a Hamiltonian [22]

 {Q,Q†}=2H, (13) {Q,Q}={Q†,Q†}=0, (14) [Q,H]=[Q†,H]=0. (15)

A simple realization is:

 Q=ψ†(−ip+W),Q†=ψ(ip+W), (16)

where is the canonical momentum operator; and are fermionic operators with anti commutation relation

 {ψ,ψ†}=1, (17)

and is an arbitrary potential (the superpotential).

A realization using Pauli matrices is:

 ψ=12(σ1−iσ2),ψ†=12(σ1+iσ2), (18)

 B=12[ψ†,ψ]=12σ3, (19)

where is the generator of transformations , with eigenvalues and .

In the Schrödinger picture the supercharges are realized as operators in , with :

 Q=ψ†(−ddx+W(x)), (20)

and

 Q†=ψ(ddx+W(x)), (21)

 H=12{Q,Q†}=12(−d2dx2+W2(x)−2W′(x)B). (22)

The Hamiltonian operates on 2-spinors

 |ϕ⟩=(ϕ1ϕ2), (23)

of which one component can be attributed to fermion number 1 and the other 0. Imposing conformal symmetry leads to an unique choice of [23, 13], namely

 W(x)=fx, (24)

with a dimensionless constant .

Introducing the spinor operators

 S=ψ†x,S†=ψx, (25)

one can construct the larger graded algebra [24] (superconformal algebra), which contains the conformal algebra with the dilatation generator and the special conformal transformation generator . The extended algebraic structure is

 12{Q,Q†} = H,12{S,S†}=K, (26) {Q,S†} = f−B+2iD, (27) {Q†,S} = f−B−2iD, (28)

where the operators

 H = 12(−d2dx2+f2+2Bfx2), (29) K = 12x2, (30) D = i4(ddxx+xddx), (31)

satisfy the conformal algebra

 [H,D]=iH,[H,K]=2iD,[K,D]=−iK. (32)

The anti-commutators of all the other generators vanish: .

Fubini and Rabinovici considered several ways to construct a new compact quantum mechanical evolution operator inside the superconformal algebra. The most straightforward way is to directly follow the procedure of dAFF [7] and construct a linear combination of the (old) Hamiltonian and the generator of special conformal transformations [23, 13]. There is, however, the interesting possibility of constructing a new Hamiltonian using the superposition of generalized supercharges within the extended graded algebra [13] and thus preserving supersymmetry. This is the procedure we shall follow here. To this end, we slightly generalize the definitions of FR [13] and introduce a new supercharge as a linear combination of the generators and

 Rλ=Q+λS. (33)

This leads, in analogy to the dAFF procedure in conformal quantum mechanics, to the introduction of a constant with nonzero dimensions; in fact, since has dimension , and has dimension , must therefore have dimension .

One can now construct a new evolution operator inside the superconformal algebra in terms of the new supercharge :

 {Rλ,R†λ}=G, (34) {Rλ,Rλ}={R†λ,R†λ}=0, (35) [Rλ,G]=[R†λ,G]=0. (36)

We find

 G=2H+2λ2K+2λ(f\bf I−B), (37)

which is a compact operator for .

The supercharge operator transforms a state into the state with different fermion number (See Appendix B). By construction, the evolution operator commutes with ; it thus follows that the states and have identical eigenvalues. In fact, if is an eigenstate of with ,

 G|ϕE⟩=E|ϕE⟩, (38)

then , and thus is also an eigenstate of with the same eigenvalue.

The new Hamiltonian is diagonal. In the Schrödinger representation:

 G11 = ⎛⎝−d2dx2+λ2x2+2λf−λ+4(f+12)2−14x2⎞⎠, (39) G22 = ⎛⎝−d2dx2+λ2x2+2λf+λ+4(f−12)2−14x2⎞⎠, (40)

with For and the spectra of both operators are identical:

 En=4(n+f+12)λ. (41)

Comparing (9, 10) with (39, 40) we recover the result of Ref. [12], namely that the modified Hamiltonian G of superconformal quantum mechanics is the same as the Hamiltonian derived in LF holographic QCD, provided we identify , the eigenfunction of , with the positive chirality wave function , identify , the eigenfunction of , with ; and take and . The consequences of this remarkable result have been discussed extensively in Ref. [12].

In Ref. [12] the operator (19) was identified in the light-front with the Dirac matrix which acts on physical spinors. In that paper we showed that the supercharges relate the chirality-plus component of a baryonic wave function with the chirality-minus component of the same baryonic state. In the usual applications of supersymmetry, however, the supercharges connect bosonic to fermionic states. We therefore shall explore in the next section the possibility to relate mesonic with baryonic wave functions by the supercharges within the extended graded algebra. In this case, the supercharges act on some internal space. The supercharges in [12] and those used in the following are therefore only formally related. The bosonic operators and , however, have in both cases the same physical meaning. In particular, we will show that the and equations (39) and (40) match our light-front holographic equations for both the pion and nucleon trajectories. The extension of this superconformal connection to the - families will also be discussed.

## 4 Baryon-Meson Supersymmetry

### 4.1 The Superpartner of the Nucleon Trajectory

In the case of baryons, the assignment of the leading-twist parameter in Eqs. (9, 10), as given in Table 1 [12], successfully describes the structure of the light baryon orbital and radial excitations 777The ‘leading-twist’ assignment referred to here is the effective twist of the baryonic quark-cluster system; it is thus equal to two. This is in distinction to the usual application of twist for hard exclusive processes which emerges when the baryon cluster is resolved at high momentum transfer and is thus equal to the total number of components.The assignment for the lowest trajectory, the nucleon trajectory, is straightforward and follows from the stability of the ground state – the proton – and the mapping to LF quantized QCD.

The bound-state equations for the nucleon trajectory are (cf. Eqs. (9, 10)):

 (−d2dζ2+λ2Bζ2+2λB(LB+1)+4L2B−14ζ2)ψ+J = M2ψ+J, (42) (−d2dζ2+λ2Bζ2+2λBLB+4(LB+1)2−14ζ2)ψ−J = M2ψ−J. (43)

We will now search for the meson supersymmetric partners of the nucleon trajectory. We choose as starting point the leading-twist chirality component which satisfies (42). With the identifications and , the plus chirality component is also an eigenfunction of , Eq. (40). This identification allow us to define an effective “baryon number” as a convenient convention to label our “meson” and “baryon” states. In terms of the operator

 NB=12−B, (44)

with eigenvalues

 NB|ϕ⟩M = 0, (45) NB|ϕ⟩B = |ϕ⟩B, (46)

where has only a lower component () and only an upper component ():

 |ϕ⟩B=(0ϕ2),|ϕ⟩M=(ϕ10). (47)

Therefore, the supersymmetric partner (39) should describe a meson trajectory. Indeed, the Hamiltonian with the above mentioned substitutions agrees with the bound-state equation (5) for mesons with , provided we identify and set . The lowest state on the mesonic trajectory, with     – the pion – is massless in the chiral limit. It corresponds to a negative value of , namely and thus its baryonic partner would have , which is an unphysical state. As discussed in Appendix A, this remarkable result, also follows directly from the superconformal algebra. As shown there, the operator which transforms a mesonic state into its baryonic supersymmetric counterpart, annihilates the meson state if .

We have thus derived the astonishing result that the pion has no supersymmetric partner even though no explicit breaking of supersymmetry has been introduced. Since the supercharges , which connect mesonic and baryonic wave functions, commute with the Hamiltonian (34 - 36), it follows that if is a mesonic state with eigenvalue , then there exists also a baryonic state with the same eigenvalue . Indeed

 G|ϕ⟩B=GR†w|ϕ⟩M=R†wG|ϕ⟩M=E|ϕ⟩B. (48)

However, for the specific eigenvalue we can have the trivial solution

 |ϕ(E=0)⟩B=(00). (49)

This remarkable feature underlines the special role played by the pion in light-front holographic QCD. As a unique state of zero energy, it plays the same role as the unique vacuum state in a supersymmetric quantum field theory [22] 888In our assignment the Witten index [25] for has the value . It has the same value for  [13]..

In is interesting to note that the case of negative was not considered in [13], since the classical potential has no stable ground state for . Nevertheless, the lowest lying bound state of with has the normalizable wave function . This situation is reminiscent of the AdS/QCD correspondence: angular momentum corresponds to a tachyonic AdS mass (See Eq. (3)), but nonetheless the Breitenlohner-Freedman stability bound [26] is still satisfied.

We thus obtain from superconformal quantum mechanics a very satisfactory result: both the nucleon and the mesons lie on linear trajectories with the same slope and the same radial and orbital excitation energies. The lowest lying state on the meson trajectory is the massless pion. In superconformal quantum mechanics it corresponds to the value , and therefore it has no supersymmetric partner.

In the framework of superconformal quantum mechanics all eigenstates with eigenvalues different from zero have supersymmetric partners. We emphasize that the pion with and zero mass is unique: it is annihilated by the fermion-number changing supercharge , and it therefore has no supersymmetric partner (See Fig. 1). This is in accordance with the spectroscopy derived from light-front holographic QCD, where the partners have the masses and respectively 999This result follows from (12) and (6) with and , respectively.. If one takes in LF holographic QCD, which is automatic in the superconformal theory, the spectral results are then identical for .

The predictions of supersymmetric quantum mechanics are based on the fact that the supercharge operator transforms baryon states with angular momentum into their mesonic superpartners with angular momentum . The operator operates in the opposite direction. We thus have a complete correspondence between light-front holographic QCD and supersymmetric quantum mechanics. The pion has a very special role: its existence is predicted by the superconformal algebra, and according to the formalism, it is massless and has no supersymmetric partner.

The superconformal predictions presented in Fig. 1 should be understood as a zeroth-order approximation. There are, however, several phenomenological corrections to this initial approximation. First, the slope of the trajectory is not exactly identical to the slope of the nucleon trajectory: for the mesons GeV, whereas for the nucleons GeV [20]. This makes the heavier than its supersymmetric partner, the nucleon. In terms of LFHQCD this indicates that for this internal spin configuration, the confining force between the spectator and the cluster in the baryon is weaker than between the constituents of the meson; this makes the meson a more compact object since . Second, the negative parity nucleon states are systematically higher than the nucleons with positive parity, a fact which in LF holographic QCD has been taken into account phenomenologically by the half-integer twist assignment given in Table 1. It is expected that this effect could be explained by the different quark configurations and symmetries of the baryon wave function [27, 28, 29] 101010In Ref. [29] it is suggested that this parity level-shift effect in baryons could be a consequence of the tunneling of the spectator quark into the cluster..

The nucleon-meson superpartner pairs are plotted in Fig. 2 with their measured masses. The observed difference in the squared masses of the supersymmetric partners indicates that the most important breaking of supersymmetry is due to the difference between and . Only confirmed PDG states have been included [30].

### 4.2 The Mesonic Superpartners of the Delta Trajectory

The essential physics derived from the superconformal connection of nucleons and mesons follows from the action of the fermion-number changing supercharge operator . As we have discussed in the previous section, this operator transforms a baryon wave function with angular momentum into a superpartner meson wave function with angular momentum (See Appendix B), a state with the identical eigenvalue – the hadronic mass squared. We now check if this relation holds empirically for other baryon trajectories.

We first observe that baryons with positive parity and internal spin , such as the , and baryons with with negative parity and internal spin , such as the , lie on the same trajectory; this corresponds to the phenomenological assignment , given in Table 1. From (12) we obtain the spectrum 111111For the -states this assignment agrees with the results of Ref. [31].

 M2(+)n,LB,S=32=M2(−)n,LB,S=12=4(n+LB+32)λB. (50)

If we now apply the superconformal relation and we predict a meson trajectory with eigenvalues

 M2n,LM=4(n+LM+12)λM, (51)

which is, precisely, the expression for the spectrum of the -meson (6) for . Again, one sees that the lowest-lying mesonic state, in this case the meson, has no superpartner, since would be negative.

Since the phenomenological value of for the trajectory is close to that of the trajectory, and (See Ref. [20]), one can expect good agreement for the masses of the supersymmetric partners. This is indeed the case, as can be seen from Fig. 3, where we have included the confirmed and , , vector-meson states from Ref. [30].

Using the assignment from (Table 1) and the comparison of Eqs. (9) with (40) (or (10) with (39)), we obtain the relation for the superconformal relation . Thus from (39) we obtain the LF-Hamiltonian for the superpartner vector meson trajectory

 G11=⎛⎝−d2dζ2+λ2Mζ2+2λM(LM−1)+4(LM+12)2−14ζ2⎞⎠, (52)

with . This expression is to be compared with the light-from holographic Hamiltonian which follows from (5) for and :

 HLF=(−d2dζ2+λ2Mζ2+2λMLM+4L2M−14ζ2). (53)

Thus, by extending the meson-baryon connection for baryons with we obtain an identical expression for the vector meson-spectrum, but with a different LF Hamiltonian. This somewhat less satisfactory feature of the - relations is reflected in the transformation under the supercharge (Appendix B). The meson wave function , that is the eigenfunction of with , is not annihilated by the action of (96). Indeed the two Hamiltonians and , (39) and (40) respectively, are identical for . Thus in this case, the unphysical value of the angular momentum, , is the only reason to exclude the baryonic superpartner of the . This is in contrast to the case of the pion, where the fermion-number changing operator actually annihilates the pion wave function (94), since it is a zero mass eigenmode.

## 5 Summary and Conclusions

Conformal and superconformal Quantum Mechanics, together with light-front holographic QCD, has revealed the importance of conformal symmetry and its breaking for understanding the confinement mechanism of QCD.

If one introduces the mass scale scale for hadrons using the method developed by de Alfaro, Fubini and Furlan [7], one obtains a confining theory for mesons while retaining a conformally invariant action. If one applies the DAFF procedure to light-front Hamiltonian theory, the form of the LF potential is uniquely fixed to that of a harmonic oscillator in the invariant LF radial variable  [8]. It predicts color confinement and linear Regge meson trajectories with the same slope in the radial and orbital excitations and . If one compares the construction of the confining LF potential with the Hamiltonian obtained in light-front holographic QCD, then the dilaton factor in the AdS action is uniquely fixed [4, 5]. The appearance of the extra spin-dependent constant term in the LF potential is a consequence of the specific embedding of the LF wave equations in AdS for arbitrary integer-spin [6]. This extra term is essential for agreement with experiment, including the prediction of a massless pion in the chiral limit.

In the case of half-integer spin, the dilaton in the AdS action does not lead to confinement for baryons since such a term can be absorbed into the wave function. Confinement thus requires the addition of a Yukawa-like term in the half-integer spin Lagrangian. However, this apparent deficiency of the AdS theory is cured [12] by the application of superconformal quantum mechanics.

Superconformal quantum mechanics can be constructed by restricting the superpotential in Witten’s construction [22] to a conformally invariant expression [13, 23]. Remarkably, it is possible to introduce a mass scale into the quantum mechanical evolution equations, without violating supersymmetry, by introducing a new supercharge which is a linear combination of generators of the super conformal algebra [13]. Furthermore, by connecting the resulting wave equations to the light-front holographic formalism, one fixes not only the confining term for baryons and mesons for all spins, but also the constant terms in the LF potential. The resulting spectra reproduces the principal observed features of mesonic and baryonic Regge trajectories: the resulting trajectories are linear, and the spacing of the radial excitations equals the spacing of the orbital ones. Furthermore, the baryon masses depend only on the LF angular momentum , but not on the total spin , as observed in experiment.

There are striking phenomenological similarities between the baryon and meson spectra which would not be expected from the underlying quark degrees of freedom, given that in QCD the valence state in the meson case consists of confined excitations, and baryons are normally considered bound states. However, the observed Regge trajectories are linear in the squared mass for both cases, with equal spacings of the orbital and angular excitations – both features which are typical for the proto-string theory such as the Veneziano model [32]. These essential features also follow from the light-front clustering properties of the semiclassical approximation to strongly coupled QCD and its holographic embedding in AdS space. In this approximation a nucleon is effectively a quark-diquark object, and it is also described by a one-dimensional effective theory. Furthermore, the coefficients of the confining term for mesons and baryons agree within , although they would seem to be completely unrelated. These similarities suggest that supersymmetric relations are responsible for these remarkable features.

In Ref. [12] superconformal quantum mechanics was used to describe baryonic states. There, the supercharges were shown to relate the positive and negative chirality components of the baryon wave functions, consistent with parity conservation. In this paper we have shown that supercharges, constructed formally as in [12], can also be used to relate hadronic states with different fermion number. This leads to remarkable relations between the spectroscopy of baryons and mesons, thus extending the applicability of light-front superconformal quantum mechanics to hadronic physics.

An important feature of the Hamiltonian operators (39, 40), which act on the two components of a supermultiplet , is the difference in the singular term of the potential. For one component of the Hamiltonian, it is ; for the other component, it is . This has the consequence that the power behavior of the wave function at the origin (twist) differs by one unit for the two components. In in light-front holographic QCD this implies a difference of the LF angular momentum by one unit . Comparing the spectra of the nucleon and the trajectory one indeed observes this degeneracy (See Fig. 2). The leading-twist wave function of the baryons is identified with the component of the supermultiplet , and the wave function of the mesons is identified with the component . As a consequence, the shared symmetric features of mesons and baryons are in fact a consequence of the properties of the superconformal algebra.

The problem for supersymmetry posed by the pion, which is massless in the chiral limit, and therefore can have no baryonic superpartner, is solved in a simple way: The value of the dimensionless constant of the conformal potential (24) has the value . The supercharge , (33), which transforms the meson into the baryonic partner, annihilates the pion state, and therefore there cannot be a baryonic partner. The case was not considered by Fubini and Rabinovici [13], since the classical potential in this case has no lower limit. Nevertheless, the pion wave function is regular at the origin and normalizable.

We have previously demonstrated a correspondence between superconformal quantum mechanics and light-front holographic QCD; however, this demonstration requires both a dilaton term in the bosonic AdS action, as well as a Yukawa-like interaction term . One must also assume in LFHQCD the same positive value of in both terms. In contrast, in the superconformal theory, the equality of of for mesons and baryon is exact. (Phenomenologically, this relation is broken since GeV for the and GeV for the nucleon trajectory (See Fig. 2).

We have also applied the same procedure to the and the -trajectories. The wave functions of the -trajectory are identified with the component , and the component of the super multiplet is identified with the -states. As for the case of the -nucleon connection, the properties of the fermion-changing supercharge imply that the meson angular momentum is one unit larger than the baryon angular momentum , consistent with the Hamiltonians (39, 40). One indeed obtains excellent agreement between the spectra of the mesonic and baryonic states (See Fig. 3). The values of and are nearly degenerate as predicted by superconformal quantum mechanics.

There is, however, a problem with the - connection in that half-integer twist is apparently required. For the trajectory the observed spectrum corresponds to half-integer twist , which also implies half-integer twist for the mesons on the trajectory. Although the spectra of this half-integer twist obtained with the superconformal Hamiltonian operator (39) correspond fully to those obtained by LFHQCD (and experiment), the wave functions do not; they differ by a factor . Related to this problem is the fact that the supercharge does not annihilate the wave function, but it formally leads to a baryonic state with the same mass. However, this state is excluded as a physical state, since it would have the angular momentum .

It should be noted that the semiclassical equations of light-front holographic QCD and superconformal quantum mechanics are intended to be a zeroth order approximation to the complex problem of bound states in QCD. We also emphasize that the quantum mechanical supersymmetric relations derived here are not a consequence of a supersymmetry of the underlying quark and gluon fields; they are instead a consequence of the superconformal-confining dynamics of the semi-classical theory and the clustering inherent in light-front holographic QCD.

In this paper we have concentrated on the consequences of superconformal algebra for the spectral properties of meson and baryons. Since the meson and baryon wave functions are also related, there are also interesting dynamical consequences; e.g., for elastic and transition form factors. The wave function is predicted to be identical to the non-leading-twist wave function of the nucleon, which in turn is related to the leading-twist wave function via a parity transform – see [12]; therefore, at low resolution the form factors of the nucleon and the are related. Another dynamical consequence of the model is that for high resolution, at large momentum transfer when the baryon cluster is resolved into its individual constituents, the twists of the superpartners are equal: the higher value of of the meson, , is compensated by the additional constituent in the baryon.

Acknowledgements

We thank Adi Armoni for helpful comments. The work of SJB was supported by the Department of Energy contract DE–AC02–76SF00515.

## Appendix A Other Possible Evolution Operators

Fubini and Rabinovici have discussed three different ways of constructing compact Hamiltonians from the superconformal algebra. Some care should be taken, however, in transferring their interpretation to our application. The emphasis in [7] and later in [23, 13] was on quantum mechanics as a one dimensional field theory and the investigation of the vacuum structure in this field theory. Therefore they only the case with a stable classical potential, implying , was considered. In our search for semiclassical bound-state equations, however, the lowest state is a hadronic state. Furthermore, in the field theoretical investigations of FR the dimensionless constant is an arbitrary positive parameter, each value of representing a different field theory with a different vacuum. In our investigations, where the procedure of dAFF [7] and its extension by FR  [13] has been embedded in LFHQCD the dimensionless constant determines the angular momentum and we are confined to the series of discrete values representing the orbital excitations. Nevertheless, it is informative to discuss the three different ways to construct the compact Hamiltonian representing hadronic bound states also from our perspective. For generality purposes we use in the appendices for the dimensionful constants the symbols and , which can be positive or negative. In our applications to meson and baryon spectroscopy we are restricted .

The simplest way to construct a Hamiltonian with discrete spectrum in the frame of the superconformal graded algebra is to apply directly the method of dAFF [7]. This yields the Hamiltonian [23, 13], again in the slightly generalized notation

 G0={Q,Q†}+w2K. (54)

Both supersymmetry and dilatation symmetry are broken here. The two components of the eigen-spinor of have different spectra

 (G0)11ϕ1 = (4n+2f+3)|w|ϕ1, (55) (G0)22ϕ2 = (4n+2f+1)|w|ϕ2, (56)

and thus supersymmetry is broken from the onset for all levels. This approach would yield a LF potential , without any additional constants which occur in LFHQCD (See 5, 9, 10), and which are phenomenologically very important.

On the other hand, the approach where supersymmetry is conserved by constructing a new Hamiltonian from the spinor operator , a superposition of the supercharges and within the superalgebra [13],

 Rw=Q+wS, (57)

conserves supersymmetry for , since commutes with the evolution operator

 G(w)={Rw,R†w}. (58)

Therefore is an eigenstate of with identical eigenvalue as the eigenstate .

The spectra of for real values of and are:

 E1 = (4n+2)|w|+2∣∣∣f+12∣∣∣|w|+2(f−12)w, (59) E2 = (4n+2)|w|+2∣∣∣f−12∣∣∣|w|+2(f+12)w, (60)

where are eigenvalues of representing mesonic states and the eigenvalues of for baryons. For and the spectra are independent of :

 E1 = 4(n+1)|w|, (61) E2 = 4n|w|, (62)

and therefore cannot lead to angular excitations of the corresponding LF Hamiltonians. For and , we have

 E1 = 4nw, (63) E2 = 4(n+1)w. (64)

There exists no baryonic state

 |ϕ⟩=(0ϕB), (65)

with zero energy. The reason for this seeming contradiction with the above mentioned commutation relation, lies in the fact that the operator annihilates the mesonic state (See (94)).

## Appendix B Transformation Operators and Quantum Mechanical Evolution

The generalized hypercharge has the commutation relations

 [G(w),Rv] = −2(w−v)R−w, (66) [G(w),R†v] = 2(w−v)R†−w, (67)

with the new Hamiltonian .

For the commutator vanishes, therefore if is an eigenstate of , also is an eigenstate with the same eigenvalue. Therefore the spinor supercharge transforms the baryonic superpartner with angular momentum , into the mesonic one with angular momentum . The operator acts in the opposite direction.

For , however, we have the typical commutation behavior of a raising and lowering operator, respectively:

 [G(w),R−w] = −4wR−w, (68) [G(w),R†−w] = 4wR†−w. (69)

That is, if is eigenfunction of with eigenvalue , then is eigenstate with the energy . This means that a baryonic state with angular momentum and radial excitation is transformed into a mesonic state with angular momentum and radial excitation , which has the same energy as the baryonic state with angular momentum and radial excitation .

There is also a bosonic raising operator, that is, a raising operator which does not change fermion number. It is composed of the bosonic operators of the graded algebra. Generalizing again slightly the operators introduced by FR in Ref. [13]

 Lv=H+v2K+2ivD, (70)

one obtains from the algebra (26) the commutation relations

 [G(w),Lw] = 4wLw, (71) [G(w),L−w] = −4wLw. (72)

These relations imply that also is a raising operator, which transforms a baryon with into a baryon with and the same with the mesons. Since it is composed of operators of the conformal group, it can also be applied to the lowest mesonic state, although there is no supersymmetric partner.

Since the hypercharges change the angular momentum by one unit, it is tempting to look for an operator which also leads to angular excitations. Such an operator which increases the angular momentum by one unit is easily constructed and has the form:

 Λw={Q,ψ}+w{S†,S}+1ζψ†ψ. (73)

If is an eigenstate to the Hamiltonian operator constructed with , then is eigenstate to , constructed with . This operator is, however, not an element of the superconformal algebra. The action of the different operators in the baryon-meson system is illustrated in Fig. 4.

### b.1 Quantum-Mechanical Evolution

In this paper, as in [12], we have concentrated on algebraic aspects and its consequences for the spectra. We now briefly discuss the quantum-mechanical time evolution. The Hamiltonian of unbroken superconformal quantum mechanics, Eq. (22), is the translation operator for the time variable

 iddt|ϕ⟩=H|ϕ⟩. (74)

The quantum-mechanical evolution of the operator (37)

 G=2H+2w2K+2λ(f\bf I−B), (75)

follows from the action of the generators and on the state . We have (See Appendix C in Ref. [20]),

 e−iHϵ|ϕ(t)⟩ = |ϕ(t)⟩+ddt|ϕ(t)⟩ϵ+O(ϵ2), (76) e−iKϵ|ϕ(t)⟩ = |ϕ(t)⟩+ddt|ϕ(t)⟩ϵt2+O(ϵ2). (77)

There follows

 (78)

where the new evolution parameter is related to in (74) by

 dτ=dt2(1+λt2), (79)

as in dAFF [7]. From the eigenvalue equation follows the stationary state solution

 |ϕE(τ)⟩=|ϕE(0)⟩e−i((E\bf\footnotesize I% −2λ(f\bf\footnotesize I−B))τ. (80)

### b.2 Operators in Matrix Form

It is sometimes convenient to work with a special matrix representation of the superconformal algebra. For convenience we give here an explicit realization in the Schrödinger picture. Define

 q=−ddx+fx,q†=ddx+fx. (81)

Then we can write the spinor operators and as

 Q=(0q00),Q†=(00q†0), (82)

and

 S=(0x00),S†=(00x0). (83)

The Hamiltonian in matrix form is

 2H=(qq†00q†q