# Supersymmetry across the Hadronic Spectrum

Authors: Hans Günter Dosch
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

Semiclassical light-front bound-state equations for hadrons are presented and compared with experiment. The essential dynamical feature is the holographic approach; that is, the hadronic equations in four-dimensional Minkowski space are derived as holograms of classical equations in a 5-dimensional anti-de Sitter space. The form of the equations is constrained by the imposed superconformal algebra, which fixes the form of the light-front potential. If conformal symmetry is strongly broken by heavy quark masses, the combination of supersymmetry and the classical action in the 5-dimensional space still fixes the form of the potential. By heavy quark symmetry, the strength of the potential is related to the heavy quark mass. The contribution is based on several recent papers in collaboration with Stan Brodsky and Guy de Téramond.


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## Body

## 1. Introduction

Semiclassical approximations play an essential role in quantum field theories. Think of atomic or molecular physics: in spite of all refinement of renormalized QED, the semiclassical Dirac and Schrödinger equations not only are of enormous practical advantage but also are essential in order to exhibit structural properties.The search for semiclassical approximations even in strongly interacting quantum field theories received a significant advance through the Maldacena conjecture [1], which postulates the correspondence of a superconformal gauge quantum field theory in 4 dimensions with a classical gravitational theory in a 5-dimensional anti-de Sitter space (AdS5). Here a light-front Hamiltonian for meson and boson wave functions is presented, which is based on the holographic AdS/CFT correspondence and receives essential dynamical constraints from an implemented superconformal algebra.The kinematical frame is based on the light-front (LF) quantization at fixed LF timex+=x0+x3; I refer for the essential features and the importance of this frame to the contribution of Stan Brodsky in the same volume. In the two constituent sectors, an essential variable is the boost invariant transverse separation ζ=x1x2b→⊥,2-b→⊥,1, where xi and b→⊥,i, i=1,2, are the longitudinal momentum fractions and the transverse coordinates of the two constituents, respectively.Since in the holographic correspondence there is only one variable to describe the internal structure of hadrons, namely, the coordinate of the 5th dimension, hadrons with more than two constituents have to be treated as clusters. In that case, one introduces the effective longitudinal momentum fractionxaeff=∑i=1Naxi, where Na is the number of constituents in the cluster a, and correspondingly one introduces an effective transverse coordinate b→⊥,aeff=∑i=1Naxib→⊥,i/xaeff. The effective boost invariant separation is then ζ=xaeffxbeffb→⊥,beff-b→⊥,aeff. There is no theoretical limit for the number of the constituents in the cluster.In the conformal limit (massless quarks), the LF Hamiltonian in the two-particle sector does not contain the longitudinal variablexi or xieff explicitly:(1)H=-d2dζ2+4L2-14ζ2+Uζ;here, L is the LF angular momentum. In this approach, the squared hadron masses are the eigenvalues of the LF Hamiltonian.In order to get some insight into the LF potentialUζ, which is in principle determined by the QCD Lagrangian, we turn to the dynamical scheme of AdS5/CFT correspondence.AdS5 is a maximally symmetric 5-dimensional space with the metric(2)ds2=R2z2dx02-dx12-dx22-dx32-dz2,where the coordinate of the 5th dimension z is the holographic variable.From the AdS5 action, one derives the field equations of AdS, which are interpreted as bound-state wave equations. As noticed and exploited first by Brodsky and de Téramond [2], the structure of the field equations from the AdS5 action is equal to the structure of the LF Hamilton, if one identifies the holographic variable z with the boost invariant separation ζ. From unperturbed AdS, one obtains a vanishing LF potential. This is no wonder: the space AdS5 has maximal symmetry that in turn leads to a conformal symmetry in the holographically corresponding 4-dimensional quantum field theory and no scale can appear.In order to introduce interaction, one has to break the maximal symmetry. This is done by introducing a dilaton termeφz into the AdS action. Due to the occurrence of covariant derivatives in the AdS Lagrangian, the derivation of the Euler-Lagrange equations is rather tedious, but finally one obtains for a meson wave function [3] Φμ1⋯μJ=ϵμ1⋯μJJϕJ,Lζ the bound-state equation:(3)-d2dζ2+4L2-14ζ2+UζϕJζ=M2ϕJζwithUζ=12φ′′ζ+14φ′z2+2J-32ζφ′ζ;here, ϵμ1⋯μJJ is the polarization tensor of a meson with total angular momentum J; the LF angular momentum L is related to the AdS mass by mR2=-2-J2+L2.; the holographic variable z is identified with the LF separation ζ.For Baryons, a dilaton factoreφζ does not lead to an interaction and there one has to introduce an additional Yukawa term u¯Vζu in the 5-dimensional fermionic AdS action. The wave function is split into a positive-chirality and a negative-chirality component:(4)uμ1⋯μJ-1/2p=1+γ52u⋯pψB+ζ+1-γ52u⋯pψB-ζand one obtains the wave equations [3] with the potential U±=V2±V′+1+2L/ζV:(5)-d2dζ2+4L+12-14ζ2+U-ζψB-ζ=M2ψB-ζ,(6)-d2dζ2+4L2-14ζ2+U+ζψB+ζ=M2ψB+ζ.For phenomenological reasons one has chosen ad hoc [4–8] (7)eφζ=eλMz2,Vζ=λBz;which leads to the potential for(8)mesons:Uζ=λM2ζ2+2λMJ-1,(9)baryons:U±ζ=λB2ζ2+2LλF+λF12±12.These potentials lead to linear Regge trajectories and a massless pion. For the Delta trajectory, however, one has to choose formally half-integer twist (i.e., to add 1/2 to the angular momentum L) in order to get agreement with the data.


## 2. Constraints Given by the Superconformal Graded Algebra

Although one obtains in that way a satisfactory description of several essential features of hadron spectroscopy, there are some points that are not so satisfactory. First, and this is the most important shortcoming, the special choice of maximal symmetry breaking given in (7) is completely arbitrary and has no theoretical foundation. Second, the symmetry breaking for mesons and baryons is completely unrelated, and finally the remarkable fact that the Regge slopes of mesons and baryons are very similar, which implies that λM≈λB in (7), is in this context completely fortuitous.In the search for further dynamical constraints, we go back to the original Maldacena correspondence [1] of a classical 5-dimensional theory and a 4-dimensional quantum field theory. There the latter is heavily constrained by symmetries: it is superconformal; that is, it obeys supersymmetry and conformal symmetry. Since our goal is to find a semiclassical approximation to QCD, it is not too far-fetched to investigatesuperconformal quantum mechanics (QM).Supersymmetric QM is very simple [9]. It is built on a graded algebra consisting of two fermionic operators (supercharges), Q and Q†, and a bosonic operator (the Hamiltonian), H, with the anticommutation and commutation relations:(10)Q,Q†=H,Q,Q=Q†,Q†=0;Q,H=Q†,H=0.A matrix realization in a Hilbert space with two components is(11)Q=0q00,Q†=00q†0,(12)q=-ddx+fx+Vx,q†=ddx+fx+Vx,where f is a dimensionless constant and V is the arbitrary superpotential.If the superpotentialV=0, then there is no scale present and also conformal symmetry holds: one can extend the supersymmetric (graded) algebra 9 to a superconformal one by introducing the additional supercharges S and S†, which can be viewed as a square root of the special conformal transformation K, since S,S†=K; for more details, see [10, 11]. In matrix notation, one can write(13)S=0x00,S†=00x0.Following the procedure introduced by de Alfaro et al. [12] for the conformal algebra and extended by and Fubini and Rabinovici [13] to the superconformal algebra, one can construct a new Hamiltonian G inside the superconformal algebra:(14)G=Rλ,Rλ†withRλ=Q+λS,Rλ†=Q†+λS†. In this generalization of the Hamiltonian, a scale λ appears naturally, since the supercharges S and Q have different physical dimensions.The new Hamiltonian is diagonal with(15)G11=-d2dx2+4f+1/22-14x2+λ2x2+2λf-12,(16)G22=-d2dx2+4f-1/224x2+λ2x2+2λf+12.By construction, the eigenvalues of the two operators are identical and the operator Rλ† transforms an eigenstate of G11 with eigenvalue E≠0 into an eigenfunction of G22.If we identifyf=L+1/2 and x=ζ, it turns out that the potentials occurring in the new Hamiltonian in (15) and (16) are exactly the potentials following from the ad hoc introduced Yukawa term in (7) in the fermion action. Furthermore, the operator G11 is the Hamiltonian of a meson with L=J=f+1/2 (see (8)). And G22 is the Hamiltonian of the baryon component ψB+ with L=f-1/2 (see (9)). That is, the meson eigenfunction of G11 with LF LM=f+1/2 has the same eigenvalue as a baryon with LB=f-1/2, eigenfunction of G22. The supercharge Rλ† transforms the meson wave function into a wave function of the mass-degenerate baryon. A meson with LM=f+1/2 is superpartner of baryon with LB=f-1/2; mesons with LM=0 (π,e.g.) have no superpartner, since the supercharge Rλ† annihilates its wave function.


## 3. Spin Effects

In the superconformal quantum mechanical approach, no internal quark spin effects are foreseen, in contrast to the AdS5 approach for mesons. The meson Hamiltonian derived in the frame of the superconformal QM, G11, differs from the one derived from AdS by the constant λs, where s is the total quark spin of the meson. Supersymmetry demands the occurrence of the same term also for baryons; that is, we arrive at the following expression for the supersymmetric Hamiltonian for mesons and baryons:(17)G=Rλ,Rλ†+λsI,where for π and its SUSY partner, namely, the nucleon, we have s=0, and for ρ and its SUSY partner Δ, we have s=1. Generally, for baryons, s is the lowest possible internal spin of the cluster. The results for the π-N and the ρ-Δ trajectories are displayed in Figure 1; they show especially for Δ and ρ the nearly perfectly realized symmetry.Figure 1
Theπ-N trajectory and the ρ-Δ trajectory from [10].


## 4. Completing the Supersymmetric Multiplet

The supersymmetric quadruplet containing the meson and the baryon is not yet complete, since the partner of negative chirality component of the baryon,ψB-, is missing. We can complete the supersymmetric multiplet by applying the fermion operator Rλ† to that negative-chirality component baryon wave function and thus obtain a bosonic state. This operator, which transforms the meson wave function in the positive-chirality baryon wave function, can be interpreted as transforming a quark into an antidiquark cluster in the colour representation 3. Therefore the operator Rλ† applied to the negative-chirality component of a baryon will give a wave function ϕT=Rλ†ψB-, which can be interpreted as that of a tetraquark consisting of a diquark cluster and an antidiquark cluster. The SUSY quadruplet can be arranged in a 2×2 matrix in the following way: ϕMLB+1ψB-LB+1ψB+LBϕTLB.Candidates for complete supermultiplets with fitting quantum numbersIJP areb1123511+ and N+,-9401/21/2+ and as tetraquark f098000+,a2132012+ and Δ+,-12303/23/2+ and as tetraquark a1126011+.A special feature of tetraquarks is that states consisting of a spin-singlet and a spin-triplet cluster are degenerate inG parity.


## 5. Beyond the Chiral Limit

It is reasonable to assume that small quark masses do not affect superconformal dynamics; that is, the LF potentials remain unchanged and only the invariant mass is affected; that is, the chiral expression∑ik⊥i2/xi has to be replaced by ∑ik⊥i2+mi2/xi. To first approximation, this gives an additive term to the squared hadron mass [14–16] for mesons MM2=4λn+LM+2λs+ΔM22m1,m2 and for baryons: MB2=4λn+LB+1+2λs+ΔM32m1,m2,m3, where ΔM2m1,…,mn=λ2/FdF/dλ with Fλ=∫01⋯∫dx1⋯dxne-1/λ∑inmi2/xi‍δ∑inxi-1.Fromπ and K mass, the light quark masses are determined to mq=0.046 GeV and ms=0.357 GeV; the resulting spectra are shown in Figure 2.Figure 2
TheK, Λ; K∗, Σ∗; η′, Ξ; and ϕ, Ξ∗ trajectories, from [14].


## 6. Strong Breaking of Conformal Symmetry by Heavy Quark Masses

In heavy-light hadrons, where one constituent is heavy and the other is light, we have still ultrarelativistic kinematics, but conformal symmetry is strongly broken. Supersymmetry, however, can still survive and can give, together with AdS/CFT breaking, crucial dynamical constraints.Let us go back to the SUSY algebra in (11) and (12). From H=Q,Q†, we can express the mesonic LF potential in terms of Vx. For a meson with JM=LM=L+1, we obtain(18)USusyζ=V2+2L+1ζV-V′.With a dilaton term eφζ, we obtain, from the AdS/CFT action for the same meson,(19)Udilζ=14φ′2+2LM-32ζφ′+12φ′′.It has been shown in [14] that the two expressions are only compatible if(20)φζ=12λAζ2+B,Vζ=12λAζ.Thus, the combination of SUSY with AdS constraints implies that the LF potential even for strongly broken conformal invariance is harmonic; the strength, however, is not determined. The resulting hadronic spectrum is for mesons M2=4λQn+L+μ22 and for baryons M2=4λQn+L+1+μ32.In Figure3, the heavy-light hadrons together with linear trajectories are displayed. In all cases where several recurrences are observed, the data are within the expected accuracy compatible with the predicted harmonic potential. The approximate masses of many states, which are still to be observed, can be predicted in that way [14]. The average values for the scale parameters for hadrons, which contain one c-quark or b-quark, are λc=0.71±0.04 GeV and λb=1.06±0.1 GeV.Figure 3
Trajectories for heavy-light hadrons with strongly broken conformal symmetry, from [14].
(a)
(b)
(c)
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## 7. Constraints from Heavy Quark Symmetry

If the mass of the heavy quarkMQ goes to ∞, it decouples and heavy quark symmetry (HQS) holds [17]. This has as a consequence that the decay constant of the heavy-light pseudoscalar meson, fM, scales in that limit with the inverse square root of the meson mass [17, 18]:(21)MMfM⟶C.With mass dependence of the wave functions modified according to the modification of the invariant mass discussed above, we can include the mass dependence into the wave function: (22)ψn,Lm=1Nme-1/2λ∑i=1nmi2/xiψn,L0,where ψn,L0 is the wave function in the case of massless quarks.From this wave function, one derives the decay constant:(23)fM=1∫01dxe-mQ2/λ1-xλπ∫01dxe-mQ2/2λ1-xx1-x.Performing an integration by steepest descent, one obtains(24)fM=6πe1+erf⁡12λ3/2mQ2.Taking into account the fact that, in the limit of mQ→∞, the heavy quark mass and the meson mass become equal, comparison of this equation with (21) yields that λQ~MM. Thus, HQS demands that, for heavy quark masses, the scale parameter λ increase. It relates λb with λc even in a quantitative way, as can be seen from Figure 4.Figure 4
The scale parameterλ for different pseudoscalar mesons versus the meson mass. The solid line is the functional dependence λ~MM as predicted by HQS.


## 8. Conclusions

We were motivated by the Maldacena conjecture, which relates a classical gravitational theory in AdS5 to a superconformal colour gauge QFT with infinitely many colours. Our approach, which aimed at a semiclassical approximation, led to superconformal quantum mechanics on the light front. The form of the light-front potential is determined by the superconformal algebra. This model describes essential features of light hadron spectra: linear trajectories in orbital and radial excitations and a massless pion. The observed similarity of meson and baryon spectra is a consequence of the supersymmetry between the wave functions. We also predict tetraquarks with definite masses and quantum numbers.The supersymmetry seems to survive even in the case where conformal symmetry is strongly broken by heavy quark masses. Here, the supersymmetry, together with the embedding into AdS5, implies a harmonic potential λζ2 also for heavy-light hadrons, but the scale parameter λ increases with heavy quark mass, in quantitative accordance with heavy quark symmetry.Our model does not contain colour degrees of freedom, but the fact that all states have zero width is well compatible with theNC→∞ limit of QCD; the accuracy, which is generally about 10%, is also compatible with the application of this limit to an NC=3 theory. The model can also be applied beyond spectroscopy [8, 19], but in this case additional assumptions and input are necessary.



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