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Observed Supersymmetry and Parity Doubling of Excited Hadrons from a Simple Spectrum-Generating Algebra

We find remarkable agreement with the observed excitations of hadrons with a simple three parameter mass relation of the SU(3) subgroup of the underlying U(15/30) graded Lie group. The baryons are the appropriate supersymmetric partners of the mesons. An interesting feature, which is a focus of current interest, is that the baryons and isobars show parity doubling. Significantly, the ground state baryons and mesons have no place in the fit, so that the parity doubling is indicated only when excitation energy is available. This has correspondence with the parity doubling seen in recent lattice calculations, when thermal excitation is present. The agreement with experiment is comparable to the semi-relativistic quark model which fits with many force parameters and where the observed parity doubling is accidental. The gross splitting of the levels is the same for the strange and the non-strange sectors, suggesting flavour independence.


Introduction.
Dynamical groups and spectrum generating algebras are much used in the current literature for classifying states of a composite system in terms of underlying group symmetry of the Hamiltonian. The pioneers in this field were Barut [1], Barut and Bohm [2] and Dothan, Gell Mann and Ne'emna [3]. The subject has been reveiwed in detail recently in two volumes edited by Bohm, Ne'eman and Barut [4].
Lattice calculations and other QCD motivated models exist and they indicate that the phase transition to free quarks and gluons is a weak first order or second order one. So there is no dissolution of the hadrons, but rather one sees presence of parity doublets [5].
This was soon confirmed by Gottleib et al [6] In this connection we must stress the early work of Barut [1] where he had looked at parity doubled states in the conformal O(4,2) model. The conformal model has been revived in the version of string theory given by Kutasov and Seiberg [7] and leads to a rich phenomenology as shown by the Freund and Rosner [8], Dey, Dey and Tomio [10], Cudell and Dienes [9] and Mustafa et al. [11]. This is based on old wisdom of the Regge model giving same trajectory for mesons and baryons. According to Kutasov and Seiberg, the appearance of the destabilizing tachyons in a string theory severely constrains the difference of the densities of bosons and fermions in that theory. Their result shows that tachyon elimination does not require full-fledged supersymmetry. Cancellation between the boson and fermion states is all that is needed. It turns out that though the density of states of mesons and baryons each rises exponentially with energy, their difference rises only like a low power of energy. The present model uses a much simpler compact group structure and our only motivation is to prompt more calculations based on the phenomenological success of the model proposed. We retain the same common feature of the two very different calculations [1] and [5], namely that the ground state does not show parity doubling but the excited states of the baryons do.

The Model
Looking at the baryon spectrum we see that the lowest states, i.e. the nucleon octet (N, Λ, Σ, Ξ), have no odd parity partners. But when some excitation energy is available, we encounter parity doubling. It is no surprising that one can get this at finite T, in lattice as we have already discussed. The Laplace transform of the finite temperature partition function in fact gives the excitation spectrum at zero temperature [10]. Chiral symmetry is realized at such high excitation within 5-10 %. In the non-relativistic and the semi-relativistic models, the occurrence of parity doubled states is contrived : the perturbative hyperfine interaction is adjusted to bring down the even parity to match with the odd levels, sometimes invoking multi-shell configurations [12], sometimes deformation [13].
In an earlier paper [14], we had fitted mesons and baryons in a simple model using the supersymmetric graded Lie group U(15/30). We add the odd parity baryons into our scheme, in view of the present interest. In the meson sector the interest is of a different nature. Recently many more new mesons sates have been discovered and these do not all fit into any single model. We discuss our fit in terms of the new experimental data.
Our model is based on excitations of bosons and fermions in the s, d and g shells of some effective potential, in terms of a U(15/30) graded Lie group. The reduction of this group into simpler structures, in particular to the SU(3) scheme, so well known in Nuclear Physics [15] have been worked out [16], [17]. In [14] we had given a mass formula which fitted more than 60 mesons and baryons using the classification given by Yu. We now add more than 40 new states.
The scheme we follow is extremely simple with states belonging to a representation where the total number of particles is three. This implies that the mesons are different from other models, here they are two fermion, one boson states : ff b. The alternative is to use the meson as a two fermion and the baryon as a quark-diquark state, and The spin zero mesons may belong to partition ν = [3], [21]. Spin one mesons and octet baryons (spin 1/2) can belong to the latter partition. This and the allowed values of (λ, µ) are given in Table 1. The isobars and the J = 1 mesons can be placed in the partiton ν = [111]. The allowed (λ, µ) are also given in Table 1.
The hadron mass M is then given by In a simple model like ours it is not justified to try for accuracy, particularly in view of the typical width of 50 MeV for the states. However one can associate α with the inverse of a moment of inertia parameter, and since the masses and the radii vary for different bandheads, given by a set (λ, µ), we have allowed α to vary a little.

Results and discussion
We now turn to baryons which are given in Tables 2 to 5. Unlike [12] all levels are fitted and the fit is generally similar. The Roper resonance N(1440) and the N(1710) are fitted very well ( Table 2). The placement of 1710 in the second band is consistent with its very different properties : it has a gamma decay width consistent with zero and ten times less than the Roper. It also has a very large two pion decay rate and very little Nπ unlike the Roper. Altogether 18 positiv parity baryons are fitted in Table 2.
The quality of the fit is very good, the difference from experiment being 5.7%. All the 4 well-established Λ-s and Σ-s are fitted along with three two-star Σ-s. It is interesting to note that in Table 2, and indeed also in the subsequent tables, the strange baryons fit along side nucelon excitations. On the other hand recall, that the splitting between the ground state multiplet is dependent on the mass of particular quark flavours, for example in the recent Shuryak and Rosner model [20]. Thus one might say that the gross spacing of hadron excitations is independent of flavour, but the hyperfine splitting between the states, particularly the ground states, is known to depend on the masses of its quark content.
We include the odd parity baryons in our fit ( The phenomenology may find justification only in an effective potential based on noncompact conformal group structure as indicated in Barut's work [1]. Notice that the spectrum given by him has parity doubling, not in the ground state, but starting from the first excited state. We have placed some of the odd parity baryons as L = 0 excitations of odd-parity quasiparticles in Table 3. This gives much better fit and fits in with the fact that Λ(1405) and the N(1535) are different from the 1520 Λ or N. This is supported by the rather unusual 100 % Σ π decay of the first one and the 45-55% Nη decay of the second one.
The reader who does not like such exotic quasiparticles may group them together with the (L = 1) 1520 Λ and N.
The number of odd parity baryons fitted in Table 3 is 24, the fit is comparable to that of Table 2. In fact using all the 42 states the fit improves marginally to 5.03%.
We fit 8 positive and 6 negative parity baryons in Tables 4, 5 for which S = 3/2 in eq. (2) and α is taken to be 30 MeV . The fit to experiment is 2.93%. The experimental states consist of 9 well establsihed and 5 states with (**) status.
In Table 6  In [19] there tables, although the 1810 has been seen by three different groups. The 1565 has also been confirmed now but may be a diquark-antidiquark state as already mentioned [18].
In Table 8  namely the oscillator term and the quadrupole term. Also experience with nuclear physics shows its usefulnessness in effective potential models. Finally one can look for decay systematics, but for this one has to impose a model ground state structure, since the ground states are left out of the scheme. A conformal model which incorporates the ground state in consistent way may be able to achieve this.