BLED WORKSHOPS IN PHYSICS VOL. 1, NO. 1 Proceedins of the Mini-Workshop Few-Quark Problems (p. 64) Bled, Slovenia, July 8-15, 2000 Few-body problems inspired by hadron spectroscopy Jean-Marc Richard? Institut des Sciences Nucléaires Université Joseph Fourier – CNRS-IN2P3 53, avenue des Martyrs, F-38026 Grenoble Cedex, France Abstract. I discuss some results derived in very simplifiedmodels of hadron spectroscopy, where a static potential is associated with non-relativistic kinematics. Several regularity patterns of the experimental spectrum are explained in such simple models. It is un- derlined that some methods developed for hadronic physics have applications in other fields, in particular atomic physics. A few results can be extended to cases involving spin- dependent forces or relativistic kinematics. 1 Introduction As discussed in several contributions at this nice workshop, the dynamics of light quarks is far from being simple, with non-perturbative effects even at short dis- tances, and highly-relativistic motion of the constituents inside hadrons. Never- theless, it is interesting to consider a fictitious world, with the hadron spectrum governed by a simple Hamiltonian where a non-relativistic kinematics is supple- mented by a static, flavour-independent potential. The regularities derived from the properties of the Schrödinger equation are similar to these observed in the actual spectrum. This suggests that the actual QCD theory of quark confinement should exhibit similar regularities. One should also notice that several results derived in the context of quark models of hadrons have been successfully applied to other few-body problems, in particular in atomic physics. Another challenge consists of extending theorems on level order, convexity, etc., to less naive Hamiltonians with spin-dependent forces and relativistic kine- matics. Some of the first results will be mentioned. 2 Results on mesons The discovery of and  resonances and their excitations has stimulated many studies in the quark model. In particular, the successful description of these spec- tra by the same potential has motivated investigations on the consequences of flavour independence. The rigorous results have been summarized in the reviews by Quigg and Rosner and byMartin andGrosse. A few examples are given below, dealing with energy levels.? E-mail: jmrichar@isn.in2p3.fr Few-body problems inspired by hadron spectroscopy 65 All potentials models reproduce the observed pattern of quarkonium thatE(1P) < E(2S). Note the notation adopted here, (n; `), in terms of which the principal quantum number of atomic physics is n + `. It has been proved thatE(n + 1; `) > E(n; ` + 1) if V > 0, and the reverse if V < 0. The Coulomb degeneracy is recovered as a limiting case. The sign of  reflects whether the chargeQ(r) seen at distance r grows (asymptotic freedom), decreases or remains constant (Gauss theorem). This “Coulomb theorem” can be applied successfully tomuonic atoms, which are sensitive to the size of the nucleus (Q(r)%), and to alkaline atoms whose last electron penetrates the inner electron shells (Q(r)&). Another theorem describes how the harmonic oscillator (h.o.) degeneracyE(n+ 1; `) = E(n; `+ 2) is broken. A strict inequality is obtained if the sign of V 00 is constant. In both the complete Hamiltonian p21=(2m1) + p22=(2m2) + V(r12) or its re- duced version p2=(2) + V(r), the individual inverse masses mi or the inverse reduced mass enter through a positive operator p2, and linearly. It results that each energy level is an increasing function of this inverse massm-1i or -1, and that the ground-state energy (or the sum of first levels) is a concave function of this variable. There are many applications. For instance, for the ground-state of the meson with charm and beauty,(bs̄) + ( ̄) - ( s̄) < (b ̄) < (bb̄+ ̄)=2: (1) 3 Level order of baryon spectra For many years, the only widespread knowledge of the 3-body problem was the harmonic oscillator. This remains true outside the few-body community. The dis- cussion on baryon excitations is thus often restricted to situations where V =P v(rij), with v(r) = Kr2 + Æv, and Æv treated as a correction. First-order perturbation theory is usually excellent, especially if the oscilla- tor strength K is variationally adjusted to minimise the magnitude of the correc- tions. However, when first-order perturbation is shown (or claimed) to produce a crossing of levels, one is reasonably worried about higher-order terms, and a more rigorous treatment of the energy spectrum becomes desirable. A decomposition better than V =PKr2ij + Æv is provided by the generalised partial-wave expansion V = V0() + ÆV; (2) where  / (r212 + r223 + r231)1=2 is the hyperradius. The last term ÆV gives a very small correction to the first levels. With the hyperscalar potential V0 only, the wave function reads = -5=2u()P[L℄( ), where the last factor contains the “grand-angular” part. The energy and the hyperradial part are governed byu 00() - `(` + 1)2 u() +m[E -V0()℄u() = 0; (3) very similar to the usual radial equations of the 2-body problem, except that the effective angular momentum is now ` = 3=2 for the ground-state and its radial 66 J.-M. Richard excitations and ` = 5=2 for the first orbital excitation with negative parity. The Coulomb theorem holds for non-integer `. If V > 0, then E(2S) > E(1P), i.e., the Roper comes above the orbital excitation. Note that a three-body potential cannot be distinguished from a simple pairwise interaction once it is reduced to its hyperscalar component V0 by suitable angular integration. It also results from numerical tests that relativistic kinematics does not change significantly the relativemagnitude of orbital vs. radial excitation energies. The splitting of levels in the nearly hyperscalar potential (2) is very similar to the famous pattern of the N = 2 h.o. multiplet, except that the Roper is disen- tangled. A similar result is found for higher negative-parity excitation: the splitN = 3 levels of the nearly harmonic model are separated into a radially excitedL = 1 and a set of split L = 3 levels. 4 Tests of flavour independence for baryons The analogue for baryons of the inequality between bb̄, ̄ and b ̄ reads.(QQq) + (Q 0Q 0q)  2(QQ 0q): (4) Unlike the meson case, it requires mild restrictions on the potentials. For instance, the equal spacing rule -- = - = - is understood as follows: the central force gives a concave behaviour, with for instance - - <  -, but a quasi perfect linearity is restored by the spin–spin interaction which acts more strongly on light quarks. A similar scenario holds for the Gell- Mann–Okubo formula. Inequalities can also be written for baryons with heavy flavour, some of them being more accessible than others to experimental checks in the near future. Ex- amples are 3(b s)  (bbb) + ( ) + (sss); (5)2(b q)  (bbq) + ( q); 2( qq)  ( q) + (qqq): 5 Baryons with two heavy flavours There is a renewed interest in this subject. The recent observation of the (b ̄) mesons demonstrates our ability to reconstruct hadrons with two heavy quarks from their decay products. Baryons with two heavy quarks (QQ 0q) are rather fascinating: they combine the adiabatic motion of two heavy quarks as in J= and mesons with the highly relativistic motion of a light quark as in flavoured mesons D or B. Thewave function of (QQq) exhibits a clear diquark clusteringwith r(QQ)r(Qq) for the average distances. This does not necessarily mean that for a given potential model, a naive two-step calculation is justified. Here I mean: estimate first the (QQ) mass using the direct potential v(QQ) only, and then solve the[(QQ)-q℄ 2-body problemusing a point-like diquark. If v is harmonic, one would Few-body problems inspired by hadron spectroscopy 67 miss a factor 3=2 in the effective spring constant of the (QQ) system, and thus a factor (3=2)1=2 in its excitation energy. On the other hand, it has been checked that the Born–Oppenheimer approx- imation works extremely well for these (QQq) systems, even when the quark mass ratio Q=q is not very large. This system is the analogue of H+2 in atomic physics. 6 The search of multiquarks A concept of “order” or “disorder” might be introduce to study multiquark sta- bility. This is related to the breaking of permutation symmetry. Consider for in- stanceH4(x) = 4Xi=1 p2i2m + (1 - 2x)(V12+V34) + (1 + x)(V13+V14+V23+ V24)= HS + xHMS; (6) where the parameter x measures the departure from a fully symmetric interac- tion. From the variational principle, the ground-state energy E(x) is maximal atx = 0. In most cases, E(x)will be approximately parabolic, so the amount of bind- ing below E(0) is measured by jxj. In simple colour models of multiquark confinement, the analogue of jxj is larger for the threshold (two mesons) that for a (q̄q̄qq) composite. So a stable multiquark is unlikely. For the (Q̄Q̄qq) systems presented by our slovenian hosts, and discussed earlier by Ader et al., Stancu and Brink, and others, there is another asymme- try, in the kinetic energy, which now favours multiquark binding. So there is a competition with the colour-dependent potential. The methods developed for quark studies has been applied for systematic investigations of the stability of three-charge and four-charge systems in atomic physics. Bibliography A more comprehensive account of these considerations, including references to original papers or to recent review articles will be found in the Proceedings of the Few-Body Conference held at Evora, Portugal, in September 2000 (to appear as a special issue of Nuclear Physics A). I would like to thank again the organizers of this Workshop for the very pleasant and stimulating environment.