SOFT QCD LIGHT QUARK PHYSICS with CHROMODYNAMICS Nathan
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247 SOFT QCD LIGHT QUARK PHYSICS WITH CHROMODYNAMICS Nathan Isgur Department of Physics University of Toronto Toronto, Canada MSS 1A7 The addition of dynamical ingredients suggested by QCD dramatically improves the ability of quark models to describe soft hadronic phenomena . This approach is discussed in terms of a particularly simple non-relativistic potential model which has been used to study the spectroscopy, decays, and static properties of baryons and mesons . The application of the model to such QCD exotica as multiquark states and glueballs is also mentioned. In addition, an attempt is made to at least partially rationalise the success of the model in terms of more fundamental physics. 248 I. Introduction to the Model Why Potential Models? A. The successes of quarkonium models for the cc and bb systems , based on a non-relativistic potential model with flavour and spin independent con finement and one gluon exchange, at least raise the question of where , as a function of quark mass, such models become useless. �he models I will discuss here are,in response to this question, based on an optimistic extension of the physics expected from QCD for heavy quark sys tems into the light quark domain . We shall see that the models not only provide a useful framework within which to describe the physics of such systems , but moreover that this extension has, at least at some level , been successful. The reasons for this success are only partly understood; a possible beginning for an interpretation is given below. While there are many possible variants, I will concentrate on the l,Z,3) version of such models that I know best It has three main ingredients: confinement , "point-like" constituent quarks, and one gluon exchange . B. How Do Potentials Confine Colour? It is non-trivial to construct a potential which will, for example, 4) confine qqq but not qq. One solution is to assume---b ased on a colour electric flux tube mechanism for confinement---a colour-dependent two body potential between quarks i and j of the form (1) with V(r ) a flavour and spin independent confining potential and with the ij prescription that antiquark potentials follow by the replacement c * _,_ (--t ) . The resulting model allows only colour singlet hadrons to t t = 1) exist , but has the property that wh ile the confinement of qq and qqq is automatic, the existence of more complicated colour singlet bound states (like qqqq) becomes a dynamical issue . The model is also very economical in that it relates the physics of baryons , mesons , and multiquark systems ; this feature will be discussed below. 249 C. What Are the Constituent Quarks? The quarks of the model are not the current quarks of the QCD Lagrangian, but rather a set of "dressed" constituent quarks appropriate to the distance scale of light hadrons , These quarks (which are assumed to be approximately Pointlike) have masses which are (in part by their nature and in . part because they are at this stage also probably repositories of some of our ignorance about the potential, relativistic corrections, etc.) ill-known, but the values m m 0.33 GeV (2) u d m - m MeV (3) d u 6 m 0.55 GeV (4) s m 1. 75 GeV (5) c etc. , are typical . These masses play a crucial role in the model because their differences are the principal origin of flavour symmetry breaking. Note that if we assume that the quarks in a proton are in a Gaussian wave function with 2 the proton 's observed charge radius , then with these masses � 100 MeV so that these light quark models are probably not hopelessly relativistic . � D. Where Are the Gluons? The last principal ingredient of the model is one gluon exchange . The mo st important effects of this type are due to colour hyperfine (magnetic dipole-magr,etic dipole) interactions which in lowest order in " s are given by ij ll --hyp - s.·s.J } l J (6) where 2a s in baryon 3m m a i j (7) 4a s in a meson 3m m i j The first term of ' called the contact term, is , in electromagnetism, l\iyp responsible for the 21 centimetre line of hydrogen; the second (tensor) term is the usual interaction of a magnetic dipole with a magnetic dipole field. The most important characteristics of H are : 1) it is short range in h y §__ 2) it gives a fixed relation ( �) etween the strengths of the contact 3 S.l ·S.J , and tensor terms , 3) it has a strength inversely proportional to the product � 250 of the quark masses, and it violates SU (6) and in particular automatically 4) gives m m and m p > n � > �· One gluon exchange also gives ris·e to spin-orbit interactions from the interaction of a moving quark colour magnetic moment with colour electric fields . As in the case of electromagnetism, this effect tends to be masked by Thomas precession (the relativistic precession of an accelerated spin) , but in this case the suppression of spin-orbit effects is (in the case of qq with m ) by a factor of (1-l;z- f) where the and the "f" come from Thomas mq >> q "'>" precession in the Coulomb-like and confinement potentials, respectively. In the light quark hadrons , the quark wave function overlaps very strongly with the confinement region so that f is very large and spin-orbit effects are 5) suppressed , Finally, one gluon exchange gives rise to a variety of spin independent effects which can normally be absorbed into the unknown confinement potential . Most prominent among these is, of course, the Coulomb-like l/r potential itself . II. Baryons First A. Why Baryons First? We begin our discussion of the model with the qqq baryon sector even though the qq meson sector is simpler to treat theoretically . There are several reasons for doing this; probably the most important of these is that the baryons are much better known experimentally than the corresponding mesons (a consequence of their accessibility as s-channel resonances) . Not only are more resonances known , but their masses and widths are better known and there is a wealth of information on their (signed) decay amplitudes . Specifically, the P-wave mesons remain very poorly known (think of the A and the while 1 E) , all seven expected S=O P-wave baryons are known and have reasonably well measured amplitudes for both electromagnetic and strong channels (typically 5 to 10 amplitudes per resonance). Aside from their being more extensively known, baryons probably have other advantages in practice . For one thing, the quarks in a baryon appear to be somewhat more non-relativistic than those in a meson, making their treatment more reliable. It is also possible that three bodies are sufficient to make the effective potential seen by a quark in a baryon significantly "smoother" than in a meson, thereby making baryons less sensitive to precise knowledge of the potentials. Finally, baryons do not have the isoscalar mixing problem (to be described below) which , while interesting, renders I=O meson data unreliable for spectroscopic studies and leaves very few 251 mesons indeed to compare with a potential mo del. B. What is the Model for Baryons and Its Solution? The application of the model to baryons is quite straightforward in 6 7) 8 9 10) . its simplest form ' . Building on the pre-QCD analyses ' ' (which were on their own terms already very successful) we take the simple Hamiltonian p ,2 H 3 = E ( m E i 2 . (8) i=l + i > + i<J and solve it approximately by �setting and treating � and the anharmonicity U as perturbations . Note that V here yp contains both the "true" confinement potential and pieces of the one gluon exchange potential (so that U may contain l/r terms , linear terms , etc.) ; also note that spin orbit effects have been completely dropped . We have already mentioned that Thomas precession is expected to make spin-orbit effects smaller, but the adequacy of completely neglecting them in baryons is not well understood. Evidence from me sons , where the suppression is more easily studied, 5) does , however, tend to support this approach . The approximate solution of this model Hamiltonian is simple. In the harmonic limit with m = m m and m m' (the most general case required for 1 2 = = 3 u, d, and s quarks in the approximation m = m ), u d p 2 p 2 p A 2 2 H +--+ -- + - k ( p + A (10) 2m 2m;i. 23 ) p where _,. _,. _,. p = l r - r (11) /2 ( l 2 _,. 1 _,. _,. _,. A r r 2r (12) = /6 ( l + 2 3 and mp m (13) m (14) A 2m+m'3mm1 so that the unperturbed spectra with up to two units of excitation are those of Figure I. One interesting -- and important -- feature is that the solutions of 6) the confinement problem maximally violate SU(6) in excited baryons . For + example, the p-A basis = 2 eig�nfunction p! is a 45° mixture of the LP . 12 2 + 1 2 2 [56,2+ ) wave function (p + A ) and the [70,2 ) wave function /2 + + rz (p A ) + + 252 (r S=n,-J I: Jigure the unperturbed solutions to the harmonic confinement problem (with arbitrary scales set to give the same spacing in each sector) . of SU(6) . Clearly, to the extent that this phenomenon is important , an SU(6) (or even SU (3)) analysis of excited baryons will fail. We next take these states and perturb them with the anharmonic term U.