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Phenomenological Aspects of Nonrelativistic Potential Models

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Phenomenological Aspects of Nonrelativistic Potential Models HEPHY/PUB-527 Q^UWThPh-1989-71 J December 1989 PHENOMENOLOGICAL ASPECTS OF NONRELATIVISTIC POTENTIAL MODELS Wolfgang LUCHA Institut für Hochenergiephysik Österreichische Akademie der Wissenschaften A-1050 Wien, Austria Franz F. SCHOBERL Institut für Theoretische Physik Universität Wien A-1090 Wien, Austria Abstract This review reports on the description of hadrons as bound states of quarks by nonrel- ativistic potential models. It contains a brief sketch of the way in which information on the form of the inter-quark potential may be gained from quantum chromodynamics, proofs of some general theorems related to the potential-model approach, a discussion of the significance of the treatment of bound states consisting of relativistically-moving constituents by the nonreiativistic Schrodinger formalism, as well as a brief survey of the motivations for the various proposed potential models. Finally, it illustrates the application of the developed theoretical framework at a few selected examples. i Contents 1 Introduction 1 2 Basics 2 2.1 Gauge Theories 2 2.2 Quantum Chromodynamics 8 2.3 Asymptotic Freedom 9 3 Quark Forces in Hadrons 12 3.1 The Potential as Nonrelativistic Limit 12 3.2 The Spin Structure of the Quark Interaction 16 3.3 Generalized Breit-Fermi Hamiltonian 18 4 The Prototype 21 4.1 The Funnel Potential 21 4.2 Spin Structure of the Funnel Potential 22 5 Some General Results 25 5.1 Scaling Behaviour of the Schrödinger Equation .... 26 5.2 Linear Regge Trajectories 29 5.3 The Wave Function at the Origin 36 5.4 Some Typical Potentials 39 5.5 The Virial Theorem 41 5.6 The Feynman-Hellmann Theorem 44 5.7 The Dependence of the Meson Mass on the Quark Mass 45 6 Reliability of a Nonrelativistic Description 48 7 Various Potential Models 51 8 Applications of Potential Models 59 8.1 Mass Spectroscopy of Hadrons 60 8.2 Mass Differences between Spin-Singlet and Spin-Triplet States 61 8.3 Electromagnetic Mass Differences 65 ii 8.4 The Electric Polarizability of Hadrons 67 A Feynman Rules for a General Gauge Theory 75 B Some Further Formulae for Spectroscopy 81 C The Richardson Potential in Configuration Space 82 1 1. Introduction Quantum chromodynamics (QCD), the quantum field theory of strong interactions, shows two remarkable features - which it (presumably) shares with all non-abelian gauge theories: asymptotic freedom - the behaviour of the theory approaches in the high-energy limit the one of a non-interacting theory - and colour confinement - all observable states are colour singlets. Unfortunately, confinement is a phenomenon beyond reach of perturbation theory. As a consequence, the binding force between quarks cannot be derived from "first principles", that is, from the underlying quantum field theory, QCD. The way out which comes closest to intuition is employment of nonrelativistic potential models. This report summarizes the present state-of-the-art concerning the potential-model approach to the description of hadrons as bound states of quarks (for some other recent reviews see, for instance, [1,2]). However, we do not attempt to present a complete collection of the results obtained in this field as well as of the corresponding references. Our intention rather is to address also non-experts in potential mod­ els or even experimentalists working in phenomenology. We therefore decided to include also brief sketches of some relevant derivations in­ stead of just quoting the result and to demonstrate the possibilities for applying the formalism at a few selected but illustrative examples. The topics covered in this report may be found from the Table of Contents. Since, in the above spirit, we try to restrict ourselves to the most simple cases, we do not consider baryons or hadron decays nor do we enlarge on semi-relativistic extensions of the nonrelativistic potential-model approach. Finally, we do not touch upon level ordering since there are rumours that a comprehensive review on this subject is in preparation [3]. 2 2. Basics 2.1 Gauge Theories There are essentially two reasons for dealing with (non-abelian) gauge theories. First of all, the most general renormalizable quantum field theory is (equivalent to) a spontaneously broken gauge theory based on a compact Lie group [4,5]. Secondly, all known interactions, strong, electromagnetic as well as weak, may be described by gauge theories. We will treat here only quantum electrodynamics (QED) and quan­ tum chromodynamics (QCD). Consequently, it is sufficient to restrict our discussion to gauge theories involving only Dirac fermions, which means a twofold degeneracy of the fermion mass matrix when for­ mulating the theory in terms of Majorana fermions. Furthermore, in this case there is no need to introduce scalar bosons or to consider spontaneous symmetry breaking. A gauge theory is characterized by the requirement of local sym­ metry with respect to a (non-abelian) Lie group G. A Lie group G is defined by the commutation relations of its generators Ta, a b c [T ,T ]=ifabeT , (2.1) where fai,c is the structure constant tensor of the group. The group invariants for a (possible reducible) representation R are defined in terms of the generators Ta as usual: The quadratic Casimir operator Ci{R) is defined by a ZC2(R,r)Erk := (T RT°R)ik , (2.2) where Ea denotes the projector onto the irreducible representation R„ in the decomposition R = (Bo-Rv The second-order Dynkin index T{R) is defined by a T(R)Sab:=Tr(T RTR) , T0R)=£T(i?„) . (2.3) 3 These invariants are related to each other by the dimension of the group, d(G), and the dimension of the representation R, d(R), ac­ cording to T(ÄMG) = I;C2(Ä,MJW • (2-4) Specified to the adjoint (or regular) representation G, 1 (^c)fcc = -rfabe , (2.5) the above definitions read c C2{G)6ab := {T%T G)ab = facdfbcd = Tr(T£T£) =: T(G)6ab , (2.6) i.e., C2(G)=T(G). For simplicity we introduce in the following for objects Xa, a = 1,2,..., d(G), - like fields transforming according to the adjoint rep­ a resentation (2.5) - the matrix notation X = XaT . Thence J^Ir(XY) = XaYa , (2.7) where T(R) is the Dynkin index defined in (2.3). A gauge transfor­ mation then reads U — exp(—iA), A = A„Ta. The vector tp = {^i} of fermions of the theory under consideration thus transforms according to 1 ^ _> tf = uj) , $->fi = $U- . (2.8) The starting point of our considerations is the Lagrangian of free Dirac spinors, £ = ^(i#-m)^ . (2.9) This Lagrangian is obviously invariant under global symmetry trans­ formations, dpA — 0, if the fermion mass matrix m = {m,*} satisfies U"lmU = tri, i.e., if it commutes with all generators Ta of the gauge group, [Ta, m] = 0 V a. Consequently, according to the second lemma of Schur, the mass matrix has to be proportional to the unit matrix, m oc 1, in every irreducible representation Ra. In contrast to that, the kinetic term irjj^j) is not invariant under local symmetry transforma­ tions, A = A(x), since in this case the derivation of the fermion field generates an additional term, d^ -> d„ip' = Udpip -f {dßU)tl> . (2.10) 4 Nevertheless, the requirement of local gauge invariance may be ful­ a filled by introducing vector gauge fields V(i = V"T with the transfor­ mation behaviour 1 V„ -> VJ = t/(^ + -dti)U- , (2.11) where g denotes the gauge coupling constant. Furthermore, every par­ tial derivative dp has to be replaced by the (gauge) covariant derivative D^^d^- igV» . (2.12) In particular, this covariant derivative reads Dp — dpf>ab - gfabcVf; (2.13) for the adjoint representation (2.5), and D^X^d^X-ig^X) (2.14) when applied to matrices defined above. The transformation behaviour of the covariant derivative D^ip is then identical to the transformation behaviour of the fields t/> themselves, D^-^{D^)' = UD^ . (2.15) This can easily be checked with the help of the relation (d^U-^U + U-%U = 0 (2.16) resulting from U~XU — 1. The above transformation behaviour of the covariant derivative ensures the local gauge invariance of the La- grangian CF = $(iB-m)if) . (2.17) In order to be able to regard the vector fields as genuine dynamical degrees of freedom of the theory, a corresponding kinetic term has to be present in the Lagrangian. Otherwise the gauge bosons would only be auxiliary fields which might be eliminated from the theory by their (then purely algebraic) equation of motion. The kinetic term for vector fields (in the case of non-abelian symmetries sometimes called "Yang-Mills term") is 1 t 5 with the field strength tensor F^ = FJJT = djrv - d^V» - ig[V,t, K) (2.19) or FZ, = dly;-drV; + gfaiev}v; . (2.20) The homogeneous transformation behaviour of the field strength, re­ sulting from (2.11), F^^F^^UF^U'1 , (2.21) gt» aiantees the gauge invariance of the kinetic term Cy of the vector bosons, TriF^F'") -* TriF^F*"") = Tr^F") . (2.22) The covariant derivative D^ and the field strength tensor Fpi/ are connected via the relation [Dft,Dv] = -igFflv . (2.23) A mass term for vector bosons £M = \Ky;Vf (2.24) is forbidden by the requirement of gauge invariance. The gauge fields V£ are therefore (apparently) massless. In summary, the most general gauge invariant Lagrangian for Dirac spinors, but without scalar bosons, reads C = £y + £p = -ü^^iF^Fn + tW-rnW l a = - -F^F^ + ^^gYaT -m)rl> . (2.25) The requirement of local gauge invaiiance implies the existence of (ap­ parently) massless gauge bosons.
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