Spectroscopy without Quarks: a Skyrme-model Sampler*

A potpourri of Skyrme-model results for the meson-baryon system is surveyed. This talk is devoted to a study of meson-nucleon scattering in skyrmion models of the nucleon. [l”’ We shall focu s o n the characteristic energy-range of the baryon resonances, typically 1.5-2.5 GeV. This is well beyond the point where the chiral Lagrangians are conventionally applied; nor is it known at present how to apply QCD directly in this domain. Thus it is especially interesting to see what insights emerge in this regime from skyrmion physics. The results presented here will only be valid to leading order in l/N,, where Ne is the number of colors of the underlying gauge theory.“’ The object of our investigations will be effective Lagrangians (Skyrme’s included) of the form L, = g Tr (a,Uaj‘Ut) + . *. . 16 (1) The leading term is the usual 2-flavor or 3-flavor nonlinear sigma model, depending on whether 27 E SU(2) or U E SU(3). The dots stand for higher-derivative terms, which are not usually exploited in traditional soft-pion physics. Nevertheless, they are needed to stabilize a soliton, or “skyrmion,” whose topological charge (following Skyrme) is interpreted as baryon number. The standard identification of the pion field in (1) in the baryon-number-Osector of the 2-flavor theory is via: U(z) = exp($Z(z) -3). r (2) Thus the pions can be thought of as “small fluctuations” about the trivial vacuum U(z) 1. It is a straightforward procedure to introduce additional fields into (1) in such a way as to preserve chiral invariance. I” In particular, the traditional approach to studying the coupling of pions to the nucleon isodoublet N is to set &N = 2 Tr (a,U#‘Ut) + N(i7pDp n)N + gADpjim Nf’ypr5N. (3) Here D is the covariant derivative appropriate to the manifold G/H, where G = sU(2)1, x sum -and H = SU(2)i,,qi,. From this Lagrangian, all soft-pion theorems pertaining to the ?rN interaction, such as Weinberg’s calculation of the S-wavescattering lengths,“’ can be derived. * Work supported by the Department of Energy, contract DE AC03 76SF00515. Invited talk presented at the 2nd Conference on the Interactions Between Particle and Nuclear Physics, Lake Louise, Canada, May 23-31, 1986 It is the moral of this talk that the purely mesonic Lagrangian (1) contains at least as much information as does (3)! Not only does (1) properly encompass soft-pion physics, as Schnitzer has shown,“’ but in addition-well beyond the soft-pion regimeit yields surprisingly accurate predictions concerning the spectrum of nucleon and A resonances and the qualitative behavior of the large majority of ?rN and EN partialwave amplitudes. The study of meson-nucleon scattering in skyrmion models involves splitting the Goldstone field ii into two pieces: a spatially-varying c-number piece, i.e., the skyrmion, and a fluctuating piece, which we identify with physical mesons. [‘I The skyrmion will be assumed to be of the hedgehog form: Uo(Z) = exp(iF(r)3. f?). (4 Calculating the T-matrix then reduces to a problem of potential scattering, from which partial-wave phase-shifts can be extracted in the usual manner. In addition, it is necessary to fold in a little group theory, as we now describe. For simplicity, let us focus on the non-strange processes TN + TN , ?rN + 7rA andrA+rA. The quantum numbers needed to describe such processes are: the initial and final pion angular momenta L and L'; the initial and final spin (or isospin) representation of the baryon s and s’, which equal t for nucleons and $ for A’s; and the total pion-baryon isospin and angular momentum I and J. The T-matrix describing such processes can then be shown to be:i9”01 T({LsIJ} + {L’s’I’J’}) = ~,,~~,z,bv~b., (-lf-’ L+1 x &2s + 1)(2s’ + 1) c (2K + 1) K=L-1 {sff;;}{f--:)TKLILm (5) The expressions in curly brackets are 6&symbols. The quantities ‘TKL,L, which are functions of pion energy, are the “reduced amplitudes” of the model, obtainable numerically from a phase-shift analysis about the skyrmion. The Kronecker 6’s express the reassuring fact that I and J are conserved in these models, as they ought to be. Although, in Eq. (5), K plays the part of a dummy index, it actually has an interesting physical interpretation. Specifically, K can be viewed as the vector sum of the pion’s angular momentum and isospin in the unphysical frame in which the pion scatters, not from a nucleon, but rather from an unrotated hedgehog soliton. This frame is “unphysical” in that a nucleon properly corresponds to a rotating hedgehog soliton in the skyrmion approach.Ial Pleasingly, all the model-dependence in (5) arising from the details of the terms indicated by dots in the Lagrangian (1) is subsumed in the reduced amplitudes TKL#L; the 6&symbols, in contrast, follow purely from the assumed hedgehog symmetry of the skyrmion. Equation (5) is thus analogous to the Wigner-Eckart theorem in that a large number of physical matrix elements (the T’s) are expressed in terms of a substantially smaller set of reduced matrix elements (the 7~‘s) weighted by appropriate group-theoretical coefficients. One can carry the analogy further by finding those special linear combinations (analogous to the Gell-Mann-Okubo formula) for which the model-dependent righthand side of (5) cancels out; the net result will be a set of energy-independent linear