On the Decay of Soliton Excitations

In field theory the scattering about spatially extended objects, such as solitons, is commonly described by small amplitude fluctuations. Since soliton configurations often break internal symmetries, excitations exist that arise from quantizing the modes that are introduced to restore these symmetries. These modes represent collective distortions and cannot be treated as small amplitude fluctuations. Here we present a method to embrace their contribution to the scattering matrix. In essence this allows us to compute the decay widths of such collective excitations. As an example we consider the Skyrme model for baryons and explain that the method helps to solve the long–standing Yukawa problem in chiral soliton models. 1. Statement of the problem Phase shifts, δ, are essential tools to probe external forces in field theory because – among other applications – they describe the response to background potentials. In turn they play an important role for determining Casimir (or vacuum polarization) energies cf. ref. [1], Evac ∼ ∫ dk 2π ωk d dk δ(k) + Ec.t. (1) of extended objects, such as solitons Φcl. Usually these phase shifts are computed from small amplitude fluctuations about Φcl. Solitons often break symmetries of the fundamental theory. For example, the Skyrme hedgehog soliton is not rotationally invariant. This applies to both, coordinate and flavor rotations. In turn this gives rise to large amplitude fluctuations in these directions. The symmetries are restored by canonically quantizing collective coordinates that parameterize the orientation of the soliton in the corresponding spaces. The so–constructed states correspond to resonances of the soliton and their Yukawa exchanges contribute significantly to the scattering data. Yet, this important contribution is not necessarily captured by the small amplitude fluctuations. Here I will exemplify their incorporation within the Skyrme model. There is a serious problem for computing properties of resonances soliton models. Commonly the coupling of resonances to mesons is described by a Yukawa interaction of the generic structure Γint[ψB′ , ψB, φ] ∼ g ∫ dx ψ̄B′ φψB , (2) On the Decay of Soliton Excitations 2 where B′ is the resonance that might decay into the (ground) state B and meson φ and g is a coupling constant. It is crucial that this interaction is linear in φ! If φ is pseudoscalar, this interaction yields the decay width (residuum of the scattering amplitude) Γ(B′ → Bφ) ∝ g2|~ pφ|, with ~ pφ being the momentum of the outgoing meson. Soliton models, however, are based on action functionals of only meson degrees of freedom, Γ = Γ[Φ]. They contain classical (static) soliton solutions, Φsol, that are identified as baryons whose interaction with the mesons is described by the (small) meson fluctuations about the soliton: Φ = Φsol+φ. By pure definition of the stationary condition, the expansion of Γ[Φ] about Φsol does not have a term that is linear in φ to be interpreted as Yukawa interaction, eq. (2). This puzzle has become famous as the Yukawa problem in soliton models. Hence the resonance properties must be extracted from meson baryon scattering amplitudes. In soliton models two–meson processes acquire contributions from the second order term Γ = 1 2 φ · δ Γ[Φ] δ2Φ ∣∣ Φ=Φsol · φ . (3) This also represents an expansion in N , the number of internal degrees of freedom (color in strong interactions): Γ = O(N) and Γ = O(N0). Terms O(φ3) vanish when N → ∞. Thus Γ contains all large–N information about the contribution of resonances to scattering data. The large–N expansion is systematic but a low order truncation is not necessarily reliable at the physical point and it is very challenging to reliably compute subleading contributions. Presumably resonance exchanges contribute significantly in that regime. To probe the reliability of the computed resonance contributions we transform the above statement into a consistency condition: For N → ∞ any valid computation of hadronic decay widths in soliton models must identically match the result obtained from Γ. Unfortunately, the most prominent baryon resonance, the ∆ isobar, becomes degenerate with the nucleon as N → ∞. It is stable in that limit and its decay is not subject to the just described litmus–test. The situation is more interesting in soliton models for flavor SU(3). In the so–called rigid rotator approach (RRA), that generates baryon states as (flavor) rotational excitations of the soliton, exotic resonances emerge that dwell in the anti–decuplet representation of flavor SU(3)[2]. The most discussed (and disputed) such state is the Θ pentaquark with zero isospin and strangeness S = +1. When N → ∞ the mass difference between anti–decuplet states and the nucleon does not vanish. So the properties of Θ predicted from any model treatment must also be seen in the quantizing of the strangeness degrees of freedom based on the harmonic approximation, Γ. This (seemingly alternative) quantization is called the bound state approach (BSA) for reasons that will become obvious later. The above discussed litmus– test requires that the BSA and RRA give identical results for Θ as N → ∞. This did not seem to be true and it was argued that the prediction of pentaquarks would be a mere artifact of the RRA [3]. We will show that this conclusion is premature. Furthermore the comparison between the BSA and RRA provides an unambiguous computation of pentaquark widths: It differs substantially from approaches [4] that adopted transition On the Decay of Soliton Excitations 3 operators for Θ → KN from the axial current. This presentation is based on ref. [5] which should be consulted for further details.