Quantum soliton scattering manifolds

Abstract

We consider the quantum multisoliton scattering problem. For BPS theories one truncates the full field theory to the moduli space, a finite dimensional manifold of energy minimising field configurations, and studies the quantum mechanical problem on this. Non-BPS theories -the generic case -have no such obvious truncation. We define a quantum soliton scattering manifold as a configuration space which satisfies asymptotic completeness and respects the underlying classical dynamics of slow moving solitons. Having done this, we present a new method to construct such manifolds. In the BPS case the dimension of the n-soliton moduli space M n is n multiplied by the dimension of M 1 . We show that this scaling is not necessarily valid for scattering manifolds in non-BPS theories, and argue that it is false for the Skyrme and baby-Skyrme models. In these models, we show that a relative phase difference can generate a relative size difference during a soliton collision. Asymptotically, these are zero and non-zero modes respectively and this new mechanism softens the dichotomy between such modes. Using this discovery, we then show that all previous truncations of the 2-Skyrmion configuration space are unsuitable for the quantum scattering problem as they have the wrong dimension. This gives credence to recent numerical work which suggests that the low-energy configuration space is 14dimensional (rather than 12-dimensional, as previously thought). We suggest some ways to construct a suitable manifold for the 2-Skyrmion problem, and discuss applications of our new definition and construction for general soliton theories.

A consistent two-skyrmion configuration space from instantons

“…This suggests that size fluctuations are low energy excitations and should be included in any two-skyrmion configuration space. An analogous mode can be seen in the baby Skyrme model, where it can be understood in terms of rational map solutions of the nonlinear sigma model [12]. The pairing of perturbations can be understood by considering the symmetry of the torus.…”

“…The 2-skyrmion configuration space should contain degrees of freedom which describe two well separated skyrmions, and hence it should be at least 12-dimensional. Recently, it was shown that two skyrmions with equal initial sizes can change their relative size after scattering [12]. Hence size fluctuations are low energy excitations and should be included in the configuration space.…”

“…Specifically we show that a consistent 16-dimensional 2-skyrmion configuration space can be constructed. This contains the asymptotic region describing well separated skyrmions, which is required for quantum scattering [12]. To solve the quantum two skyrmion problem one then needs to construct this space in detail and solve Schrödinger's equation on it.…”

“…We have given the first construction of a manifold of low energy 2-skyrmions which includes enough degrees of freedom to describe well separated solutions, and their scatterings. This means our space is a Quantum Soliton Scattering Manifold (as defined in [12]) and can be used to describe nucleon-nucleon scattering. To proceed further, one must calculate the potential and metric on our proposed manifold.…”

Baby Skyrmion crystals

“…There is a rather nice way to graphically represent the phase of a baby Skyrmion [34] using a HSV color model, which is almost analogous to the Runge color sphere coloring in the Skyrme model. We begin by plotting the energy density and color it using the stereographic coordinate W , given in (13).…”