[Paper Review] Non-existence of Skyrmion-Skyrmion and Skrymion-anti-Skyrmion static equilibria

G. W. Gibbons, Claude Warnick+1 more|arXiv (Cornell University)|May 14, 2010

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proves that classical static Skyrmion-Skyrmion and Skyrmion-anti-Skyrmion configurations cannot exist in equilibrium due to the sign of the normal stress tensor component across a reflection plane: repulsion between Skyrmions and attraction between Skyrmions and anti-Skyrmions, regardless of symmetry details. The result holds for the massive Skyrme model and extends to multi-Skyrmion and self-gravitating systems.

ABSTRACT

We consider classical static Skyrmion-anti-Skyrmion and Skyrmion-Skyrmion configurations, symmetric with respect to a reflection plane, or symmetric up to a $G$-parity transformation respectively. We show that the stress tensor component completely normal to the reflection plane, and hence its integral over the plane, is negative definite or positive definite respectively. Classical Skyrmions always repel classical Skyrmions and classical Skyrmions always attract classical anti-Skyrmions and thus no static equilibrium, whether stable or unstable, is possible in either case. No other symmetry assumption is made and so our results also apply to multi-Skyrmion configurations. Our results are consistent with existing analyses of Skyrmion forces at large separation, and with numerical results on Skymion-anti-Skyrmion configurations in the literature which admit a different reflection symmetry. They also hold for the massive Skyrme model. We also point out that reflection symmetric self-gravitating Skyrmions or black holes with Skyrmion hair cannot rest in symmetric equilibrium with self-gravitating anti-Skyrmions.

Motivation & Objective

• To determine whether static equilibrium configurations can exist between classical Skyrmions and anti-Skyrmions under reflection or G-parity symmetry.
• To resolve the long-standing question of whether Skyrmions can form bound states or stable configurations with each other or their antiparticles.
• To generalize existing results on Skyrmion forces at large separations to arbitrary static configurations without assuming specific field profiles.
• To extend the analysis to self-gravitating Skyrmions and black holes with Skyrmion hair, assessing their equilibrium possibilities.

Proposed method

• Analysis of the stress-energy tensor component normal to a reflection plane in static, symmetric Skyrmion configurations.
• Use of symmetry constraints—reflection symmetry for Skyrmion-Skyrmion, G-parity symmetry for Skyrmion-anti-Skyrmion—combined with energy-momentum tensor properties.
• Proof that the normal stress component is negative definite for Skyrmion-Skyrmion and positive definite for Skyrmion-anti-Skyrmion systems.
• Integration of the normal stress component over the reflection plane to infer net force direction.
• Application of the result to the massive Skyrme model, confirming robustness beyond the standard model.
• Extension to self-gravitating systems by considering the stress tensor in the context of general relativity and Skyrmion hair on black holes.

Experimental results

Research questions

• RQ1 Can static equilibrium exist between two classical Skyrmions under reflection symmetry?
• RQ2 Can static equilibrium exist between a Skyrmion and an anti-Skyrmion under G-parity symmetry?
• RQ3 What is the sign of the normal stress tensor component across the reflection plane in these configurations, and what does it imply for net force?
• RQ4 Does the result depend on the specific form of the Skyrme Lagrangian or hold generally, including in the massive Skyrme model?
• RQ5 Can self-gravitating Skyrmions or black holes with Skyrmion hair rest in symmetric equilibrium with their anti-particle counterparts?

Key findings

• The normal component of the stress tensor is negative definite for Skyrmion-Skyrmion configurations, implying net repulsion and thus no static equilibrium.
• The normal stress component is positive definite for Skyrmion-anti-Skyrmion configurations, implying net attraction and hence no static equilibrium.
• The non-existence of static equilibria holds without assuming specific field profiles, relying only on symmetry and stress tensor properties.
• The result is robust in the massive Skyrme model, confirming consistency with known long-distance force behavior.
• Self-gravitating Skyrmions or black holes with Skyrmion hair cannot rest in symmetric equilibrium with self-gravitating anti-Skyrmions due to the same stress tensor sign constraints.