[Paper Review] Force between Kinks with Long-range Tails

N. S. Manton|arXiv (Cornell University)|Oct 1, 2018

Experimental and Theoretical Physics Studies5 references4 citations

TL;DR

This paper investigates the force between kinks with long-range tails in a scalar field theory with an octic potential featuring quartic and quadratic minima. Using an adiabatic ansatz and a modified first-order Bogomolny equation, it derives that the force is repulsive and decays with the fourth power of the separation, confirming a previously estimated scaling law with a precise coefficient.

ABSTRACT

In a scalar field theory that has a symmetric octic potential with a quartic minimum and two quadratic minima, kink and mirror kink solutions have long-range tails. We calculate the force between these kinks when their long-range tails overlap. This is a nonlinear problem, solved using an adiabatic ansatz for the accelerating kinks that leads to a modified, first-order Bogomolny equation. We find the force is repulsive and decays with the fourth power of the kink separation.

Motivation & Objective

• To determine the force between kinks with long-range, 1/x-type tails in a scalar field theory with a non-quadratic minimum.
• To resolve the breakdown of standard force calculation methods (e.g., linear superposition) when tails are long-ranged.
• To establish consistency across multiple approaches—Noether's theorem, energy minimization, and effective equations of motion—for the kink-kink interaction.
• To clarify the definition of kink center in asymmetric, long-tailed profiles to ensure unambiguous separation and force measurement.
• To provide a quantitative coefficient for the 1/r⁴ decay of the force, previously only estimated numerically.

Proposed method

• Employing an adiabatic ansatz for accelerating kinks to model time-dependent dynamics with overlapping long-range tails.
• Deriving a modified first-order Bogomolny equation to describe the accelerating kink profile under the influence of the other kink’s tail.
• Using the Noether formula for momentum change to compute the force via the energy-momentum tensor at a point between the kinks.
• Applying linear superposition of tails in the intermediate region, valid due to the linearity of the static field equation in that regime.
• Estimating the force from the energy difference of a static, two-kink configuration as a function of separation.
• Deriving an effective equation of motion for kink positions, with the force term derived from energy or momentum balance.

Experimental results

Research questions

• RQ1 What is the functional form of the force between two kinks with long-range, 1/x-type tails in a scalar field theory with a non-quadratic minimum?
• RQ2 How does the standard method for computing kink-kink forces fail when tails are long-ranged, and what modifications are required?
• RQ3 Can multiple independent methods—Noether’s theorem, energy minimization, and effective dynamics—yield consistent results for the long-range force?
• RQ4 What is the precise coefficient of the 1/r⁴ decay in the force, and how sensitive is it to the definition of kink center?
• RQ5 Does the effective equation of motion derived from the force remain valid for slowly separating kinks at large distances?

Key findings

• The force between two kinks with long-range tails is repulsive, consistent with the expectation that kinks repel.
• The force decays with the fourth power of the separation, confirming the 1/r⁴ scaling previously estimated by González and Estrad-Sarlabous.
• The numerical coefficient of the 1/r⁴ force is determined through the adiabatic ansatz and Noether momentum method, though with some uncertainty.
• The effective equation of motion for kink separation is derived as m ddot{c} = C e^{-2c} for short-range tails, but for long-range tails, the force scales as 1/c⁴.
• The kink center must be carefully defined—specifically at the point where d²W/dϕ² = 0—for consistent force calculations, as alternative definitions alter the force coefficient.
• The results are robust across multiple methods: Noether’s theorem, energy-based force estimation, and effective dynamics, all yielding consistent 1/r⁴ scaling.