[Paper Review] Some novel considerations about the collective coordinates approximation for the scattering of $ϕ^4$ kinks

C. F. S. Pereira, Gabriel Luchini+2 more|arXiv (Cornell University)|Apr 1, 2020

Nonlinear Photonic Systems4 citations

TL;DR

This paper presents a fully analytical derivation of the effective Lagrangian in the collective coordinates approximation for kink-antikink scattering in the 1+1 dimensional $\phi^4$ model, resolving inconsistencies in prior literature by rigorously computing non-trivial integrals via complex analysis. The key contribution is a finite, well-behaved expression for the kinetic term coefficient $Q(a)$, which approaches $-2$ as $a \to 0$, contrasting with previous divergent results and significantly improving the accuracy of the effective theory.

ABSTRACT

The collective coordinates approximation for the kink/anti-kink scattering in the $1+1$ dimensional $ϕ^4$ model is considered and we discuss how the results found in the current literature on the topic can be improved by giving the analytical expression for the lagrangian in the space of parameters. A comprehensive discussion of the role of the collective coordinates approximation in this particular situation is given for completeness.

Motivation & Objective

• To provide a complete and analytically rigorous derivation of the effective Lagrangian in the collective coordinates approximation for $\phi^4$ kink-antikink scattering.
• To resolve longstanding inconsistencies in the literature regarding the behavior of key integrals, particularly the coefficient $Q(a)$, which previous works claimed diverged as $a \to 0$.
• To improve the accuracy of the collective coordinates method by systematically computing all required integrals using complex contour integration.
• To establish a reliable foundation for studying resonant scattering phenomena in non-integrable field theories through a well-defined effective theory.

Proposed method

• The method employs complex analysis to compute highly non-trivial integrals arising in the collective coordinates Lagrangian, particularly those involving hyperbolic functions of the kink separation parameter $a$.
• For each integral, the real-line integral is embedded into a contour integral over a rectangular path in the complex plane, exploiting periodicity and decay properties of hyperbolic functions.
• The residues at poles located at $z = i\pi/2$ and $z = i\pi/2 + 2a$ are computed to evaluate the contour integral, yielding exact analytical expressions.
• The procedure is applied to key terms such as $Q(a)$ and $C(a)$, which govern the kinetic and coupling terms of the collective coordinates $a$ (center position) and $\xi$ (internal excitation mode).
• The method ensures consistency and avoids approximations or omissions that plagued earlier approaches, particularly the unjustified neglect of divergent-looking terms.
• The final effective Lagrangian is derived in closed form, with all integrals evaluated exactly using residue calculus, providing a self-consistent framework for dynamics.

Experimental results

Research questions

• RQ1 Why do previous collective coordinate approximations for $\phi^4$ kink-antikink scattering yield inconsistent or divergent results for the coefficient $Q(a)$ as $a \to 0$?
• RQ2 Can the full effective Lagrangian for the collective coordinates be derived analytically without numerical or heuristic approximations?
• RQ3 How does the inclusion of the internal excitation mode $\xi$ affect the dynamics, and what is its exact coupling to the center-of-mass coordinate $a$?
• RQ4 What is the true behavior of the kinetic term coefficient $Q(a)$ in the limit of small kink separation ($a \to 0$)?
• RQ5 Can complex analysis techniques be systematically applied to resolve the technical challenges in computing the integrals that define the effective theory?

Key findings

• The coefficient $Q(a)$, which governs the kinetic term for the internal excitation mode $\xi$, is found to be finite and equal to $12a\textrm{csch}(2a) - 12\textrm{csch}(2a)\coth(2a) + 24a\textrm{csch}^3(2a)$, with $\lim_{a\to 0} Q(a) = -2$, contradicting previous claims of divergence.
• The term $C(a)$, responsible for the coupling between the center-of-mass coordinate $a$ and the internal mode $\xi$, is analytically computed as $\pi\sqrt{3/2}\tanh(a)\operatorname{sech}^2(a)$, providing a precise expression for the interaction potential.
• The use of complex contour integration with residue calculus yields exact results, eliminating the need for numerical approximations or heuristic truncations of integrals.
• The analytical derivation confirms that the effective Lagrangian is well-defined and finite for all $a > 0$, resolving a major technical obstacle in the collective coordinates method.
• The improved analytical framework significantly enhances the accuracy of the collective coordinate approximation, bringing it much closer to full numerical simulations of kink-antikink scattering.
• The results validate the importance of including the internal excitation mode in the effective theory, as its proper treatment is essential for capturing resonant scattering behavior.