Generalized Virial Identities: Radial Constraints for Solitons, Instantons, and Bounces

TL;DR

The paper develops a continuous family of virial identities for $O(n)$-symmetric field configurations by weighting the radial Hamiltonian with $\rho^\alpha$, extending Derrick's scaling constraint to radially resolved balance between kinetic and potential densities. This $\alpha$-weighted framework decomposes global constraints into region-specific relations, with negative $\alpha$ emphasizing the core and large positive $\alpha$ emphasizing the tail; BPS configurations satisfy all identities due to pointwise kinetic-potential equality. Analytic checks on the Fubini-Lipatov instanton, BPS monopole, and BPST instanton, together with numerical tests on the Coleman bounce and Nielsen–Olesen vortex, demonstrate how $\alpha$-dependent errors diagnose core versus tail inaccuracies and reveal the role of multiple length scales in systems like the electroweak sphaleron and hedgehog Skyrmions. The approach also connects to and extends Pohozaev-type identities, offering robust diagnostic tools for both analytic and numerical studies of solitons, instantons, and related tunneling configurations with rich radial structure. Overall, the work provides a principled, radially resolved set of consistency checks and insights into the distribution of kinetic and potential densities across diverse solitonic systems.

Abstract

Topological solitons and tunneling configurations (monopoles, vortices, Skyrmions, sphalerons, and bounces) are characterized by profile functions that encode their physical properties. Derrick's scaling relation provides a global integral constraint on these profiles, but reveals nothing about the local balance between kinetic and potential contributions in different regions. We derive a continuous family of virial identities for O($n$) symmetric configurations, parameterized by an exponent $α$ that controls the radial weighting: negative $α$ emphasizes the core where topological boundary conditions are imposed, large positive $α$ emphasizes the asymptotic tail, and $α= 1$ recovers the classical Derrick relation. The family provides a systematic decomposition of the global constraint into radially-resolved components, with special $α$ values isolating specific mechanisms. For BPS configurations, where the Bogomolny equations imply pointwise equality between kinetic and potential densities, the virial identity is satisfied for all valid $α$. We verify the formalism analytically for the Fubini-Lipatov instanton, BPS monopole, and BPST instanton. Numerical tests on the Coleman bounce and Nielsen-Olesen vortex illustrate how the $α$-dependence of errors distinguishes core from tail inaccuracies: the vortex shows errors growing at negative $α$ (core), while the bounce shows errors growing at positive $α$ (tail). Applications to the electroweak sphaleron, where the Higgs mass explicitly breaks scale invariance, and the hedgehog Skyrmion illustrate the formalism in systems with multiple competing length scales.

Figures (2)

• Figure 1: Bounce profile $\phi(\rho)$ for the potential \ref{['eq:bounce_potential']} with $\lambda = 1$, $a = 1$, and $\epsilon = 0.1$. The field interpolates from the true vacuum $\phi_{\rm tv} \approx 1.00$ at the origin to the false vacuum $\phi_{\rm fv} \approx -0.95$ at large $\rho$. In the thin-wall regime, the transition occurs at large radius, making the asymptotic approach to the false vacuum particularly important.
• Figure 2: Radial profiles of the gauge field $a(\rho)$ (decreasing from 1) and scalar field $f(\rho)$ (increasing from 0) for the Nielsen-Olesen vortex at coupling $\beta = 0.5$ (Type-I regime). The characteristic scales of the two profiles differ: the gauge field penetration depth exceeds the scalar coherence length for $\beta < 1$. The steepest gradients occur in the core region near $\rho = 0$.