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

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.