[Paper Review] The Gravity of Dark Vortices: Effective Field Theory for Branes and Strings Carrying Localized Flux

C. P. Burgess, Ross Diener+1 more|arXiv (Cornell University)|Jun 26, 2015

Cosmology and Gravitation Theories31 references5 citations

TL;DR

This paper develops an effective field theory for dark vortices—cosmic strings or branes carrying localized flux that also permeates the bulk—by incorporating off-diagonal kinetic mixing between internal and external U(1) gauge fields. The key result is that gravitational observables like tension, defect angle, and on-vortex curvature depend only on low-energy parameters (tension and flux), not microscopic details, and brane-localized flux does not gravitate but only renormalizes tension independently of magnetic fields.

ABSTRACT

A Nielsen-Olesen vortex usually sits in an environment that expels the flux that is confined to the vortex, so flux is not present both inside and outside. We construct vortices for which this is not true, where the flux carried by the vortex also permeates the `bulk' far from the vortex. The idea is to mix the vortex's internal gauge flux with an external flux using off-diagonal kinetic mixing. Such `dark' vortices could play a phenomenological role in models with both cosmic strings and a dark gauge sector. When coupled to gravity they also provide explicit ultra-violet completions for codimension-two brane-localized flux, which arises in extra-dimensional models when the same flux that stabilizes extra-dimensional size is also localized on space-filling branes situated around the extra dimensions. We derive simple formulae for observables such as defect angle, tension, localized flux and on-vortex curvature when coupled to gravity, and show how all of these are insensitive to much of the microscopic details of the solutions, and are instead largely dictated by low-energy quantities. We derive the required effective description in terms of a world-sheet brane action, and derive the matching conditions for its couplings. We consider the case where the dimensions transverse to the bulk compactify, and determine how the on- and off-vortex curvatures and other bulk features depend on the vortex properties. We find that the brane-localized flux does not gravitate, but just renormalizes the tension in a magnetic-field independent way. The existence of an explicit UV completion puts the effective description of these models on a more precise footing, verifying that brane-localized flux can be consistent with sensible UV physics and resolving some apparent paradoxes that can arise with a naive (but commonly used) delta-function treatment of the brane's localization within the bulk.

Motivation & Objective

• To construct a UV-complete effective field theory for vortices that carry both internal flux and permeate external flux into the bulk via kinetic mixing.
• To resolve inconsistencies in delta-function treatments of brane-localized flux in extra-dimensional models.
• To derive universal formulae for gravitational observables—tension, defect angle, curvature—dependent only on low-energy parameters.
• To clarify how brane-localized flux affects geometry without contributing to gravity, contrary to naive expectations.
• To provide a consistent framework for dark photon models and codimension-two brane-world scenarios with flux stabilization.

Proposed method

• Derive a world-sheet brane action with leading-order terms in tension and flux coupling: $ S_b = -T_b \int \omega + \zeta_b \int \star A + \cdots $.
• Use off-diagonal kinetic mixing to couple the vortex’s internal gauge field to an external U(1) field, enabling flux to extend into the bulk.
• Construct explicit solutions in higher-dimensional gravity with compact transverse dimensions, solving Einstein equations with localized sources.
• Perform a derivative expansion to identify leading-order gravitational responses, isolating tension and flux as dominant parameters.
• Derive matching conditions for brane couplings and compute on- and off-vortex curvatures in warped compactifications.
• Analyze the limit of small warping and derive corrections to metric components using coordinate transformations to angular variables.

Experimental results

Research questions

• RQ1 How does the gravitational back-reaction of a vortex change when it carries both internal flux and external flux permeating the bulk?
• RQ2 Why is the gravitational response of such a vortex locally independent of its flux content, despite flux contributing to energy density?
• RQ3 Can brane-localized flux in codimension-two brane-world models be consistently UV-completed without violating gravity constraints?
• RQ4 How do on-vortex and off-vortex curvatures depend on vortex properties like tension and flux in compactified extra dimensions?
• RQ5 What is the role of kinetic mixing in enabling bulk-permeating flux and stabilizing the geometry around the vortex?

Key findings

• The gravitational response of dark vortices is insensitive to the amount of localized flux; only tension and flux coupling matter, not microscopic details.
• The defect angle and on-vortex curvature are determined solely by the tension $ T_b $ and flux $ \zeta_b $, with no dependence on magnetic field strength.
• Brane-localized flux does not gravitate directly but renormalizes the tension in a magnetic-field-independent way.
• In the small-warping limit, the metric takes a rugby-ball-like form with radius $ r $ and defect parameter $ \alpha $, both dependent on $ T_b $ and $ \zeta_b $.
• The warp factor changes across the bulk by an amount $ \Delta W \propto (T_+ - T_-) $, with corrections scaling as $ \kappa^2 L_{\pm} $, showing sensitivity to tension differences.
• The effective theory resolves paradoxes in delta-function treatments by providing a UV-complete, smooth solution where flux and tension are consistently coupled to gravity.