Running with rugby balls: bulk renormalization of codimension-2 branes

Abstract

We compute how one-loop bulk effects renormalize both bulk and brane effective interactions for geometries sourced by codimension-two branes. We do so by explicitly integrating out spin-zero, -half and -one particles in 6-dimensional Einstein-Maxwell-Scalar theories compactified to 4 dimensions on a flux-stabilized 2D geometry. (Our methods apply equally well for D dimensions compactified to D − 2 dimensions, although our explicit formulae do not capture all divergences when D > 6.) The renormalization of bulk interactions are independent of the boundary conditions assumed at the brane locations, and reproduce standard heat-kernel calculations. Boundary conditions at any particular brane do affect how bulk loops renormalize this brane's effective action, but not the renormalization of other distant branes. Although we explicitly compute our loops using a rugby ball geometry, because we follow only UV effects our results apply more generally to any geometry containing codimension-two sources with conical singularities. Our results have a variety of uses, including calculating the UV sensitivity of one-loop vacuum energy seen by observers localized on the brane. We show how these one-loop effects combine in a surprising way with bulk back-reaction to give the complete low-energy effective cosmological constant, and comment on the relevance of this calculation to proposed applications of codimension-two 6D models to solutions of the hierarchy and cosmological constant problems.

Distributed SUSY breaking: dark energy, Newton’s law and the LHC

“…In particular, as shown in more detail below, for the most supersymmetric 5 situations we find the typical contribution of a 2 For extra dimensions that are a two-sphere and fields with KK spectrum m 2 = ( + 1)/r 2 we take mKK = 1/r even though the lowest nonzero KK mass would be m1 = √ 2/r. 3 See our companion paper [10] for a take on the long-standing question of what this theory looks like from a four-dimensional perspective. 4 For two spherical extra dimensions of radius r our conventions are that M 2 p = 4πr 2 M 4 g , where Mp := (8πGN ) −1/2 = 2.44 × 10 18 GeV is the rationalized 4D Planck Mass.…”

“…As a result, the most important quantum corrections are those bulk loops that can induce a brane-φ coupling [3,4,9]. Ref.…”

“…Ref. [3,4] shows by explicit calculation how bulk supersymmetry acts to suppress the bulk-loop contributions to the vacuum energy. As is true for many higher-dimensional supergravities, the classical bulk lagrangian can be written in the 'string-frame' form [20][21][22] …”

“…The remarkable part is that this expectation is wrong [3,4], ultimately because it makes the mistaken assumption that supersymmetry must be linearly realized on the branes. To investigate this, imagine adding some brane particles and doing so in a way that is not…”

“…In a nutshell, the supersymmetry-breaking story implies (and explicit calculations bear out -see below and in [3,4]) that for generic brane configurations (assuming no direct brane couplings to φ or to massive multiplets) the low-energy 4D cosmological constant is of order…”