Casimir Energy and Brane Stability
Richard Obousy · Gerald Cleaver
Abstract and summary · read the original at the source · arXiv non-exclusive distribution licence
In one page
An extra spatial dimension has to be held at a definite size, or the theory containing it predicts nothing: left free it either collapses towards the Planck length or inflates away. Obousy and Cleaver argue that the vacuum can do the holding by itself. Working in the Randall-Sundrum picture, where our universe is one of two branes at the ends of a short fifth dimension, they compute the Casimir energy — the same shape-dependent vacuum energy measured between two mirrors — for each kind of field allowed to roam in the bulk, and add the contributions up as a potential that depends on the size of the dimension. The Standard Model fermions plus the Higgs are not enough, they find: that potential has no minimum, so nothing holds the dimension still. Add one light exotic fermion and a stable minimum appears. Add a scalar coupled to a vector that points along the fifth dimension and the depth of the well, and the radius it sits at, become adjustable by a single number set by that vector’s vacuum value.
Why it matters hereChapter 2 treats the vacuum as a real medium whose energy depends on the boundaries around it, and this paper takes that literally enough to let the vacuum set the size of a dimension. That is the load-bearing step under chapter 4: if the radius of a compact dimension fixes the local vacuum energy, and a field can move that radius, then the same quantity that drives cosmic expansion becomes something engineering could act on.
What it claims
01Casimir energy is an attractive stabilisation mechanism for a compact extra dimension precisely because it is an inherent property of the quantum vacuum and does not have to be added by hand; negative energy is a component generic to realistic stabilisation schemes, and the Casimir energy of fields in the bulk supplies it without new machinery.Section 1, Introduction
Published and peer-reviewed02Computing the one-loop effective potential with zeta-function regularisation, the Schwinger proper-time method and the Jacobi theta identity, the Casimir energies of the three generations of Standard Model fermions together with the Higgs field produce no stable minimum at all, and the result is unchanged across the whole experimentally and theoretically allowed range of Higgs masses; further field content is therefore necessary.Section 3; Section 5.1; Figure 1
Published and peer-reviewed03Adding a single light exotic antiperiodic massive fermion — mass 0.02 in units normalised to the Z-boson mass, motivated by proposals for a light sterile bulk neutrino — does produce a stable minimum: below a critical radius near 0.4 the fifth dimension tends to grow and above it tends to shrink, so the separation settles there. That minimum sits at negative energy density, an anti-de Sitter solution, and a positive contribution from the brane tension can lift the whole potential so the minimum lies above zero.Section 5.2; Figure 2
Published and peer-reviewed04A massless scalar coupled in the bulk to a Lorentz-violating vector field whose vacuum expectation value points along the fifth dimension changes the spacing of the Kaluza-Klein tower and so enhances the Casimir energy by a factor set by one parameter, the ratio of that vector’s vacuum value to the mass scale. With the Standard Model fields present and the scalar periodic, increasing that parameter deepens the minimum and moves the stabilised radius to progressively smaller values — the size of the fifth dimension is set by the value of a field rather than chosen by hand, which turns the usual fine-tuning into the natural minimisation of a potential.Section 4, Equations (4.1)–(4.4); Section 5.4; Figure 5
Published and peer-reviewed05A minimum at positive energy density, a de Sitter solution of the kind our expanding universe requires, needs at least two additional exotic bulk fields: an antiperiodic massive fermion and a massive periodic scalar, taken here at masses 1.1 and 1.8 in normalised units. Their masses can be tuned so that the minimum lies extremely close to zero potential, which gives a route to the accepted value of the cosmological constant, and the configuration also permits vacuum tunnelling out of the well into an eternally inflating extra dimension.Section 5.5; Figure 7; Equation (5.6)
Published and peer-reviewed06In a toy anisotropic cosmology whose entire energy content is higher-dimensional Casimir energy, with the compact dimension already at its minimum, accelerated expansion of the three large dimensions follows a simple inequality between the higher-dimensional energy density and the pressures in the large and the compact directions — for two compact dimensions, the energy density must be at least one thirteenth of three times the large-dimension pressure minus six times the compact-dimension pressure. The authors present this as a link between dark energy, the hierarchy problem and bulk field content under one framework.Section 6, Equations (6.1)–(6.10); Section 7, Discussion
What to watch
Read it · abstract
Abstract
We investigate the role of Casimir energy as a mechanism for brane stability in five-dimensional models with the fifth dimension compactified on an S^1/Z_2 orbifold, which includes the Randall-Sundrum two brane model (RS1). We employ a ζ-function regularization technique utilizing the Schwinger proper time method and the Jacobi theta function identity to calculate the one-loop effective potential. We show that the combination of the Casimir energies of a scalar Higgs field, the three generations of Standard Model fermions and one additional massive non-SM scalar in the bulk produce a non-trivial minimum of the potential. In particular, we consider a scalar field with a coupling in the bulk to a Lorentz violating vector particle localized to the compactified dimension. Such a scalar may provide a natural means of the fine-tuning needed for stabilization of the brane spearation. Lastly, we briefly review the possibility that Casimir energy plays a role in generating the currently observed epoch of cosmological inflation by examining a simple five-dimensional anisotropic metric.
Richard Obousy and Gerald Cleaver, Casimir Energy and Brane Stability, arXiv:0810.1096, version 2 of 10 November 2008, preprints BU-HEPP-08-15 and CASPER-08-04; published in the Journal of Geometry and Physics 61, 577 (2011). The research was funded in part by Baylor University research committee grant 0301533BP.
(Abstract only — see the rights note above. On this site, Obousy’s doctoral thesis developing the same calculation is at /library/stm-583b9f421d, the earlier Obousy and Cleaver proposal that turns this vacuum energy into a drive is at /library/stm-7dfda7640a, their fuller statement of it is at /library/stm-14b3884bc2, the Defense Intelligence Reference Document built on all three is at /library/stm-2a2a21e516, Elizalde on the zeta-function machinery used here is at /library/stm-1c774a8666, and Milton’s survey of what is settled and what is still argued about in Casimir physics is at /library/stm-719d2e0a0b.)
The way in
https://arxiv.org/abs/0810.1096LICENCE. Posted to arXiv as 0810.1096 on 7 October 2008, version 2 of 10 November 2008, preprint numbers BU-HEPP-08-15 and CASPER-08-04, from the Center for Astrophysics, Space Physics and Engineering Research in the Department of Physics at Baylor University, Waco, Texas. The posting carries the arXiv non-exclusive distribution licence, which is not a Creative Commons licence, so the sheet stays abstract-only and no text beyond the abstract is reproduced. The paper was afterwards published as Journal of Geometry and Physics volume 61, pages 577 to 588, 2011, doi 10.1016/j.geomphys.2010.11.006, held by Elsevier. TEXT READ. The complete arXiv version 2 was retrieved and read on 2026-09-08, so the claims below are located to the paper’s own numbered sections, equations and figures. ABSTRACT. The abstract below is the authors’ own. Two renderings are the site’s: the orbifold is written S^1/Z_2 as the paper’s own title page prints it rather than with the backslash the arXiv record carries, because a backslash is markup on this page; and the authors’ spelling of separation in the last-but-one sentence is left exactly as they wrote it.
How to cite it
Richard Obousy, Gerald Cleaver (2008) Casimir Energy and Brane Stability. doi:10.1016/j.geomphys.2010.11.006
Where it sits in the curriculum