Casimir Energy and Brane Stability

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1. Casimir 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

2. Computing 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

3. Adding 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

4. A 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

5. A 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)

6. In 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

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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

• Casimir physics and vacuum-force engineering
• The metric, warp drives and wormholes