Casimir energy and radius stabilization in five and six dimensional orbifolds

TL;DR

The paper computes one-loop Casimir energies on five- and six-dimensional orbifolds and studies whether they can stabilize the compactification radius. It finds that massless-field Casimir energy alone is typically repulsive in the motivated models, but brane-localized kinetic terms, massive fields, and tuned brane/cosmological counterterms can produce minima.

Key findings

• For \(S^1/Z_2\), neglecting fixed-point terms, the Casimir energy is one half that of \(S^1\). The massless periodic scalar result is \(V_{+,{\rm scalar}}(\phi)=-\frac{3\zeta(5)}{64\pi^2}\frac{1}{\phi^2 L^4}\), with ratios \(V_{+,{\rm graviton}}=5V_{+,{\rm scalar}}\), \(V_{+,{\rm fermion}}=-4V_{+,{\rm scalar}}\), \(V_{+,{\rm vector}}=3V_{+,{\rm scalar}}\), \(V_{+,{\rm gravitino}}=-8V_{+,{\rm scalar}}\); antiperiodic scalar gives \(V_{-,{\rm scalar}}=-\frac{15}{16}V_{+,{\rm scalar}}\) and antiperiodic fermion \(V_{-,{\rm fermion}}=\frac{15}{4}V_{+,{\rm scalar}}\).
• Massive fields can stabilize the radius if their Casimir contribution is repulsive and dominates at small radii, while the massless contribution is attractive and the bulk cosmological constant is fine-tuned positive; the radius is then typically of order the inverse mass.
• Brane-localized kinetic terms can also stabilize the radion. With one boundary term vanishing, a minimum occurs at a radius of order the nonzero \(c_i\); with both present, a local minimum persists for asymmetry \(c_0/c_L\sim 0.3\), but there is no minimum if \(c_0=c_L\). NDA-sized terms can yield a radius several times the cutoff, and larger terms can yield a larger radius while perturbation theory remains consistent.
• In the 6D toroidal case, for repulsive total Casimir energy the fixed-area minimum is at \(\tau=e^{2\pi i/3}\). Applying the results to the 5D supersymmetric model of ref. [22], the massless-field radion potential is repulsive, \(V_{\rm massless}(\phi)=-1364V_{+,{\rm scalar}}(\phi)=+\frac{1023\zeta(5)}{16}\frac{1}{\phi^2 L^4}\); a stable minimum with vanishing cosmological constant requires positive bulk cosmological constant and negative brane tensions, which appears problematic for supersymmetry.

Methods

• One-loop Casimir energies are computed using \(\zeta\)-function regularization on \(S^1/Z_2\) and \(T^2/Z_k\) orbifolds, for gravitational, massless, and massive fields with periodic or antiperiodic boundary conditions.
• The radius potential is analyzed including bulk cosmological constant, brane-tension counterterms, and brane-localized kinetic/mass terms; normalization conditions \(V(\phi)=V'(\phi)=0\) are imposed to seek minima with vanishing four-dimensional cosmological constant.
• An exact tree-level propagator with brane kinetic terms is derived to check perturbativity, and the general results are applied to the 5D supersymmetric electroweak model of ref. [22] and the 6D composite-Higgs model of ref. [23].

Limitations

• The paper states that it neglects warping/backreaction from brane tensions and bulk cosmological constant counterterms, assumes brane tensions are small enough, and treats only static minima; cosmological evolution is left for future work.
• It states that deciding whether the required stabilization is viable in the supersymmetric 5D case requires a full embedding in 5D supergravity, since the needed bulk counterterm is not supersymmetric in the simplest 5D supergravities.
• The text also notes that the field-theory orbifolds considered cannot at present be derived from known string constructions and are treated only as low-energy effective theories up to a cutoff; additionally, the 5D SUSY model has a second degenerate minimum at \(\phi=\infty\), and for negative bulk cosmological constant the true minimum would be AdS, raising tunneling concerns.