Fermions and Kaluza-Klein vacuum decay: A toy model

TL;DR

The paper studies a (1+1)-dimensional toy model of Kaluza-Klein vacuum decay with twisted fermions and computes the ratio of fermion determinants in bounce and vacuum backgrounds. Contrary to expectations, the determinant is finite for \(0 \le q < 1/2\), suggesting twisted fermions do not stabilize the \(M^4 \times S^1\) Kaluza-Klein vacuum.

Key findings

• The fermion determinant ratio \(\Lambda(q)\) is finite for arbitrary \(q\) in the studied range \(0 \le q < 1/2\); the authors state this is their main result.
• The vacuum-vacuum contribution is \(\ln \frac{\det D_{\rm vacuum}^q}{\det D_{\rm vacuum}^0} = -\frac{T}{R}(q^2-q)\).
• The bounce-bounce \(a\)-dependent contribution is \((q-q^2)\ln(a/2)\), which cancels the vacuum term using \(\ln(a/2)=T/R\), leaving an infrared-finite result.
• The final finite expression is given by Eq. (50), with the leading analytic part \(\gamma(q-q^2)+\frac{1}{12}(2\gamma-1+\psi(1+q)+\psi(2-q))\) plus a convergent sum.

Methods

• Toy model: replace the five-dimensional Kaluza-Klein bounce by a two-dimensional cigar-shaped Euclidean metric, Eq. (5), and the vacuum by \(R^1 \times S^1\), Eq. (6).
• Massless fermions with twisted periodicity \(\psi(\theta)=e^{-2i\pi q}\psi(\theta+2\pi)\) are studied; conformal flatness maps the bounce determinant to a plane calculation.
• The determinant ratio is split as in Eq. (12); the vacuum Casimir energy is computed by regularized sums, and the bounce determinant is evaluated using the Gelfand-Yaglom formalism and \(\zeta\)-function methods.

Limitations

• The paper explicitly leaves the case \(q=1/2\) for future work because the solutions with \(n=-1\) are not square integrable.
• The analysis is a toy model in two dimensions with massless fermions, not the full five-dimensional Kaluza-Klein theory; the authors present it as capturing relevant features but not a complete proof.
• The calculation uses an IR cutoff \(T\) sent to infinity and a regularization parameter \(\varepsilon\); the text notes subtleties in demonstrating finiteness of the \(q=0\) term, handled in the Appendix.