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Fermions and Kaluza-Klein vacuum decay: A toy model
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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.
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0 Valid · 3 Not valid · 6 Not checked · 24 Own results and interpretations
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, the contribution (44) cancels out the second term in Eq. ( 12), which is given by Eq. ( 17).”
§ Fermions and Kaluza-Klein vacuum decay: a toy model > + (n -q)e -ε(n-q) -ne -εn (43)
plainstates a cause- Asserted: passes — The authors put it forward as true.
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A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We find that the terms that diverge as ε → 0 actually cancel out due to the identity”
§ Fermions and Kaluza-Klein vacuum decay: a toy model > + (n -q)e -ε(n-q) -ne -εn (43)
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“So, the Casimir energy of periodic massless fermions equals E = 1 6R .”
§ Fermions and Kaluza-Klein vacuum decay: a toy model > + (n -q)e -ε(n-q) -ne -εn (43)
plainstates an association- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, ln det D bounce 0 = -T 6R + finite .”
§ Fermions and Kaluza-Klein vacuum decay: a toy model > + (n -q)e -ε(n-q) -ne -εn (43)
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This expression shows that there is no UV divergence in ln det D bounce 0 , as expected.”
§ Fermions and Kaluza-Klein vacuum decay: a toy model > + (n -q)e -ε(n-q) -ne -εn (43)
boosted- Asserted: passes — The authors put it forward as true.
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- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
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In the paper
§ Fermions and Kaluza-Klein vacuum decay: a toy model > 2.2.2 Sum over angular spectral numbers
“-ln Γ(n + q)e ε(n+q) -ln Γ(n + 1 -q)e ε(n+1-q) (42”
§ Fermions and Kaluza-Klein vacuum decay: a toy model > 2.2.2 Sum over angular spectral numbers
“ln a 2 n+q-1 Γ -1 (n + q) e -ε(n+q-1) + ln a 2 -(n-1) Γ(n) e -ε(n-1) + ∞ n ′ =1 ln a 2 n ′ -q Γ -1 (n ′ -q + 1) e -ε(n ′ -q)”
Citations
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Bubbles of nothing and supersymmetric compactifications
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Energy of the boundary of spacetime
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Metadata
- Authors
- Rubakov, V. A., & Kuznetsov, M. Y.
- Affiliations
- Lomonosov Moscow State University (RU)
- Moscow Institute of Physics and Technology (RU)
- Corresponding author
- Rubakov, V. A.
- ORCID
- Kuznetsov, M. Yu.
- Venue
- Theoretical and Mathematical Physics
- Published
- 2013-04
- Volume
- 175
- Issue
- 1
- Pages
- 489-498
- ISSN
- 0040-5779 (ISSN-L)
- 1573-9333
- OpenAlex
- W1993743207
- Cited by
- 2
- References
- 31
- Citations by year
- 2023: 1 · 2016: 1
- FWCI
- 0.13
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Theoretical and Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Russian Foundation for Basic Research
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Particle physics theoretical and experimental studies
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Noncommutative and Quantum Gravity Theories
- Keywords
- vacuum decay, fermions, periodic boundary conditions, fermion determinant, (1+1)-dimensional model, compactified extra dimensions, vacuum stability
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