Abstract
We derive a closed-form false vacuum decay rate at one loop in the thin wall limit, where the true and false vacua are nearly degenerate. We obtain the bounce configuration in D dimensions, together with the Euclidean action with a higher order correction, counter-terms and renormalization group running. We extract the functional determinant via the Gel’fand-Yaglom theorem for low and generic orbital multipoles. The negative and zero eigenvalues appear for low multipoles and the translational zeroes are removed. We compute the fluctuations for generic multipoles, multiply and regulate the orbital modes. We find an explicit finite renormalized decay rate in D = 3, 4 and give a closed-form expression for the finite functional determinant in any dimension.
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13 July 2022
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP07(2022)085
29 November 2022
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP11(2022)157
References
J. Langer, Theory of the condensation point, Annals Phys. 41 (1967) 108 [281 (2000) 941].
I.Y. Kobzarev, L.B. Okun and M.B. Voloshin, Bubbles in Metastable Vacuum, Sov. J. Nucl. Phys. 20 (1975) 644 [INSPIRE].
S.R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D 15 (1977) 2929 [Erratum ibid. 16 (1977) 1248] [INSPIRE].
S.R. Coleman, The Uses of Instantons, Subnucl. Ser. 15 (1979) 805 [INSPIRE].
S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge, U.K. (1985), [DOI] [INSPIRE].
C.G. Callan Jr. and S.R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D 16 (1977) 1762 [INSPIRE].
S.R. Coleman and F. De Luccia, Gravitational Effects on and of Vacuum Decay, Phys. Rev. D 21 (1980) 3305 [INSPIRE].
C.L. Hammer, J.E. Shrauner and B. DeFacio, Alternate Derivation of Vacuum Tunneling, Phys. Rev. D 19 (1979) 667 [INSPIRE].
A. Andreassen, D. Farhi, W. Frost and M.D. Schwartz, Direct Approach to Quantum Tunneling, Phys. Rev. Lett. 117 (2016) 231601 [arXiv:1602.01102] [INSPIRE].
A. Andreassen, D. Farhi, W. Frost and M.D. Schwartz, Precision decay rate calculations in quantum field theory, Phys. Rev. D 95 (2017) 085011 [arXiv:1604.06090] [INSPIRE].
S. Weinberg, Mass of the Higgs Boson, Phys. Rev. Lett. 36 (1976) 294 [INSPIRE].
A.D. Linde, Dynamical Symmetry Restoration and Constraints on Masses and Coupling Constants in Gauge Theories, JETP Lett. 23 (1976) 64 [Pisma Zh. Eksp. Teor. Fiz. 23 (1976) 73] [INSPIRE].
P.H. Frampton, Consequences of Vacuum Instability in Quantum Field Theory, Phys. Rev. D 15 (1977) 2922 [INSPIRE].
P.H. Frampton, Vacuum Instability and Higgs Scalar Mass, Phys. Rev. Lett. 37 (1976) 1378 [Erratum ibid. 37 (1976) 1716] [INSPIRE].
G. Isidori, G. Ridolfi and A. Strumia, On the metastability of the standard model vacuum, Nucl. Phys. B 609 (2001) 387 [hep-ph/0104016] [INSPIRE].
A. Andreassen, W. Frost and M.D. Schwartz, Scale Invariant Instantons and the Complete Lifetime of the Standard Model, Phys. Rev. D 97 (2018) 056006 [arXiv:1707.08124] [INSPIRE].
M. Quirós, Finite temperature field theory and phase transitions, hep-ph/9901312 [INSPIRE].
J.I. Kapusta and C. Gale, Finite-temperature field theory: Principles and applications, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2011), [DOI] [INSPIRE].
M. Laine and A. Vuorinen, Basics of Thermal Field Theory, vol. 925, Springer (2016), [DOI] [arXiv:1701.01554] [INSPIRE].
A.D. Linde, Fate of the False Vacuum at Finite Temperature: Theory and Applications, Phys. Lett. B 100 (1981) 37 [INSPIRE].
A.D. Linde, Decay of the False Vacuum at Finite Temperature, Nucl. Phys. B 216 (1983) 421 [Erratum ibid. 223 (1983) 544] [INSPIRE].
I. Affleck, Quantum Statistical Metastability, Phys. Rev. Lett. 46 (1981) 388 [INSPIRE].
D. Croon, O. Gould, P. Schicho, T.V.I. Tenkanen and G. White, Theoretical uncertainties for cosmological first-order phase transitions, JHEP 04 (2021) 055 [arXiv:2009.10080] [INSPIRE].
A. Ekstedt, Higher-order corrections to the bubble-nucleation rate at finite temperature, Eur. Phys. J. C 82 (2022) 173 [arXiv:2104.11804] [INSPIRE].
O. Gould and J. Hirvonen, Effective field theory approach to thermal bubble nucleation, Phys. Rev. D 104 (2021) 096015 [arXiv:2108.04377] [INSPIRE].
J. Löfgren, M.J. Ramsey-Musolf, P. Schicho and T.V.I. Tenkanen, Nucleation at finite temperature: a gauge-invariant, perturbative framework, arXiv:2112.05472 [INSPIRE].
J. Hirvonen, J. Löfgren, M.J. Ramsey-Musolf, P. Schicho and T.V.I. Tenkanen, Computing the gauge-invariant bubble nucleation rate in finite temperature effective field theory, arXiv:2112.08912 [INSPIRE].
E. Witten, Cosmic Separation of Phases, Phys. Rev. D 30 (1984) 272 [INSPIRE].
C.J. Hogan, Gravitational radiation from cosmological phase transitions, Mon. Not. Roy. Astron. Soc. 218 (1986) 629 [INSPIRE].
A. Kosowsky, M.S. Turner and R. Watkins, Gravitational waves from first order cosmological phase transitions, Phys. Rev. Lett. 69 (1992) 2026 [INSPIRE].
A. Kosowsky, M.S. Turner and R. Watkins, Gravitational radiation from colliding vacuum bubbles, Phys. Rev. D 45 (1992) 4514 [INSPIRE].
C. Grojean and G. Servant, Gravitational Waves from Phase Transitions at the Electroweak Scale and Beyond, Phys. Rev. D 75 (2007) 043507 [hep-ph/0607107] [INSPIRE].
M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, Gravitational waves from the sound of a first order phase transition, Phys. Rev. Lett. 112 (2014) 041301 [arXiv:1304.2433] [INSPIRE].
M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, Numerical simulations of acoustically generated gravitational waves at a first order phase transition, Phys. Rev. D 92 (2015) 123009 [arXiv:1504.03291] [INSPIRE].
M. Hindmarsh, S.J. Huber, K. Rummukainen and D.J. Weir, Shape of the acoustic gravitational wave power spectrum from a first order phase transition, Phys. Rev. D 96 (2017) 103520 [Erratum ibid. 101 (2020) 089902] [arXiv:1704.05871] [INSPIRE].
D. Cutting, M. Hindmarsh and D.J. Weir, Gravitational waves from vacuum first-order phase transitions: from the envelope to the lattice, Phys. Rev. D 97 (2018) 123513 [arXiv:1802.05712] [INSPIRE].
D. Cutting, M. Hindmarsh and D.J. Weir, Vorticity, kinetic energy, and suppressed gravitational wave production in strong first order phase transitions, Phys. Rev. Lett. 125 (2020) 021302 [arXiv:1906.00480] [INSPIRE].
LIGO Scientific collaboration, Advanced LIGO, Class. Quant. Grav. 32 (2015) 074001 [arXiv:1411.4547] [INSPIRE].
VIRGO collaboration, Advanced Virgo: a second-generation interferometric gravitational wave detector, Class. Quant. Grav. 32 (2015) 024001 [arXiv:1408.3978] [INSPIRE].
LISA collaboration, Laser Interferometer Space Antenna, arXiv:1702.00786 [INSPIRE].
C. Caprini et al., Science with the space-based interferometer eLISA. II: Gravitational waves from cosmological phase transitions, JCAP 04 (2016) 001 [arXiv:1512.06239] [INSPIRE].
C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update, JCAP 03 (2020) 024 [arXiv:1910.13125] [INSPIRE].
S. Kawamura et al., The Japanese space gravitational wave antenna: DECIGO, Class. Quant. Grav. 28 (2011) 094011 [INSPIRE].
J. Crowder and N.J. Cornish, Beyond LISA: Exploring future gravitational wave missions, Phys. Rev. D 72 (2005) 083005 [gr-qc/0506015] [INSPIRE].
V. Corbin and N.J. Cornish, Detecting the cosmic gravitational wave background with the big bang observer, Class. Quant. Grav. 23 (2006) 2435 [gr-qc/0512039] [INSPIRE].
J.M. Cline, Baryogenesis, hep-ph/0609145 [INSPIRE].
T. Vachaspati, Magnetic fields from cosmological phase transitions, Phys. Lett. B 265 (1991) 258 [INSPIRE].
G. Sigl, A.V. Olinto and K. Jedamzik, Primordial magnetic fields from cosmological first order phase transitions, Phys. Rev. D 55 (1997) 4582 [astro-ph/9610201] [INSPIRE].
A. De Simone, G. Nardini, M. Quirós and A. Riotto, Magnetic Fields at First Order Phase Transition: A Threat to Electroweak Baryogenesis, JCAP 10 (2011) 030 [arXiv:1107.4317] [INSPIRE].
A.G. Tevzadze, L. Kisslinger, A. Brandenburg and T. Kahniashvili, Magnetic Fields from QCD Phase Transitions, Astrophys. J. 759 (2012) 54 [arXiv:1207.0751] [INSPIRE].
J. Ellis, M. Fairbairn, M. Lewicki, V. Vaskonen and A. Wickens, Intergalactic Magnetic Fields from First-Order Phase Transitions, JCAP 09 (2019) 019 [arXiv:1907.04315] [INSPIRE].
T.A. Chowdhury, M. Nemevšek, G. Senjanović and Y. Zhang, Dark Matter as the Trigger of Strong Electroweak Phase Transition, JCAP 02 (2012) 029 [arXiv:1110.5334] [INSPIRE].
S. Fabian, F. Goertz and Y. Jiang, Dark matter and nature of electroweak phase transition with an inert doublet, JCAP 09 (2021) 011 [arXiv:2012.12847] [INSPIRE].
M.J. Baker, J. Kopp and A.J. Long, Filtered Dark Matter at a First Order Phase Transition, Phys. Rev. Lett. 125 (2020) 151102 [arXiv:1912.02830] [INSPIRE].
A. Azatov, M. Vanvlasselaer and W. Yin, Dark Matter production from relativistic bubble walls, JHEP 03 (2021) 288 [arXiv:2101.05721] [INSPIRE].
V. Brdar, A.J. Helmboldt and J. Kubo, Gravitational Waves from First-Order Phase Transitions: LIGO as a Window to Unexplored Seesaw Scales, JCAP 02 (2019) 021 [arXiv:1810.12306] [INSPIRE].
V. Brdar, L. Graf, A.J. Helmboldt and X.-J. Xu, Gravitational Waves as a Probe of Left-Right Symmetry Breaking, JCAP 12 (2019) 027 [arXiv:1909.02018] [INSPIRE].
M.J. Baker, M. Breitbach, J. Kopp and L. Mittnacht, Detailed Calculation of Primordial Black Hole Formation During First-Order Cosmological Phase Transitions, arXiv:2110.00005 [INSPIRE].
G.H. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys. 5 (1964) 1252 [INSPIRE].
S.R. Coleman, V. Glaser and A. Martin, Action Minima Among Solutions to a Class of Euclidean Scalar Field Equations, Commun. Math. Phys. 58 (1978) 211 [INSPIRE].
K. Blum, M. Honda, R. Sato, M. Takimoto and K. Tobioka, O(N) Invariance of the Multi-Field Bounce, JHEP 05 (2017) 109 [Erratum ibid. 06 (2017) 060] [arXiv:1611.04570] [INSPIRE].
F.C. Adams, General solutions for tunneling of scalar fields with quartic potentials, Phys. Rev. D 48 (1993) 2800 [hep-ph/9302321] [INSPIRE].
U. Sarid, Tools for tunneling, Phys. Rev. D 58 (1998) 085017 [hep-ph/9804308] [INSPIRE].
S. Fubini, A New Approach to Conformal Invariant Field Theories, Nuovo Cim. A 34 (1976) 521 [INSPIRE].
L.N. Lipatov, Divergence of the Perturbation Theory Series and the Quasiclassical Theory, Sov. Phys. JETP 45 (1977) 216 [INSPIRE].
F. Loran, Fubini vacua as a classical de Sitter vacua, Mod. Phys. Lett. A 22 (2007) 2217 [hep-th/0612089] [INSPIRE].
K.-M. Lee and E.J. Weinberg, Tunneling Without Barriers, Nucl. Phys. B 267 (1986) 181 [INSPIRE].
A. Ferraz de Camargo, R.C. Shellard and G.C. Marques, Vacuum Decay in a Soluble Model, Phys. Rev. D 29 (1984) 1147 [INSPIRE].
A. Aravind, B.S. DiNunno, D. Lorshbough and S. Paban, Analyzing multifield tunneling with exact bounce solutions, Phys. Rev. D 91 (2015) 025026 [arXiv:1412.3160] [INSPIRE].
K. Dutta, C. Hector, P.M. Vaudrevange and A. Westphal, More Exact Tunneling Solutions in Scalar Field Theory, Phys. Lett. B 708 (2012) 309 [arXiv:1110.2380] [INSPIRE].
V. Guada and M. Nemevšek, Exact one-loop false vacuum decay rate, Phys. Rev. D 102 (2020) 125017 [arXiv:2009.01535] [INSPIRE].
M.J. Duncan and L.G. Jensen, Exact tunneling solutions in scalar field theory, Phys. Lett. B 291 (1992) 109 [INSPIRE].
A. Amariti, Analytic bounces in d dimensions, arXiv:2009.14102 [INSPIRE].
K. Dutta, C. Hector, T. Konstandin, P.M. Vaudrevange and A. Westphal, Validity of the kink approximation to the tunneling action, Phys. Rev. D 86 (2012) 123517 [arXiv:1202.2721] [INSPIRE].
A. Masoumi, K.D. Olum and J.M. Wachter, Approximating tunneling rates in multi-dimensional field spaces, JCAP 10 (2017) 022 [arXiv:1702.00356] [INSPIRE].
G. Pastras, Exact Tunneling Solutions in Minkowski Spacetime and a Candidate for Dark Energy, JHEP 08 (2013) 075 [arXiv:1102.4567] [INSPIRE].
V. Guada, A. Maiezza and M. Nemevšek, Multifield Polygonal Bounces, Phys. Rev. D 99 (2019) 056020 [arXiv:1803.02227] [INSPIRE].
V. Guada, M. Nemevšek and M. Pintar, FindBounce: Package for multi-field bounce actions, Comput. Phys. Commun. 256 (2020) 107480 [arXiv:2002.00881] [INSPIRE].
J.R. Espinosa, A Fresh Look at the Calculation of Tunneling Actions, JCAP 07 (2018) 036 [arXiv:1805.03680] [INSPIRE].
J.R. Espinosa and T. Konstandin, A Fresh Look at the Calculation of Tunneling Actions in Multi-Field Potentials, JCAP 01 (2019) 051 [arXiv:1811.09185] [INSPIRE].
R. Jinno, Machine learning for bounce calculation, arXiv:1805.12153 [INSPIRE].
W.-Y. Ai, B. Garbrecht and C. Tamarit, Functional methods for false vacuum decay in real time, JHEP 12 (2019) 095 [arXiv:1905.04236] [INSPIRE].
M.P. Hertzberg and M. Yamada, Vacuum Decay in Real Time and Imaginary Time Formalisms, Phys. Rev. D 100 (2019) 016011 [arXiv:1904.08565] [INSPIRE].
J. Braden, M.C. Johnson, H.V. Peiris, A. Pontzen and S. Weinfurtner, New Semiclassical Picture of Vacuum Decay, Phys. Rev. Lett. 123 (2019) 031601 [arXiv:1806.06069] [INSPIRE].
C.L. Wainwright, CosmoTransitions: Computing Cosmological Phase Transition Temperatures and Bubble Profiles with Multiple Fields, Comput. Phys. Commun. 183 (2012) 2006 [arXiv:1109.4189] [INSPIRE].
A. Masoumi, K.D. Olum and B. Shlaer, Efficient numerical solution to vacuum decay with many fields, JCAP 01 (2017) 051 [arXiv:1610.06594] [INSPIRE].
P. Athron, C. Balázs, M. Bardsley, A. Fowlie, D. Harries and G. White, BubbleProfiler: finding the field profile and action for cosmological phase transitions, Comput. Phys. Commun. 244 (2019) 448 [arXiv:1901.03714] [INSPIRE].
R. Sato, SimpleBounce: a simple package for the false vacuum decay, Comput. Phys. Commun. 258 (2021) 107566 [arXiv:1908.10868] [INSPIRE].
R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, U.S.A. (1965). Emended by D F Styer, Dover, Mineola, New York, U.S.A. (2010).
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific (2009) [DOI].
R.V. Konoplich, Calculation of Quantum Corrections to Nontrivial Classical Solutions by Means of the Zeta Function, Theor. Math. Phys. 73 (1987) 1286 [INSPIRE].
G. Münster and S. Rotsch, Analytical calculation of the nucleation rate for first order phase transitions beyond the thin wall approximation, Eur. Phys. J. C 12 (2000) 161 [cond-mat/9908246] [INSPIRE].
B. Garbrecht and P. Millington, Green’s function method for handling radiative effects on false vacuum decay, Phys. Rev. D 91 (2015) 105021 [arXiv:1501.07466] [INSPIRE].
I.M. Gelfand and A.M. Yaglom, Integration in functional spaces and it applications in quantum physics, J. Math. Phys. 1 (1960) 48 [INSPIRE].
R.H. Cameron and W.T. Martin, Evaluation of various Wiener integrals by use of certain Sturm-Liouville differential equations, Bull. Am. Math. Soc. 51 (1945) 73.
E.W. Montroll, Markoff chains, Wiener integrals, and quantum theory, Commun. Pure Appl. Math. 5 (1952) 415.
K. Kirsten, Spectral functions in mathematics and physics, AIP Conf. Proc. 484 (1999) 106 [hep-th/0005133] [INSPIRE].
K. Kirsten, Spectral functions in mathematics and physics, Chapman & Hall/CRC Press, Boca Raton, FL, U.S.A. (2002).
G.V. Dunne, Functional determinants in quantum field theory, J. Phys. A 41 (2008) 304006 [arXiv:0711.1178] [INSPIRE].
G.V. Dunne and H. Min, Beyond the thin-wall approximation: Precise numerical computation of prefactors in false vacuum decay, Phys. Rev. D 72 (2005) 125004 [hep-th/0511156] [INSPIRE].
G.V. Dunne and K. Kirsten, Functional determinants for radial operators, J. Phys. A 39 (2006) 11915 [hep-th/0607066] [INSPIRE].
J. Hur and H. Min, A Fast Way to Compute Functional Determinants of Radially Symmetric Partial Differential Operators in General Dimensions, Phys. Rev. D 77 (2008) 125033 [arXiv:0805.0079] [INSPIRE].
M. Endo, T. Moroi, M.M. Nojiri and Y. Shoji, On the Gauge Invariance of the Decay Rate of False Vacuum, Phys. Lett. B 771 (2017) 281 [arXiv:1703.09304] [INSPIRE].
M. Endo, T. Moroi, M.M. Nojiri and Y. Shoji, False Vacuum Decay in Gauge Theory, JHEP 11 (2017) 074 [arXiv:1704.03492] [INSPIRE].
S. Chigusa, T. Moroi and Y. Shoji, Precise Calculation of the Decay Rate of False Vacuum with Multi-Field Bounce, JHEP 11 (2020) 006 [arXiv:2007.14124] [INSPIRE].
G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432 [Erratum ibid. 18 (1978) 2199] [INSPIRE].
S.R. Coleman, Quantum Tunneling and Negative Eigenvalues, Nucl. Phys. B 298 (1988) 178 [INSPIRE].
J. Baacke and G. Lavrelashvili, One loop corrections to the metastable vacuum decay, Phys. Rev. D 69 (2004) 025009 [hep-th/0307202] [INSPIRE].
A. Jevicki, Treatment of Zero Frequency Modes in Perturbation Expansion About Classical Field Configurations, Nucl. Phys. B 117 (1976) 365 [INSPIRE].
H.J. Lee, Negative Modes in Vacuum Decay, Ph.D. thesis, Columbia U., 2014. https://doi.org/10.7916/D84X55Z3 [INSPIRE].
M. Matteini, M. Nemevšek and L. Ubaldi, to appear.
D.R.T. Jones, The Two loop β-function for a G1 × G2 gauge theory, Phys. Rev. D 25 (1982) 581 [INSPIRE].
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Ivanov, A., Matteini, M., Nemevšek, M. et al. Analytic thin wall false vacuum decay rate. J. High Energ. Phys. 2022, 209 (2022). https://doi.org/10.1007/JHEP03(2022)209
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DOI: https://doi.org/10.1007/JHEP03(2022)209