Abstract
We clarify issues of convexity, gauge-dependence and radiative corrections in relation to tunneling rates. Despite the gauge dependence of the effective action at zero and finite temperature, it is shown that tunneling and nucleation rates remain independent of the choice of gauge-fixing. Taking as a starting point the functional that defines the transition amplitude from a false vacuum onto itself, it is shown that decay rates are exactly determined by a non-convex, false vacuum effective action evaluated at an extremum. The latter can be viewed as a generalized bounce configuration, and gauge-independence follows from the appropriate Nielsen identities. This holds for any election of gauge-fixing that leads to an invertible Faddeev-Popov matrix.
References
R. Jackiw, Functional evaluation of the effective potential, Phys. Rev. D 9 (1974) 1686 [INSPIRE].
L. Dolan and R. Jackiw, Gauge invariant signal for gauge symmetry breaking, Phys. Rev. D 9 (1974) 2904 [INSPIRE].
N.K. Nielsen, On the gauge dependence of spontaneous symmetry breaking in gauge theories, Nucl. Phys. B 101 (1975) 173 [INSPIRE].
R. Fukuda and T. Kugo, Gauge invariance in the effective action and potential, Phys. Rev. D 13 (1976) 3469 [INSPIRE].
B.L. Voronov, P.M. Lavrov and I.V. Tyutin, Canonical transformations and the gauge dependence in general gauge theories (in Russian), Yad. Fiz. 36 (1982) 498 [INSPIRE].
R. Kobes, G. Kunstatter and A. Rebhan, Gauge dependence identities and their application at finite temperature, Nucl. Phys. B 355 (1991) 1 [INSPIRE].
P.M. Lavrov and I.V. Tyutin, On structure of renormalization in gauge theories (in Russian), Yad. Fiz. 34 (1981) 277 [INSPIRE].
P.M. Lavrov and I.V. Tyutin, On generating functional of vertex functions in the Yang-Mills theories (in Russian), Yad. Fiz. 34 (1981) 850 [INSPIRE].
I.J.R. Aitchison and C.M. Fraser, Gauge invariance and the effective potential, Annals Phys. 156 (1984) 1 [INSPIRE].
D. Johnston, Nielsen identities in the ’t Hooft gauge, Nucl. Phys. B 253 (1985) 687 [INSPIRE].
J.R.S. Do Nascimento and D. Bazeia, Gauge invariance of the effective potential, Phys. Rev. D 35 (1987) 2490 [INSPIRE].
C. Contreras and L. Vergara, The Nielsen identities for the generalized R ξ gauge, Phys. Rev. D 55 (1997) 5241 [Erratum ibid. D 56 (1997) 6714] [hep-th/9610109] [INSPIRE].
O.M. Del Cima, D.H.T. Franco and O. Piguet, Gauge independence of the effective potential revisited, Nucl. Phys. B 551 (1999) 813 [hep-th/9902084] [INSPIRE].
L.P. Alexander and A. Pilaftsis, The one-loop effective potential in non-linear gauges, J. Phys. G 36 (2009) 045006 [arXiv:0809.1580] [INSPIRE].
D. Johnston, Coleman-Weinberg, Nielsen and daisies, Phys. Lett. B 186 (1987) 185 [INSPIRE].
D. Metaxas and E.J. Weinberg, Gauge independence of the bubble nucleation rate in theories with radiative symmetry breaking, Phys. Rev. D 53 (1996) 836 [hep-ph/9507381] [INSPIRE].
D. Metaxas, Derivative expansion and gauge independence of the false vacuum decay rate in various gauges, Phys. Rev. D 63 (2001) 085009 [hep-ph/0011015] [INSPIRE].
D. Binosi, J. Papavassiliou and A. Pilaftsis, Displacement operator formalism for renormalization and gauge dependence to all orders, Phys. Rev. D 71 (2005) 085007 [hep-ph/0501259] [INSPIRE].
M. Garny and T. Konstandin, On the gauge dependence of vacuum transitions at finite temperature, JHEP 07 (2012) 189 [arXiv:1205.3392] [INSPIRE].
A. Andreassen, W. Frost and M.D. Schwartz, Consistent use of effective potentials, Phys. Rev. D 91 (2015) 016009 [arXiv:1408.0287] [INSPIRE].
A. Andreassen, W. Frost and M.D. Schwartz, Consistent use of the Standard Model effective potential, Phys. Rev. Lett. 113 (2014) 241801 [arXiv:1408.0292] [INSPIRE].
S.R. Coleman and E.J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7 (1973) 1888 [INSPIRE].
H.H. Patel and M.J. Ramsey-Musolf, Baryon washout, electroweak phase transition and perturbation theory, JHEP 07 (2011) 029 [arXiv:1101.4665] [INSPIRE].
L. Di Luzio and L. Mihaila, On the gauge dependence of the Standard Model vacuum instability scale, JHEP 06 (2014) 079 [arXiv:1404.7450] [INSPIRE].
A.V. Bednyakov, B.A. Kniehl, A.F. Pikelner and O.L. Veretin, Stability of the electroweak vacuum: gauge independence and advanced precision, Phys. Rev. Lett. 115 (2015) 201802 [arXiv:1507.08833] [INSPIRE].
Z. Lalak, M. Lewicki and P. Olszewski, Hints of BSM physics in the SM effective potential, PoS(CORFU2014)106 [arXiv:1505.05505] [INSPIRE].
J.R. Espinosa et al., The cosmological Higgstory of the vacuum instability, JHEP 09 (2015) 174 [arXiv:1505.04825] [INSPIRE].
D.E. Morrissey and M.J. Ramsey-Musolf, Electroweak baryogenesis, New J. Phys. 14 (2012) 125003 [arXiv:1206.2942] [INSPIRE].
J. Baacke and K. Heitmann, Gauge invariance of the one loop effective action of the Higgs field in the SU(2) Higgs model, Phys. Rev. D 60 (1999) 105037 [hep-th/9905201] [INSPIRE].
N.K. Nielsen, Removing the gauge parameter dependence of the effective potential by a field redefinition, Phys. Rev. D 90 (2014) 036008 [arXiv:1406.0788] [INSPIRE].
L. Delle Rose, C. Marzo and A. Urbano, On the fate of the Standard Model at finite temperature, JHEP 05 (2016) 050 [arXiv:1507.06912] [INSPIRE].
P.H. Frampton, Vacuum instability and Higgs scalar mass, Phys. Rev. Lett. 37 (1976) 1378 [Erratum ibid. 37 (1976) 1716] [INSPIRE].
C.M. Bender, F. Cooper, B. Freedman and R.W. Haymaker, Tunneling and the low momentum expansion of the effective action, Nucl. Phys. B 256 (1985) 653 [INSPIRE].
E.J. Weinberg, Vacuum decay in theories with symmetry breaking by radiative corrections, Phys. Rev. D 47 (1993) 4614 [hep-ph/9211314] [INSPIRE].
B. Garbrecht and P. Millington, Constraining the effective action by a method of external sources, Nucl. Phys. B 906 (2016) 105 [arXiv:1509.07847] [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].
B. Garbrecht and P. Millington, Self-consistent solitons for vacuum decay in radiatively generated potentials, Phys. Rev. D 92 (2015) 125022 [arXiv:1509.08480] [INSPIRE].
J. Iliopoulos, C. Itzykson and A. Martin, Functional methods and perturbation theory, Rev. Mod. Phys. 47 (1975) 165 [INSPIRE].
E.J. Weinberg and A.-Q. Wu, Understanding complex perturbative effective potentials, Phys. Rev. D 36 (1987) 2474 [INSPIRE].
L. O’Raifeartaigh, A. Wipf and H. Yoneyama, The constraint effective potential, Nucl. Phys. B 271 (1986) 653 [INSPIRE].
R. Fukuda and E. Kyriakopoulos, Derivation of the effective potential, Nucl. Phys. B 85 (1975) 354 [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].
B.S. DeWitt, Quantum theory of gravity. 2. The manifestly covariant theory, Phys. Rev. 162 (1967) 1195 [INSPIRE].
Y. Fujimoto, L. O’Raifeartaigh and G. Parravicini, Effective potential for nonconvex potentials, Nucl. Phys. B 212 (1983) 268 [INSPIRE].
C.M. Bender and F. Cooper, Failure of the naive loop expansion for the effective potential in ϕ 4 field theory when there is ‘broken symmetry’, Nucl. Phys. B 224 (1983) 403 [INSPIRE].
F. Cooper and B. Freedman, Renormalizing the effective potential for spontaneously broken gϕ 4 field theory, Nucl. Phys. B 239 (1984) 459 [INSPIRE].
K. Tabata and I. Umemura, Convexity of the effective potential, Prog. Theor. Phys. 74 (1985) 1360 [INSPIRE].
M. Hindmarsh and D. Johnston, Convexity of the effective potential, J. Phys. A 19 (1986) 141 [INSPIRE].
J. Alexandre, Spontaneous symmetry breaking and linear effective potentials, Phys. Rev. D 86 (2012) 025028 [arXiv:1205.1160] [INSPIRE].
J. Alexandre and A. Tsapalis, Maxwell construction for scalar field theories with spontaneous symmetry breaking, Phys. Rev. D 87 (2013) 025028 [arXiv:1211.0921] [INSPIRE].
G. ’t Hooft, The background field method in gauge field theories, (1975) [INSPIRE].
D.G. Boulware, Gauge dependence of the effective action, Phys. Rev. D 23 (1981) 389 [INSPIRE].
L.F. Abbott, The background field method beyond one loop, Nucl. Phys. B 185 (1981) 189 [INSPIRE].
R.W. Haymaker and J. Perez-Mercader, Convexity of the effective potential, Phys. Rev. D 27 (1983) 1948 [INSPIRE].
G. Degrassi et al., Higgs mass and vacuum stability in the Standard Model at NNLO, JHEP 08 (2012) 098 [arXiv:1205.6497] [INSPIRE].
M.B. Einhorn and D.R.T. Jones, The effective potential, the renormalisation group and vacuum stability, JHEP 04 (2007) 051 [hep-ph/0702295] [INSPIRE].
S.R. Coleman, The fate of the false vacuum. 1. Semiclassical theory, Phys. Rev. D 15 (1977) 2929 [Erratum ibid. D 16 (1977) 1248] [INSPIRE].
L.H. Chan, Effective action expansion in perturbation theory, Phys. Rev. Lett. 54 (1985) 1222 [Erratum ibid. 56 (1986) 404] [INSPIRE].
O. Cheyette, Derivative expansion of the effective action, Phys. Rev. Lett. 55 (1985) 2394 [INSPIRE].
R.V. Konoplich, Calculation of quantum corrections to nontrivial classical solutions by means of the zeta function, Theor. Math. Phys. 73 (1987) 1286 [Teor. Mat. Fiz. 73 (1987) 379] [INSPIRE].
C. Scrucca, Advanced quantum field theory, http://itp.epfl.ch/page-60691-en.html.
Z. Lalak, M. Lewicki and P. Olszewski, Gauge fixing and renormalisation scale independence of tunneling rate in Abelian Higgs model and in the Standard Model, arXiv:1605.06713 [INSPIRE].
A. Andreassen, D. Farhi, W. Frost and M.D. Schwartz, A direct approach to quantum tunneling, arXiv:1602.01102 [INSPIRE].
M. Quirós, Finite temperature field theory and phase transitions, in High energy physics and cosmology. Proceedings, Summer School, Trieste Italy June 29-July 17 1998, pg. 187 [hep-ph/9901312] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1510.07613
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Plascencia, A.D., Tamarit, C. Convexity, gauge-dependence and tunneling rates. J. High Energ. Phys. 2016, 99 (2016). https://doi.org/10.1007/JHEP10(2016)099
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2016)099
Keywords
- Gauge Symmetry
- Nonperturbative Effects
- Spontaneous Symmetry Breaking