Abstract
We revisit the perturbative expansion at high temperature and investigate its convergence by inspecting the renormalisation scale dependence of the effective potential. Although at zero temperature the renormalisation group improved effective potential is scale independent at one-loop, we show how this breaks down at high temperature, due to the misalignment of loop and coupling expansions. Following this, we show how one can recover renormalisation scale independence at high temperature, and that it requires computations at two-loop order. We demonstrate how this resolves some of the huge theoretical uncertainties in the gravitational wave signal of first-order phase transitions, though uncertainties remain stemming from the computation of the bubble nucleation rate.
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References
LIGO Scientific, Virgo collaboration, Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
C. Caprini and D.G. Figueroa, Cosmological backgrounds of gravitational waves, Class. Quant. Grav. 35 (2018) 163001 [arXiv:1801.04268] [INSPIRE].
C. Caprini et al., Detecting gravitational waves from cosmological phase transitions with LISA: an update, JCAP 03 (2020) 024 [arXiv:1910.13125] [INSPIRE].
M.B. Hindmarsh, M. Lüben, J. Lumma and M. Pauly, Phase transitions in the early universe, SciPost Phys. Lect. Notes 24 (2021) 1 [arXiv:2008.09136] [INSPIRE].
LISA collaboration, Laser Interferometer Space Antenna, arXiv:1702.00786 [INSPIRE].
S. Kawamura et al., The Japanese space gravitational wave antenna: DECIGO, Class. Quant. Grav. 28 (2011) 094011 [INSPIRE].
G.M. Harry, P. Fritschel, D.A. Shaddock, W. Folkner and E.S. Phinney, Laser interferometry for the big bang observer, Class. Quant. Grav. 23 (2006) 4887 [Erratum ibid. 23 (2006) 7361] [INSPIRE].
W.-H. Ruan, Z.-K. Guo, R.-G. Cai and Y.-Z. Zhang, Taiji program: gravitational-wave sources, Int. J. Mod. Phys. A 35 (2020) 2050075 [arXiv:1807.09495] [INSPIRE].
NANOGrav collaboration, The NANOGrav 12.5 yr data set: search for an isotropic stochastic gravitational-wave background, Astrophys. J. Lett. 905 (2020) L34 [arXiv:2009.04496] [INSPIRE].
A. Addazi, Y.-F. Cai, Q. Gan, A. Marciano and K. Zeng, NANOGrav results and dark first order phase transitions, arXiv:2009.10327 [INSPIRE].
Y. Nakai, M. Suzuki, F. Takahashi and M. Yamada, Gravitational waves and dark radiation from dark phase transition: connecting NANOGrav pulsar timing data and Hubble tension, Phys. Lett. B 816 (2021) 136238 [arXiv:2009.09754] [INSPIRE].
H.-H. Li, G. Ye and Y.-S. Piao, Is the NANOGrav signal a hint of dS decay during inflation?, Phys. Lett. B 816 (2021) 136211 [arXiv:2009.14663] [INSPIRE].
W. Ratzinger and P. Schwaller, Whispers from the dark side: confronting light new physics with NANOGrav data, SciPost Phys. 10 (2021) 047 [arXiv:2009.11875] [INSPIRE].
CMS collaboration, Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
ATLAS collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
D.E. Morrissey and M.J. Ramsey-Musolf, Electroweak baryogenesis, New J. Phys. 14 (2012) 125003 [arXiv:1206.2942] [INSPIRE].
M.J. Ramsey-Musolf, The electroweak phase transition: a collider target, JHEP 09 (2020) 179 [arXiv:1912.07189] [INSPIRE].
J. Kozaczuk, M.J. Ramsey-Musolf and J. Shelton, Exotic Higgs boson decays and the electroweak phase transition, Phys. Rev. D 101 (2020) 115035 [arXiv:1911.10210] [INSPIRE].
M. Quirós, Finite temperature field theory and phase transitions, talk given at the ICTP Summer School in High-Energy Physics and Cosmology, Trieste, Italy (1999) [hep-ph/9901312] [INSPIRE].
J.M. Cline, Baryogenesis, talk given at the Les Houches Summer School — Session 86: Particle Physics and Cosmology: The Fabric of Spacetime, July 31–August 25, Les Houches, France (2006) [hep-ph/0609145] [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].
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].
P. Basler and M. Mühlleitner, BSMPT (Beyond the Standard Model Phase Transitions): a tool for the electroweak phase transition in extended Higgs sectors, Comput. Phys. Commun. 237 (2019) 62 [arXiv:1803.02846] [INSPIRE].
P. Athron, C. Balázs, A. Fowlie and Y. Zhang, PhaseTracer: tracing cosmological phases and calculating transition properties, Eur. Phys. J. C 80 (2020) 567 [arXiv:2003.02859] [INSPIRE].
S.J. Huber and M.G. Schmidt, Electroweak baryogenesis: concrete in a SUSY model with a gauge singlet, Nucl. Phys. B 606 (2001) 183 [hep-ph/0003122] [INSPIRE].
C. Grojean, G. Servant and J.D. Wells, First-order electroweak phase transition in the standard model with a low cutoff, Phys. Rev. D 71 (2005) 036001 [hep-ph/0407019] [INSPIRE].
S.W. Ham, Y.S. Jeong and S.K. Oh, Electroweak phase transition in an extension of the standard model with a real Higgs singlet, J. Phys. G 31 (2005) 857 [hep-ph/0411352] [INSPIRE].
D. Bödeker, L. Fromme, S.J. Huber and M. Seniuch, The baryon asymmetry in the standard model with a low cut-off, JHEP 02 (2005) 026 [hep-ph/0412366] [INSPIRE].
L. Fromme, S.J. Huber and M. Seniuch, Baryogenesis in the two-Higgs doublet model, JHEP 11 (2006) 038 [hep-ph/0605242] [INSPIRE].
C. Delaunay, C. Grojean and J.D. Wells, Dynamics of non-renormalizable electroweak symmetry breaking, JHEP 04 (2008) 029 [arXiv:0711.2511] [INSPIRE].
J.R. Espinosa and M. Quirós, Novel effects in electroweak breaking from a hidden sector, Phys. Rev. D 76 (2007) 076004 [hep-ph/0701145] [INSPIRE].
S. Profumo, M.J. Ramsey-Musolf and G. Shaughnessy, Singlet Higgs phenomenology and the electroweak phase transition, JHEP 08 (2007) 010 [arXiv:0705.2425] [INSPIRE].
A. Noble and M. Perelstein, Higgs self-coupling as a probe of electroweak phase transition, Phys. Rev. D 78 (2008) 063518 [arXiv:0711.3018] [INSPIRE].
J.R. Espinosa, T. Konstandin, J.M. No and M. Quirós, Some cosmological implications of hidden sectors, Phys. Rev. D 78 (2008) 123528 [arXiv:0809.3215] [INSPIRE].
K. Funakubo and E. Senaha, Electroweak phase transition, critical bubbles and sphaleron decoupling condition in the MSSM, Phys. Rev. D 79 (2009) 115024 [arXiv:0905.2022] [INSPIRE].
J.M. Cline, G. Laporte, H. Yamashita and S. Kraml, Electroweak phase transition and LHC signatures in the singlet Majoron model, JHEP 07 (2009) 040 [arXiv:0905.2559] [INSPIRE].
J. Kehayias and S. Profumo, Semi-analytic calculation of the gravitational wave signal from the electroweak phase transition for general quartic scalar effective potentials, JCAP 03 (2010) 003 [arXiv:0911.0687] [INSPIRE].
J.R. Espinosa, T. Konstandin, J.M. No and G. Servant, Energy budget of cosmological first-order phase transitions, JCAP 06 (2010) 028 [arXiv:1004.4187] [INSPIRE].
J.R. Espinosa, T. Konstandin and F. Riva, Strong electroweak phase transitions in the standard model with a singlet, Nucl. Phys. B 854 (2012) 592 [arXiv:1107.5441] [INSPIRE].
G. Gil, P. Chankowski and M. Krawczyk, Inert dark matter and strong electroweak phase transition, Phys. Lett. B 717 (2012) 396 [arXiv:1207.0084] [INSPIRE].
D.J.H. Chung, A.J. Long and L.-T. Wang, 125 GeV Higgs boson and electroweak phase transition model classes, Phys. Rev. D 87 (2013) 023509 [arXiv:1209.1819] [INSPIRE].
L. Leitao, A. Megevand and A.D. Sanchez, Gravitational waves from the electroweak phase transition, JCAP 10 (2012) 024 [arXiv:1205.3070] [INSPIRE].
G.C. Dorsch, S.J. Huber and J.M. No, A strong electroweak phase transition in the 2HDM after LHC8, JHEP 10 (2013) 029 [arXiv:1305.6610] [INSPIRE].
S. Profumo, M.J. Ramsey-Musolf, C.L. Wainwright and P. Winslow, Singlet-catalyzed electroweak phase transitions and precision Higgs boson studies, Phys. Rev. D 91 (2015) 035018 [arXiv:1407.5342] [INSPIRE].
D. Curtin, P. Meade and C.-T. Yu, Testing electroweak baryogenesis with future colliders, JHEP 11 (2014) 127 [arXiv:1409.0005] [INSPIRE].
M. Jiang, L. Bian, W. Huang and J. Shu, Impact of a complex singlet: electroweak baryogenesis and dark matter, Phys. Rev. D 93 (2016) 065032 [arXiv:1502.07574] [INSPIRE].
N. Blinov, J. Kozaczuk, D.E. Morrissey and C. Tamarit, Electroweak baryogenesis from exotic electroweak symmetry breaking, Phys. Rev. D 92 (2015) 035012 [arXiv:1504.05195] [INSPIRE].
J. Kozaczuk, Bubble expansion and the viability of singlet-driven electroweak baryogenesis, JHEP 10 (2015) 135 [arXiv:1506.04741] [INSPIRE].
V. Vaskonen, Electroweak baryogenesis and gravitational waves from a real scalar singlet, Phys. Rev. D 95 (2017) 123515 [arXiv:1611.02073] [INSPIRE].
P. Basler, M. Krause, M. Muhlleitner, J. Wittbrodt and A. Wlotzka, Strong first order electroweak phase transition in the CP-conserving 2HDM revisited, JHEP 02 (2017) 121 [arXiv:1612.04086] [INSPIRE].
A. Beniwal, M. Lewicki, J.D. Wells, M. White and A.G. Williams, Gravitational wave, collider and dark matter signals from a scalar singlet electroweak baryogenesis, JHEP 08 (2017) 108 [arXiv:1702.06124] [INSPIRE].
C.-W. Chiang, M.J. Ramsey-Musolf and E. Senaha, Standard model with a complex scalar singlet: cosmological implications and theoretical considerations, Phys. Rev. D 97 (2018) 015005 [arXiv:1707.09960] [INSPIRE].
P. Basler, M. Mühlleitner and J. Wittbrodt, The CP-violating 2HDM in light of a strong first order electroweak phase transition and implications for Higgs pair production, JHEP 03 (2018) 061 [arXiv:1711.04097] [INSPIRE].
M. Chala, C. Krause and G. Nardini, Signals of the electroweak phase transition at colliders and gravitational wave observatories, JHEP 07 (2018) 062 [arXiv:1802.02168] [INSPIRE].
P.S.B. Dev, F. Ferrer, Y. Zhang and Y. Zhang, Gravitational waves from first-order phase transition in a simple axion-like particle model, JCAP 11 (2019) 006 [arXiv:1905.00891] [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. Papaefstathiou and G. White, The electro-weak phase transition at colliders: confronting theoretical uncertainties and complementary channels, JHEP 05 (2021) 099 [arXiv:2010.00597] [INSPIRE].
M. Carena, Z. Liu and Y. Wang, Electroweak phase transition with spontaneous Z2-breaking, JHEP 08 (2020) 107 [arXiv:1911.10206] [INSPIRE].
J.R. Espinosa and M. Quirós, Improved metastability bounds on the standard model Higgs mass, Phys. Lett. B 353 (1995) 257 [hep-ph/9504241] [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].
S.J. Huber, T. Konstandin, G. Nardini and I. Rues, Detectable gravitational waves from very strong phase transitions in the general NMSSM, JCAP 03 (2016) 036 [arXiv:1512.06357] [INSPIRE].
R.-G. Cai, M. Sasaki and S.-J. Wang, The gravitational waves from the first-order phase transition with a dimension-six operator, JCAP 08 (2017) 004 [arXiv:1707.03001] [INSPIRE].
A. Braconi, M.-C. Chen, G. Gaswint and G. Gaswint, Revisiting electroweak phase transition with varying Yukawa coupling constants, Phys. Rev. D 100 (2019) 015032 [arXiv:1810.02522] [INSPIRE].
C.-W. Chiang and B.-Q. Lu, First-order electroweak phase transition in a complex singlet model with ℤ3 symmetry, JHEP 07 (2020) 082 [arXiv:1912.12634] [INSPIRE].
M. Carena, C. Krause, Z. Liu and Y. Wang, A new approach to electroweak symmetry non-restoration, arXiv:2104.00638 [INSPIRE].
L. Dolan and R. Jackiw, Symmetry behavior at finite temperature, Phys. Rev. D 9 (1974) 3320 [INSPIRE].
J.I. Kapusta, Quantum chromodynamics at high temperature, Nucl. Phys. B 148 (1979) 461 [INSPIRE].
R.R. Parwani, Resummation in a hot scalar field theory, Phys. Rev. D 45 (1992) 4695 [Erratum ibid. 48 (1993) 5965] [hep-ph/9204216] [INSPIRE].
P.B. Arnold and O. Espinosa, The effective potential and first order phase transitions: Beyond leading-order, Phys. Rev. D 47 (1993) 3546 [Erratum ibid. 50 (1994) 6662] [hep-ph/9212235] [INSPIRE].
Z. Fodor and A. Hebecker, Finite temperature effective potential to order g4, λ2 and the electroweak phase transition, Nucl. Phys. B 432 (1994) 127 [hep-ph/9403219] [INSPIRE].
W. Buchmüller, Z. Fodor and A. Hebecker, Thermodynamics of the electroweak phase transition, Nucl. Phys. B 447 (1995) 317 [hep-ph/9502321] [INSPIRE].
K. Farakos, K. Kajantie, K. Rummukainen and M.E. Shaposhnikov, 3D physics and the electroweak phase transition: perturbation theory, Nucl. Phys. B 425 (1994) 67 [hep-ph/9404201] [INSPIRE].
K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, Generic rules for high temperature dimensional reduction and their application to the standard model, Nucl. Phys. B 458 (1996) 90 [hep-ph/9508379] [INSPIRE].
E. Braaten and A. Nieto, Effective field theory approach to high temperature thermodynamics, Phys. Rev. D 51 (1995) 6990 [hep-ph/9501375] [INSPIRE].
K. Farakos, K. Kajantie, K. Rummukainen and M.E. Shaposhnikov, 3D physics and the electroweak phase transition: a framework for lattice Monte Carlo analysis, Nucl. Phys. B 442 (1995) 317 [hep-lat/9412091] [INSPIRE].
J.O. Andersen and M. Strickland, Resummation in hot field theories, Annals Phys. 317 (2005) 281 [hep-ph/0404164] [INSPIRE].
M. Laine and A. Vuorinen, Basics of thermal field theory, vol. 925, Springer (2016), 10.1007/978-3-319-31933-9 [arXiv:1701.01554] [INSPIRE].
J. Ghiglieri, A. Kurkela, M. Strickland and A. Vuorinen, Perturbative thermal QCD: formalism and applications, Phys. Rept. 880 (2020) 1 [arXiv:2002.10188] [INSPIRE].
P.M. Schicho, T.V.I. Tenkanen and J. Österman, Robust approach to thermal resummation: Standard Model meets a singlet, arXiv:2102.11145 [INSPIRE].
K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, The electroweak phase transition: a nonperturbative analysis, Nucl. Phys. B 466 (1996) 189 [hep-lat/9510020] [INSPIRE].
K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, Is there a hot electroweak phase transition at m(H) larger or equal to m(W)?, Phys. Rev. Lett. 77 (1996) 2887 [hep-ph/9605288] [INSPIRE].
K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, A nonperturbative analysis of the finite T phase transition in SU(2) × U(1) electroweak theory, Nucl. Phys. B 493 (1997) 413 [hep-lat/9612006] [INSPIRE].
J.O. Andersen, Dimensional reduction of the two Higgs doublet model at high temperature, Eur. Phys. J. C 11 (1999) 563 [hep-ph/9804280] [INSPIRE].
M. Losada, The two loop finite temperature effective potential of the MSSM and baryogenesis, Nucl. Phys. B 537 (1999) 3 [hep-ph/9806519] [INSPIRE].
M. Losada, Mixing effects in the finite temperature effective potential of the MSSM with a light stop, Nucl. Phys. B 569 (2000) 125 [hep-ph/9905441] [INSPIRE].
M. Laine and M. Losada, Two loop dimensional reduction and effective potential without temperature expansions, Nucl. Phys. B 582 (2000) 277 [hep-ph/0003111] [INSPIRE].
A. Gynther, Electroweak phase diagram at finite lepton number density, Phys. Rev. D 68 (2003) 016001 [hep-ph/0303019] [INSPIRE].
A. Gynther and M. Vepsäläinen, Pressure of the standard model at high temperatures, JHEP 01 (2006) 060 [hep-ph/0510375] [INSPIRE].
A. Gynther and M. Vepsäläinen, Pressure of the standard model near the electroweak phase transition, JHEP 03 (2006) 011 [hep-ph/0512177] [INSPIRE].
M. Laine and M. Meyer, Standard model thermodynamics across the electroweak crossover, JCAP 07 (2015) 035 [arXiv:1503.04935] [INSPIRE].
T. Gorda, A. Helset, L. Niemi, T.V.I. Tenkanen and D.J. Weir, Three-dimensional effective theories for the two Higgs doublet model at high temperature, JHEP 02 (2019) 081 [arXiv:1802.05056] [INSPIRE].
L. Niemi, H.H. Patel, M.J. Ramsey-Musolf, T.V.I. Tenkanen and D.J. Weir, Electroweak phase transition in the real triplet extension of the SM: Dimensional reduction, Phys. Rev. D 100 (2019) 035002 [arXiv:1802.10500] [INSPIRE].
K. Kainulainen, V. Keus, L. Niemi, K. Rummukainen, T.V.I. Tenkanen and V. Vaskonen, On the validity of perturbative studies of the electroweak phase transition in the two Higgs doublet model, JHEP 06 (2019) 075 [arXiv:1904.01329] [INSPIRE].
L. Niemi, M.J. Ramsey-Musolf, T.V.I. Tenkanen and D.J. Weir, Thermodynamics of a two-step electroweak phase transition, Phys. Rev. Lett. 126 (2021) 171802 [arXiv:2005.11332] [INSPIRE].
O. Gould, Real scalar phase transitions: a nonperturbative analysis, JHEP 04 (2021) 057 [arXiv:2101.05528] [INSPIRE].
L. Niemi, P. Schicho and T.V.I. Tenkanen, Singlet-assisted electroweak phase transition at two loops, arXiv:2103.07467 [INSPIRE].
R. Jackiw, Functional evaluation of the effective potential, Phys. Rev. D 9 (1974) 1686 [INSPIRE].
S.R. Coleman and E.J. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7 (1973) 1888 [INSPIRE].
B.M. Kastening, Renormalization group improvement of the effective potential in massive ϕ4 theory, Phys. Lett. B 283 (1992) 287 [INSPIRE].
M. Bando, T. Kugo, N. Maekawa and H. Nakano, Improving the effective potential, Phys. Lett. B 301 (1993) 83 [hep-ph/9210228] [INSPIRE].
C. Ford, D.R.T. Jones, P.W. Stephenson and M.B. Einhorn, The effective potential and the renormalization group, Nucl. Phys. B 395 (1993) 17 [hep-lat/9210033] [INSPIRE].
M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, Reading U.S.A. (1995).
J. Ellis, TikZ-Feynman: Feynman diagrams with TikZ, Comput. Phys. Commun. 210 (2017) 103 [arXiv:1601.05437] [INSPIRE].
J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002) 1.
X.-p. Sun, Monte Carlo studies of three-dimensional O(1) and O(4) ϕ4 theory related to BEC phase transition temperatures, Phys. Rev. E 67 (2003) 066702 [hep-lat/0209144] [INSPIRE].
J.I. Kapusta and C. Gale, Finite-temperature field theory: principles and applications, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2011) [INSPIRE].
P. Basler, M. Muhlleitner and J. Müller, BSMPT v2 a tool for the electroweak phase transition and the baryon asymmetry of the universe in extended Higgs sectors, arXiv:2007.01725 [INSPIRE].
A. Ekstedt and J. Löfgren, A critical look at the electroweak phase transition, JHEP 12 (2020) 136 [arXiv:2006.12614] [INSPIRE].
J.O. Andersen, The Screening mass squared in hot scalar theory to order g-5 using effective field theory, Phys. Rev. D 57 (1998) 5004 [hep-ph/9708276] [INSPIRE].
J.P. Blaizot, E. Iancu and A. Rebhan, On the apparent convergence of perturbative QCD at high temperature, Phys. Rev. D 68 (2003) 025011 [hep-ph/0303045] [INSPIRE].
M. Laine and Y. Schröder, Quark mass thresholds in QCD thermodynamics, Phys. Rev. D 73 (2006) 085009 [hep-ph/0603048] [INSPIRE].
A.V. Manohar, Introduction to effective field theories, arXiv:1804.05863 [INSPIRE].
T. Brauner, T.V.I. Tenkanen, A. Tranberg, A. Vuorinen and D.J. Weir, Dimensional reduction of the Standard Model coupled to a new singlet scalar field, JHEP 03 (2017) 007 [arXiv:1609.06230] [INSPIRE].
H.-K. Guo, K. Sinha, D. Vagie and G. White, The benefits of diligence: how precise are predicted gravitational wave spectra in models with phase transitions?, arXiv:2103.06933 [INSPIRE].
R. Jinno, S. Lee, H. Seong and M. Takimoto, Gravitational waves from first-order phase transitions: towards model separation by bubble nucleation rate, JCAP 11 (2017) 050 [arXiv:1708.01253] [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].
F. Giese, T. Konstandin and J. van de Vis, Model-independent energy budget of cosmological first-order phase transitions — A sound argument to go beyond the bag model, JCAP 07 (2020) 057 [arXiv:2004.06995] [INSPIRE].
H.-K. Guo, K. Sinha, D. Vagie and G. White, Phase transitions in an expanding universe: stochastic gravitational waves in standard and non-standard histories, JCAP 01 (2021) 001 [arXiv:2007.08537] [INSPIRE].
F. Giese, T. Konstandin, K. Schmitz and J. Van De Vis, Model-independent energy budget for LISA, JCAP 01 (2021) 072 [arXiv:2010.09744] [INSPIRE].
A. Azatov and M. Vanvlasselaer, Bubble wall velocity: heavy physics effects, JCAP 01 (2021) 058 [arXiv:2010.02590] [INSPIRE].
J.S. Langer, Statistical theory of the decay of metastable states, Annals Phys. 54 (1969) 258 [INSPIRE].
J. Langer, Metastable states, Physica 73 (1974) 61.
D. Bödeker and G.D. Moore, Electroweak bubble wall speed limit, JCAP 05 (2017) 025 [arXiv:1703.08215] [INSPIRE].
M. Barroso Mancha, T. Prokopec and B. Swiezewska, Field-theoretic derivation of bubble-wall force, JHEP 01 (2021) 070 [arXiv:2005.10875] [INSPIRE].
Z.-G. Mou, P.M. Saffin and A. Tranberg, Simulations of a bubble wall interacting with an electroweak plasma, JHEP 02 (2021) 189 [arXiv:2006.13620] [INSPIRE].
S. Höche, J. Kozaczuk, A.J. Long, J. Turner and Y. Wang, Towards an all-orders calculation of the electroweak bubble wall velocity, JCAP 03 (2021) 009 [arXiv:2007.10343] [INSPIRE].
B. Laurent and J.M. Cline, Fluid equations for fast-moving electroweak bubble walls, Phys. Rev. D 102 (2020) 063516 [arXiv:2007.10935] [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].
A. Papaefstathiou and G. White, The electro-weak phase transition at colliders: confronting theoretical uncertainties and complementary channels, JHEP 05 (2021) 099 [arXiv:2010.00597] [INSPIRE].
C.-x. Zhai and B.M. Kastening, The free energy of hot gauge theories with fermions through g5, Phys. Rev. D 52 (1995) 7232 [hep-ph/9507380] [INSPIRE].
E. Braaten and A. Nieto, Free energy of QCD at high temperature, Phys. Rev. D 53 (1996) 3421 [hep-ph/9510408] [INSPIRE].
K. Kajantie, M. Laine, K. Rummukainen and Y. Schröder, The pressure of hot QCD up to g6 ln(1/g), Phys. Rev. D 67 (2003) 105008 [hep-ph/0211321] [INSPIRE].
A. Gynther, M. Laine, Y. Schröder, C. Torrero and A. Vuorinen, Four-loop pressure of massless O(N) scalar field theory, JHEP 04 (2007) 094 [hep-ph/0703307] [INSPIRE].
J.O. Andersen, L. Kyllingstad and L.E. Leganger, Pressure to order g8logg of massless ϕ4 theory at weak coupling, JHEP 08 (2009) 066 [arXiv:0903.4596] [INSPIRE].
A.D. Linde, Infrared problem in thermodynamics of the Yang-Mills gas, Phys. Lett. B 96 (1980) 289 [INSPIRE].
K. Kajantie, M. Karjalainen, M. Laine and J. Peisa, Three-dimensional U(1) gauge + Higgs theory as an effective theory for finite temperature phase transitions, Nucl. Phys. B 520 (1998) 345 [hep-lat/9711048] [INSPIRE].
M. Laine and K. Rummukainen, Two Higgs doublet dynamics at the electroweak phase transition: a nonperturbative study, Nucl. Phys. B 597 (2001) 23 [hep-lat/0009025] [INSPIRE].
M. Laine, G. Nardini and K. Rummukainen, Lattice study of an electroweak phase transition at mh ≃ 126 GeV, JCAP 01 (2013) 011 [arXiv:1211.7344] [INSPIRE].
D. Croon, V. Sanz and G. White, Model discrimination in gravitational wave spectra from dark phase transitions, JHEP 08 (2018) 203 [arXiv:1806.02332] [INSPIRE].
M. Sher, Electroweak Higgs potentials and vacuum stability, Phys. Rept. 179 (1989) 273 [INSPIRE].
S.Y. Lee and A.M. Sciaccaluga, Evaluation of higher order effective potentials with dimensional regularization, Nucl. Phys. B 96 (1975) 435 [INSPIRE].
A.K. Rajantie, Feynman diagrams to three loops in three-dimensional field theory, Nucl. Phys. B 480 (1996) 729 [Erratum ibid. 513 (1998) 761] [hep-ph/9606216] [INSPIRE].
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Gould, O., Tenkanen, T.V.I. On the perturbative expansion at high temperature and implications for cosmological phase transitions. J. High Energ. Phys. 2021, 69 (2021). https://doi.org/10.1007/JHEP06(2021)069
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DOI: https://doi.org/10.1007/JHEP06(2021)069