Electroweak Vacuum Stability and Renormalized Vacuum Field Fluctuation Hiroki Matsui KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan The Graduate University of Advanced Studies (Sokendai),Tsukuba, Ibaraki 305-0801, Japan, matshiro@post.kek.jp Based mainly on: K, Kohri and H, Mastui , Phys.Rev. D94 (2016) no.10, 103509, arXiv:1607.08133 (to appear in JCAP), arXiv:1704.06884, arXiv:1708.????? Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 1 / 24
Introduction Electroweak Vacuum Metastability Electroweak Vacuum Metastability The recent LHC experiments of the Higgs boson mass m h = 125 . 09 ± 0 . 21 ( stat ) ± 0 . 11 ( syst ) GeV and the top quark mass m t = 172 . 44 ± 0 . 13 ( stat ) ± 0 . 47 ( syst ) GeV suggest that the electroweak vacuum is metastable and finally cause a catastrophic vacuum decay through quantum tunneling. 0.8 6 8 10 180 No EW vacuum 10 7 10 8 200 10 4 6 12 14 16 10 18 14 10 12 8 6 10 4 Instability 10 9 8 18 16 Instability Gauge coupling g 2 � M Pl � � g 1 � M Pl � 10 4 12 68 10 14 178 6 7 8 910 12 1416 19 10 10 0.6 Top pole mass M t in GeV 10 4 10 11 150 Meta � stability Top pole mass M t in GeV 176 10 12 SM Non � perturbativity 6 10 13 10 16 0.4 5 Instability 174 100 � I � 10 4 GeV Meta � 1,2,3 Σ Stability Meta � stability stability 172 10 19 0.2 50 Stability 170 10 18 10 14 Stability 8 10 12 14 16 18 0 0.0 168 0 100 200 � 0.06 � 0.04 � 0.02 0.00 0.02 0.04 0.06 50 150 120 122 124 126 128 130 132 Higgs coupling Λ � M Pl � Higgs pole mass M h in GeV Higgs pole mass M h in GeV [D. Buttazzo, G. Degrassic, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, A. Strumia, JHEP 1312 (2013)] Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 2 / 24
Introduction Electroweak Vacuum Metastability Electroweak Vacuum Stability during Large-scale Inflation In de Sitter space or inflationary Universe, the vacuum field fluctuation δφ 2 � � enlarges in proportion to the Hubble scale H . δφ 2 � 1 / 2 ≈ O ( H ) � Λ I ≈ 10 11 GeV = � ⇒ CATASTROPHE !? If the inflationary vacuum fluctuation δφ 2 � � overcomes the barrier of V eff ( φ ) , it can trigger off a false vacuum decay. The preheating or reheating case are also trouble. [ J. R. Espinosa, G. F. Giudice, and A. Riotto, JCAP 0805, 002 (2008), M. Herranen, T. Markkanen, S. Nurmi, and A. Rajantie, Phys.Rev.Lett.113 211102 (2014), A. Hook, J. Kearney, B. Shakya, and K. M. Zurek, JHEP 01,061 (2015) ] Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 3 / 24
Introduction Electroweak Vacuum Metastability Electroweak Vacuum Stability around Evaporating Black Hole δφ 2 � � In Schwarzschild space, proportional to inverse of the black-hole mass M BH and approximately approach the Hawking thermal fluctuation near the black-hole event horizon. δφ 2 � 1 / 2 ≈ O ( 1 / M BH ) � Λ I ≈ 10 11 GeV = � ⇒ CATASTROPHE !? Recent discussion about the vacuum stability around the evaporating (primordial) black hole which can be formulated by large density fluctuations in the early universe, has been growing. [ P. Burda, R. Gregory, and I. Moss, Phys.Rev.Lett.115,071303 (2015), N. Tetradis, JCAP 1609,036 (2016), D. Gorbunov, D. Levkov, and A. Panin, arXiv:1704.05399, K. Mukaida and M. Yamada, arXiv:1706.04523 ] Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 4 / 24
Introduction Electroweak Vacuum Metastability Electroweak Vacuum Metastability in Curved Background We assume the bare Higgs potential in the curved spacetime. V ( φ ) ≡ 1 φ 2 + λ m 2 + ξ R 4 φ 4 + · · · � � 2 where ξ is the non-minimal curvature coupling. The Friedmann- Lemaitre-Robertson-Walker (FLRW) metric and the Schwarzschild metric can be written by dr 2 � d θ 2 + sin 2 θ d ϕ 2 �� ds 2 = − dt 2 + a ( t ) 2 1 − Kr 2 + r 2 � dr 2 � 1 − 2 M BH � ds 2 = − dt 2 + d θ 2 + sin 2 θ d ϕ 2 � 1 − 2 M BH / r + r 2 � r Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 5 / 24
Introduction Electroweak Vacuum Metastability The Vacuum Field Fluctuation in Curved Spacetime δφ 2 � � In quantum field theory (QFT), the two-point correlation function δφ 2 � � corresponding to the vacuum field fluctuation − → ∞ becomes (quadratically or logarithmically) divergent. � � ∞ dk d 3 k | δφ k ( t , x ) | 2 = � δφ 2 � k ∆ 2 δφ ( k ) − = → ∞ . 0 These UV divergences must be removed by using the regularization or the renormalization method, and we take out the dynamical field fluctuation being depend on the curved background, i.e. H , R , R abcd R abcd . In QFT, there are various regularization methods like cutoff regularization, dimensional regularization, ζ -function regularization, lattice regularization, point-splinting regularization (QFT in curved spacetime), adiabatic regularization (QFT in curved spacetime) etc. Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 6 / 24
Introduction Electroweak Vacuum Metastability The Vacuum Field Fluctuation in De-Sitter (Inflation) Spacetime δφ 2 � � The vacuum field fluctuation of the Higgs field can be written by � ∞ � 2 � 1 1 + 2 | β k | 2 + α k β ∗ δφ 2 � dkk 2 Ω − 1 k δϕ 2 � k + α ∗ k β k δϕ ∗ = 4 π 2 a 2 ( η ) k k 0 where α k ( η ) and β k ( η ) are the Bogoliubov coefficients. δφ 2 � ( s ) + δφ 2 � ( d ) δφ 2 � � � � = � ∞ δφ 2 � ( s ) = 1 dkk 2 Ω − 1 � 4 π 2 a 2 ( η ) k 0 � ∞ δφ 2 � ( d ) = 1 dkk 2 Ω − 1 � { 2 n k + 2 Re z k } 4 π 2 a 2 ( η ) k 0 where we introduce n k = | β k | 2 which can be interpreted as the particle number k δϕ 2 density and z k = α k β ∗ k . Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 7 / 24
Introduction Electroweak Vacuum Metastability The Vacuum Field Fluctuation in De-Sitter (Inflation) Spacetime δφ 2 � ( s ) correspond to the Minkowskian divergences � The divergences of δφ 2 � ( s ) = M 2 ( φ ) � � M 2 ( φ ) � � − 1 ǫ − log 4 π − γ − 3 � × ln 16 π 2 µ 2 2 M 2 ( φ ) = m 2 + 3 λφ 2 + ( ξ − 1 / 6 ) R These divergences can be canceled by the counterterms δ m 2 , δξ and δλ . ren = M 2 ( φ ) � M 2 ( φ ) � � − 3 � δφ 2 � ( s ) � ln 16 π 2 µ 2 2 δφ 2 � ( s ) � From the above renormalized vacuum field fluctuations ren , we can construct the one-loop effective potential on the curved spacetime 4 φ 4 + M 2 ( φ ) � M 2 ( φ ) V eff ( φ ) = 1 2 m 2 φ 2 + 1 � � − 3 � 2 ξ R φ 2 + λ ln 16 π 2 µ 2 2 Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 8 / 24
Introduction Electroweak Vacuum Metastability The Vacuum Field Fluctuation in De-Sitter (Inflation) Spacetime The adiabatic regularization is the powerful method to obtain the dynamical vacuum fluctuation. � ∞ δφ 2 � ( d ) = δφ 2 � ( s ) = 1 δφ 2 � dkk 2 Ω − 1 � � � − { 2 n k + 2 Re z k } 4 π 2 a 2 ( η ) k 0 � � ∞ � ∞ 1 � dk 2 k 2 | δχ k | 2 − dkk 2 Ω − 1 = 4 π 2 a 2 ( η ) k 0 0 where we must determine exactly δχ ( η ) . In de Sitter spacetime, the dynamical δφ 2 � ( d ) can be summarized as � (renormalized) vacuum field fluctuation H 3 t / 4 π 2 ( M ( φ ) = 0 ) δφ 2 � ( d ) ≃ 3 H 4 / 8 π 2 M 2 ( φ ) � ( M ( φ ) ≪ H ) H 2 / 24 π 2 ( M ( φ ) � H ) Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 9 / 24
Introduction Electroweak Vacuum Metastability The Electroweak Vacuum Stability in Inflationary Universe In curved spacetime, the effective Higgs potential must include the dynamical δφ 2 � ( d ) � vacuum field fluctuation V eff ( φ ) = 1 2 m 2 φ 2 + 1 2 ξ R φ 2 + λ 4 φ 4 + λ δφ 2 � ( d ) φ 2 � 2 9 � log M 2 � � n i i ( φ ) 64 π 2 M 4 + i ( φ ) − C i µ 2 i = 1 δφ 2 � ( d ) + κ ′ i ( φ ) = κ i φ 2 + κ i M 2 � i + θ i R We can take the renormalization scale µ 2 ≈ φ 2 + R + δφ 2 � ( d ) . � ⇒ λ ( µ ) φ 4 1 / 2 δφ 2 � ( d ) ) � Λ I ≈ 10 11 GeV = � µ ≈ ( R + < 0 4 [ M. Herranen, T. Markkanen, S. Nurmi, and A. Rajantie, Phys.Rev.Lett. 113,211102 (2014), K, Kohri and H, Mastui , arXiv:1607.08133 (to appear in JCAP), arXiv:1704.06884 ] Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 10 / 24
The Electroweak Vacuum Stability in Inflationary Universe The Electroweak Vacuum Stability in Inflationary Universe The Electroweak Vacuum Stability in Inflationary Universe δφ 2 � � The inflationary vacuum fluctuation of the Higgs field destabilize the effective Higgs potential V eff ( φ ) . Hiroki Matsui (KEK Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan) Electroweak Vacuum Stability August 2, 2017 11 / 24
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