Electroweak Vacuum Instability and Renormalized Higgs Field Vacuum Fluctuations in the Inﬂationary Universe

Prepared for submission to JCAP

Kazunori Kohri,a,b Hiroki Matsuib

aInstitute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba 305-0801, Japan bThe Graduate University for Advanced Studies (SOKENDAI), 1-1 Oho, Tsukuba 305-0801, Japan E-mail: [email protected], [email protected]

Abstract. In this work, we investigated the electroweak vacuum instability during or after inflation. In the inflationary Universe, i.e., de Sitter space, the vacuum field fluctuations δφ2 enlarge in proportion to the Hubble scale H2. Therefore, the large inflationary vacuum fluctuations of the Higgs field δφ2 are potentially catastrophic to trigger the vacuum transition to the negative-energy Planck-scale vacuum state and cause an immediate collapse of the Universe. However, the vacuum field fluctuations δφ2 , i.e., the vacuum expectation values have an ultraviolet divergence, and therefore a renormalization is necessary to estimate the physical effects of the vacuum transition. Thus, in this paper, we revisit the electroweak vacuum instability from the perspective of quantum field theory (QFT) in curved space-time, and discuss the dynamical behavior of the homogeneous Higgs field φ determined by the effective potential Veff (φ) 2 in curved space-time and the renormalized vacuum fluctuations δφ ren via adiabatic arXiv:1607.08133v2 [hep-ph] 7 Aug 2017 regularization and point-splitting regularization. We simply suppose that the Higgs field only couples the gravity via the non-minimal Higgs-gravity coupling ξ(µ). In this scenario, the electroweak vacuum stability is inevitably threatened by the dynamical behavior of the homogeneous Higgs field φ, or the formations of AdS domains or bubbles unless the Hubble scale is small enough H < ΛI . Contents

1 Introduction1

2 Higgs eﬀective potential in curved space-time via adiabatic expansion method2

3 Renormalized vacuum ﬂuctuations via adiabatic regularization7 3.1 Massless minimally coupled cases8 3.2 Massive non-minimally coupled cases 11

4 Renormalized vacuum ﬂuctuations via point-splitting regularization 13

5 Electroweak vacuum instability from dynamical behavior of homogeneous Higgs ﬁeld and renormalized Higgs vacuum ﬂuctuations 14

6 Conclusion 23

1 Introduction

The recent measurements of the Higgs boson mass mh = 125.09 ± 0.21 (stat) ± 0.11 (syst) GeV [1–4] and the top quark mass mt = 173.34 ± 0.27 (stat) GeV [5] suggest that the current electroweak vacuum state of the Universe is not stable, and finally cause a catastrophic vacuum decay through quantum tunneling [6–8] although the cosmological timescale for the quantum tunneling decay is longer than the age of the Universe [9–12]. In de Sitter space, especially the inflationary Universe, however, the curved background enlarges the vacuum field fluctuations δφ2 in proportion to the Hubble scale H2. Therefore, if the large inflationary vacuum fluctuations δφ2 of the Higgs field overcomes the barrier of the standard model Higgs effective potential Veff (φ), it triggers off a catastrophic vacuum transition to the negative Planck-energy true vacuum and cause an immediate collapse of the Universe [13–26]. The vacuum field fluctuations δφ2 , i.e., the vacuum expectation values are for- mally given by Z Z ∞ 2 3 2 dk 2 δφ = d k|δφk (η, x)| = ∆δφ (η, k) . (1.1) 0 k

2 where ∆δφ (η, k) is defined as the power spectrum of quantum vacuum fluctuations. As well-known facts in quantum field theory (QFT), the vacuum expectation values δφ2 have an ultraviolet divergence (quadratic or logarithmic) and therefore a regularization is necessary. The quadratic divergence corresponds to the normal contribution from the fluctuations of the vacuum in Minkowski space, and it can be eliminated by standard renormalization in flat spacetime. The logarithmic divergence, however, appears as a consequence of the expansion of the Universe, and has the physical contributions to

– 1 – the origin of the primordial perturbations or the backreaction of the inflaton field. We usually eliminate this logarithmic ultraviolet divergence by simply neglecting the modes with k > aH. That corresponds to the stochastic Fokker-Planck (FP) equation, which treats the inflationary field fluctuations generally originating from long wave modes, i.e., the IR parts [27]. Recent works [22–26] for the electroweak vacuum stability during inflation based on the stochastic Fokker-Planck (FP) equation. However, from the viewpoint of QFT, we must treat carefully short wave modes as well as long wave modes [28, 29] and it is necessary to renormalize the vacuum expectation values δφ2 in the curved space-time in order to obtain exact physical contributions [30, 31]. Thus, in this paper, we revisit the electroweak vacuum instability from the legitimate perspective of QFT in curved space-time. In the first part of this paper, we derive the one-loop Higgs effective potential in curved space-time via the adiabatic expansion method. In the second part, we discuss the renormalized field vacuum fluctuations δφ2 in de Sitter space by using adiabatic regularization and point-splitting regularization. In the third part, we investigate the electroweak vacuum instability during or after inflation from the global and homogeneous Higgs field φ, and the renor- 2 malized vacuum fluctuations δφ ren corresponding to the local and inhomogeneous Higgs field fluctuations. The behavior of the homogeneous Higgs field φ is determined by the effective potential Veff (φ) in curved space-time, and then, the excursion of the homogeneous Higgs field φ to the negative Planck-energy vacuum state can terminate inflation and triggers off a catastrophic collapse of the Universe. The local and in- 2 homogeneous Higgs fluctuations described by δφ ren generate catastrophic Anti-de Sitter (AdS) domains or bubbles and finally cause a vacuum transition. In this work, we improve our previous work [25], provide a comprehensive study of the phenomenon and reach new conclusions. In addition, we persist in the zero-temperature field theory, leaving the generalization to the finite-temperature case and discussion of the thermal History of the metastable Universe for a forthcoming work. This paper is organized as follows. In Section2 we derive the Higgs effective potential in curved space-time by using the adiabatic expansion method. In Section3 we discuss the problem of renormalization to the vacuum fluctuations in de Sitter space by using adiabatic regularization. In Section4 we consider the renormalized vacuum fluctuations via point-splitting regularization and show that the renormalized expectation values via point-splitting regularization is consistent with the previous results via adiabatic regularization. In Section5 we discuss the behavior of the global Higgs field φ and the vacuum transitions via the renormalized Higgs field vacuum fluctuations 2 δφ ren, and investigate the electroweak vacuum instability during inflation or after inflation. Finally, in Section6 we draw the conclusion of our work.

2 Higgs eﬀective potential in curved space-time via adiabatic expansion method

The behavior of the homogeneous Higgs field φ on the entire Universe is determined by the effective potential Veff (φ) in curved space-time. In this section, we review the standard derivation of the one-loop effective potential in curved space-time via

– 2 – the adiabatic expansion method [32–36], and then, show how the effective potential Veff (φ) of curved space-time is changed from the Minkowski space-time, where we use the notations and the conventions of Ref.[32, 33]. For simplicity, we consider a flat Robertson-Walker background not including metric perturbations. Thus, the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is given by

a2 (t) g = diag −1, , a2 (t) r2, a2 (t) r2 sin2 θ , (2.1) µν 1 − Kr2 where a = a (t) is the scale factor with the cosmic time t and K is the curvature parameter, where positive, zero, and negative values correspond to closed, flat, and hyperbolic space-time, respectively. For the spatially flat Universe, we take K = 0. Then, the scalar curvature is obtained as " # a˙ 2 a¨ a00 R = 6 + = 6 . (2.2) a a a3 where the conformal time η has been introduced and is defined by dη = dt/a. In the de Sitter space-time, the scale factor becomes a (t) = eHt or a (η) = −1/Hη, the scalar curvature is estimated to be R = 12H2 in the de Sitter Universe. The action for the Higgs field with the potential V (φ) in curved space-time is given by Z √ 1 S [φ] = − d4x −g gµν∇ φ∇ φ + V (φ) , (2.3) 2 µ ν where we assume the simple form for the Higgs potential as

1 λ V (φ) = m2 + ξR φ2 + φ4. (2.4) 2 4 Thus, the Klein-Gordon equation for the Higgs ﬁeld are given by

0 φ (η, x) + V (φ (η, x)) = 0, (2.5)

µν where denotes the generally covariant d’Alembertian operator, = g ∇µ∇ν = √ √ µ 1/ −g∂µ ( −g∂ ) and ξ is the non-minimal Higgs-gravity coupling constant. There are two popular choices for ξ, i.e., minimal coupling (ξ = 0) and conformal coupling (ξ = 1/6), which is conformally invariant in the massless limit. However, the non- minimal Higgs-gravity coupling ξ is inevitably generated through the loop corrections. In the quantum field theory, we treat the Higgs field φ (η, x) as the operator acting on the states. We assume that the vacuum expectation value of the Higgs field is φ (η) = h0| φ (η, x) |0i. In this case, the Higgs field φ (η, x) can be decomposed into a classic component and a quantum component as

φ (η, x) = φ (η) + δφ (η, x) , (2.6)

– 3 – where h0| δφ (η, x) |0i = 0. In the one-loop approximation, we can obtain the following equations 1 φ + V 0 (φ) + V 000 (φ) δφ2 = 0, (2.7) 2 00 δφ + V (φ) δφ = 0, (2.8) where the mass of the quantum ﬁeld δφ is written by

V 00 (φ) = m2 + 3λφ2 + ξR. (2.9)

The quantum field δφ is decomposed into each k mode, Z 3 † ∗ δφ (η, x) = d k akδφk (η, x) + akδφk (η, x) , (2.10) where eik·x δφk (η, x) = δχk (η) , (2.11) (2π)3/2pC (η) with C (η) = a2 (η). Thus, the Higgs field vacuum fluctuations δφ2 , i.e., the expectation field values can be written as Z 2 3 2 h0| δφ |0i = d k|δφk (η, x)| , (2.12) Z ∞ 1 2 2 = 2 dkk |δχk| , (2.13) 2π C (η) 0 where δφ2 has an ultraviolet divergence (quadratic and logarithmic) and requires a regularization, e.g. cut-off regularization or dimensional regularization. From Eq. (2.8), the Klein-Gordon equation for the quantum field δχ is written by

00 2 δχk + Ωk (η) δχk = 0. (2.14) Here, we use the adiabatic (WKB) approximation to obtain the mode function. For the lowest-order approximation, the time-dependent mode function is given by

1 Z δχk = p exp −i Ωk (η) dη , (2.15) 2Ωk (η)

2 2 2 2 where Ωk (η) = k +C (η)(m + 3λφ + (ξ − 1/6) R). More precisely, we must consider the higher order approximation and include the exact effects of the particle produc- tions for the one-loop effective potential (see Ref.[34] for the details). However, we can simply include such effects by adding the backreaction term of the vacuum filed fluctu- 2 ations, i.e., δφ ren. Furthermore, we comment the condition of the adiabatic (WKB) 2 0 2 approximation (Ωk > 0 and |Ω k/Ωk| 1). This condition breaks during inflation for the massless scalar field, or in the parametric resonance of the preheating (see, e.g. Ref.[37]). In these cases, the IR parts at k < aH breaks the WKB approximation and

– 4 – we can expect enormous particle production. However, the UV parts at k > aH are not affected by the cosmological dynamics of the Universe, and the effective potential generally originates from the UV parts, i.e., short wave modes. Therefore, as well as the radiation dominated and the matter dominated eras, we can adopt the adiabatic expansion method for the effective potential in de Sitter space by taking into account the IR backreaction effects (see Ref.[34] for the details). As a consequence, we must 2 add the backreaction term, i.e., renormalized field vacuum fluctuations δφ ren (see Section3 for the details) to the effective potential in curved space-time. Then, we write the expectation field value as follows 1 Z ∞ k2 h0| δφ2 |0i = dk . (2.16) 2 2 p 2 2 2 2 4π a 0 k + (m + 3λφ + (ξ − 1/6) R) a The one-loop contribution to the Higgs effective potential is given by 1 V 000 (φ) h0| δφ2 |0i 2 d 1 Z Λ = dk k2pk2 + (m2 + 3λφ2 + (ξ − 1/6) R) a2 , dφ 4π2a4 dV (φ) = 1 , (2.17) dφ where we take an ultraviolet cut-off as Λ in order to regularize quadratic or logarithmic divergence. For convenience, we rewrite the classic Higgs field equation as follows

0 0 φ + V (φ) + V1 (φ) = 0. (2.18) In order to obtain the eﬀective potential, we calculate exactly the integral 1 Z Λ V (φ) = dk k2pk2 + (m2 + 3λφ2 + (ξ − 1/6) R) a2, (2.19) 1 4π2a4 " 1 = Λ 2Λ2 + m2 + 3λφ2 + (ξ − 1/6) R a2 32π2a4 ×pΛ2 + (m2 + 3λφ2 + (ξ − 1/6) R) a2 + m2 + 3λφ2 + (ξ − 1/6) R2 a4 !# (m2 + 3λφ2 + (ξ − 1/6) R)1/2 a × ln . (2.20) Λ + pΛ2 + (m2 + 3λφ2 + (ξ − 1/6) R) a2 In the limit Λ → ∞, we can obtain the following expression

Λ4 (m2 + 3λφ2 + (ξ − 1/6) R)Λ2 (m2 + 3λφ2 + (ξ − 1/6) R)2 Λ2 V (φ) = + − ln 1 16π2a4 16π2a2 64π2 µ2 (m2 + 3λφ2 + (ξ − 1/6) R)2 (m2 + 3λφ2 + (ξ − 1/6) R) a2 + ln 64π2 µ2 (m2 + 3λφ2 + (ξ − 1/6) R)2 + C , (2.21) 64π2 i

– 5 – where we introduced the renormalization scale µ and the constant Ci = (1/2 − 2 ln 2) depends on the regularization method and the renormalization scheme. Here, we focus on the divergent contribution to the effective potential, which is given by Λ4 (m2 + 3λφ2 + (ξ − 1/6) R)Λ2 (m2 + (ξ − 1/6) R)2 Λ2 V (φ) = + − ln Λ 16π2a4 16π2a2 64π2 µ2 3λΛ2 6λ(m2 + (ξ − 1/6) R) Λ2 9λ2 Λ2 + − ln φ2 − ln φ4. (2.22) 16π2a2 64π2 µ2 64π2 µ2 For convenience, we replace Λ → aΛ, µ/a → µ and the divergent contribution can be written as Λ4 (m2 + 3λφ2 + (ξ − 1/6) R)Λ2 (m2 + (ξ − 1/6) R)2 Λ2 V (φ) = + − ln Λ 16π2 16π2 64π2 µ2 3λΛ2 6λ(m2 + (ξ − 1/6) R) Λ2 9λ2 Λ2 + − ln φ2 − ln φ4. (2.23) 16π2 64π2 µ2 64π2 µ2 We can obviously remove all the divergences (quartic, quadratic and logarithmic) by absorbing the counter-terms as follows: 1 1 1 V (φ) = V (φ) + V (φ) + δ + δ φ2 + δ φ2 + δ φ4. (2.24) eff 1 cc 2 m 2 ξ 4 λ We obtain the Higgs effective potential in curved space-time as follows: 1 1 λ V (φ) = m2φ2 + ξRφ2 + φ4 (2.25) eff 2 2 4 (m2 + 3λφ2 + (ξ − 1/6) R)2 m2 + 3λφ2 + (ξ − 1/6) R + ln − C , 64π2 µ2 i which is consistent with the results by using the heat kernel method [38, 39]. The effective potential in curved background has been thoroughly investigated in the liter- atures [18, 40–54] and there are a variety of ways to derive the effective potential in curved spacetime. Now, we can read off the scale dependence of m2, ξ and λ from the Eq. (2.25), and the β functions are given by dλ 18λ2 βλ ≡ = , (2.26) d ln µ (4π)2 dξ 6λ βξ ≡ = (ξ − 1/6) , (2.27) d ln µ (4π)2 dm2 6λm2 βm2 ≡ = . (2.28) d ln µ (4π)2 Finally, we can rewrite the classical Higgs field equation by using the effective potential in curved space-time as follows: 0 φ + Veff (φ) = 0. (2.29) The effective potential given by Eq. (2.25) does not include the exact particle produc- tions and we must consider the backreaction term from the renormalized field vacuum 2 fluctuations δφ ren to improve the effective potential in curved space-time.

– 6 – 3 Renormalized vacuum ﬂuctuations via adiabatic regularization

The main difficulty with the vacuum instability in de Sitter space comes from the vacuum field fluctuations on the dynamical background. The renormalized vacuum 2 fluctuations δφ ren originate from the dynamical particle production effects, and it corresponds to the local and inhomogeneous Higgs field fluctuations. In this section, we 2 discuss the renormalized vacuum fluctuations δφ ren by using adiabatic regularization. The adiabatic regularization [55–60] is the extremely powerful method to remove the ultraviolet divergence from the expectation field value δφ2 and obtain the renormalized finite value, which has physical contribution. For convenience, we write the equation of the field δχ for conformal time coordinate η given by

00 2 δχk + Ωk (η) δχk = 0, (3.1) 2 2 2 2 2 2 where Ωk (η) = ωk (η) + C (η)(ξ − 1/6) R and ωk (η) = k + C (η)(m + 3λφ ). More precisely, the self-coupling term 3λφ2 includes the backreaction effect, i.e., 3λφ2 + 2 2 2 2 3λ δφ ren where we shift the Higgs field φ → φ + δφ ren. Therefore, the vacuum field fluctuations on the dynamical background become complicated and intricate in contrast with static space-time. For simplicity, we neglect the self-coupling term 3λφ2 in Section3 and Section4. The Wronskian condition is given as

∗ ∗ δχkδχk − δχkδχk = i, (3.2) which ensures the canonical commutation relations for the ﬁeld operator δχ below † † ak, ak0 = ak, ak0 = 0, (3.3) † 0 ak, ak0 = δ (k − k ) . (3.4)

The adiabatic vacuum |0iA is the vacuum state which is annihilated by all the operators ak and deﬁned by choosing χk (η) to be a positive-frequency WKB mode. The adiabatic (WKB) approximation to the time-dependent mode function is written by 1 Z δχk = p exp −i Wk (η) dη , (3.5) 2Wk (η) where 00 0 2 2 2 1 Wk 3 (Wk) Wk = Ωk − + 2 . (3.6) 2 Wk 4 Wk We can obtain the WKB solution by solving Eq. (3.6) with an iterative procedure and 0 the lowest-order WKB solution Wk is given by 02 2 Wk = Ωk. (3.7) 1 The ﬁrst-order WKB solution Wk is given by 2 1 (W 0)00 3 W 00 W 12 = Ω2 − k + k . (3.8) k k 2 W 0 4 0 2 k (Wk )

– 7 – For the high-order with the nearly conformal case ξ ' 1/6, we can obtain the following expression 2 4 2 2 3 (ξ − 1/6) 0 2 m C 0 2 5m C D Wk ' ωk + 2D + D − 3 D + D + 5 4ωk 8ωk 32ω 2 m C 000 0 02 0 2 4 + 5 D + 4D D + 3D + 6D D + D 32ωk 4 2 m C 00 02 02 4 − 7 28D D + 19D + 122D + 47D 128ωk 6 3 8 4 4 221m C 0 2 4 1105m C D + 9 D D + D − 11 256ωk 2048ωk

(ξ − 1/6) 000 00 02 − 3 3D + 3D D + 3D 8ωk 2 m C 00 02 0 2 4 + (ξ − 1/6) 5 30D D + 18D + 57D D + 9D 32ωk 4 2 75m C 0 2 4 − (ξ − 1/6) 7 2D D + D 128ωk 2 (ξ − 1/6) 02 0 2 4 − 3 36D + 36D D + 9D , (3.9) 32ωk 0 2 where D = C /C. The vacuum expectation values δφ ad by using the adiabatic vacuum sate |0iA can be written by Z Λ 2 2 2 1 k δφ ad = A 0|δφ |0 A = 2 dk. (3.10) 4π C (η) 0 Wk The adiabatic regularization is not the method of regularizing divergent integrals as cut-oﬀ regularization or dimensional regularization. Thus, Eq. (3.10) includes the divergences which need to be removed by these regularization. However, the divergences in the exact expression δφ2 , which come from the large k modes, are the same as the 2 divergences in the adiabatic expression δφ ad. Thus, we can remove the divergences 2 2 by subtracting the adiabatic expression δφ ad from the original expression δφ as follows: 2 2 2 δφ ren = δφ − δφ ad, (3.11) "Z Λ Z Λ 2 # 1 2 2 k = 2 2k |δχk| dk − dk . (3.12) 4π C (η) 0 0 Wk This method has been shown to be equivalent to the point-splitting regularization [61, 62], which has been used in a large number of space-time background.

3.1 Massless minimally coupled cases In this subsection, we discuss the vacuum expectation values in the massless minimally coupled case (m = 0 and ξ = 0). In the massless minimally coupling case, the power

– 8 – spectrum on super-horizon scales is given by

H 2 ∆2 (k) = , (3.13) δφ 2π

2 3 2 2 where ∆δφ (k) = k |δφk| /2π originates from the facts that the inflationary quantum field fluctuations are constant on super-horizon scales. If we take aH as the UV cut-off and H as the IR cut-off, the vacuum expectation values δφ2 are simply given by

Z aH 3 2 dk 2 H δφ = ∆δφ (k) = 2 t. (3.14) H k 4π

Here, we review the renormalization of the vacuum expectation values δφ2 with m = 0 and ξ = 0 by using the adiabatic regularization, where we use the result of Ref.[59, 60], and show that Eq. (3.14) is consistent with the results via the adiabatic regularization. In this case, the mode function δχk (η) can be exactly given by

∗ δχk (η) = akδϕk (η) + bkδϕk (η) , (3.15) where r 1 1 δϕ (η) = e−ikη 1 + . (3.16) k 2k ikη In the massless minimally coupled case, the vacuum expectation values δφ2 have not only ultraviolet divergences but also infrared divergences. To avoid the infrared divergences, we assume that the Universe changes over from the radiation-dominated phase to the de Sitter phase as the following

( η 2 − , (η < η0) η0 a (η) = η (3.17) , (η > η0) η0 where η0 = −1/H and, during the radiation-dominated Universe (η < η0), we choose the mode function √ −ikη δχk = e / 2k. (3.18) 0 which is in-vacuum state. Requiring the matching conditions δχk (η) and δχk (η) at η = η0, we can obtain the coeﬃcients of the mode function

H H2 2ik k2 a = 1 + − , b = a + + O . (3.19) k ik 2k2 k k 3H H2

For small k modes in the de Sitter Universe (η > η0), we have " # 1 2 H2η2 2 k2 |δχ |2 = + 2 + + O + ··· . (3.20) k 2k 3Hη 6 H2

– 9 – 2 2 Therefore, we obviously have no infrared divergences because of k |δχk| ∼ O (k). For large k modes, we can obtain the mode function 1 1 H2 H3 |δχ |2 = 1 + − cos (2k (1/H + η)) + O + ··· . (3.21) k 2k k2η2 k2 k3 Here, we consider the following adiabatic (WKB) solution

2 00 4 0 2 1 0 2 1 m C 5 m (C ) Wk = ωk − 2D + D − 3 + 5 , (3.22) 8ωk 8 ωk 32 ωk 1 1 m2C00 5 m4 (C0)2 = ωk − 2 − 3 + 5 . (3.23) η ωk 8 ωk 32 ωk We can obtain 1 1 1 1 m2C00 5 m4 (C0)2 ' + 2 3 + 5 − 7 . (3.24) Wk ωk η ωk 8 ωk 32 ωk 2 √The condition√ of the adiabatic (WKB) approximation, i.e., Ωk > 0 requires k > 2/ |η| = 2aH as the cut-oﬀ of k mode. Therefore, Eq. (3.12) can be given as follows:

"Z Λ Z Λ 2 # 2 1 2 2 k δφ ren = lim 2 2k |δχk| dk − √ dk , (3.25) m→0 4π C (η) 0 2/|η| Wk "Z Λ Z Λ 2 Z Λ 2 1 2 2 k k = lim 2k |δχk| dk − dk − dk m→0 2 √ √ 2 3 4π C (η) 0 2/|η| ωk 2/|η| η ωk # m2C00 Z Λ k2 5m4(C0)2 Z Λ k2 + √ 5 dk + √ 7 dk . (3.26) 8 2/|η| ωk 32 2/|η| ωk For large k modes, we can subtract the ultraviolet divergences as 1 Z Λ k2 lim 2 √ k − dk = 0. (3.27) m→0 4π C (η) 2/|η| ωk 1 Z Λ 1 k2 lim − dk = 0. (3.28) m→0 2 2 √ 3 4π η C (η) 2/|η| k ωk Furthermore, we can eliminate the following divergences m2C00 Z Λ k2 5m4(C0)2 Z Λ k2 lim dk = lim dk = 0. (3.29) m→0 √ 5 m→0 √ 7 8 2/|η| ωk 32 2/|η| ωk Therefore, we can obtain the following expression √ Z 2/|η| 2 1 2 2 δφ ren = 2 k |δχk| dk 2π C (η) 0 η2H2 Z ∞ H2 + 2 √ − 2 cos (2k (1/H + η)) + ··· kdk. (3.30) 4π 2/|η| k

– 10 – At the late cosmic-time (η ' 0 that is Ntot = Ht 1), we have the following approximation √ 2 2 Z 2/|η| 2 η H 2 2 δφ ren ' 2 k |δχk| dk, 2π 0 √ 1 Z H H2 Z 2/|η| 1 ' 2 kdk + 2 dk, (3.31) 9π 0 4π H k

2 2 2 2 2 3 where we approximate |δχk| = 2/9η H k for small k modes and |δχk| = 1/2η k for large k modes. We can ﬁnally obtain

H2 H2 1 H3 δφ2 ' + log 2 + Ht ' t, (3.32) ren 18π2 4π2 2 4π2 which coincides with Eq. (3.14) via the physical cut-oﬀ.

3.2 Massive non-minimally coupled cases In this subsection, we consider the massive non-minimally coupled case (m 6= 0 and ξ 6= 0). At ﬁrst, we discuss the renormalized vacuum ﬂuctuations for m H.1 In m H case, the power spectrum on super-horizon scales can be written as

H 2 k 3−2ν ∆2 (k) = , (3.33) δφ 2π aH where ν = p9/4 − m2/H2. If we take aH as the UV cut-oﬀ and H as the IR cut-oﬀ, the vacuum expectation values δφ2 can be given by

Z aH 4 2 dk 2 3H δφ = ∆δφ (k) = 2 2 . (3.34) H k 8π m

2 Here, we brieﬂy discuss the renormalized vacuum ﬂuctuations δφ ren for m H by using the adiabatic regularization. The UV parts can be eliminated by using the adiabatic regularization, i.e., the following integral of Eq. (3.12) can converge

Z Λ 2 1 2 2 1 δφ div = 2 √ k |δχk| − dk, (3.35) 2π C (η) 2/|η| 2Wk where we take the mode function corresponding to the exact Bunch-Davies vacuum state given by rπ δχ = η1/2H(1) (kη) , (3.36) k 4 ν

1In de Sitter space, the scalar curvature becomes R = 12H2, and therefore the non-minimal coupling ξ provides the eﬀective mass-term m2 = 12H2ξ.

– 11 – (1) with Hν is the Hankel function of the first kind. By discarding the convergent terms, whose orders are O(m2), we can obtain the following expression √ Z H Z 2/|η| 2 1 2 2 1 2 2 δφ ren ' 2 k |δχk| dk + 2 k |δχk| dk, (3.37) 2π C (η) 0 2π C (η) H as well as the massless minimally coupled case. At the late-time limit (η ' 0 or Ht 1), the first integral converges to zero (see Ref.[59, 60] for the details) and obtain the following approximation √ Z 2/|η| 2 1 2 2 δφ ren ' 2 k |δχk| dk, 2π C (η) H 4 3H h 2 i ' 1 − e−2m t/3H . (3.38) 8π2m2 which coincides with Eq. (3.34) by using the physical cut-off. 2 Then, we briefly discuss the renormalized vacuum fluctuations δφ ren for m H by using the adiabatic regularization. In this case, the power spectrum on super- horizon scales is approximately given by [63]

H 2 H k 3 ∆2 (k) = . (3.39) δφ 2π m aH

For the very massive case m H, the amplitude of the power spectrum is suppressed and the spectrum on long wave modes rapidly drops down. Therefore, the massive field vacuum fluctuations are more inhomogeneous fluctuations and then, break the scale invariance of the spectrum of perturbations. In this case, we must pay attention 2 to the UV cut-off. For the very massive case m H, the adiabatic conditions Ωk > 0 satisfy for all k modes and we can estimate the renormalized vacuum fluctuations by eliminating the lowest-order adiabatic (WKB) approximation

δφ2 ' δφ2 − δφ2 , (3.40) ren Wk Ωk " # 1 Z Λ k2 Z Λ k2 = 2 dk − dk . (3.41) 4π C (η) 0 Wk 0 Ωk

By using Eq. (3.9), we can simply estimate the dominated terms as follows: " # 1 m2C00 Z Λ k2 5m4(C0)2 Z Λ k2 lim dk − dk Λ→∞ 2 5 7 4π C (η) 8 0 ωk 32 0 ωk " # 1 m2C00 Λ3 5m4(C0)2 5m2CΛ3 + 2Λ5 = lim − , Λ→∞ 4π2C (η) 8 3m2C(Λ2 + m2C)3/2 32 15m4C2(Λ2 + m2C)5/2 " # 1 1C0 2 C00 1 a00 R = − − = = . (3.42) 96π2C (η) 2 C C 48π2 a3 288π2

– 12 – 4 Renormalized vacuum ﬂuctuations via point-splitting regularization

The point-splitting regularization is the method of regularizing divergences as the point separation in the two-point function, and has been studied in detail in Ref.[30, 64, 65]. In this section, we review the renormalized vacuum ﬂuctuations by using the point-splitting regularization, and compare the results in the previous section. The regularized vacuum expectation values are expressed as [30] R 1 1 δφ2 = − 16π22 + + m2 + ξ − R reg 576π2 16π2 6 (4.1) 2µ2 R 3 3 ln + ln + 2γ − 1 + ψ + ν + ψ − ν . 12 µ2 2 2 where we take Bunch-Davies vacuum state, is the regularization parameter which corresponds with the point separation, µ is the renormalization scale, γ is Euler’s constant and ψ (z) = Γ0 (z) / Γ(z) is the digamma function. By using m2 and ξ renormalization, we have the following expression 1 12m2 δφ2 = − m2 ln ren 16π2 µ2 (4.2) 1 R 3 3 + m2 + ξ − R ln + ψ + ν + ψ − ν . 6 µ2 2 2

3 2 where the additive constant ψ 2 ± ν has been chosen so that δφ ren = 0 at the radiation-dominated Universe R = 0. In the massive ﬁeld theory, we can remove simply the renormalization scale µ to set µ2 = 12m2 1 1 R 3 3 δφ2 = m2 + ξ − R ln + ψ + ν + ψ − ν (4.3) ren 16π2 6 12m2 2 2 In the massless and nearly conformal coupling case m = 0 and ξ ' 1/6, we cannot 3 remove the renormalization scale µ and ψ 2 ± ν may be absorbed by the non-minimal coupling renormalization ξ − 1/6 R δφ2 = R ln . (4.4) ren 16π2 µ2 In the massless conformal coupling case m = 0 and ξ = 1/6, we can simply obtain the following expression 2 R H2 δφ2 = = . (4.5) ren 576π2 48π2 In the minimal coupling case ξ = 0 and we take massless limit m → 0 R2 3H4 δφ2 → = . (4.6) ren 384π2m2 8π2m2 2 2 2 This is equal to the thermal ﬂuctuations δφ ren = T /12 with the Gibbons-Hawking temperature T = H/2π experienced by a point observer inside the de Sitter horizon (see, e.g. Ref.[66, 67]).

– 13 – which corresponds with Eq. (3.38). Then, the digamma function ψ (z) for z 1 can be approximated as [65]

3 11 127 Re ψ + iz = log z + − + ··· . (4.7) 2 24z2 960z4

In the massive case m H,

H2 3 3 ln + ψ + ν + ψ − ν m2 2 2 H2 3 m 3 m ≈ ln + ψ + i + ψ − i , (4.8) m2 2 H 2 H 11 H2 127 H4 ≈ − + ··· . (4.9) 12 m2 480 m4 Therefore, the renormalized vacuum ﬂuctuations of the massive Higgs ﬁeld for m H is given as follows:

1 1 11 H2 127 H4 δφ2 = m2 + ξ − R − + ··· , (4.10) ren 16π2 6 12 m2 480 m4 ' O H2 , (4.11) which is consistent with Eq. (4.12). Therefore, the renormalized vacuum ﬂuctuations 2 δφ ren via the point-splitting regularization is equivalent to the results of Eq. (3.42) via the adiabatic regularization. Finally, we summarize the renormalized vacuum ﬂuc- 2 tuations δφ ren via the adiabatic regularization and the point-splitting regularization as follows:  H3t/4π2, (m = 0)  2 4 2 2 δφ ren ' 3H /8π m , (m H) (4.12)  2 2 H /24π . (m & H)

5 Electroweak vacuum instability from dynamical behavior of homogeneous Higgs ﬁeld and renormalized Higgs vacuum ﬂuctuations

In this section, we apply the results of Section2, Section3 and Section4 to the in- vestigation of the electroweak vacuum instability during inflation (de Sitter space) or after inflation. The vacuum instability of the electroweak false vacuum on the dynamical background is determined by the behavior of the homogeneous Higgs field φ and the inhomogeneous Higgs field fluctuations, i.e., the renormalized vacuum fluctuations 2 δφ ren. The behavior of the global and homogeneous Higgs field φ is governed by the effective Klein-Gordon equation

0 φ + Veﬀ (φ) = 0. (5.1)

– 14 – We can rewrite the eﬀective Klein-Gordon equation as the following

¨ ˙ 0 φ (t) + 3Hφ (t) + Veﬀ (φ (t)) = 0. (5.2)

0 2 If we approximate the effective potential as Veff (φ) = −meff φ, the behavior of the coherent Higgs field φ (t) is described as follows: ( √ meff t 1 −3H+ 9H2+4m2 t e (meff H) , φ (t) ∝ e 2 eff ' (5.3) m2 t/3H e eff (meff . H) .

The one-loop standard model Higgs eﬀective potential Veﬀ (φ) of curved space- time [18] in the ’t Hooft-Landau gauge and the MS scheme, is given as follows:

1 1 λ(µ) V (φ) = m2(µ)φ2 + ξ(µ)Rφ2 + φ4 (5.4) eﬀ 2 2 4 9 2 X ni M (φ) + M 4 (φ) log i − C , 64π2 i µ2 i i=1 where 2 2 0 Mi (φ) = κiφ + κi + θiR. (5.5)

0 The coeﬃcients ni, κi, κi and θi are given by Table I of Ref.[18]. Furthermore, the β-function for the non-minimal coupling ξ (µ) in the standard model ignoring gravity, is given by 1 2 3 02 9 2 βξ = (ξ − 1/6) 6λ + 3y − g − g . (5.6) (4π)2 t 4 4

The running of the non-minimal coupling ξ (µ) can be obtained by integrating βξ

1 1 ξ (µ) = + ξ − F (µ) , (5.7) 6 EW 6 where F (µ) depend on the renormalization scale µ. If we have the nearly minimal −2 coupling ξEW . O (10 ) at the electroweak scale, the running non-minimal coupling ξ (µ) becomes negative at some scale [18]. On the other hand, we can take the initial condition of the running non-minimal coupling ξ (µ) at the Planck scale [24]. If the 3 Hubble scale is larger than the instability scale i.e., H > ΛI and ξ (H) < 0, the 0 Higgs effective potential in de Sitter space becomes negative, i.e., Veff (φ) . 0, and the homogeneous Higgs field φ on the entire Universe rolls down to the negative-energy Planck-scale true vacuum.

3 We deﬁne the instability scale ΛI as the derivative of the standard model Higgs eﬀective potential in Minkowski space-time becomes negative at the scale and the current experiments of the Higgs boson mass, mh = 125.09 ± 0.21 (stat) ± 0.11 (syst) GeV [1–4] and top quark mass, mt = 10 11 173.34 ± 0.27 (stat) ± 0.71 (syst) GeV [5] suggest the instability scale ΛI = 10 ∼ 10 GeV [68].

– 15 – However, as mentioned in Section2, we must shift the Higgs field φ2 → φ2 + 2 δφ ren in order to include the backreaction term via the renormalized vacuum fluctuations and the one-loop standard model Higgs effective potential in curved space-time is modified as follows: 1 1 1 λ(µ) V (φ) = m2(µ)φ2 + ξ(µ)Rφ2 + λ(µ) δφ2 φ2 + φ4 (5.8) eff 2 2 2 ren 4 9 2 X ni M (φ) + M 4 (φ) log i − C . 64π2 i µ2 i i=1 with 2 2 2 0 Mi (φ) = κiφ + κi δφ ren + κi + θiR. (5.9) Here, we must choose the appropriate scale µ in order to suppress the higher order 2 2 corrections of log Mi (φ)/µ . In the case of flat spacetime R = 0, it is known that µ2 = φ2 is a good choice to suppress the high order log-corrections. In de Sitter space, 2 2 2 2 2 2 however, we must choose µ = φ + R + δφ ren = φ + 12H + δφ ren to suppress 2 2 1/2 the log-corrections. Therefore, if we have µ ' 12H + δφ ren > ΛI , the quartic term λ(µ)φ4/4 makes negative contribution to the effective potential. The Higgs field phenomenologically acquires various effective masses during inflation or after inflation, e.g. the inflaton-Higgs coupling λφS provides an extra contri- 2 2 bution to the Higgs mass meff = λφSS where S is the inflaton field. However, in this work, we restrict our attention to the simple case that the Higgs field only couples to the gravity via the non-minimal Higgs-gravity coupling ξ(µ) and we disregard other inflationary effective mass-terms. For convenience, we only consider the inflationary 2 2 effective mass-term meff = ξ(µ)R = 12ξ(µ)Hinf and use the results of Eq. (4.12), the renormalized vacuum field fluctuations are given by ( H2 /32π2ξ(µ), (ξ(µ) O (10−1)) δφ2 ' inf (5.10) ren 2 2 −1 Hinf /24π . (ξ(µ) & O (10 ))

2 2 1/2 4 Here, if we have µ ' 12Hinf + δφ ren > ΛI , we can infer the sign of Veff (φ) 2 2 by using the relations ξ(µ)R < λ(µ) δφ ren . If we assume ξ(µ)R = ξ(µ)12Hinf , 2 2 2 λ(µ) δφ ren ' λ(µ)Hinf /32π ξ(µ) and λ(µ) ' −0.01, we obtain the constraint on −3 p 2 the non-minimal coupling ξ(µ) . O (10 ) where Hinf > 32π ξ (µ)ΛI . In this case, the homogeneous Higgs field φ goes out to the negative Planck-energy vacuum state, and therefore, the excursion of the homogeneous Higgs field φ to the Planck-scale true vacuum can terminate inflation and cause an immediate collapse of the Universe. 2 On the other hand, the inhomogeneous Higgs field fluctuations δφ ren can cause the vacuum transition of the Universe [13–25]. If the inhomogeneous and local Higgs

4 2 2 1/2 4 In the case 12Hinf + δφ ren < ΛI , the quartic term λ(µ)φ /4 makes positive contribution to the effective potential unless φ > ΛI . Therefore, the homogeneous Higgs field φ cannot classically go out to the Planck-scale vacuum state. However, it is possible to generate AdS domains or bubbles via the inhomogeneous Higgs field fluctuations shown in (5.17).

– 16 – field gets over the hill of the effective potential, the local Higgs fields classically roll down into the negative Planck-energy true vacuum and catastrophic Anti-de Sitter (AdS) domains are formed. Note that not all AdS domains formed during inflation threaten the existence of the Universe [22, 24], which highly depends on the evolution of the AdS domains at the end of inflation (see Ref.[24] for the details). The AdS domains can either shrink or expand, eating other regions of the electroweak vacuum. Although the high-scale inflation can generate more expanding AdS domains than shrinking domains during inflation, such domains never overcome the inflationary expansion of the Universe, i.e., one AdS domain cannot terminate the inflation of the Universe 5. However, after inflation, some AdS domains expand and consume the entire Universe. Therefore, the existence of AdS domains on our Universe is catastrophic, and so we focus on the conditions not to be generated during or after inflation. We assume that the probability distribution function of the Higgs field fluctuations is Gaussian, i.e.

2 ! 2 1 φ P φ, δφ = exp − 2 . (5.11) ren q 2 2 δφ ren 2π δφ ren

By using Eq. (5.11), the probability that the electroweak vacuum survives can be obtained as

Z φmax 2 P (φ < φmax) ≡ P φ, δφ ren dφ, (5.12) −φmax   φmax = erf   . (5.13) q 2 2 δφ ren where φmax defined as the Higgs effective potential Veff (φ) given by Eq. (5.8) takes its maximal value 6.

5 The expansion of AdS domains or bubbles never takes over the expansion of the inflationary dS space [24], and therefore, it is impossible that one AdS domain terminates the inflation of the Universe. However, if the non-inflating domains or the AdS domains dominates all the space of the Universe [69], the inflating space would crack, and the inflation of all the space of the Universe finally comes to an end. 6 The Higgs effective potential with the large effective mass-term can be approximated by

2! 1 2 2 1 φ Veff (φ) ' meff φ 1 − , 2 2 φmax where φmax is given by s m2 φ = − eff . max λ (µ)

Our approximation φmax ' 10 meﬀ is numerically valid for the one-loop eﬀective potential.

– 17 – On the other hand, the probability that the inhomogeneous Higgs ﬁeld falls into the true vacuum can be expressed as   φmax P (φ > φmax) = 1 − erf   , (5.14) q 2 2 δφ ren

q 2 φ2 2 δφ ren − max 2hδφ2i ' e ren . (5.15) πφmax The vacuum decay probability is given by

3Nhor e P (φ > φmax) < 1, (5.16) where e3Nhor corresponds to the physical volume of our universe at the end of the inﬂation, and we take the e-folding number Nhor ' NCMB ' 60. Imposing Eq. (5.15) on the condition shown in (5.16), we obtain the relation

2 δφ ren 1 2 < . (5.17) φmax 6Nhor Now, we consider the condition shown in (5.17) by using the Higgs effective potential in de-Sitter space-time given by Eq. (5.8) and the Higgs field vacuum fluctuations by Eq. (4.12). In the same way as our previous work [25], we can numerically obtain the −2 p 2 constraint of the non-minimal coupling ξ(µ) . O (10 ) where Hinf > 32π ξ (µ)ΛI . Therefore, the catastrophic Anti-de Sitter (AdS) domains or bubbles from the high- scale inflation can be avoided if the relatively large non-minimal Higgs-gravity coupling (or e.g. inflaton-Higgs coupling λφS) is introduced. Here, we summarize the conclusions obtained in this section as follows:

−3 p 2 • In ξ(µ) . O (10 ) and Hinf > 32π ξ (µ)ΛI , the Higgs effective potential 2 during inflation is destabilized by the backreaction term λ(µ) δφ ren, which overcome the stabilization term ξ(µ)R. In this case, the effective potential be- 0 comes negative, i.e. Veff (φ) . 0, the excursion of the homogeneous Higgs field φ to the negative Planck-energy vacuum state terminates the inflation of the Universe and cause a catastrophic collapse.

−3 −2 p 2 • In O (10 ) . ξ(µ) . O (10 ) and Hinf > 32π ξ (µ)ΛI , The inflationary effec- 2 tive mass ξ(µ)R = 12ξ(µ)Hinf can raise and stabilize the Higgs effective potential during inflation. Thus, the dangerous motion of the homogeneous Higgs field φ 2 cannot occur. However, the inhomogeneous Higgs field fluctuations δφ ren generate the catastrophic Anti-de Sitter (AdS) domains or bubbles, which finally cause the vacuum transition of the Universe.

−2 p 2 • In ξ(µ) & O (10 ) or Hinf < 32π ξ (µ)ΛI , the Higgs eﬀective potential sta- bilizes and the catastrophic Anti-de Sitter (AdS) domains or bubbles are not formed during inﬂation.

– 18 – However, after inflation, the effective mass-terms ξ(µ)R via the non-minimal Higgs-gravity coupling drops rapidly and sometimes become negative. Therefore, the effect of the stabilization via ξ(µ)R disappears and the Higgs effective poten- 2 tial becomes rather unstable due to the terms of λ(µ) δφ ren or ξ(µ)R with µ ' 2 1/2 R + δφ ren > ΛI . Furthermore, the non-minimal Higgs-gravity coupling can generate the large Higgs field vacuum fluctuations via tachyonic resonance [25, 70, 71], thus, the Higgs effective potential is destabilized, or the catastrophic Anti-de Sitter (AdS) domains or bubbles are formed during subsequent preheating stage. In the rest of this section, we briefly discuss the instability of the electroweak false vacuum after inflation. Just after inflation, the inflaton field S begins coherently oscillating near the minimum of the inflaton potential Vinf (S) and produces extremely a huge amount of massive bosons via the parametric or the tachyonic resonance. This temporal non- thermal stage is called preheating [37], and is essentially different from the subsequent stages of the reheating and the thermalization. For simplicity, we approximate the inflaton potential as the quadratic form 1 V (S) = m2 S2. (5.18) inf 2 S In this case, the inflaton field S classically oscillates as r 8 Mpl S (t) = Φ sin (mSt), Φ = , (5.19) 3 mSt

18 where the reduced Planck mass is Mpl = 2.4 × 10 GeV. If the inflaton field S dominates the energy density and the pressure of the Universe, i.e., during inflation or preheating stage, the scalar curvature R(t) is written by

h i 1 ˙ 2 R (t) = 2 4Vinf (S) − S , (5.20) Mpl 2 2 mSΦ 2 ' 2 3 sin (mSt) − 1 . (5.21) Mpl

When the inflaton field S oscillates as Eq. (5.19), the effective mass ξ(µ)R drasti- cally changes between positive and negative values. Therefore, the Higgs field vacuum fluctuations grows extremely rapidly via the tachyonic resonance, which is called geometric preheating [72, 73]. The general equation for k modes of the Higgs field during preheating is given as follows:

d2 a3/2δφ 2 k k 0 1 3 ˙ 2 + 2 + Veﬀ (φ) + 2 − ξ S dt a Mpl 8 (5.22) ! 1 3 3/2 − 2 − 4ξ V (S) a δφk = 0. Mpl 4

– 19 – Eq. (5.22) can be reduced to the following Mathieu equation

d2 a3/2δφ k + (A − 2q cos 2z) a3/2δφ = 0, (5.23) dz2 k k where we take z = mSt and Ak and q are given as

2 0 2 k Veff (φ) Φ Ak = 2 2 + 2 + 2 ξ, (5.24) a mS mS 2Mpl 3Φ2 1 q = 2 ξ − . (5.25) 4Mpl 4 The solutions of the Mathieu equation via the non-minimal coupling in Eq. (5.23) show the tachyonic (broad) resonance when q & 1 or the narrow resonance when q < 1. In 2 2 the tachyonic resonance regime, where q & 1, i.e. Φ ξ & Mpl, the tachyonic resonance 2 extremely amplifies the Higgs vacuum fluctuations at the end of inflation as δφ ren 2 O (H (t)). In the context of preheating, Ak and q are z-dependent function due to the expansion of the Universe, making it very difficult to derive analytical estimation −6 2 (see e.g. Ref.[71]). If we take mS ' 7 × 10 Mpl assuming chaotic inflation with a quadratic potential, we can numerically obtain the condition of the tachyonic resonance as ξ(µ) & O (10) (see e.g. Ref.[25] for the details). In the narrow resonance regime, 2 2 where q < 1, i.e. Φ ξ < Mpl, the tachyonic resonance cannot occur, and therefore, the 2 2 Higgs vacuum fluctuations after inflation decrease as δφ ren ' O (H (t)) due to the expansion of the Universe. Here, we briefly summarize the results of the Higgs field vacuum fluctuations after inflation as follows: ( δφ2 O (H2 (t)) , Φ2ξ M 2 ren & pl (5.26) 2 2 2 2 δφ ren ' O (H (t)) . Φ ξ < Mpl

2 1/2 In the same way as the inflation stage, if we have µ ' R + δφ ren > ΛI , we shall 2 infer the sign of Veff (φ) by using the inequality ξ(µ)R (t) < λ(µ) δφ ren , the scalar curvature |R (t)| ' 3H2 (t) and the self-coupling λ(µ) ' −0.01. If the tachyonic reso- 0 nance happens, it is clear that the effective potential becomes negative, i.e. Veff (φ) . 0, and then, the homogeneous Higgs field φ goes out to the negative Planck-energy vacuum state. On the other hand, in the narrow resonance, it cannot happen the same 2 catastrophe, where ξ(µ)R (t) < λ(µ) δφ ren , because the inhomogeneous Higgs field 2 2 fluctuations after inflation are given as δφ ren ' O (H (t)) due to the expansion of the Universe, and the scalar curvature decreases as |R (t)| ' 3H2 (t). Therefore, the narrow resonance does not destabilize the effective potential during preheating. However, we recall that the scalar curvature R (t) shown in (5.21) oscillates during each cycle t ' 1/mS. The stabilization of ξ(µ)R (t) to the coherent Higgs field generally does not work at the end of the inflation, because ξ(µ)R (t) changes sign during each oscillation cycle. If the oscillation time-scale t ' 1/mS is relatively long, the curvature term ξ(µ)R (t) can accelerate catastrophic motion of the coherent Higgs field φ (t) immediately after the end of the inflation. Here, we briefly discuss the development

– 20 – of the coherent Higgs field φ (t) at the end of the inflation. In one oscillation time- 2 scale t ' 1/mS, we can simply approximate the effective mass as meff ' ξ(µ)R (t) ≈ 2 −3ξ(µ)Hend. By using Eq. (5.3), the classical behavior of the coherent Higgs field φ (t) at the end of the inflation can be approximated as

2 (3ξ(µ)H )t/3Hend φ (t) ' φend · e end , (5.27) 2 (3ξ(µ)H /3HendmS ) ' φend · e end , (5.28) (ξ(µ)Hend/mS ) ' φend · e , (5.29) where the coherent Higgs field φend at the end of inflation is generally not zero, and 2 2 corresponds to the Higgs field vacuum fluctuations δφ ren ' O (Hend) at the end of 7 p inflation. Therefore, if we have φ (t) > φmax , i.e. Hend/mS & (log 10 3ξ (µ))/ξ (µ), the almost coherent Higgs fields φ (t) produced at the end of the inflation go out to the negative Planck-vacuum state and cause a catastrophic collapse of the Universe. Furthermore, the inflation produces an enormous amount of causally disconnected horizon-size domains and our observable Universe contains e3Nhor of them. Thus, we 2 can consider one domain which has the large Higgs field fluctuations 6Nhor δφ ren by using Eq. (5.17). The classical motion of the coherent Higgs field on such domain can be given as the following

q 2 2 (3ξ(µ)Hend)t/3Hend φ (t) ' 6Nhor δφ ren · e , (5.30)

(ξ(µ)Hend/mS ) ' 10Hend · e , (5.31) where we take the e-folding number Nhor = 60. Therefore, if we have φ (t) > φmax i.e., p Hend/mS & (log 3ξ (µ))/ξ (µ), the coherent Higgs field φ (t) on such domain goes out to the negative Planck-vacuum state and forms the catastrophic AdS domains or bubbles, which finally cause the vacuum transition of the Universe. That conclusion depends strongly on the non-minimal coupling ξ(µ), the oscillation time-scale t ' 1/mS and the Hubble scale Hend at the end of the inflation. Thus, the large non-minimal Higgs-gravity coupling ξ(µ) can destabilize the behavior of the coherent Higgs field after the end of the inflation. However, if the curvature oscillation is very fast, the curvature mass-term ξ(µ)R (t) cannot generate the exponential growth of the coherent Higgs field φ (t) after inflation. After inflation, the inflaton field S oscillates and produces a huge amount of elementary particles. These particles produced during preheating stage interact with each other and eventually form a thermal plasma. We comment that thermal effects during reheating stage raise the effective Higgs potential via the extra effective mass 2 2 meff = O (T ). The one-loop thermal corrections to the Higgs effective potential is given as follows:

4 Z ∞ √ X niT 2 − k2+M 2(φ)/T 2 ∆V (φ, T ) = dkk ln 1 ∓ e i . (5.32) eﬀ 2π2 i=W,Z,t,φ 0

7 p p By using φmax ' 10meﬀ , we can simply approximate φmax ' 10 ξ(µ) |R (t)| ' 10Hend 3ξ(µ)

– 21 – It is possible to lift up the Higgs effective potential throughout the thermal phase via the large thermal effective mass. However, it is impossible to prevent the exponential growth of the coherent Higgs field φ (t) after inflation, or the large Higgs vacuum fluctuations via the tachyonic resonance during preheating by the thermal effects, because there can exist a considerable time lag for the production of the thermal bath on the oscillation of the inflaton. Furthermore, the thermal fluctuations of the Higgs field 2 2 hφ iT ' T /12 might generate the catastrophic AdS domains or bubbles if T > ΛI (see e.g. Ref.[25, 74–77] for the details). Thus, the thermal effects cannot generally guarantee the stability of the electroweak vacuum in the inflationary Universe. Here, we summarize the conclusions obtained by the above discussion as follows: p • In Hend > ΛI and Hend/mS & (log 10 3ξ (µ))/ξ (µ), the almost coherent Higgs fields φ (t) generated at the end of the inflation exponentially grow and finally go out to the Planck-energy vacuum state, which leads to the catastrophic collapse of the Universe. p • In Hend > ΛI and Hend/mS & (log 3ξ (µ))/ξ (µ), the coherent Higgs field φ (t) on one horizon-size domain exponentially grows at the end of the inflation and forms catastrophic AdS domains or bubbles, which finally cause the vacuum transition of the Universe.

2 2 • In the tachyonic resonance regime Φ ξ & Mpl, the Higgs field fluctuations ex- 2 2 tremely increase as δφ ren O (H (t)). Therefore, the effective potential 0 becomes negative i.e. Veff (φ) . 0, and the excursion of the homogeneous Higgs field φ (t) to the negative Planck-energy vacuum state occurs during preheating stage and cause the catastrophic collapse of the Universe.

2 2 • In the narrow resonance regime Φ ξ < Mpl, the Higgs field fluctuations decrease 2 2 as δφ ren ' O (H (t)) due the expansion of the Universe, and therefore, it is improbable to destabilize the effective potential during preheating stage.

−2 The relative large non-minimal Higgs-gravity coupling as ξ(µ) & O (10 ) can stabilize the effective Higgs potential and suppress the formations of the catastrophic AdS domains or bubbles during inflation. However, after inflation, the effective mass-term ξ(µ)R via the non-minimal coupling drops rapidly, sometimes become negative and lead to the exponential growth of the coherent Higgs field φ (t) at the end of inflation, or the large Higgs vacuum fluctuations via the tachyonic resonance during preheating stage. Therefore, the non-minimally coupling ξ(µ) cannot prevent the catastrophic scenario during or after inflation. After all, if we have large Hubble scale H > ΛI , meaning the relatively large tensor-to-scalar ratio rT from the polarization measurements of the cosmic microwave background radiation, the safety of our electroweak vacuum is inevitably threatened during inflation or after inflation by the behavior of the homogeneous Higgs field φ or the generations of the catastrophic AdS domains or bubbles. We can simply avoid this situation by assuming the inflaton-Higgs couplings λφS [15], the inflationary stabilizations [78, 79], or the high-order corrections from GUT

– 22 – or Planck-scale new physics [80–84] etc. In any case, however, the electroweak vacuum instability from inﬂation gives tight constraints on the beyond the standard model.

6 Conclusion

In this work, we have investigated the electroweak vacuum instability during or after inflation. In the inflationary Universe, i.e., de Sitter space, the vacuum field fluctuations δφ2 enlarge in proportion to the Hubble scale H2. Therefore, the large inflationary vacuum fluctuations of the Higgs field δφ2 is potentially catastrophic to trigger the vacuum decay to a negative-energy Planck-scale vacuum state and cause an immediate collapse of the Universe. However, the vacuum field fluctuations δφ2 , i.e., the vacuum expectation values have a ultraviolet divergence, which is a well-known fact in quantum field theory, and therefore a renormalization is necessary to estimate the physical effects of the vacuum transition. Thus, we have revisited the electroweak vacuum instability during or after inflation from the legitimate perspective of QFT in curved space-time. We have discussed dynamics of homogeneous Higgs field φ determined by the effective potential Veff (φ) in curved space-time and the renormalized 2 vacuum fluctuations δφ ren by using adiabatic regularization and point-splitting regularization, where we assumed the simple scenario that the Higgs field only couples the gravity via the non-minimal Higgs-gravity coupling ξ(µ). In this scenario, we conclude that the Hubble scale must be smaller than H < ΛI , or the Higgs effective potential in curved space-time is stabilized below the Planck scale by a new physics beyond the standard model. Otherwise, our electroweak vacuum is inevitably threatened by the catastrophic behavior of the homogeneous Higgs field φ or the formations of AdS domains or bubbles during or after inflation.

Acknowledgments

This work is supported in part by MEXT KAKENHI No.15H05889 and No.JP16H00877 (K.K.), and JSPS KAKENHI Nos.26105520 and 26247042 (K.K.).

References

[1] collaboration, G. Aad et al., Combined Measurement of the Higgs ATLAS, CMS √ Boson Mass in pp Collisions at s = 7 and 8 TeV with the ATLAS and CMS Experiments, Phys. Rev. Lett. 114 (2015) 191803,[1503.07589]. [2] ATLAS collaboration, G. Aad et al., Measurements of Higgs boson production and couplings in diboson final states with the ATLAS detector at the LHC, Phys.Lett. B726 (2013) 88–119,[1307.1427]. [3] CMS collaboration, S. Chatrchyan et al., Measurement of the properties of a Higgs boson in the four-lepton final state, Phys.Rev. D89 (2014) 092007,[1312.5353]. [4] P. P. Giardino, K. Kannike, I. Masina, M. Raidal and A. Strumia, The universal Higgs fit, JHEP 1405 (2014) 046,[1303.3570].

– 23 – [5] T. ATLAS, CDF, CMS and D0 Collaborations, First combination of Tevatron and LHC measurements of the top-quark mass, ArXiv e-prints (Mar., 2014) , [1403.4427]. [6] I. Yu. Kobzarev, L. B. Okun and M. B. Voloshin, Bubbles in Metastable Vacuum, Sov. J. Nucl. Phys. 20 (1975) 644–646. [7] S. R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D15 (1977) 2929–2936. [8] C. G. Callan, Jr. and S. R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D16 (1977) 1762–1768. [9] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice et al., Higgs mass and vacuum stability in the Standard Model at NNLO, JHEP 1208 (2012) 098, [1205.6497]. [10] G. Isidori, G. Ridolfi and A. Strumia, On the metastability of the standard model vacuum, Nucl.Phys. B609 (2001) 387–409,[hep-ph/0104016]. [11] J. Ellis, J. Espinosa, G. Giudice, A. Hoecker and A. Riotto, The Probable Fate of the Standard Model, Phys.Lett. B679 (2009) 369–375,[0906.0954]. [12] J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori, A. Riotto et al., Higgs mass implications on the stability of the electroweak vacuum, Phys.Lett. B709 (2012) 222–228,[1112.3022]. [13] J. Espinosa, G. Giudice and A. Riotto, Cosmological implications of the Higgs mass measurement, JCAP 0805 (2008) 002,[0710.2484]. [14] M. Fairbairn and R. Hogan, Electroweak Vacuum Stability in light of BICEP2, Phys.Rev.Lett. 112 (2014) 201801,[1403.6786]. [15] O. Lebedev and A. Westphal, Metastable Electroweak Vacuum: Implications for Inflation, Phys.Lett. B719 (2013) 415–418,[1210.6987]. [16] A. Kobakhidze and A. Spencer-Smith, Electroweak Vacuum (In)Stability in an Inflationary Universe, Phys.Lett. B722 (2013) 130–134,[1301.2846]. [17] K. Enqvist, T. Meriniemi and S. Nurmi, Generation of the Higgs Condensate and Its Decay after Inflation, JCAP 1310 (2013) 057,[1306.4511]. [18] M. Herranen, T. Markkanen, S. Nurmi and A. Rajantie, Spacetime curvature and the Higgs stability during inflation, Phys.Rev.Lett. 113 (2014) 211102,[1407.3141]. [19] A. Kobakhidze and A. Spencer-Smith, The Higgs vacuum is unstable, 1404.4709. [20] K. Kamada, Inflationary cosmology and the standard model Higgs with a small Hubble induced mass, Phys. Lett. B742 (2015) 126–135,[1409.5078]. [21] K. Enqvist, T. Meriniemi and S. Nurmi, Higgs Dynamics during Inflation, JCAP 1407 (2014) 025,[1404.3699]. [22] A. Hook, J. Kearney, B. Shakya and K. M. Zurek, Probable or Improbable Universe? Correlating Electroweak Vacuum Instability with the Scale of Inflation, JHEP 1501 (2015) 061,[1404.5953]. [23] J. Kearney, H. Yoo and K. M. Zurek, Is a Higgs Vacuum Instability Fatal for High-Scale Inflation?, 1503.05193.

– 24 – [24] J. R. Espinosa, G. F. Giudice, E. Morgante, A. Riotto, L. Senatore et al., The cosmological Higgstory of the vacuum instability, 1505.04825. [25] K. Kohri and H. Matsui, Higgs vacuum metastability in primordial inflation, preheating, and reheating, 1602.02100. [26] W. E. East, J. Kearney, B. Shakya, H. Yoo and K. M. Zurek, Spacetime Dynamics of a Higgs Vacuum Instability During Inflation, 1607.00381. [27] A. D. Linde, D. A. Linde and A. Mezhlumian, From the Big Bang theory to the theory of a stationary universe, Phys. Rev. D49 (1994) 1783–1826,[gr-qc/9306035]. [28] L. Parker, Amplitude of Perturbations from Inflation, hep-th/0702216. [29] I. Agullo, J. Navarro-Salas, G. J. Olmo and L. Parker, Remarks on the renormalization of primordial cosmological perturbations, Phys. Rev. D84 (2011) 107304,[1108.0949]. [30] A. Vilenkin and L. H. Ford, Gravitational Effects upon Cosmological Phase Transitions, Phys. Rev. D26 (1982) 1231. [31] J. P. Paz and F. D. Mazzitelli, Renormalized Evolution Equations for the Back Reaction Problem With a Selfinteracting Scalar Field, Phys. Rev. D37 (1988) 2170–2181. [32] A. L. Maroto and F. Prada, Higgs effective potential in a perturbed Robertson-Walker background, Phys. Rev. D90 (2014) 123541,[1410.0302]. [33] F. D. Albareti, A. L. Maroto and F. Prada, Gravitational perturbations of the Higgs field, 1602.02776. [34] A. Ringwald, Evolution Equation for the Expectation Value of a Scalar Field in Spatially Flat Rw Universes, Annals Phys. 177 (1987) 129. [35] S. Sinha and B. L. Hu, Symmetry Behavior in Cosmological Space-times: Effect of Slowly Varying Background Fields, Phys. Rev. D38 (1988) 2423–2433. [36] W.-H. Huang, Quantum field effects on cosmological phase transition in anisotropic space-times, Class. Quant. Grav. 10 (1993) 2021–2033,[gr-qc/0401046]. [37] L. Kofman, A. D. Linde and A. A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D56 (1997) 3258–3295,[hep-ph/9704452]. [38] L. Parker and D. J. Toms, New form for the coincidence limit of the feynman propagator, or heat kernel, in curved spacetime, Phys. Rev. D 31 (Feb, 1985) 953–956. [39] I. Jack and L. Parker, Proof of summed form of proper-time expansion for propagator in curved space-time, Phys. Rev. D 31 (May, 1985) 2439–2451. [40] D. J. Toms, The Effective Action and the Renormalization Group Equation in Curved Space-time, Phys. Lett. B126 (1983) 37–40. [41] I. L. Buchbinder and S. D. Odintsov, EFFECTIVE POTENTIAL IN A CURVED SPACE-TIME, Sov. Phys. J. 27 (1984) 554–558. [42] B. L. Hu and D. J. O’Connor, Effective Lagrangian for λφ4 Theory in Curved Space-time With Varying Background Fields: Quasilocal Approximation, Phys. Rev. D30 (1984) 743. [43] J. Balakrishnan and D. J. Toms, Gauge independent effective action for minimally coupled quantum fields in curved space, Phys. Rev. D46 (1992) 4413–4420.

– 25 – [44] T. Muta and S. D. Odintsov, Model dependence of the nonminimal scalar graviton effective coupling constant in curved space-time, Mod. Phys. Lett. A6 (1991) 3641–3646. [45] K. Kirsten, G. Cognola and L. Vanzo, Effective Lagrangian for selfinteracting scalar field theories in curved space-time, Phys. Rev. D48 (1993) 2813–2822, [hep-th/9304092]. [46] E. Elizalde and S. D. Odintsov, Renormalization group improved effective potential for gauge theories in curved space-time, Phys. Lett. B303 (1993) 240–248, [hep-th/9302074]. [47] E. Elizalde and S. D. Odintsov, Renormalization group improved effective Lagrangian for interacting theories in curved space-time, Phys. Lett. B321 (1994) 199–204, [hep-th/9311087]. [48] E. Elizalde and S. D. Odintsov, Renormalization group improved effective potential for interacting theories with several mass scales in curved space-time, Z. Phys. C64 (1994) 699–708,[hep-th/9401057]. [49] E. Elizalde and S. D. Odintsov, Renormalization group improved effective potential for finite grand unified theories in curved space-time, Phys. Lett. B333 (1994) 331–336, [hep-th/9403132]. [50] E. Elizalde, K. Kirsten and S. D. Odintsov, Effective Lagrangian and the back reaction problem in a selfinteracting O(N) scalar theory in curved space-time, Phys. Rev. D50 (1994) 5137–5147,[hep-th/9404084]. [51] E. V. Gorbar and I. L. Shapiro, Renormalization group and decoupling in curved space, JHEP 02 (2003) 021,[hep-ph/0210388]. [52] E. V. Gorbar and I. L. Shapiro, Renormalization group and decoupling in curved space. 2. The Standard model and beyond, JHEP 06 (2003) 004,[hep-ph/0303124]. [53] E. V. Gorbar and I. L. Shapiro, Renormalization group and decoupling in curved space. 3. The Case of spontaneous symmetry breaking, JHEP 02 (2004) 060, [hep-ph/0311190]. [54] O. Czerwinska, Z. Lalak and L. Nakonieczny, Stability of the effective potential of the gauge-less top-Higgs model in curved spacetime, JHEP 11 (2015) 207,[1508.03297]. [55] T. S. Bunch, Adiabatic regularisation for scalar fields with arbitrary coupling to the scalar curvature, Journal of Physics A: Mathematical and General 13 (1980) 1297. [56] S. A. Fulling, L. Parker and B. L. Hu, Conformal energy-momentum tensor in curved spacetime: Adiabatic regularization and renormalization, Phys. Rev. D 10 (Dec, 1974) 3905–3924. [57] S. Fulling and L. Parker, Renormalization in the theory of a quantized scalar field interacting with a robertson-walker spacetime, Annals of Physics 87 (1974) 176 – 204. [58] L. Parker and S. A. Fulling, Adiabatic regularization of the energy-momentum tensor of a quantized field in homogeneous spaces, Phys. Rev. D 9 (Jan, 1974) 341–354. [59] J. Haro, Calculation of the renormalized two-point function by adiabatic regularization, Theor. Math. Phys. 165 (2010) 1490–1499. [60] J. Haro, Topics in Quantum Field Theory in Curved Space, 1011.4772.

– 26 – [61] N. D. Birrell, The application of adiabatic regularization to calculations of cosmological interest, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 361 (1978) 513–526, [http://rspa.royalsocietypublishing.org/content/361/1707/513.full.pdf]. [62] P. R. Anderson and L. Parker, Adiabatic Regularization in Closed Robertson-walker Universes, Phys. Rev. D36 (1987) 2963. [63] A. Riotto, Inflation and the theory of cosmological perturbations, in Astroparticle physics and cosmology. Proceedings: Summer School, Trieste, Italy, Jun 17-Jul 5 2002, pp. 317–413, 2002. hep-ph/0210162. [64] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 1984. [65] T. S. Bunch and P. C. W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A360 (1978) 117–134. [66] G. Acquaviva, L. Bonetti, L. Vanzo and S. Zerbini, Vacuum fluctuation of conformally coupled scalar field in FLRW space-times, Phys. Rev. D89 (2014) 084031,[1402.0718]. [67] N. Obadia, How hot are expanding universes?, Phys. Rev. D78 (2008) 083532, [0804.2890]. [68] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala et al., Investigating the near-criticality of the Higgs boson, JHEP 1312 (2013) 089,[1307.3536]. [69] Y. Sekino, S. Shenker and L. Susskind, On the Topological Phases of Eternal Inflation, Phys.Rev. D81 (2010) 123515,[1003.1347]. [70] M. Herranen, T. Markkanen, S. Nurmi and A. Rajantie, Spacetime curvature and Higgs stability after inflation, 1506.04065. [71] Y. Ema, K. Mukaida and K. Nakayama, Fate of Electroweak Vacuum during Preheating, 1602.00483. [72] B. A. Bassett and S. Liberati, Geometric reheating after inflation, Phys. Rev. D58 (1998) 021302,[hep-ph/9709417]. [73] S. Tsujikawa, K.-i. Maeda and T. Torii, Resonant particle production with nonminimally coupled scalar fields in preheating after inflation, Phys. Rev. D60 (1999) 063515,[hep-ph/9901306]. [74] G. W. Anderson, New Cosmological Constraints on the Higgs Boson and Top Quark Masses, Phys. Lett. B243 (1990) 265–270. [75] P. B. Arnold and S. Vokos, Instability of hot electroweak theory: bounds on m(H) and M(t), Phys. Rev. D44 (1991) 3620–3627. [76] J. R. Espinosa and M. Quiros, Improved metastability bounds on the standard model Higgs mass, Phys. Lett. B353 (1995) 257–266,[hep-ph/9504241]. [77] L. Delle Rose, C. Marzo and A. Urbano, On the fate of the Standard Model at finite temperature, 1507.06912. [78] Y. Ema, K. Mukaida and K. Nakayama, Electroweak Vacuum Stabilized by Moduli during/after Inflation, 1605.07342.

– 27 – [79] M. Kawasaki, K. Mukaida and T. T. Yanagida, A Simple Cosmological Solution to the Higgs Instability Problem in the Chaotic Inﬂation and Formation of Primordial Black Holes, 1605.04974. [80] V. Branchina and E. Messina, Stability, Higgs Boson Mass and New Physics, Phys. Rev. Lett. 111 (2013) 241801,[1307.5193]. [81] Z. Lalak, M. Lewicki and P. Olszewski, Higher-order scalar interactions and SM vacuum stability, JHEP 05 (2014) 119,[1402.3826]. [82] V. Branchina, Stability of the EW vacuum, Higgs boson, and new physics, in Proceedings, 49th Rencontres de Moriond on QCD and High Energy Interactions, pp. 25–28, 2014. 1405.7864. [83] V. Branchina, E. Messina and A. Platania, Top mass determination, Higgs inﬂation, and vacuum stability, JHEP 09 (2014) 182,[1407.4112]. [84] V. Branchina, E. Messina and M. Sher, Lifetime of the electroweak vacuum and sensitivity to Planck scale physics, Phys. Rev. D91 (2015) 013003,[1408.5302].

– 28 –