sign in sign up
<<
Home , Inflaton

PHYSICAL REVIEW D 98, 015020 (2018)

Light completing Higgs

Hyun Min Lee* Department of Physics, Chung-Ang University, Seoul 06974, Korea

(Received 22 February 2018; published 16 July 2018)

We consider the extension of the with a light inflaton where both the unitarity problem in Higgs inflation and the vacuum instability problem are resolved. The linear nonminimal coupling of the inflaton to leads to a significant kinetic mixing between the inflaton and the such that perturbative unitarity is restored up to the Planck scale. We show the correlation between the unitarity scale and inflationary observables in this model and discuss how the effective Higgs inflation appears.

DOI: 10.1103/PhysRevD.98.015020

I. INTRODUCTION scale-invariant completion does not change the inflaton potential in the region in which the inflationary observables Cosmic inflation [1,2] solves the horizon problem and are evaluated [10]. On the other hand, the new physics scale explains isotropy, homogeneity, the flatness of the pffiffiffi M = ξ Universe, etc., in cosmology (see, for instance, is saturated to P in the gaugepffiffiffi sector, so the semi- ξ Refs. [3,4] for a recent review). The observation of cosmic classical expansion in powers of H=MP or the inflation microwave background (CMB) anisotropies [5–7] is well potentialp mightffiffiffi pffiffiffi depend on unknown new physics entering ξ ∼ ξ consistent with the existence of the early period of slow-roll at MP= H [3]. inflation and quantum fluctuations during inflation seed the Furthermore, the quartic self-coupling of the Higgs large-scale structure of the Universe. runs to smaller but positive values at high energies, although Predictions of large field models for inflation can be its precise value depends on the top pole mass and the – sensitive to unknown high-scale physics unless there is a strong coupling at a low energy [11 13].Inthiscontext,there reason to have the power expansion of inflaton potential at is an interesting possibility that an inflection point in the any new physics scale under control such that the semi- Higgs potential at high energies can be used for inflation [14]. classical approximation is justified during inflation. The However, if a large pole mass is taken, the vacuum identification of new physics scales depends on the inflaton instability scale is much lower than the typical inflation scale ¼ 173 2 field values and the kinetic terms during inflation, so power in Higgs inflation. For instance, for mt . GeV, ¼ 125 α ð Þ¼0 1183 counting for the effective field theory should be applied together with mh GeV and s MZ . ,the Λ ¼ 4 1010 with care to capture physics during inflation. vacuum instability scale is given by I × GeV It is a legitimate question whether there is a way to obtain [13], so the would not be appropriate for a a viable inflation model with the known or testable dominant component of the inflaton in this case. and interactions at a low energy. Higgs inflation [8] uses the In this paper, we propose a simple extension of the SM Standard Model (SM) Higgs boson as the inflaton, but the with a light singlet as a dominant component of validity of the inflaton potential from electroweak all the inflaton. We introduce nonminimal couplings of the ξ ξ the way to large field values during inflation is challenging. singlet scalar to gravity at both linear ( 1) and quadratic ( 2) The nonminimal coupling of the Higgs boson to gravity orders and discuss the roles of the singlet field as an – must be of the order of ξ ∼ 104, violating perturbative ultraviolet (UV) completion of Higgs inflation [15 17] as well as for solving the vacuum instability problem. A large unitarity at MP=ξ in the SM vacuum, comparable to the pffiffiffiffiffi linear nonminimal coupling with ξ1 ∼ ξ2 allows for a Hubble scale H ∼ MP=ξ during inflation [9]. But the new physics scale becomes field dependent during inflation [10] significant kinetic mixing of the singlet field with the and gets larger to the Planck scale in the inflaton sector, so a graviton, ensuring perturbative unitarity up to the Planck scale. In this model, we show that the linear nonminimal coupling makes a crucial difference from Higgs inflation. *[email protected]

Published by the American Physical Society under the terms of II. LIGHT SINGLET INFLATON the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to The most general Lagrangian with the SM Higgs boson the author(s) and the published article’s title, journal citation, ϕ and a light singlet sigma field σ coupled to gravity is the and DOI. Funded by SCOAP3. following:

2470-0010=2018=98(1)=015020(6) 015020-1 Published by the American Physical Society HYUN MIN LEE PHYS. REV. D 98, 015020 (2018)   L 1 1 1 2 2 2 1 3 2 2 1 2 pffiffiffiffiffiffi ¼ M Ωðσ; ϕÞR − ð∂μσÞ − ð∂μϕÞ − V ðσ; ϕÞ; L 0 ¼ − 1 þ ξ ð∂μσÞ − ð∂μϕÞ : ð6Þ −g 2 P 2 2 J kin; 2 2 1 2 ð1Þ Then, redefining the sigma field by   where the frame function and the scalar potential are given, 3 1=2 respectively, by σ˜ ¼ 1 þ ξ2 σ ð Þ 2 1 ; 7 ξ σ 1 Ωðσ ϕÞ¼1 þ 1 þ ðξ σ2 þ ξ ϕ2Þ ð Þ ; 2 2 ϕ ; 2 we obtain the leading derivative interaction terms MP MP 1 1 1 1 1 1 L ¼ σ˜ð∂ σ˜Þ2 þ σ˜ 2ð∂ σ˜Þ2 þ ϕ2ð∂ σ˜Þ2 ðσ ϕÞ¼ 2 ϕ2 þ λ ϕ4 þ 2σ2 − μσϕ2 int Λ μ Λ2 μ Λ2 μ VJ ; 2 mϕ 4 ϕ 2 mσ 1 2 3 1 1 1 1 1 1 2 2 2 2 2 3 2 2 4 þ σ˜ð∂μϕÞ þ σ˜ ð∂μϕÞ þ ϕ ð∂μϕÞ þ ασ þ λσϕσ ϕ þ λσσ : ð3Þ Λ 2 2 3 2 4 4 Λ5 Λ6 1 1 − ϕð∂ σ˜Þð∂μϕÞ − σϕ˜ ð∂ σ˜Þð∂μϕÞþ ð Þ This is a generalized form of the UV complete Higgs μ 2 μ ; 8 Λ7 Λ inflation proposed in Refs. [15,17] or Starobinsky models 8 for inflation in Refs. [16,18]. We take the mass parameters where the ellipses are higher-dimensional terms and the σ μ α for in the above scalar potential (3), i.e., mσ, , ,tobeof cutoff scales in the leading terms read the order of the weak scale for a light inflaton. A model with a singlet scalar field similar to ours was considered in 3 2 3=2 2ð1 þ 2 ξ1Þ Λ1 ≡ ; ð Þ Ref. [19], but the effect of the linear nonminimal coupling ξ ð1 þ 3ξ2 − 6ξ Þ 9 was not discussed. The detailed discussion on the light 1 1 2 inflaton with a small nonminimal coupling, ξ2 < 1 and pffiffiffi 2ð1 þ 3 ξ2Þ ξ1 ¼ 0, can be found in Ref. [20]. jΛ j ≡ 2 1 ð Þ 2 2 3 2 2 1=2 ; 10 Assuming ξ2, ξϕ > 0, we have only to impose the jξ1ð1 þ 2 ξ1Þþξ2ð1 þ 3ξ1 − 6ξ2Þj condition for stable gravity as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ 3 ξ2Þ 2 2 Λ ≡ 2 1 ð Þ ξ1 < 4ðξ2 þ ξϕτ Þ; ð4Þ 3 2 ; 11 ξϕð1 þ 3ξ1Þ with τ ¼ ϕ=σ, because otherwise the effective Planck mass qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 squared could become negative during the cosmological 2 1 þ 2 ξ1 evolution. Thus, Eq. (4) leads to the upper bound on the Λ4 ¼ ; ð12Þ ξ1 linear nonminimal coupling ξ1 for stable gravity in the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi entire field space. 2ð1 þ 3 ξ2Þ From Eq. (1), setting M ¼ 1 and performing the metric 2 1 P Λ5 ¼ ; ð13Þ E 2 2 2 rescaling by gμν ¼ gμν=Ω with Ω ¼ 1þξ1σ þξ2σ þξϕϕ , ξ1 þ ξ2 we get the Einstein frame Lagrangian as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L 1 1 3 jΛ6j¼ ; ð14Þ 2 2 jξ ð1 − 6ξ Þj pffiffiffiffiffiffiffiffi ¼ RðgEÞ − ð∂μσÞ − ð∂μ log ΩÞ ϕ ϕ −gE 2 2Ω 4 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − ð∂ ϕÞ2 − ðσ ϕÞ ð Þ 1 þ 3 ξ2 2Ω μ VE ; ; 5 2 1 Λ7 ¼ ; ð15Þ 3ξϕξ1 where the Einstein frame potential is given by VEðσ; ϕÞ¼ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi VJ=Ω . 3 2 1 þ 2 ξ1 Λ8 ¼ : ð16Þ 6ξϕξ2 III. UNITARITY AND HIGGS INFLATION We consider the unitarity scale in our model with the Therefore, taking ξϕ ¼ Oð1Þ to avoid the unitarity problem light inflaton and discuss the effective Higgs inflation after due to Higgs interactions such as Λ6;7, we can read off the the sigma field is integrated out. Near the true minimum minimum cutoff scales from Λ1;2;5;8. In Fig. 1, we depict the with σ, ϕ ≪ 1, we get the quadratic kinetic terms in Eq. (5) minimum cutoff scale jΛ2j in units of MP in the plane of ξ2 approximated to and ξ1. Then, we find that the cutoff scale of the order of the

015020-2 LIGHT INFLATON COMPLETING HIGGS INFLATION PHYS. REV. D 98, 015020 (2018)

2 1 1 1000 ¼ 2 ϕ2 þ λ ϕ4 ð Þ Veff 2 mϕ 4 eff 20

100 with

λσϕξ2 a 2 ξ ≡ ξ − ð Þ ϕ;eff ϕ ; 21 10 λσ

1 λ2 λ ≡ λ − σϕ ð Þ 1 eff ϕ : 22 2 1 λσ

a 0.01 2 2 Therefore, for ϕ ≪ jmσ=λσϕj, a large effective nonminimal 0.10 –1 a 0.1 2 10 ξ ξ ≫ 1 coupling eff appears for 2 [15] and the Higgs quartic

–2 –3 –4 coupling gets a tree-level shift due to the scalar threshold, 2 10 2 10 2 10 0.01 curing the vacuum instability problem [13,21]. In the limit 1 10 100 1000 104 105 ϕ2 ≫ j 2 λ j of mσ= qσϕ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, the Higgs field follows the sigma field 2 2 satisfying σ ≈ −λσϕϕ =λσ. Then, for ξ2 ≫ ξϕjλσ=λσϕj, FIG. 1. Contours of jΛ2j in units of M in the plane of ξ2 and ξ1 P the effective frame function (19) becomes in black solidpffiffiffiffiffi lines. We overlaid in blue dashed lines the contours of a ¼ ξ1= ξ2. pffiffiffiffiffi η ϕ2 ξ ϕ2 Ω ≈ 1 þ þ ϕ;eff ð Þ eff 2 23 MP MP

Planck scale requires a sizable linear nonminimal coupling, pffiffiffiffiffiffiffiffiffiffi λ ξ pffiffiffiffiffi η ≈ pξ1ffiffiffi ξ ξ ≈− σϕ 2 0 λ 0 ¼ Oð1Þ ξ ∼ ξ with ϕ;eff and ϕ;eff λ > for σϕ < .As namely, a or 1 2. ξ2 σ Consequently, mass parameters in the scalar potential (3) compared to the original Higgs inflation, the resulting are not constrained by unitarity, as far as they are below the frame function is augmented byp a nonanalyticffiffiffiffiffi form of the scale of unitarity violation in Higgs inflation. Thus, the nonminimal coupling to gravity, ϕ2R. As the sigma field sigma field mass can be of the order of the weak scale or pffiffiffiffiffi theory with ξ1 ∼pffiffiffiffiffiffiffiffiffiffiξ2 is unitary up to the Planck scale, lower, being consistent with the UV complete inflation, η ∼ ξ resulting in ϕ;eff, the effective Higgs inflation keeps unlike the cases with only either ξ2 [15] or ξ1 [17], where the new singlet scalar must be much heavier than the weak a similar prediction for inflation as in the sigma field theory. scale to get a large vacuum expectation value (VEV) from the renormalizable scalar potential [15] or match the cosmic IV. INFLATON DYNAMICS WITH SIGMA FIELD background explorer normalization with the singlet mass Now we discuss the general inflation vacua at large fields term [17]. in our model. As far as the stable gravity condition (4) is In order to discuss the effective Higgs inflation, we plug satisfied, the frame function Ω is always positive during into Eq. (1) the equation of motion for σ with 2 2 inflation. For ξ1σ þ ξ2σ þ ξϕϕ ≫ 1 during inflation, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Ω ≈ ξ1σ þ ξ2σ þ ξϕϕ 2 2 we get and introduce a new set −mσ − λσϕϕ σ ¼ ; ð17Þ of fields by λσ p2ffiffi χ 2 2 e 6 ¼ ξ1σ þ ξ2σ þ ξϕϕ ; ð24Þ where the dimensionful interactions are ignored in the scalar potential (3). As a result, we obtain the effective ϕ τ ¼ ð Þ Lagrangian for the Higgs inflation from σ : 25 pffiffi L 1 1 2 χ ξ2 eff 2 2 3 ≫ 1 pffiffiffiffiffiffi ¼ M Ω ðϕÞR − ð∂μϕÞ − V ðϕÞ; ð18Þ Then, since e 4ðξ þξ τ2Þ at large fields, we get the −g 2 P eff 2 eff 2 ϕ approximate relation between σ and redefined fields χ τ where the effective frame function and Higgs potential and as become, respectively, 1   pffiffi χ 2 e 6 a −p1ffiffi χ a −p2ffiffi χ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ ≈ 1 − 6 þ 6 ð Þ 2 1 2 e e 26 2 2 2 2 ðξ þ ξ τ Þ = 2 8 ξ ξ −mσ − λσϕϕ ξϕ ϕ 2 ϕ Ω ¼ 1 − 2mσ þ 1 þ ;eff ð Þ eff 2 2 ; 19 λσM M λσ M P P P with

015020-3 HYUN MIN LEE PHYS. REV. D 98, 015020 (2018)

ξ ≡ 1 ð Þ breaking at a low energy. The resulting a 2 1=2 : 27 ðξ2 þ ξϕτ Þ for the viable inflation becomes   As a consequence, the scalar potential in the Einstein frame 1 λ2 ð Þ∶ ¼ λ − σϕ ð Þ becomes i V0 2 σ ; 32 4ξ2 λϕ

1 4 2 −p2ffiffi χ −2 4 6 λ VEðχÞ¼ ðλϕτ þ 2λσϕτ þ λσÞð1 þ e Þ σ σ 4 ð Þ; ð Þ∶ V0 ¼ : ð Þ ii iv 4ξ2 33 −p1ffiffi χ 2 −p2ffiffi χ 2 ≈ V0ð1 − 2ae 6 − ð2 þ a Þe 6 Þð28Þ In all the above cases, the sigma quartic self-coupling with contributes dominantly to the inflaton vacuum energy. τ 4 2 We note that the physical mass of the field is given as λϕτ þ 2λσϕτ þ λσ 2 2 ≡ ð Þ follows: (i) mτ ≈ ð−2λσϕÞ=ξ2, or (ii), (iv) mτ ≈ λσϕ=ξ2,so V0 2 2 : 29 4ðξ2 þ ξϕτ Þ 2 2 2 mτ ≫ H ∼ λσ=ξ2. Therefore, the dynamics of the τ field can be safely ignored during inflation. The results are in Here, the condition of stable gravity (4) requires that agreement with the related analytic and numerical analyses j j 2 a < . for the effective single-field inflation in similar models as in ξ ≪ 2ðξ þ ξ τ2Þσ On the other hand, for 1 2 ϕ , the kinetic Refs. [22,23]. terms for σ and ϕ in Eq. (5) can be rewritten in terms of χ Consequently, from Eqs. (28) and (30), we obtain the and τ [22] as follows: effective Lagrangian for the inflaton χ as follows:   L 1 1 1 þ τ2 L 1 1 pffiffiffiffiffiffiffiffikin ≈ 1 þ ð∂ χÞ2 eff 2 2 μ pffiffiffiffiffiffiffiffi ¼ Rðg Þ − ð∂μχÞ − V ðχÞ: ð34Þ −g 2 6 ξ2 þ ξϕτ E E E −gE 2 2 τðξ − ξ Þ 1 2 ϕ μ þ pffiffiffi ð∂ χÞð∂ τÞ 1 2 2 μ −pffiffi χ 6 ðξ2 þ ξϕτ Þ Therefore, for a ≳ e 6 ∼ 0.1 during inflation, which 10−2 ξ2 τ2 þ ξ2 makes the cutoff scale higher than MP even for a large 1 ϕ 2 2 4 þ ð∂ τÞ : ð Þ ξ2 ∼ 10 ,asshowninFig.1, the linear nonminimal coupling 2 ðξ þ ξ τ2Þ3 μ 30 2 ϕ modifies the inflaton potential significantly, as compared with the case with quadratic nonminimal coupling only. Here, we note that, for ξ2 þ ξϕ ≫ 1 or ξ2 ¼ ξϕ or hτi¼0, We remark that the linear nonminimal coupling ξ1 in χ we can safely ignore the kinetic mixing term between and Eq. (1) can be eliminated by redefining the σ field with τ. The results coincide with those in Ref. [22] for ξ1 ¼ 0. ξ σ¯ ¼ σ þ 1 M , and then the frame function (2) and the We turn to the stabilization of τ from the scalar potential 2ξ2 P scalar potential (3) in the Jordan frame are replaced by in Eq. (28). Ignoring the third term in Eq. (28) in stabilizing ξ τ ξ ≫ ξ ¼ Oð1Þ those with σ ¼ σ¯ − 1 M . Even in this case, small physical , for 2 ϕ and quartic couplings of the order of 2ξ2 P unity, we get the conditions for the inflation vacua [22] as masses for the singlet scalar are kept. The large VEVof the sffiffiffiffiffiffiffiffiffiffiffi shifted field σ¯ leads to a large kinetic mixing between the λ singlet scalar and the graviton in the Jordan frame or a large ð Þ τ ¼ − σϕ∶ λ 0 λ 0 i λ ϕ > ; σϕ < ; rescaling of the singlet scalar kinetic term in the Einstein ϕ 2 2 ξ1 frame [15,16]. For instance, for ϕ ¼ 0 and ξ2σ¯ ≫ 1 − 4ξ2 ðiiÞ τ ¼ 0∶ λϕ > 0; λσϕ > 0; duringqffiffi inflation, the canonical inflaton field becomes ðiiiÞ τ ¼ ∞∶ λϕ < 0; λσϕ < 0; 3 2 χ ¼ 2 lnðσ¯ =ξ2Þ for the σ¯ field. Then, from the λσ term ð Þ τ ¼ 0 ∞∶ λ 0 λ 0 ð Þ iv ; ϕ < ; σϕ > : 31 in this field basis, we obtain the same effective Lagrangian for the inflaton with τ ¼ 0 as Eq. (34). In the first two cases, we need the Higgs quartic coupling to From the effective inflaton Lagrangian in Eq. (34), the be positive during inflation: The former is the sigma-Higgs slow-roll parameters during inflation are given by mixed inflation, and the latter is the pure sigma inflation. In   the third case, as the Higgs quartic coupling is required to 2 1 2 2 −p2ffiffi χ −p1ffiffi χ a be negative as λϕ < 0, V0 < 0, so it is not possible to get a ϵ ¼ ð2 þ a Þ e 6 e 6 þ ; ð35Þ 3 2 þ a2 de Sitter vacuum for inflation. But, in the fourth case, even for λϕ < 0, the inflation could be driven by the sigma field   2 2 −p1ffiffi χ −p1ffiffi χ a τ ¼ 0 η ¼ − ð2 þ Þ 6 6 þ ð Þ at the metastable vacuum with , so it could lead to a a e e 2 : 36 viable cosmology with correct electroweak symmetry 3 2ð2 þ a Þ

015020-4 LIGHT INFLATON COMPLETING HIGGS INFLATION PHYS. REV. D 98, 015020 (2018)

N=50, 60 As a result, the spectral index is given by ns ¼ 0.020 1–6ϵ þ 2η , where denotes the evaluation of the Planck 1 slow-roll parameters, (35) and (36), at the horizon exit. The tensor-to-scalar ratio is also given by r ¼ 16ϵ with Eq. (35) at the horizon exit. The measured spectral index 0.015 and the bound on the tensor-to-scalar ratio are given by ns ¼ 0.9652 0.0047 and r<0.10 at 95% C.L., respec- þ tively, from Planck TT, TE, EE low P [6]. r 0.010 N=50 Moreover, the number of e-foldings required to solve the horizon problem can be calculated as follows: N=60 Z χ i dχ 0.005 N ¼ pffiffiffiffiffi χ 2ϵ f    2 3ð2 þ Þ 1 1 a a pffiffi χ a pffiffi χ ≈ e 6 − ln 1 þ e 6 ; 0.950 0.955 0.960 0.965 0.970 0.975 a2 2 þ a2 2 þ a2 ns ð37Þ FIG. 3. Spectral index ns vs tensor-to-scalar ratio r for a ¼ 0–2. We have chosen N ¼ 50 and 60 in blue and black lines, χ where i;f are the inflaton values at the beginning and respectively. The Planck 1σ band is shown in green. end of inflation and we can take χi ¼ χ. In Fig. 2, we show the slow-roll parameters evaluated at the horizon exit as a However, for a sizable a or ξ1, the inflationary observ- function of a for N ¼ 50, 60. Thus, we find that η is ables are modified by the linear nonminimal coupling insensitive to a but ϵ changes to a large value as a for σ. In Fig. 3, we depict the correlation between the approaches unity. spectral index and the tensor-to-scalar ratio for a varying p1ffiffi 2 1 2 − χ ≡ ξ ðξ þ ξ τ Þ = ¼ 50 For a ≪ e 6 , i.e., ξ1 ≪ 1, the results with quadratic a 1= 2 ϕ for N and 60 in blue and black nonminimal couplings only are recovered, namely, N ≈ lines, respectively. As ϵ increases for a sizable a, the 2 4 2 3 pffiffi χ 4 −pffiffi χ 4 −pffiffi χ 6 ϵ ≈ 6 η ≈− 6 ϵ ≈ tensor-to-scalar ratio varies significantly from the one for 4 e , 3 e , and 3 e . Then, we get ¼ 0 015 ¼ 50 3 1 the original Higgs inflation [8] up to r . for N 2 and η ≈− , so the spectral index and the tensor-to- 4N N and r ¼ 0.010 for N ¼ 60. Thus, the primordial gravity ≈ 1 − 2 ≈ 12 scalar ratio become ns N and r N2, respectively [8]. waves are at the detectable level in future CMB experi- ments such as LiteBIRD [24].

N 50, 60 0.050 V. CONCLUSIONS We have presented the inflation model with a light singlet field containing both linear and quadratic non- 0.010 minimal couplings to gravity. We showed that the inflaton (– * ) potential is determined by the quartic couplings and the 0.005 nonminimal couplings only. We found that the linear *

, nonminimal coupling for the singlet field makes the model ) * unitary up to the Planck scale and allows for a sizable - ( 0.001 deviation in the tensor-to-scalar ratio from Higgs inflation. The light singlet inflaton can be probed by low-energy 5.×10–4 phenomena such as direct production at the Large Collider through Higgs interactions.

ACKNOWLEDGMENTS 1.×10–4 0.0 0.5 1.0 1.5 2.0 a I thank Gian F. Giudice for valuable discussion. The work is supported in part by the Basic Science Research FIG. 2. Slow-roll parameters as a function of a ≡ ξ1= Program through the National Research Foundation of 2 1=2 ðξ2 þ ξϕτ Þ . We have chosen N ¼ 50 and 60 in solid and Korea (NRF) funded by the Ministry of Education, Science dashed lines, respectively. and Technology Grant No. NRF-2016R1A2B4008759.

015020-5 HYUN MIN LEE PHYS. REV. D 98, 015020 (2018)

[1] A. H. Guth, Phys. Rev. D 23, 347 (1981); A. D. Linde, Phys. [13] J. Elias-Miro, J. R. Espinosa, G. F. Giudice, H. M. Lee, and Lett. 108B, 389 (1982); A. Albrecht and P. J. Steinhardt, A. Strumia, J. High Energy Phys. 06 (2012) 031. Phys. Rev. Lett. 48, 1220 (1982). [14] K. Allison, J. High Energy Phys. 02 (2014) 040; A. Salvio, [2] A. D. Linde, Phys. Lett. 129B, 177 (1983). Phys. Lett. B 727, 234 (2013); Y. Hamada, H. Kawai, K. y. [3] C. P. Burgess, arXiv:1711.10592. Oda, and S. C. Park, Phys. Rev. Lett. 112, 241301 (2014);F. [4] D. Baumann, arXiv:0907.5424. Bezrukov and M. Shaposhnikov, Phys. Lett. B 734, 249 [5] P. A. R. Ade et al. (Planck Collaboration), Astron. (2014); S. M. Choi and H. M. Lee, Eur. Phys. J. C 76, 303 Astrophys. 594, A13 (2016). (2016); J. Fumagalli and M. Postma, J. High Energy Phys. [6] P. A. R. Ade et al. (Planck Collaboration), Astron. 05 (2016) 049. Astrophys. 594, A20 (2016). [15] G. F. Giudice and H. M. Lee, Phys. Lett. B 694, 294 (2011); [7] P. A. R. Ade et al. (BICEP2 and Planck Collaborations), H. M. Lee, Phys. Lett. B 722, 198 (2013). Phys. Rev. Lett. 114, 101301 (2015). [16] G. F. Giudice and H. M. Lee, Phys. Lett. B 733, 58 (2014); [8] F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659, H. M. Lee, Eur. Phys. J. Web Conf. 168, 06001 (2018). 703 (2008). [17] J. L. F. Barbon, J. A. Casas, J. Elias-Miro, and J. R. Espinosa, [9] C. P. Burgess, H. M. Lee, and M. Trott, J. High Energy J. High Energy Phys. 09 (2015) 027. Phys. 09 (2009) 103; J. L. F. Barbon and J. R. Espinosa, [18] A. Kehagias, A. M. Dizgah, and A. Riotto, Phys. Rev. D 89, Phys. Rev. D 79, 081302 (2009); C. P. Burgess, H. M. Lee, 043527 (2014). and M. Trott, J. High Energy Phys. 07 (2010) 007; M. P. [19] Y. Ema, M. Karciauskas, O. Lebedev, S. Rusak, and M. Hertzberg, J. High Energy Phys. 11 (2010) 023. Zatta, arXiv:1711.10554. [10] F. Bezrukov, A. Magnin, M. Shaposhnikov, and S. [20] F. Bezrukov and D. Gorbunov, J. High Energy Phys. 07 Sibiryakov, J. High Energy Phys. 01 (2011) 016. (2013) 140. [11] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. [21] O. Lebedev, Eur. Phys. J. C 72, 2058 (2012). Giudice, G. Isidori, and A. Strumia, J. High Energy Phys. 08 [22] O. Lebedev and H. M. Lee, Eur. Phys. J. C 71,1821 (2012) 098; D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. (2011). Giudice, F. Sala, A. Salvio, and A. Strumia, J. High Energy [23] J. Garcia-Bellido, J. Rubio, M. Shaposhnikov, and D. Phys. 12 (2013) 089. Zenhausern, Phys. Rev. D 84, 123504 (2011); P. G. Ferreira, [12] F. Bezrukov, M. Y. Kalmykov, B. A. Kniehl, and M. C. T. Hill, and G. G. Ross, arXiv:1801.07676; P. Carrilho, Shaposhnikov, J. High Energy Phys. 10 (2012) 140; F. D. Mulryne, J. Ronayne, and T. Tenkanen, J. Cosmol. Bezrukov and M. Shaposhnikov, Zh. Eksp. Teor. Fiz. 147, Astropart. Phys. 06 (2018) 032. 389 (2015) [J. Exp. Theor. Phys. 120, 335 (2015)]. [24] A. Suzuki et al., arXiv:1801.06987.

015020-6