(null) as a uv-extension of … · 2019-03-29 · 4 2 1 e 2 3 /mpl 2 (null) simplest

Courtesy H.Oide

IBS-CTPU, PTC seminar 20/03/2019 @ IBS-CTPU, Daejeon, Korea

Kohei Kamada (RESCEU, U-Tokyo)

based on: Minxi He (Tokyo), Ryusuke Jinno (IBS-CTPU), KK, Seong Chan Park (Yonsei), Alexey Starobinsky (Landau), Jun’ichi Yokoyama (Tokyo), PLB 791(2019) 36, arXiv:1812.10099 [hep-ph]

Higgs- model as a UV-extension of Higgs inflation

and violent preheating

R2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

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1. Introduction: UV issues in Higgs inflation 2. UV extension of Higgs inflation with -term 3. Inflaton oscillation and violent preheating 4. Discussion


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Introduction

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Inflation is now a main ingredient of the standard CosmologyPremise

3H + V = 0, 3H2M2pl = V

<latexit sha1_base64="mvYK/hPHyJ1OOW5fHEhY0Ne3Vlo=">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</latexit><latexit sha1_base64="VAkYSQcAgR6TflJpqwBf+dd2fFc=">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</latexit><latexit sha1_base64="VAkYSQcAgR6TflJpqwBf+dd2fFc=">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</latexit><latexit sha1_base64="VXwCv1u67NiySpxL2bX3d24X79s=">AAACQHicbVDLSgMxFM3UV62vqks3wSIKSpmpC90Uim66ESrYB3TGIZPJtKHJzJhkhDL0P/waF270F/wDd+JKcGXazsK2HgicnHNvcu/xYkalMs13I7e0vLK6ll8vbGxube8Ud/daMkoEJk0csUh0PCQJoyFpKqoY6cSCIO4x0vYG12O//UiEpFF4p4YxcTjqhTSgGCktucXKOazD1PYjBe24T0enreOqeQbthwT5UHv3FXjjprbgMGYjfanCllssmWVzArhIrIyUQIaGW/zW7+OEk1BhhqTsWmasnBQJRTEjo4KdSBIjPEA90tU0RJxIJ53sNoJHWvFhEAl9QgUn6t+OFHEph9zTlRypvpz3xuK/nsdnfk6lHq1P/LlxVHDppDSME0VCPJ0mSBhUERynCX0qCFZsqAnCguqFIO4jgbDSmRd0UtZ8LoukVSlbZtm6NUu1qyyzPDgAh+AEWOAC1EAdNEATYPAEnsEreDNejA/j0/ialuaMrGcfzMD4+QX+Rq0+</latexit>

・Solution to horizon & flatness problem ・Seed for the structure of the Universe ・Driven (often) by potential energy of scalar field(s).

Courtesy H.Oide



Planck collaboration

3H + V = 0, 3H2M2pl = V


・Strongly favored by cosmological obs.

Courtesy H.Oide



3H + V = 0, 3H2M2pl = V



What is the scalar field (inflaton) that drove inflation?


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3H + V = 0, 3H2M2pl = V



What is the scalar field (inflaton) that drove inflation?


Why don’t we use the SM Higgs as inflaton?

Courtesy H.Oide

Question: Why have people regarded that SM Higgs is not inflaton?

New inflation?

: impossible because the potential is too steep to realize accelerating expansion of the Universe.

(’82, Linde)

Courtesy H.Oide

Question: Why have people regarded that SM Higgs is not inflaton?

Chaotic inflation?: possible to realize accelerating expansion of the Universe, but the primordial density perturbation becomes too large.

for

is inconsistent with

the observation

(’83, Linde)

P 103

Pobs 2.18 109

Courtesy H.Oide

The problem is too large primordial perturbations.

gij = a2(t)e2ij

(k)(k) =22

k3P(k)(3)(k + k)

This should be from observation. 109

Courtesy H.Oide

The problem is too large primordial perturbations.

gij = a2(t)e2ij

(k)(k) =22

k3P(k)(3)(k + k)

This should be from observation.

For slow-roll inflation, the power spectrum can be written in terms of the Hubble parameter during inflation and a slow-roll parameter

Hinf

M2

pl

2

V

V

2

P =H2

inf

82M2pl

þ

V(þ)

slow-roll inflation

reheating

Figure 2.1: This picture represents a naive picture of inflation. The slow-roll inflation isinduced by the scalar-field which has very flat potential. After inflation, the reheating erais needed for the successful inflationary scenario.

constant. Then, the universe expands as if dominated by the cosmological constant, that is,the universe behaves as ”de Sitter universe”, expanding exponentially.

η can be also rewritten as

η ≃ Vφφ

H2. (2.12)

Since Vφφ represents the effective mass squared of scalar field, m2eff , η ≪ 1 means that

m2eff ≪ H2, that is, the mass of scalar field is much smaller than the Hubble parameter

during slow-roll inflation.In addition, let us define a parameter important for later discussion, N , which represents

the amount of inflation, as

N ≡lna(tend)

a(t)=

! tend

t

H(t′)dt′ , (2.13)

where tend represents a time when inflation ends. This parameter, N , is commonly calledthe ”e-folding number”. To solve the initial condition problems of big bang universe, thee-folding number need to be typically more than about sixty (Fig. 2.2).

To realize the standard big bang universe after the inflation, some process is needed.Generally, this process is called ”reheating”, in which the inflaton decays into some particlesand the thermalization of the decay products occurs (Fig. 2.1). Thus, modern standardcosmology consist of ”hot big bang” and ”inflation”.

10

V ()

109

Courtesy H.Oide

In other words, if one can control the Higgs field dynamics to change the parameters,

- Hubble parameter during inflation - slow-roll parameter during inflation - (effective) Planck scale

the possibility of “Higgs inflation” comes back.

P =H2

inf

82M2pl

Courtesy H.Oide



It can be realized by introducing additional terms to the Higgs interaction, with taking care not to affect low-energy phenomenology.

H = (0, (h(x) + v)/

2)Unitary gauge


P =H2

inf

82M2pl

Courtesy H.Oide




H = (0, (h(x) + v)/

2)Unitary gauge


P =H2

inf

82M2pl

Non-minimal gravitational and derivative couplingcan make inflation driven by the SM Higgs possible.

Generalized Higgs inflation (’11, ’12 KK+)with the spirit of Horndesky theory.

cf. Gansukh’s latest paper 1903.05354.

Courtesy H.Oide




H = (0, (h(x) + v)/

2)Unitary gauge


P =H2

inf

82M2pl

Non-minimal gravitational and derivative couplingcan make inflation driven by the SM Higgs possible.

Generalized Higgs inflation (’11, ’12 KK+)with the spirit of Horndesky theory.

cf. Gansukh’s latest paper 1903.05354.

I believe, in the absence of any deviations from the standard prediction of inflation in the CMB observation and any new physics signature at LHC, it is important to investigate the inflation model driven by the SM Higgs in depth.

Courtesy H.Oide

Potential in Einstein frame

0

λM4/ξ2/16

λM4/ξ2/4

U(χ)

0 χend χCOBE χ

0λ v4/4

0 v

Rehe

atin

g Standard Model

DESY-Hamburg, September 27, 2011 – p. 7

’08 Bezrukov & Shaposhnikov

U() =M4

pl

42

1 e

23 /Mpl

2

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Simplest model of inflation driven by the SM Higgs, with a classical scale-invariant extension of the SM!

(’95 Cerventas-Cota & Dehnen, ’08 Bezrukov & Shaposhnikov)

Among many proposals of inflation driven by the SM Higgs, the one with a nonminimal coupling to gravity

would be the most remarkable one.

|H|2R<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Courtesy H.Oide

Potential in Einstein frame

0

λM4/ξ2/16

λM4/ξ2/4

U(χ)

0 χend χCOBE χ

0λ v4/4

0 v

Rehe

atin

g Standard Model

DESY-Hamburg, September 27, 2011 – p. 7

’08 Bezrukov & Shaposhnikov

U() =M4

pl

42

1 e

23 /Mpl

2


Among many proposals of inflation driven by the SM Higgs, the one with a nonminimal coupling to gravity

would be the most remarkable one.

|H|2R<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Simplest model of inflation driven by the SM Higgs, with a classical scale-invariant extension of the SM!

Planck Collaboration: Constraints on Inflation

0.94 0.96 0.98 1.00

Primordial tilt (ns)

0.00

0.05

0.10

0.15

0.20

Ten

sor-

to-s

cala

rra

tio

(r0.0

02)

Convex

Concave

TT,TE,EE+lowE+lensing

TT,TE,EE+lowE+lensing+BK14

TT,TE,EE+lowE+lensing+BK14+BAO

Natural inflation

Hilltop quartic model

attractors

Power-law inflation

R2 inflation

V 2

V 4/3

V

V 2/3

Low scale SB SUSYN=50

N=60

Fig. 8. Marginalized joint 68 % and 95 % CL regions for ns and r at k = 0.002 Mpc1 from Planck alone and in combination withBK14 or BK14 plus BAO data, compared to the theoretical predictions of selected inflationary models. Note that the marginalizedjoint 68 % and 95 % CL regions assume dns/d ln k = 0.

limits obtained from a CDM-plus-tensor fit. We refer the inter-ested reader to PCI15 for a concise description of the inflationarymodels studied here and we limit ourselves here to a summaryof the main results of this analysis.

– The inflationary predictions (Mukhanov & Chibisov 1981;Starobinsky 1983) originally computed for the R2 model(Starobinsky 1980) to lowest order,

ns 1 ' 2N, r '

12N2 , (48)

are in good agreement with Planck 2018 data, confirm-ing the previous 2013 and 2015 results. The 95 % CL al-lowed range 49 < N < 58 is compatible with the R2 ba-sic predictions N = 54, corresponding to Treh 109 GeV(Bezrukov & Gorbunov 2012). A higher reheating temper-ature Treh 1013 GeV, as predicted in Higgs inflation(Bezrukov & Shaposhnikov 2008), is also compatible withthe Planck data.

– Monomial potentials (Linde 1983) V() = M4Pl (/MPl)p

with p 2 are strongly disfavoured with respect to theR2 model. For these values the Bayesian evidence is worsethan in 2015 because of the smaller level of tensor modesallowed by BK14. Models with p = 1 or p = 2/3(Silverstein & Westphal 2008; McAllister et al. 2010, 2014)are more compatible with the data.

– There are several mechanisms which could lower the pre-dictions for the tensor-to-scalar ratio for a given potentialV() in single-field inflationary models. Important exam-ples are a subluminal inflaton speed of sound due to a non-standard kinetic term (Garriga & Mukhanov 1999), a non-minimal coupling to gravity (Spokoiny 1984; Lucchin et al.

1986; Salopek et al. 1989; Fakir & Unruh 1990), or an ad-ditional damping term for the inflaton due to dissipation inother degrees of freedom, as in warm inflation (Berera 1995;Bastero-Gil et al. 2016). In the following we report on theconstraints for a non-minimal coupling to gravity of the typeF()R with F() = M2

Pl + 2. To be more specific, a quartic

potential, which would be excluded at high statistical signif-icance for a minimally-coupled scalar inflaton as seen fromTable 5, can be reconciled with Planck and BK14 data for > 0: we obtain a 95 % CL lower limit log10 > 1.6 withln B = 1.6.

– Natural inflation (Freese et al. 1990; Adams et al. 1993) isdisfavoured by the Planck 2018 plus BK14 data with a Bayesfactor ln B = 4.2.

– Within the class of hilltop inflationary models(Boubekeur & Lyth 2005) we find that a quartic poten-tial provides a better fit than a quadratic one. In the quarticcase we find the 95 % CL lower limit log10(µ2/MPl) > 1.1.

– D-brane inflationary models (Kachru et al. 2003; Dvali et al.2001; Garcıa-Bellido et al. 2002) provide a good fit toPlanck and BK14 data for a large portion of their parame-ter space.

– For the simple one parameter class of inflationary potentialswith exponential tails (Goncharov & Linde 1984; Stewart1995; Dvali & Tye 1999; Burgess et al. 2002; Cicoli et al.2009) we find ln B = 1.0.

– Planck 2018 data strongly disfavour the hybrid model drivenby logarithmic quantum corrections in spontaneously brokensupersymmetric (SUSY) theories (Dvali et al. 1994), withln B = 5.0.

18

’18 Planck collaboration

It also fits the CMB data ( ) very well!!! ns r<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(’95 Cerventas-Cota & Dehnen, ’08 Bezrukov & Shaposhnikov)

Courtesy H.Oide

Obstacles?

Vacuum stability &

Perturbative unitarity

Courtesy H.Oide

Stability of Higgs potential?

h

V (h)

MplEW vacuum

Living beyond the edge: Higgs inflation and vacuum metastability

Fedor Bezrukov,1, 2, 3, Javier Rubio,4, † and Mikhail Shaposhnikov4, ‡

1Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA

2RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA

3CERN, CH-1211 Geneve 23, Switzerland

4Institut de Theorie des Phenomenes Physiques, Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland

(Dated: December 15, 2014)

The measurements of the Higgs mass and top Yukawa coupling indicate that we live in a veryspecial Universe, at the edge of the absolute stability of the electroweak vacuum. If fully stable, theStandard Model (SM) can be extended all the way up to the inflationary scale and the Higgs field,non-minimally coupled to gravity with strength , can be responsible for inflation. We show thatthe successful Higgs inflation scenario can also take place if the SM vacuum is not absolutely stable.This conclusion is based on two e↵ects that were overlooked previously. The first one is associatedwith the e↵ective renormalization of the SM couplings at the energy scale MP /, where MP is thePlanck scale. The second one is a symmetry restoration after inflation due to high temperaturee↵ects that leads to the (temporary) disappearance of the vacuum at Planck values of the Higgsfield.

I. INTRODUCTION

One of the most interesting questions in particlephysics and cosmology is the relation between the prop-erties of elementary particles and the structure of theUniverse. Some links are provided by Dark Matter andthe Baryon Asymmetry of the Universe. A number ofconstraints on hypothetical new particles can be also de-rived from Big Bang Nucleosynthesis.

The properties of the recently discovered Higgs boson[1, 2] suggest an additional and intriguing connection.Among the many di↵erent values that the Higgs masscould have taken, Nature has chosen one that allows toextend the Standard Model (SM) all the way up till thePlanck scale while staying in the perturbative regime.The behavior of the Higgs self-coupling is quite pe-culiar: it decreases with energy to eventually arrive toa minimum at Planck scale values and start increasingthereafter, cf. Fig. 1. Within the experimental and the-oretical uncertainties1 the Higgs coupling may stay posi-tive all way up till the Planck scale, but it may also crosszero at some scale µ0, which can be as low as 108 GeV,cf. Figs. 2 and 3. If that happens, our Universe becomesunstable2.

[email protected]† [email protected]‡ [email protected] The largest uncertainty comes from the determination of the topYukawa coupling. Smaller uncertainties are associated to thedetermination of Higgs boson mass and the QCD gauge coupling↵s. See Refs. [3–5] for the most refined treatments and Ref. [6]for a review.

2 The determination of the lifetime of the Universe is a rathersubtle issue that strongly depends on the high energy comple-tion of the SM. As shown in Ref. [7–9], if the gravitationalcorrections are such that the resulting e↵ective potential liesabove/below the SM one, the lifetime of our vacuum will benotably larger/smaller than the age of the Universe.

-0.04-0.02

0 0.02 0.04 0.06 0.08

0.1 0.12 0.14

100000 1e+10 1e+15 1e+20

µ, GeV

mH=125.5 GeV

yt=0.9176, mt=170.0yt=0.9235, mt=171.0yt=0.9294, mt=172.0yt=0.9359, mt=173.1yt=0.9413, mt=174.0yt=0.9472, mt=175.0

FIG. 1. Renormalization group running of the Higgs self-coupling for several values of the top quark Yukawa coupling(top pole mass) and fixed to 125.5 GeV Higgs boson mass.

The 0 3 compatibility of the data with vacuum in-stability is one of the recurrent arguments for invokingnew physics beyond the Standard Model. In particular,it is usually stated that the minimalistic Higgs inflationscenario [10], in which the Higgs field is non-minimallycoupled to gravity with strength , cannot take place ifthe Higgs self-coupling becomes negative at an energyscale below the inflationary scale.

We will show in this paper that Higgs inflation ispossible even if the SM vacuum is not absolutely sta-ble. Specifically, we will demonstrate that the renormal-ization e↵ects at the scale MP / can bring the Higgsself-coupling to positive values in the inflationary do-main. If that happens, inflation will take place with theusual chaotic initial conditions and the fate of the Uni-verse will be inevitably determined by the subsequentevolution. At the end of the exponential expansion, theHiggs field will start to oscillate around the bottom of

arX

iv:1

412.

3811

v1 [

hep-

ph]

11 D

ec 2

014

Bezrukov+, 1412.3811

There remain parameter spaces in which Higgs potential is stable, though parameter space is not so large.

Courtesy H.Oide

Stability of Higgs potential?

There remain parameter spaces in which Higgs potential is stable, though parameter space is not so large.

Bezrukov+, 1412.3811

Top properties: MASS

26

• Key SM parameter• Test EW vacuum stability

CMS all-jet (13 TeV)172.34 ±0.20 (stat+JSF)

±0.76 (syst) GeV

[GeV]fittm

100 200 300 400Dat

a/M

C

0.51

1.5

Eve

nts

/ 5 G

eV

200400600800

1000120014001600 correcttt

wrongtt

Multijet

Data

(13 TeV)-135.9 fb preliminaryCMS

CMS-PAS-TOP-17-008NEW@ ICHEP

[GeV]topm165 170 175 180 185

ATLAS+CMS Preliminary = 7-13 TeVs summary, topmLHCtopWG

shown below the line(*) Superseded by results

September 2017

World Comb. Mar 2014, [7]stattotal uncertainty

total stat

syst)± total (stat ± topm Ref.s

ATLAS, l+jets (*) 7 TeV [1] 1.35)± 1.55 (0.75 ±172.31 ATLAS, dilepton (*) 7 TeV [2] 1.50)± 1.63 (0.64 ±173.09 CMS, l+jets 7 TeV [3] 0.97)± 1.06 (0.43 ±173.49 CMS, dilepton 7 TeV [4] 1.46)± 1.52 (0.43 ±172.50 CMS, all jets 7 TeV [5] 1.23)± 1.41 (0.69 ±173.49

LHCtopWGLHC comb. (Sep 2013) 7 TeV [6] 0.88)± 0.95 (0.35 ±173.29 World comb. (Mar 2014) 1.96-7 TeV [7] 0.67)± 0.76 (0.36 ±173.34 ATLAS, l+jets 7 TeV [8] 1.02)± 1.27 (0.75 ±172.33 ATLAS, dilepton 7 TeV [8] 1.30)± 1.41 (0.54 ±173.79 ATLAS, all jets 7 TeV [9] 1.2)± 1.8 (1.4 ±175.1 ATLAS, single top 8 TeV [10] 2.0)± 2.1 (0.7 ±172.2 ATLAS, dilepton 8 TeV [11] 0.74)± 0.85 (0.41 ±172.99 ATLAS, all jets 8 TeV [12] 1.01)± 1.15 (0.55 ±173.72 ATLAS, l+jets 8 TeV [13] 0.82)± 0.91 (0.38 ±172.08

)l+jets, dil.Sep 2017(ATLAS comb. 7+8 TeV [13] 0.42)± 0.50 (0.27 ±172.51

CMS, l+jets 8 TeV [14] 0.48)± 0.51 (0.16 ±172.35 CMS, dilepton 8 TeV [14] 1.22)± 1.23 (0.19 ±172.82 CMS, all jets 8 TeV [14] 0.59)± 0.64 (0.25 ±172.32 CMS, single top 8 TeV [15] 0.95)± 1.22 (0.77 ±172.95 CMS comb. (Sep 2015) 7+8 TeV [14] 0.47)± 0.48 (0.13 ±172.44 CMS, l+jets 13 TeV [16] 0.62)± 0.63 (0.08 ±172.25

[1] ATLAS-CONF-2013-046[2] ATLAS-CONF-2013-077[3] JHEP 12 (2012) 105[4] Eur.Phys.J.C72 (2012) 2202[5] Eur.Phys.J.C74 (2014) 2758[6] ATLAS-CONF-2013-102

[7] arXiv:1403.4427[8] Eur.Phys.J.C75 (2015) 330[9] Eur.Phys.J.C75 (2015) 158[10] ATLAS-CONF-2014-055[11] Phys.Lett.B761 (2016) 350[12] arXiv:1702.07546

[13] ATLAS-CONF-2017-071[14] Phys.Rev.D93 (2016) 072004[15] EPJC 77 (2017) 354[16] CMS-PAS-TOP-17-007

LHCTopWGSummaryPlots

13 TeV, l+jets

Living beyond the edge: Higgs inflation and vacuum metastability

Fedor Bezrukov,1, 2, 3, Javier Rubio,4, † and Mikhail Shaposhnikov4, ‡

1Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA

2RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA

3CERN, CH-1211 Geneve 23, Switzerland

4Institut de Theorie des Phenomenes Physiques, Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland

(Dated: December 15, 2014)

The measurements of the Higgs mass and top Yukawa coupling indicate that we live in a veryspecial Universe, at the edge of the absolute stability of the electroweak vacuum. If fully stable, theStandard Model (SM) can be extended all the way up to the inflationary scale and the Higgs field,non-minimally coupled to gravity with strength , can be responsible for inflation. We show thatthe successful Higgs inflation scenario can also take place if the SM vacuum is not absolutely stable.This conclusion is based on two e↵ects that were overlooked previously. The first one is associatedwith the e↵ective renormalization of the SM couplings at the energy scale MP /, where MP is thePlanck scale. The second one is a symmetry restoration after inflation due to high temperaturee↵ects that leads to the (temporary) disappearance of the vacuum at Planck values of the Higgsfield.

I. INTRODUCTION

One of the most interesting questions in particlephysics and cosmology is the relation between the prop-erties of elementary particles and the structure of theUniverse. Some links are provided by Dark Matter andthe Baryon Asymmetry of the Universe. A number ofconstraints on hypothetical new particles can be also de-rived from Big Bang Nucleosynthesis.

The properties of the recently discovered Higgs boson[1, 2] suggest an additional and intriguing connection.Among the many di↵erent values that the Higgs masscould have taken, Nature has chosen one that allows toextend the Standard Model (SM) all the way up till thePlanck scale while staying in the perturbative regime.The behavior of the Higgs self-coupling is quite pe-culiar: it decreases with energy to eventually arrive toa minimum at Planck scale values and start increasingthereafter, cf. Fig. 1. Within the experimental and the-oretical uncertainties1 the Higgs coupling may stay posi-tive all way up till the Planck scale, but it may also crosszero at some scale µ0, which can be as low as 108 GeV,cf. Figs. 2 and 3. If that happens, our Universe becomesunstable2.

[email protected]† [email protected]‡ [email protected] The largest uncertainty comes from the determination of the topYukawa coupling. Smaller uncertainties are associated to thedetermination of Higgs boson mass and the QCD gauge coupling↵s. See Refs. [3–5] for the most refined treatments and Ref. [6]for a review.

2 The determination of the lifetime of the Universe is a rathersubtle issue that strongly depends on the high energy comple-tion of the SM. As shown in Ref. [7–9], if the gravitationalcorrections are such that the resulting e↵ective potential liesabove/below the SM one, the lifetime of our vacuum will benotably larger/smaller than the age of the Universe.

-0.04-0.02

0 0.02 0.04 0.06 0.08

0.1 0.12 0.14

100000 1e+10 1e+15 1e+20

µ, GeV

mH=125.5 GeV

yt=0.9176, mt=170.0yt=0.9235, mt=171.0yt=0.9294, mt=172.0yt=0.9359, mt=173.1yt=0.9413, mt=174.0yt=0.9472, mt=175.0

FIG. 1. Renormalization group running of the Higgs self-coupling for several values of the top quark Yukawa coupling(top pole mass) and fixed to 125.5 GeV Higgs boson mass.

The 0 3 compatibility of the data with vacuum in-stability is one of the recurrent arguments for invokingnew physics beyond the Standard Model. In particular,it is usually stated that the minimalistic Higgs inflationscenario [10], in which the Higgs field is non-minimallycoupled to gravity with strength , cannot take place ifthe Higgs self-coupling becomes negative at an energyscale below the inflationary scale.

We will show in this paper that Higgs inflation ispossible even if the SM vacuum is not absolutely sta-ble. Specifically, we will demonstrate that the renormal-ization e↵ects at the scale MP / can bring the Higgsself-coupling to positive values in the inflationary do-main. If that happens, inflation will take place with theusual chaotic initial conditions and the fate of the Uni-verse will be inevitably determined by the subsequentevolution. At the end of the exponential expansion, theHiggs field will start to oscillate around the bottom of

arX

iv:1

412.

3811

v1 [

hep-

ph]

11 D

ec 2

014

CMS, talk at ICHEP 2018

Courtesy H.Oide

The model is non-renormalizable. Is it well controlled?

Jordan frame: h2R

Mplh22


Einstein frame: 32

2

M2pl

h2(h)2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

conformal

trans.

Cutoff scale of the theory : Mpl


h<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>





:graviton





Mpl<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


2

M2pl


H =12

0h


Courtesy H.Oide

The model is non-renormalizable. Is it well controlled?

Jordan frame: h2R

Mplh22







:graviton

Einstein frame: 32

2

M2pl

h2(h)2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

conformal

trans.

Cutoff scale of the theory : Mpl


Typical energy scale during inflation :

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>Is the predictions in Higgs inflation unreliable?

(’09 Barbon&Espinosa, ’10 Burgess, Lee, &Trott, ’10 Hertzberg)







2

M2pl


H =12

0h


1/4inf

1/4Mpl

<latexit sha1_base64="mPjZsXTr6z+DFx+raoiuXNjopik=">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</latexit>

Courtesy H.Oide

No. The cutoff scale during inflation is background dependent and different from the one in the vacuum.

(’11 Bezrukov, Magnin, Shaposhnikov, &Sibiryakov)

L

M2pl + h2

M2pl + ( + 62)h2

h22<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

inflationary phenomenology using the e↵ective field theory parametrization of the e↵ects

which might come from the high energy theory.

One can distinguish several cuto↵ scales corresponding to violation of perturbative

unitarity in the di↵erent sectors of the model. In the gravity-scalar sector the cuto↵ is4

gs (h) '

8><

>:

MP , for h . MP

, h2

MP, for MP

. h . MPp,

ph , for h & MPp

.

(1.3)

The cuto↵ for purely gravitational interactions (graviton scattering o↵ graviton) is just the

e↵ective Planck scale (see eq. (1.1))

2Planck ' M2

P + h2 . (1.4)

The cuto↵ associated with SM gauge interactions at h > MP is

gauge ' h . (1.5)

and coincides with MP at smaller fields h.

MP/ξ

MP

MP/ξ MP/√ξ h

E

Weak coupling

Λg-s

Λgauge

Strong coupling

ΛPlanck

MP/ξ

MP

MP/ξ MP/√ξ h

E

Weak coupling

Λg-s

Λgauge

Strong coupling

ΛPlanck

Figure 1: Schematic depiction of the cut-o↵s (1.3), (1.4), and (1.5) in the Jordan (left) and Einstein(right) frames.

One of the reasons the inflation takes place while the theory remains in weakly coupling

regime is because it does not involve large momentum transfers. The validity of the theory

above momenta (h) is unclear. Thus, the question arises what happens when momentum

transfers E are between gauge (h) and Planck. Remaining within the minimal setup,5 we

can imagine two scenarios:

4We present all cuto↵s in the Jordan frame, corresponding to eq. (1.1) and use the unitary gauge,

HT =

0 , h/

p2.

5Alternatively, it is possible to attempt perturbative UV completion by introducing new states at the

cut-o↵ scale [14].

– 3 –

(’11 Bezrukov, Gorbunov, & Shaposhnikov)

Quantum fluctuations are well under-controlled during inflation!

Courtesy H.Oide

But it is not the end of the story…At the reheating stage, this issue is very subtle.





gs (h) '

8><

>:

MP , for h . MP

, h2

MP, for MP

. h . MPp,

ph , for h & MPp

.

(1.3)



2Planck ' M2

P + h2 . (1.4)


gauge ' h . (1.5)


MP/ξ

MP

MP/ξ MP/√ξ h

E

Weak coupling

Λg-s

Λgauge

Strong coupling

ΛPlanck

MP/ξ

MP

MP/ξ MP/√ξ h

E

Weak coupling

Λg-s

Λgauge

Strong coupling

ΛPlanck








HT =

0 , h/

p2.



– 3 –


At the higgs oscillation, parametric resonance of weak gauge bosons occurs and reheats the Universe.

(’09 Bezrukov, Gorbunov, & Shaposhnikov ’09 Garcia-Bellido, Figueroa, & Rubio ’16 Repond & Rubio)

For the transverse mode,

particles with are excited.

k

Mpl


Courtesy H.Oide


It is recently found that the longitudinal mode of the weak gauge bosons (or the NG mode) receives mass with spiky feature.

(’15 DeCross, Kaiser, Prabhu, Prescod-Weistein, Sfakianakis, ’16 Ema, Jinno, Mukaida, Nakayama)

m2e

(1 + 6)h2

M2pl + (1 + 6)h2


When the Higgs pass through the origin, potential energy turns to the kinetic energy, which generates the spike in the effective mass.

5!105 106 1.5!106 2!106tE

"0.2

0.2

0.4

0.6

0.8

1.0Φ

0 5!105 106 1.5!106 2!106tE

10"10

10"8

10"6

10"4

0.01mJeff2

Figure 2: Time evolution of φ and m2Jeff around the first few zero-crossings. The parameter values

are λ = 0.01 and ξ = 104, while the initial conditions are φini = MP and φini = 0. We can clearlysee that m2

Jeff blows up around the zero-crossings, thus representing a spike-like feature. We takeMP = 1 in this plot.

However, the second derivative shows a clear difference:

Ω ≃ξ(φJ φJ + φ2

J)

M2P

∼

⎧⎪⎪⎨

⎪⎪⎩

λξΦ4J

M2P

∼λΦ2

ξfor |φ| ≪

MP

ξ,

λΦ2J

ξ∼

λMPΦ

ξ2for |φ| ≫

MP

ξ.

(2.32)

The order of magnitude of Ω is different by a large factor O(ξΦ/MP) for |φ| ≪ MP/ξ and|φ| ≫ MP/ξ. We numerically follow the time evolution of Ω, Ω and Ω in Figs. 3–5. The blueline corresponds to the exact solution, while the red line does to the approximated one using

|φ|MP

≃√

3

2ln

(1 +

ξφ2J

M2P

), V (φ) ≃

λM4P

4ξ2

[

1− exp

(

−√

2

3

|φ|MP

)]2, (2.33)

in the whole region, as often used in the literature. For Ω and Ω we plot only exact solu-tions because the approximated ones show no visible difference. However, Ω shows a cleardifference when the inflaton crosses the origin. In fact, Ω shows a spike at around thezero-crossings in the exact case, while such a feature is not captured by the approximatedone♦8.

Next we consider the dynamics of φ. In this case, the spike scale appears as a suddenchange in the shape of the potential. For Φ ≫ MP/ξ, the equation of motion reads

φ ≃

⎧⎪⎪⎨

⎪⎪⎩

−λφ3 for |φ| ≪MP

ξ,

−m2oscφ for |φ| ≫

MP

ξ.

(2.34)

♦8 The value of Ω jumps when the inflaton crosses the origin as long as one uses the approximation (2.33).This is because we have to take the absolute value to keep the Z2 symmetry in Eq. (2.33).

9

tsp (

Mpl)1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

m2e


msp2e M2

pl<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(’16 Ema, Jinno, Mukaida, Nakayama)

Courtesy H.Oide



5!105 106 1.5!106 2!106tE

"0.2

0.2

0.4

0.6

0.8

1.0Φ

0 5!105 106 1.5!106 2!106tE

10"10

10"8

10"6

10"4

0.01mJeff2






J)

M2P

∼

⎧⎪⎪⎨

⎪⎪⎩

λξΦ4J

M2P

∼λΦ2

ξfor |φ| ≪

MP

ξ,

λΦ2J

ξ∼

λMPΦ

ξ2for |φ| ≫

MP

ξ.

(2.32)


|φ|MP

≃√

3

2ln

(1 +

ξφ2J

M2P

), V (φ) ≃

λM4P

4ξ2

[

1− exp

(

−√

2

3

|φ|MP

)]2, (2.33)



φ ≃

⎧⎪⎪⎨

⎪⎪⎩


ξ,


MP

ξ.

(2.34)


9


tsp (


m2e


msp2e M2


Longitudinal mode of weak gauge bosons with are excited.

k


Without backreaction


long 2M4

pl

42

inf

Mpl

2


Violent preheating!?2 102


for(’16 Ema, Jinno, Mukaida, Nakayama)

after one oscillation.

Courtesy H.Oide



5!105 106 1.5!106 2!106tE

"0.2

0.2

0.4

0.6

0.8

1.0Φ

0 5!105 106 1.5!106 2!106tE

10"10

10"8

10"6

10"4

0.01mJeff2






J)

M2P

∼

⎧⎪⎪⎨

⎪⎪⎩

λξΦ4J

M2P

∼λΦ2

ξfor |φ| ≪

MP

ξ,

λΦ2J

ξ∼

λMPΦ

ξ2for |φ| ≫

MP

ξ.

(2.32)


|φ|MP

≃√

3

2ln

(1 +

ξφ2J

M2P

), V (φ) ≃

λM4P

4ξ2

[

1− exp

(

−√

2

3

|φ|MP

)]2, (2.33)



φ ≃

⎧⎪⎪⎨

⎪⎪⎩


ξ,


MP

ξ.

(2.34)


9


tsp (


m2e


msp2e M2


Longitudinal mode of weak gauge bosons with are excited.

k


Without backreaction


long 2M4

pl

42

inf

Mpl

2


after one oscillation. Violent preheating!?

2 102<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

for





gs (h) '

8><

>:

MP , for h . MP

, h2

MP, for MP

. h . MPp,

ph , for h & MPp

.

(1.3)



2Planck ' M2

P + h2 . (1.4)


gauge ' h . (1.5)


MP/ξ

MP

MP/ξ MP/√ξ h

E

Weak coupling

Λg-s

Λgauge

Strong coupling

ΛPlanck

MP/ξ

MP

MP/ξ MP/√ξ h

E

Weak coupling

Λg-s

Λgauge

Strong coupling

ΛPlanck








HT =

0 , h/

p2.



– 3 –


But they lie in the strong coupling region…

Courtesy H.Oide

UV extension of Higgs inflation with -termR2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Courtesy H.Oide

It has been noticed that Higgs inflation + inflation

act as the UV extension of Higgs inflation

We need UV extension of Higgs inflation to understand its reheating stage. (e.g. ’11 Giudice&Lee, ’15 Barbon, Casas, Elias-Miro, Espinosa, ’18 Lee)


(’17 Wang&Wang, ’17 Ema, ’18 He, Starobinsky, &Yokoyama, ’18 Gundhi&Steinwachs)

(’17 Ema, ’18 Gorbunov&Tokareva)

Lg

=

M2

pl

2+ |H|2

R +

M2pl

12M2R2 |DµH|2 |H|4


Classically scale-invariant extension!

Courtesy H.Oide

It has been noticed that Higgs inflation + inflation

act as the UV extension of Higgs inflation

We need UV extension of Higgs inflation to understand its reheating stage. (e.g. ’11 Giudice&Lee, ’15 Barbon, Casas, Elias-Miro, Espinosa, ’18 Lee)



Lg

=

M2

pl

2+ |H|2

R +

M2pl

12M2R2 |DµH|2 |H|4


Classically scale-invariant extension!

R2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> - term gives a new scalar degree of freedom (scalaron), which pushes up the cutoff scale.

M<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

: scalaron mass

(’17 Wang&Wang, ’17 Ema, ’18 He, Starobinsky, &Yokoyama, ’18 Gundhi&Steinwachs)

Courtesy H.Oide

After conformal transformation, the system is now given by

Lg

=M2

pl

2R 1

2(µ)2 1

2e

2/3/Mpl |DµH|2 U(,H)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

U(,H) = e2

2/3/Mpl |H|4+34M2

plM2

1

1 +

2

M2pl

|H|2

e

2/3/Mpl

2


3.1 Features of the Potential

Here we give two examples for different combinations of and M in Figure 1. The potential (2.7)is invariant under ! . One can calculate the effective mass of the Higgs field, m2

, by tak-ing derivatives of the potential. The dominant contribution in small regime comes from a termproportional to , 3M2 exp

p2/3 /Mp

. For positive , Higgs field obtains a negative m

2

around the origin where its amplitude will grow exponentially. In large regime, m2 is dominated

by a term proportional to 2, 3(1 + 32M2/M

2p ) exp

2

p2/3 /Mp

2, whose coefficient is

always positive. These properties imply the existence of a local minimum on the potential for a given which corresponds to the valleys in Figure 1. Thus, independent of the initial position of , with alarge , the Higgs field will quickly fall into one of the valleys and evolve around the local minimum.If takes a small value, i.e. m

2 is small, direction will become flatter. In this case, if the initial

conditions start from a large value, it is possible for Higgs field to slowly roll down the potentialwall which is similar to the situation discussed in [23]. As for direction, it is always flat in large regime so that it has similar behavior to the scalaron in the R

2 inflation. As we shall see later, thereis a turning in the trajectory which can affect the sound speed of the curvature perturbations duringinflationary phase. The angular velocity at this turning is not large in the parameter regime we con-sider here, though. After the end of inflation, the fields will oscillate around the global minimum ofthe potential at (, ) = (0, 0) where reheating is expected to happen. According to the recent work[15], the particle production during preheating will be violent due to the appearance of non-minimalcoupling between Higgs and gravity.

Figure 1. Left: = 1000, M/Mp = 1.4 105. Right: = 3000, M/Mp = 1.9 105. The field valueshave been normalized by Mp. The value of mainly controls the position of the valleys while M determinesthe height of the plateau in the middle part and the depth of the valley.

3.2 Slow-Roll Inflation

As mentioned in previous sections, the evolution trajectory of two scalar fields are affected by thecurved nature of the field space, the potential shape and the expansion of the universe. Thus, it wouldbe more convenient to discuss the features of this trajectory by defining unit vectors T a and N

a [24]

– 4 –

’18 He, Starobinsky, &Yokoyama



Courtesy H.Oide

Lg

=M2

pl

2R 1

2(µ)2 1

2e


U(,H) = e2

2/3/Mpl |H|4+34M2

plM2

1

1 +

2

M2pl

|H|2

e

2/3/Mpl

2





p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –


Dim. 4 op. 3

M

Mpl

2

|H|4<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Dim. ≧5 op. 4

M

M2pl

2

|H|4s2 · · ·<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

For dim. 4 op. is weakly coupled and the cutoff scale fromdim. ≧5 op. is ’18 He, Starobinsky, &Yokoyama Mpl



M <

4

3Mpl




Courtesy H.Oide

Lg

=M2

pl

2R 1

2(µ)2 1

2e


U(,H) = e2

2/3/Mpl |H|4+34M2

plM2

1

1 +

2

M2pl

|H|2

e

2/3/Mpl

2





p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –



Scalaron cancels the divergencein pure Higgs inflation










2

M2pl


M2


M2


+<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>


Courtesy H.Oide

0 2 4 60

2

4

6

10-4 Mpl /M

10-3

ξ

λ = 0.01

Higgs-inflation-like regime

R2-inflation-like regime

ξ = ξc

M =Mc

ABC

ξ =4π3

MplM

ξ 2

ξc 2+Mc 2

M 2= 1

ξ 2

λ=Mpl2

3M 2

During inflation it is effectively described by a single field with

V () 34M2M2

pl

1 e

2/3/Mpl)

2


M2 M2

1 + 32M2/M2pl


Strong coupling

for sufficiently large <latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> (’17 Ema, ’18 He, Starobinsky, &Yokoyama)

’18 He, Jinno, KK, Park, Starobinsky, Yokoyama

Correct scalar perturbation

Correct amplitude of scalar perturbation is realized for

2

2c

+M2

c

M2= 1


c 4.4 103

0.01

1/2 N

54<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Mc 1.3 105

N

54

1


spectral tilt/tensor-to-scalar ratio are the same to the pure Higgs inflation/ inflationR2


ns 1 2N 0.97


r 12N2

0.003<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

N<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit># depends on reheating temp.

(’17 Ema, ’18 He, Starobinsky, &Yokoyama)

Courtesy H.Oide

Inflaton oscillation and violent preheating

Courtesy H.Oide

For sufficiently small (especially for weakly coupled case) the dynamics at inflaton oscillation is no longer single field like.





p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –


5

h field oscillates rapidly with the e↵ective mass squared M2 (for |'| ' Mpl) around the stream line at thebottom of the valley.

B. E↵ective mass for the phase direction

In order to study quantum creation of the NG modedue to the spiky behavior of its mass term , it is con-venient to define a canonically normalized scalar field cfrom the phase of the H field, H(x) = h(t)ei(x)/

p2.

Since the potential U is independent of , the relevantpart of the Lagrangian reads

pgELE 1

2

pgEe

↵'h2gµE @µ@ =1

22c

1

2a2(rc)

2+1

2

F

F2c+· · · ,

(4)where c(x) and F (t) are defined as

c(x) a3/2(t)e↵'(t)/2h(t)(x) F (t)(x), (5)

in the Friedmann background, ds2 = dt2 + a2(t)dx2,and a dot denotes di↵erentiation with respect to t. Thenwe read o↵ the e↵ective mass of the NG mode as

m2c = F (t)

F (t)= ↵

2

@U

@'+e↵'

h

@U

@h3

4

U

M2pl

+5

24

1

M2pl

'2 + e↵'h2

,

(6)where we have used the background equations (1), (2),and (3). While the last two terms are always of the orderof the Hubble parameter, the first two terms can be muchlarger when the scalar field trajectory deviates from thevalley (7). Figures 3 show the time evolution of m2

cfor our benchmark parameters (A), (B), and (C). Thee↵ective mass gets larger when ' gets negative and moreor less the spikes still appear even in the case where theR2 term is present. We can also see that the height of thespikes gets lower and their width gets wider for smallerM , when the system is more R2-inflation like. In Figs. 4,we show the heights and the widths of the spikes as afunction of M under the observational constraint (15).Here we define the width of the spikes as the full widthat half maximum.

The behavior of the spikes can be understood analyt-ically as follows. Just before the end of inflation, theenergy density of the Universe is dominated by the po-tential term (9). Since the potential energy would alsodominate the kinetic energy when the ' field climbs upthe alley ' < 0, the potential energy at h = 0 at the firstoscillation can be written as

U(', h = 0) = C2mUinf , (7)

which yields with Eq. (9)

e↵' = 1 + CmM

= 1 + Cm

sM2

pl

32M2 + M2pl

at h = 0.

(8)

30 40 50 60 70 80

-5

0

5

10

10-4 Mpl t

103h/M

pl

FIG. 2: Time evolution of the Higgs field h (top) and scalaron' (bottom) for the parameter points (A) (left), (B) (middle),and (C) (right). We fixed = 0.01. See Fig. 1 for the threeparameter points.

FIG. 3: Time evolution of the e↵ective mass squared for thephase direction m2

c for the parameter points (A) (left), (B)(middle), and (C) (right). We fixed = 0.01. The top panelsshow the evolution over the full time range shown in Fig. 2,while the bottom panels are magnifications of the top panelsaround the first peak.

5




p2.


pgELE 1

2

pgEe

↵'h2gµE @µ@ =1

22c

1

2a2(rc)

2+1

2

F

F2c+· · · ,


c(x) a3/2(t)e↵'(t)/2h(t)(x) F (t)(x), (5)


m2c = F (t)

F (t)= ↵

2

@U

@'+e↵'

h

@U

@h3

4

U

M2pl

+5

24

1

M2pl

'2 + e↵'h2

,




U(', h = 0) = C2mUinf , (7)


e↵' = 1 + CmM

= 1 + Cm

sM2

pl

32M2 + M2pl

at h = 0.

(8)

30 40 50 60

0

1

2

3

4

10-4 Mpl t

10φ/M

pl

30 40 50 60 70 80-1

0

1

2

3

4

10-4 Mpl t

10φ/M

pl




= 0.01<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Higgs

Scalaron



Mpl/M = 3 104<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>



Courtesy H.Oide

For sufficiently small (especially for weakly coupled case) the dynamics at inflaton oscillation is no longer single field like.





p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –


5




p2.


pgELE 1

2

pgEe

↵'h2gµE @µ@ =1

22c

1

2a2(rc)

2+1

2

F

F2c+· · · ,


c(x) a3/2(t)e↵'(t)/2h(t)(x) F (t)(x), (5)


m2c = F (t)

F (t)= ↵

2

@U

@'+e↵'

h

@U

@h3

4

U

M2pl

+5

24

1

M2pl

'2 + e↵'h2

,




U(', h = 0) = C2mUinf , (7)


e↵' = 1 + CmM

= 1 + Cm

sM2

pl

32M2 + M2pl

at h = 0.

(8)

30 40 50 60 70 80

-5

0

5

10

10-4 Mpl t

103h/M

pl




5




p2.


pgELE 1

2

pgEe

↵'h2gµE @µ@ =1

22c

1

2a2(rc)

2+1

2

F

F2c+· · · ,


c(x) a3/2(t)e↵'(t)/2h(t)(x) F (t)(x), (5)


m2c = F (t)

F (t)= ↵

2

@U

@'+e↵'

h

@U

@h3

4

U

M2pl

+5

24

1

M2pl

'2 + e↵'h2

,




U(', h = 0) = C2mUinf , (7)


e↵' = 1 + CmM

= 1 + Cm

sM2

pl

32M2 + M2pl

at h = 0.

(8)

30 40 50 60

0

1

2

3

4

10-4 Mpl t

10φ/M

pl

30 40 50 60 70 80-1

0

1

2

3

4

10-4 Mpl t

10φ/M

pl




= 0.01<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Higgs

Scalaron


Mpl/M = 3 104<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>




Courtesy H.Oide

Effective mass for the NG mode?

m2 =

16

U

+

e

2/3/Mpl

h

U

h 3

4U

M2pl

+5

24M2pl

2 + e

2/3/Mpl h2


# Longitudinal mode of weak gauge bosons should exhibit a similar behavior

Courtesy H.Oide

Effective mass for the NG mode?

# Longitudinal mode of weak gauge bosons should exhibit a similar behavior

m2 =

16

U

+

e

2/3/Mpl

h

U

h 3

4U

M2pl

+5

24M2pl

2 + e

2/3/Mpl h2


Figure 2: Time evolution of the Higgs field h (top) and scalaron ' (bottom) for theparameter points (A) (left), (B) (middle), and (C) (right). We fixed = 0.01. See Fig. 1 forthe three parameter points.

30 40 50 60 70 80-0.20.00.20.40.60.81.01.2

10-4 Mpl t

106m

θ c2/M

pl2

Figure 3: Time evolution of the e↵ective mass squared for the phase direction m2c for the

parameter points (A) (left), (B) (middle), and (C) (right). We fixed = 0.01. The toppanels show the evolution over the full time range shown in Fig. 2, while the bottom panelsare magnifications of the top panels around the first peak.

9

Spiky behavior still appears! ’18 He, Jinno, KK, Park, Starobinsky, Yokoyama

Courtesy H.Oide

How to understand it analytically?




p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –




Peak appears when the inflaton climbs up the alley with vanishing kinetic energy =>

Uinf<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Ualley<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Ualley Uinf<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

m2 =

16

U

+

e

2/3/Mpl

h

U

h 3

4U

M2pl

+5

24M2pl

2 + e

2/3/Mpl h2


h = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>@

This term gives much larger value than the Hubble scale when the scalaron climbs up the alley.

Courtesy H.Oide





p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –








m2 =

16

U

+

e

2/3/Mpl

h

U

h 3

4U

M2pl

+5

24M2pl

2 + e

2/3/Mpl h2



(msp )2

3MMpl


For the Higgs-inflation-like case, the height of the spike is


Courtesy H.Oide





p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –








m2 =

16

U

+

e

2/3/Mpl

h

U

h 3

4U

M2pl

+5

24M2pl

2 + e

2/3/Mpl h2


For the Higgs-inflation-like case, the height of the spike is (msp

)2

3MMpl<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Duration of spike is given by the time scale of the climbing up the alleytsp M1




Courtesy H.Oide





p2/3 /Mp


2



2p ) exp

2

p2/3 /Mp









a [24]

– 4 –








m2 =

16

U

+

e

2/3/Mpl

h

U

h 3

4U

M2pl

+5

24M2pl

2 + e

2/3/Mpl h2


For the Higgs-inflation-like case, the height of the spike is (msp

)2

3MMpl<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Duration of spike is given by the time scale of the climbing up the alleytsp M1




Figure 2: Time evolution of the Higgs field h (top) and scalaron ' (bottom) for theparameter points (A) (left), (B) (middle), and (C) (right). We fixed = 0.01. See Fig. 1 forthe three parameter points.

30 40 50 60 70 80-0.20.00.20.40.60.81.01.2

10-4 Mpl t

106m

θ c2/M

pl2

Figure 3: Time evolution of the e↵ective mass squared for the phase direction m2c for the

parameter points (A) (left), (B) (middle), and (C) (right). We fixed = 0.01. The toppanels show the evolution over the full time range shown in Fig. 2, while the bottom panelsare magnifications of the top panels around the first peak.

9


tsp M1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(msp )2

3MMpl


Courtesy H.Oide

More precisely with the parameter to fit the cosmological data

0.0 0.1 0.2 0.3 0.4 0.50.1

0.51

510

0.1

0.51

510

103 M /Mpl

106m

θ c2/M

pl2

10-4MplΔt

Figure 4: Peak amplitude and timescale of the spike in the e↵ective mass squared of theimaginary field. The green triangles and the red disks are the numerically obtained peakamplitude and timescale of the mass spike, respectively, while the brown dashed line and theblue solid line are our analytic estimates (3.10) and (3.11) with Cm ' 0.25 and Ct ' 0.2.

(3.11) any more when M significantly exceeds the spike timescale inverse in the pure-Higgscase M

pMpl. Figure 4 shows the measured peak amplitude and timescale of the spikes,

as well as our analytic estimates (3.10) and (3.11) with Cm ' 0.25 and Ct ' 0.2.6 Thegreen triangles and the red disks are the values of the amplitude and the timescale estimatedfrom the numerical time evolution, respectively, while the brown dashed line and the bluesolid line are the predictions of our analytic estimates (3.10) and (3.11), respectively. Wesee that the numerical results coincide with the analytic estimates well.

3.3 Estimate on particle production

Let us roughly estimate the energy density of the particles produced in the first spike. Weneglect particle production from the other oscillations because we are mainly interested inthe e↵ect of the strongest spike which appears in the pure-Higgs inflation. A more detailedanalysis, including the consequences of other oscillations, will be presented elsewhere [56].

For the estimate of particle production from strong spikes, we can for example refer tothe Appendix C of Ref. [57]. If we describe the strong spike with the following cosh-typespike function

m2c(t) =

m

2t

1

cosh2(t/t). (3.12)

and the produced field is in the vacuum for t ! 1, its number density after the spike is

6 A rough estimate of Cm goes as follows. For the pure Higgs or R2 inflation, the potential shape becomes/ (1 e↵')2. The slow roll condition max(||, ||) < 1 breaks down at e↵' = 2

p3 3, when the inflaton

potential energy is ' 0.287 times its value at the plateau.

10

Cm 0.25<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Ct 0.2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Smaller gives lower spike and wider duration.M<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(msp )2 = Cm

3(M2 M2

c )Mpl<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

tsp = CtM1



Courtesy H.Oide

More precisely with the parameter to fit the cosmological data

(msp )2 = Cm

3(M2 M2

c )Mpl<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

tsp = CtM1


0.0 0.1 0.2 0.3 0.4 0.50.1

0.51

510

0.1

0.51

510

103 M /Mpl

106m

θ c2/M

pl2

10-4MplΔt

Figure 4: Peak amplitude and timescale of the spike in the e↵ective mass squared of theimaginary field. The green triangles and the red disks are the numerically obtained peakamplitude and timescale of the mass spike, respectively, while the brown dashed line and theblue solid line are our analytic estimates (3.10) and (3.11) with Cm ' 0.25 and Ct ' 0.2.

(3.11) any more when M significantly exceeds the spike timescale inverse in the pure-Higgscase M

pMpl. Figure 4 shows the measured peak amplitude and timescale of the spikes,

as well as our analytic estimates (3.10) and (3.11) with Cm ' 0.25 and Ct ' 0.2.6 Thegreen triangles and the red disks are the values of the amplitude and the timescale estimatedfrom the numerical time evolution, respectively, while the brown dashed line and the bluesolid line are the predictions of our analytic estimates (3.10) and (3.11), respectively. Wesee that the numerical results coincide with the analytic estimates well.

3.3 Estimate on particle production

Let us roughly estimate the energy density of the particles produced in the first spike. Weneglect particle production from the other oscillations because we are mainly interested inthe e↵ect of the strongest spike which appears in the pure-Higgs inflation. A more detailedanalysis, including the consequences of other oscillations, will be presented elsewhere [56].

For the estimate of particle production from strong spikes, we can for example refer tothe Appendix C of Ref. [57]. If we describe the strong spike with the following cosh-typespike function

m2c(t) =

m

2t

1

cosh2(t/t). (3.12)

and the produced field is in the vacuum for t ! 1, its number density after the spike is

6 A rough estimate of Cm goes as follows. For the pure Higgs or R2 inflation, the potential shape becomes/ (1 e↵')2. The slow roll condition max(||, ||) < 1 breaks down at e↵' = 2

p3 3, when the inflaton

potential energy is ' 0.287 times its value at the plateau.

10

Cm 0.25<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Ct 0.2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Smaller gives lower spike and wider duration.M<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Too high energy scales observed in pure Higgs inflation now disappear!


Courtesy H.Oide

NG particle production? Evaluate the mode equation

c(k, ) + (k2 + m2())c(k, ) = 0


with spiky mass m2()


and evaluate the Bogoliubov coef.

c(k, ) =1

2k()

k()ei

R d k( ) + k()ei

R d k( )


Just a tunneling problem for the spiky wall in quantum mechanics.(e.g. Landau&Lifshitz)

x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

t<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Courtesy H.Oide


c(k, ) + (k2 + m2())c(k, ) = 0





c(k, ) =1

2k()

k()ei

R d k( ) + k()ei

R d k( )




2.8

Ct

0.2

4

M4 inf<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

m2(t) =

µ

2t

1cosh2(t/t)


with approximation.

After one oscillation


(’18 He, Jinno, KK, Park, Starobinsky, Yokoyama)

Courtesy H.Oide


c(k, ) + (k2 + m2())c(k, ) = 0





c(k, ) =1

2k()

k()ei

R d k( ) + k()ei

R d k( )




2.8

Ct

0.2

4

M4 inf<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

m2(t) =

µ

2t

1cosh2(t/t)


with approximation.

After one oscillation

Reheating will not complete by the violent preheating for the NG mode!


(’18 He, Jinno, KK, Park, Starobinsky, Yokoyama)

Courtesy H.Oide

Discussion

Courtesy H.Oide

- Higgs inflation with a nonminimal coupling to gravity is an interesting model. - Violent preheating for the NG mode is theoretical challenge in it. - (Classically) scale-invariant extension can push up its cutoff scale and make the model healthier. - Spiky feature in the mass of NG mode at the inflaton oscillation still appear and is really physical, but it gets lower and broader. The physical origin is a bit different. - Violent particle production occurs, but it is not sufficient to complete reheating. - Usual perturbative and non-perturbative decay of the Higgs and scalaron will reheat the Universe. - Since the inflaton oscillation is completely two-field nature, and hence results of previous studies on preheating in pure Higgs inflation cannot be applied directly.


(null) as a uv-extension of … · 2019-03-29 · 4 2 1 e 2 3 /mpl 2 (null) simplest

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