横山順一

Jun’ichi Yokoyama (RESCEU, Tokyo)

The SM Higgs particle was finally discovered !?

A candidate event from a Higgs particle observed by ATLAS (© ATLAS experiment, by courtesy of S. Asai.)

Professor Shoji Asai at the press conference together with his view graph on July 4.

ATLAS

126.5GeV 5hm

4

H

H ZZ leptons

CMS

125.3 0.6GeV 4.9hm

4

2 2

H

H ZZ l

H WW l

6% human resources, 10% $\ from Japan

http://aappsbulletin.org/

What they have found: A scalar particle which interacts with gauge fields. Higgs mechanism seems to be at work.

Coupling to quarks and leptons have not beenconfirmed yet.

It is consistent with the Standard Model Higgs,but only half of its roles have been probed.

Nevertheless, for Cosmology it is a great discovery,since no one had ever discovered any scalar field until then, which is indispensable for Inflationary Cosmology.

UT Quest

Relevance to “Superstring Cosmophysics” ?

Now that a Higgs(-like) particle is discovered,any viable superstring theory must accommodate it.

I UT Quest progress report “Generalized Higgs inflation” (Kamada, Kobayashi, Takahashi, Yamaguchi, & JY 2012)

II “Higgs condensation as an unwanted curvaton” (Kunimitsu & JY 2012)

Just after the CERN seminars in December announcing “hints of the Higgs boson”an analysis of the renormalized Higgs potential was done for m=124 ～ 126GeV.

Elias-Miro et al Phys Lett B709(2012)222

126GeV Hm For there is a finite parameter region within 2σ where the Higgs potential is stable, or its self coupling remain positive even after renormalization.

The simplest possibility tied with the recent experiment:

Easy to preach Difficult in practice

Tree level action of the SM Higgs field

Taking with φ being a real scalar field, we find

for .

This theory is the same as that proposed by Linde in 1983 to drive chaotic inflation at .PlM

126GeV Hm 0.13 at low energy scale

In order to realize we must have .

The Higgs potential is too steep as it is.

510T

T

1310

Square amplitude of curvature perturbation on scale 2

rk

2T

T

The spectral index of the curvature perturbation is given by

In potential-driven inflation models

1 6 2s V Vn 22

2, 2G

V V G

M V VM

V V

The Higgs potential is too steep as it is.Several remedies have been proposed to effectively flatten the potential.

1 The most traditional one (original idea: Spokoiny 1984 Cervantes-Cota and Dehnen 1995, ….)

introduce a large and negative nonminimal coupling to scalar curvature R

L = + L

510

2 2 2 2 212Pl Pl PlM M R M H

Pl

Effectively flatten thespectrum

2 New Higgs inflation (Germani & Kehagias 2010)

Introduce a coupling to the Einstein tensor in the kinetic term.

21 1

2 2g g w G

2

1w const

M

During inflation .

If , the normalization of the scalar field changes.

22 00

23H

w GM

H M

143

H

M is the canonically normalized scalar field.

24 4

2[ ] [ ]

4 12

MV V

H

effectively reducing the self couplingto an acceptable level.

(Actual dynamics is somewhat more complicated.)

3: Running kinetic inflation (Takahashi 2010, Nakayama & Takahashi 2010)

Introduce a field dependence to the kinetic term.

1 1( )

2 2g f

Canonical normalization of the scalar field changes the potential effectively.

( ) 1f v

( ) ( ) ,X V g X L =

Slow-roll parameters2

2, , , .

'( )

H g g X

H H gH V

, , , 1 we find slow-roll EOMsFor2 23 ( )PlM H V 33 1 ( ) 0gHH V

Background equations of motion2 2

2

3 1 (6 ) ( )

1 (3 )

3 (9 3 6 2 ) (1 2 ) ( ) 0

Pl

Pl

M H gH X V

M H gH X

gH H V

This extra friction term enhances inflation and makesit possible to drive inflation by a standard Higgs field.

Galileon-type coupling

(Kamada, Kobayashi, Yamaguchi & JY 2011)

1

2X g

Higgs G inflation

444

XX

M L

9 1COBE( ) 2.4 10 @ 0.002Mpc ( 60)k k RP N

146 134.7 10 10 GeV.PlM M

In fact, all these models can be described in the context ofthe Generalized G-inflation model seamlessly, which is the mostgeneral single-field inflation model with 2nd order field equations.

3

22

4 4

2 33

5 5

,

,

,

1, 3 2

6

X

X

K X

G X

G X R G

G X G G

2

3

4

5

L

L

L

L

Inflation from a gravity + scalar system 5

4

2i

i

S gd x

L

In fact, all these models can be described in the context ofthe Generalized G-inflation model seamlessly, which is the mostgeneral single-field inflation model with 2nd order field equations.

3

22

4 4

2 33

5 5

,

,

,

1, 3 2

6

X

X

K X

G X

G X R G

G X G G

2

3

4

5

L

L

L

L

Inflation from a gravity + scalar system 5

4

2i

i

S gd x

L

This theory includes potential-driven inflation models k-inflation model Higgs inflation model New Higgs inflation model G-inflation model

5G G

, [ ]K X X V ,K X K X

2 242 2R G

( , ) ( , )K X G X

We consider potential-driven inflation in the generalized G-inflation context.So we expand

and neglect higher orders in X.

We find the following identities. (t.d.) ＝ total derivative

As a result we can set in the Lagrangian without loss of generality.

4g g

L = + L

in the generalized G-inflation

Yet another new model:Running Einstein Inflation

 simplest case

We analyze all these possibilities on equal footing, characterizingHiggs inflation by five functions and potential.

Potential-driven slow-roll inflation

Slow-roll field equation

w/

Gravitational field equations

Cosmological PerturbationsTensor perturbation

Scalar perturbations

MCMC analysisBy Popa 2011

So far each modelhas been analyzedseparately.

G nonminimal new

We can fill the gapusing the generalized

Higgs inflation

All the Higgs inflation models are unified in the Generalized G-inflation.

Open issue: Quantum corrections in the presenceof these new interaction terms.

II Cosmology of the Higgs field

Here we study the case the Higgs field is NOT the inflaton.

Whatever the inflation model is, the Higgs field exists in the Standard Model and plays some roles.

During inflation, short-wave quantum fluctuations are continuously generated for the Higgs field and its wavelength is stretched by inflation to produce zero-modecondensate.

Higgs condensation

with Taro Kunimitsu

The most dramatic case

If the self coupling becomes negative at high energy orhigh field value, the potential becomes unstable.

𝑉 [𝜙 ]≈𝜆(𝜙)

4𝜙4

Our Universe cannot be reproduced in such a case.

(Espinosa et al 2008)

Hv

[ ]V H

4 4 2( )[ ] , (10 ) 0.

4 4V O

We consider the case λ is positive (and constant) in theregime of our interest.

The behavior of a scalar field with such a potential duringinflation has been studied using the stochastic inflation method.

(Starobinsky and JY 1994)

Decompose the scalar field as

coarse-grained mode

short-wave quantum fluctuations

: small parameter

is a solution of

namely,

with a stochastic noise term

whose correlation function is given by

A slow-roll equation for the long-wave part obtained from

Langevin equation

We can derive a Fokker-Planck equation for the one-point (one-domain) probability distribution function (PDF)

Its generic solution can be expanded as

is the complete orthonormal set of eigenfunctions of the Schrödinger-type eq.

If is finite, we find and the equilibrium PDF exists.

Each solution relaxes to the equilibrium PDF at late time which has the de Sitter invariance.

For we find an equilibrium PDF corresponding to as

First few eigenvalues (numerically obtained)

Equilibrium expectation values obtained using

2H

2H

4H

where

Higgs condensation

Properties of the Higgs condensationTwo point equilibrium temporal correlation function can be well approximated by

for

The correlation time defined by is given by

The spatial correlation function can be evaluated from it using the de Sitter invariance.

cf de Sitter invariant separation

0( ) Hta t a e

The spatial correlation length defined by therefore reads

Exponentially large !

So we can treat it as a homogeneous field when we discuss its effect on reheating.

Its energy density is small during inflation, but it remains constant until the Hubble parameter decreases to .

So the Higgs condensation or its decay products never contributes to the total energy density in the usual inflation models where inflation isfollowed by field oscillation of a massive field whose energy densitydissipates in proportion to .

At this time we find

tot

condtot

The Higgs field oscillation governed by a quartic potential dissipatesits energy density in proportion to in the same way as radiation.4 ( )a t

3( )a t

In some models, inflation may end abruptly without being followed by itscoherent field oscillation and reheating proceeds only through gravitationalparticle creation.

Such inflation models include k-inflation, (original) G-inflation, and quintessential inflation models.

A simple k-inflation example

: inflaton

If and are constants with opposite sign, the kinetic function has an attractor

solution

Then the energy density and pressure are given by

inflation with2

2 1inf 2

212 G

KH

M K

In such models the Higgs condensation contributes to the total energy density appreciably in the end.

[ ]V

Quintessential inflation(Peebles & Vilenkin 99)

(Armendariz-Picon, Damour & Mukhanov 99)

k-inflation ends when and both becomes positive. Then the kinetic energystarts to redshift quickly and only the first term of the Lagrangian becomes relevant.

k-inflation

Free scalar field

( ) Hta t e

1 3( )a t t

1 2 0K K

1 2 0K K

If this change occurs abruptly, numerical calculation shows that this transition occurs within Hubble time.

Gravitational particle production takes place due to this rapid changeof the expansion law. The energy density of a massless boson created this way is given by

1a taking at the end of inflation. is the time elapsed during smooth transition from de Sitter to a power-law expansion.t

inf 1H t

(cf Ford 87)

kination

time or scale factor

energy density

inflation

kinatic energy of the inflaton

radiation from gravitational particle production

0a6a

Let us assume there are effectively such modes and neglect the logarithmic factor. 4

inf2 4

9

32r

NH

a

4inf

2 4

9

32r

NH

a

Ra

is the tensor-scalar ratio.

time or scale factor

energy density

inflation

kinetic energy of the inflaton

radiation from gravitational particle production

radiation from Higgs condensation

4a

0a6a

0a

4aHiggs condensation

Higgs oscillation

We incorporate the Higgs condensation. When it starts oscillation @ eff( )H t m

If we try to estimate the reheat temperature from the equalitytaking the Higgs condensation into account, we find

which is to be compared with the previous estimate

Thus the Higgs condensate or its decay product contributes to the total energy density appreciably.

Its fluctuation can be important.

Higgs condensation as a curvatonHere we analyze properties of fluctuations of the Higgs condensation.

A power-law spatial correlation function like

can be obtained from a power law power spectrummaking use of the formula

We find and

1 inf2 H

It practically behaves as a massless free field.

(Enqvist & Sloth, Lyth & Wands, Moroi & Takahashi 02)

The power spectrum of (potential) energy density fluctuation is given by

Amplitude of fractional density fluctuation on scale 2

rk

(0.1)O

cond cond osc cond

tot cond osc totosc

Hc c

H

R

Its contribution to the curvature perturbation is unacceptably large.

Hubble parameter at the onset of oscillation osc effH m (Kawasaki, Kobayashi & Takahashi 11)

can be O(1)

1 2 510

ConclusionInflation models followed by kination regime with gravitational reheating is incompatible with the Higgs condensation.

f

k-inflation

Standard inflation

Stochastic gravitational wave background from inflation

Models with such anenhanced spectrumare ruled out.