Conclusions and discussion

Content

The task - the construction of a fundamental particle physics model accounting for an inflationary scenario in cosmology.

I A scalar field is very convenient for providing an inflationary stage of the cosmic expansion

I A self-interaction of a scalar field creates problems for inflation

I The inclusion of the non-minimal couplingξRφ² supplies us with an effective potential providing a slow-roll regime for the universe

I Due to quantum effects the early evolution of the universe depends not only on the inflaton-graviton sector, but is strongly effected by the particle content of the theory

I Main quantum effects are encoded in a special

combination of coupling constantsA -anomalous scaling

I The nature of an inflaton scalar field - could it be the Higgs boson ?

I Quantum effects and the renormalization group running

I Theasymptotical freedom effect for the anomalous scaling

I The cosmological model of inflation based on the non-minimally coupled Higgs boson looks as compatible with both : cosmological observations and particle physics bounds, but some details are not yet clear

2 2 4

|Φ|² =Φ^†Φ.

L_int =−X

χ

1

2λ_χχ²ϕ²−X

A

1

2g_A²A²_µϕ² −X

ψ

y_ψϕψψ.¯ Quantum one-loop correction to the potential is

X

particles

(±1)m⁴(ϕ)

64π² lnm²(ϕ)

µ² = λA

128π² ϕ⁴lnϕ² µ² +... .

A= 2 λ

ÃX

χ

λ²_χ+ 3X

A

g_A⁴−4X

ψ

y_ψ⁴

! .

In the context of quantum cosmology the positivity of the coefficient A makes one-loop wave functions of the universe (both no-boundary and tunneling) normalizable.

The anomalous scaling in the case of ξ À 1

determines the quantum rolling force in the effective

equation of the inflationary dynamics and yields the

parameters of the CMB generated during inflation.

A= 3 8λ

³

2g⁴+¡

g²+g⁰²¢₂

−16y_t⁴

´ .

In the conventional range of the Higgs mass 115 GeV≤M_H ≤180 GeV

this quantity at the electroweak scale belongs to the range

−48<A<−20

which strongly contradicts the CMB data which require

−12<A<14.

Taking into account the renormalization group running

A(t ) = 3 8λ(t)

³

2g

⁴

(t) + ¡

g

²

(t) + g

⁰²

(t) ¢

₂

− 16y

_t⁴

(t)

´ , t = ln(ϕ/µ)

we see that the value of the A on the inflationary

scale is compatible with the CMB data.

F.L. Bezrukov, A. Magnin and M. Shaposhnikov, Standard Model Higgs boson mass from inflation, arXiv:0812.4950 [hep-ph].

F. Bezrukov and M. Shaposhnikov,

Standard Model Higgs boson mass from inflation: two loop analysis,

arXiv:0904.1537 [hep-ph].

A. De Simone, M. P. Hertzberg and F. Wilczek, Running Inflation in the Standard Model, arXiv:0812.4946 [hep-ph].

One-loop approximation

S[g_µν, ϕ] = Z

d⁴x g^1/2 µ

−V(ϕ) +U(ϕ)R(g_µν)− 1

2G(ϕ) (∇ϕ)²

¶

V(ϕ) = λ

4(ϕ²−ν²)²+ λϕ⁴

128π²Alnϕ² µ², U(ϕ) = 1

2(M_P² +ξϕ²) + ϕ² 384π²

µ

Clnϕ² µ² +D

¶ , G(ϕ) = 1 + 1

192π² µ

Flnϕ² µ² +E

¶ .

ˆ

g_µν = 2U(ϕ) M_P² g_µν,

µdϕˆ dϕ

¶₂

= M_P² 2

GU+ 3U⁰² U² . Uˆ =M_P²/2, Gˆ = 1,

Vˆ( ˆϕ) = µM_P²

2

¶₂ V(ϕ) U²(ϕ)

¯¯

¯¯

¯ϕ=ϕ( ˆϕ)

. At the inflation scale with ϕ >MP/√

ξÀv and for large non-minimal couplingξ À1

Vˆ = λM_P⁴ 4ξ²

µ

1 + A 16π² lnϕ

µ

¶ .

Inflationary slow-roll parameters:

ˆ ε≡ M_P²

2 Ã1

Vˆ dVˆ dϕˆ

!₂

= 4 3

µM_P²

ξ ϕ² + A 64π²

¶₂ , ˆ

η≡ M_P² Vˆ

d²Vˆ

dϕˆ² =−4M_P² 3ξϕ² .

Their smallness determines the range of the inflationary stage ϕ > ϕ_end, terminating at the value of ε, which we chose to beˆ ˆ

ε_end= 3/4. Then the inflaton value at the exit from inflation equals

ϕ_end '2M_P/√ 3ξ.

factor e-folding numberN: ϕ²

ϕ²_I =e^x −1 +O µlnN

N

¶

, ϕ²_I = 64π²M_P² ξA , x ≡ NA

48π².

The CMB spectral index n_s, the tensor to scalar ratio r and the spectral index running α:

n_s = 1− 2 N

x e^x−1 , r = 12

N²

µ xe^x e^x −1

¶₂ , α =− 2

N²

x²e^x (e^x −1)².

Renormalization Group improvement

V(ϕ) = λ(t)

4 Z⁴(t)ϕ⁴, U(ϕ) = 1

2

³

M_P² +ξ(t)Z²(t)ϕ²

´ , G(ϕ) = Z²(t).

dλ

dt = β_λ 1−γ, dξ

dt = β_ξ 1−γ, dZ

dt = γ 1−γ. These β-functions depend on running couplings λ and ξ as well as on the rest of the coupling constants in Standard Model.

dg

dt = β_g

1−γ, dg⁰

dt = β_g⁰

1−γ, dg_s

dt = β_gs

1−γ, dy_t

dt = β_yt 1−γ.

In view of the uncertainties of the early Universe model, the inclusion of the two-loop part in the β functions seems to exceed the available precision of the theory. However, the lower Higgs mass bound is very sensitive to the initial conditions for the RG equations at the electroweak scale and to the magnitude of two-loop contributions which might result in 10 ÷ 20% variations of running couplings.

Therefore, where necessary, we will use two-loop

results for β functions.

The effect of non-minimal curvature coupling of the Higgs field

Due to the strong non-minimal coupling between graviton and Higgs-field sectors the propagator of the Higgs field is

modified by the factors(t):

s(ϕ)≡ U GU + 3U⁰²

= M_P² +ξϕ²

M_P² +ξϕ²+ 6ϕ²(ξ+ ˙ξ)².

At the electroweak scale s(t)≈1, at inflationary scale s(t)∼ ¹_ξ ¿1.

Standard Model modified by the s-factor:

γ = 1 16π²

µ9g²

4 +3g⁰² 4 −3y_t²

¶ , βλ = 1

16π²

¡24s²λ² +λA(t)¢

−4γλ, β_yt = y_t

16π² µ

−9

4g²−17

12g⁰²−8g_s²+9 2sy_t²

¶ , β_g =−20−s

6 g³ 16π², β_g⁰ = 40 +s

6 g⁰³ 16π², β_gs =− 7g_s³

16π², βξ= (6ξ+ 1)

µs²λ 8π² −γ

3

¶ .

Inflationary stage versus post-inflationary running

Thesuppression of Higgs propagators at large ϕwith s(ϕ)¿1has a drastic consequence for the RG flow during the inflation stage. The one-loop RG equations become integrable in quadratures.

The inflationary stage in units of a Higgs field e-foldings is very short.

We consider the solutions of RG equations at one-loop order and only up to termslinear in ∆t ≡t −t_end= ln(ϕ/ϕ_end).

This approximation will be justified in most of the Higgs mass range compatible with the CMB data.

λ(t) = λ

_end

1 − 4γ

_end

∆t +

^end

16π

²

∆t , ξ(t) = ξ

_end

³

1 − 2γ

_end

∆t

´

.

Here λ

_end

, γ

_end

, ξ

_end

are determined

at t

end

and A

end

= A(t

end

) is a value

of the running anomalous scaling

at the end of inflation.

The renormalization group improved potential

Vˆ = µM_P²

2

¶₂ V

U² 'M_P⁴ λ(t)

4ξ²(t) =M_P⁴ λ_end 4ξ_end²

µ

1 + A_end 16π² ln ϕ

ϕ_end

¶ .

Our ”old” formalism can be directly applied to determine the parameters of the CMB. They are mainly determined by the anomalous scalingA, this quantity should be taken at t_end. We integrate the renormalization group equations from the top quark mass scale

µ=M_t = 171Gev.

It should be determined from the CMB normalization condition for the amplitude of the power spectrum, which yields

λ_in

ξ_in² '0.5×10⁻⁹

µx_in expx_in expxin−1

¶₂

at the moment of the first horizon crossing for N = 60 which we call the “beginning” of inflation t_in.

This moment can be determined from the relation tin = lnM_P

M_t + 1 2ln 4N

3ξ_in +1

2lnexpx_in−1 x_in .

The end of inflation:

tend= lnM_P M_t + 1

2ln 4 3ξ_end.

The duration of inflation in units of inflaton field e-foldings t_in−t_end = ln(ϕ_in/ϕ_end) is very short relative to the

post-inflationary evolution tend∼35, t_in−t_end= 1

2lnN +1 2ln ξ_in

ξ_end +1

2lnexpx_in−1 x_in ' 1

2lnN ∼2.

at the electroweak scale becomes rather small at the

inflationary scale - asymptotic freedom.

Numerical analysis

The running ofA(t)depends on the behavior of λ(t). For small Higgs masses the usual RG flow leads to aninstability of the electroweak vacuum caused by negative values of λ(t)in a certain range of t.

We presentλ(t) for five values of the Higgs mass and the value of top quark mass M_t = 171 GeV. The highest Higgs massM_H = 180 GeV is chosen at the boundary of the

perturbation theory domainλ.1, the lowest one corresponds to the critical (instability bound) value

M_H^c '123.7001GeV.

M_H=123.70 M_H=140

MH=160 MH=170

0 10 20 30 40 50 60 70

0.0 0.2 0.4 0.6

t=lnHjMtL

RunningΛHtL

Runningλ(t) for five values of the Higgs mass between the instability threshold and the boundary of perturbation theory domain.

MH=123.70

M_H=130 MH=180

0 10 20 30 40 50 60 70

-60 -40 -20 0 20 40 60

t=lnHjMtL

AnomalousscalingAHtL

Running anomalous scaling for the critical Higgs mass and for two masses in the stability domain.

for the behavior of the CMB parameters. This bound depends on the initial data for weak and strong couplings and, on the top quark massM_t which is known with less precision.

Mt=169 Mt=171 Mt=173

classical

120 125 130 135

0.965 0.970 0.975 0.980 0.985 0.990

Higgs mass MHHGeVL Spectralindexns

The spectral indexn_s as a function of the Higgs massM_H for three values of the top quark massM_t.

big negative valuesA(0)<−20at the electroweak scale to small positive values at the inflation scale above t_inst. This makes the CMB data compatible with the generally accepted Higgs mass range. The knowledge of the anomalous scaling flow allows one to obtainA_end and find the parameters of the CMB power spectrum as functions of the Higgs mass.

0.94<ns(k0)<0.99.

The spectral index becomes too large only for large A_end,M_H approaches the instability bound.

M_H >M_H^CMB '124.19GeV.

This is slightly higherthan the instability bound M_H^c = 123.7001 GeV.

Running of ξ(t)

M_H=124 MH=130 MH=140 M_H=160 M_H=180

0 10 20 30 40 50 60 70

0 10 000 20 000 30 000 40 000

t=lnHjMtL

RunningΞHtL

I The model looks remarkably consistent with CMB observations in the Higgs mass range

124GeV.M_H .180GeV,

I The lower bound follows from the upper WMAP bound for the CMB spectral indexns(k0)<0.99.

I The upper bound follows from the perturbation theory requirement,λ(t).1for t = ln(ϕ/M_t)running all the way up to the inflation scale.

I Our approach represents the RG improvement of our analytical results obtained in the one-loop approximation.

I A peculiar feature of this formalism is that for large non-minimal couplingξ À1 the effect of the Standard Model particle phenomenology on the parameters of inflation is completely encoded in one quantity – the anomalous scalingA.

I The RG running raises a large negative value of A(0) at the electroweak scale to a small positivevalue at the inflation scale.

I This mechanism can be regarded asasymptotic freedom, because A/64π² determines the strength of quantum corrections in inflationary dynamics.

I The source of thisasymptotic freedom is somewhat different from that caused by the domination of vector boson loops over the fermionic and Higgs field ones in non-gravitational gauge theories. Rather it is a

suppression of the Higgs-inflaton propagatorsdue to a strong non-minimal mixing in the kinetic term of the graviton-inflaton sector.

The correct definition of the damping factor for the scalar field propagators.

Dynamics of self-sustained vacuum

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