Hot topics in Modern Cosmology Spontaneous Workshop V, Carg`ese,France

Higgs boson, renormalization group, and cosmology

A.Yu. Kamenshchik

University of Bologna and INFN, Bologna L.D. Landau Institute for Theoretical Physics, Moscow

May 9 - May 14, 2011 Based on

A.O. Barvinsky, A.Yu. Kamenshchik and A.A. Starobinsky, Inflation scenario via the Standard Model Higgs boson and LHC, JCAP 0811 (2008) 021

A.O. Barvinsky, A.Yu. Kamenshchik, C. Kiefer, A.A. Starobinsky and C. Steinwachs, Asymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field, JCAP 0912 (2009) 003

Higgs boson, renormalization group, and cosmology, arXiv: 0910.1041 [hep-ph] Content

1. Introduction 2. One-loop approximation 3. Renormalization group 4. Inflationary stage versus post-inflationary running 5. Numerical analysis 6. Conclusions and discussion Introduction

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 coupling ξRφ2 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 constants A - 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 The asymptotical 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 1 ¡ ¢ 1 λ L(g ,Φ) = M2 + ξ|Φ|2 R − |∇Φ|2 − (|Φ|2 − v 2)2, µν 2 P 2 4 |Φ|2 = Φ†Φ.

X 1 X 1 X L = − λ χ2ϕ2 − g 2A2 ϕ2 − y ϕψψ.¯ int 2 χ 2 A µ ψ χ A ψ

Quantum one-loop correction to the potential is

X m4(ϕ) m2(ϕ) λA ϕ2 (±1) ln = ϕ4 ln + ... . 64π2 µ2 128π2 µ2 particles

à ! 2 X X X A = λ2 + 3 g 4 − 4 y 4 . λ χ A ψ χ A ψ 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. For the Standard Model 3 ³ ¡ ¢ ´ A = 2g 4 + g 2 + g 02 2 − 16y 4 . 8λ t

In the conventional range of the Higgs mass 115 GeV≤ MH ≤ 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 3 ³ ¡ ¢ ´ A(t) = 2g 4(t) + g 2(t) + g 02(t) 2 − 16y 4(t) , 8λ(t) t t = ln(ϕ/µ)

we see that the value of the A on the inflationary scale is compatible with the CMB data. Our results are in a qualitative agreement with those presented in

F.L. Bezrukov, A. Magnin and M. Shaposhnikov, Standard Model Higgs boson mass from inflation, Phys. Lett. B 675, 88 (2009).

F. Bezrukov and M. Shaposhnikov, Standard Model Higgs boson mass from inflation: two loop analysis, JHEP 0907, 089 (2009).

A. De Simone, M. P. Hertzberg and F. Wilczek, Running Inflation in the Standard Model, Phys. Lett. B 678, 1 (2009). One-loop approximation

Z µ ¶ 1 S[g , ϕ] = d 4x g 1/2 −V (ϕ) + U(ϕ) R(g ) − G(ϕ)(∇ϕ)2 µν µν 2

λ λϕ4 ϕ2 V (ϕ) = (ϕ2 − ν2)2 + A ln , 4 128π2 µ2 µ ¶ 1 ϕ2 ϕ2 U(ϕ) = (M2 + ξϕ2) + C ln + D , 2 P 384π2 µ2 µ ¶ 1 ϕ2 G(ϕ) = 1 + F ln + E . 192π2 µ2 From the Jordan frame to the Einstein frame

µ ¶2 2 02 2U(ϕ) dϕˆ MP GU + 3U gˆµν = 2 gµν, = 2 . MP dϕ 2 U

ˆ 2 ˆ U = MP /2, G = 1, ¯ µ ¶2 ¯ M2 V (ϕ) ¯ Vˆ (ϕ ˆ) = P ¯ . 2 U2(ϕ) ¯ ϕ=ϕ(ϕ ˆ) √ At the inflation scale with ϕ > MP / ξ À v and for large non-minimal coupling ξ À 1 µ ¶ λM4 2M2 A ϕ Vˆ = P 1 − P + I ln , 4 ξ2 ξϕ2 16π2 µ

where inflationary anomalous scaling AI is the anomalous scaling modified by the quantum correction to the non-minimal curvature coupling: 3 ¡ ¢ A = A−12λ = 2g 4 + (g 2 + g 02)2 − 16y 4 −6λ. I 8λ t Inflationary slow-roll parameters:

à !2 µ ¶ M2 1 dVˆ 4 M2 A 2 εˆ ≡ P = P + I , 2 Vˆ dϕˆ 3 ξ ϕ2 64π2 M2 d 2Vˆ 4M2 ηˆ ≡ P = − P . Vˆ dϕˆ2 3ξϕ2

Their smallness determines the range of the inflationary stage ϕ > ϕend, terminating at the value of εˆ, which we chose to be εˆend = 3/4. Then the inflaton value at the exit from inflation equals √ ϕend ' 2MP / 3ξ. The duration of inflation which starts at ϕ in units of the scale factor e-folding number N:

2 µ ¶ 2 2 ϕ x ln N 2 64π MP 2 = e − 1 + O , ϕI = , ϕI N ξAI

NA x ≡ I . 48π2

The CMB spectral index ns , the tensor to scalar ratio r and the spectral index running α: 2 x n = 1 − , s N ex − 1 µ ¶ 12 xex 2 r = , N2 ex − 1 2 x 2ex α = − . N2 (ex − 1)2 Renormalization Group improvement

λ(t) V (ϕ) = Z 4(t) ϕ4, 4 1³ ´ U(ϕ) = M2 + ξ(t) Z 2(t) ϕ2 , 2 P G(ϕ) = Z 2(t). dλ dξ dZ = β , = β , = γ. dt λ dt ξ dt These β-functions depend on running couplings λ and ξ as well as on the rest of the coupling constants in Standard Model. 0 dg dg dgs dyt = β , = β 0 , = β , = β . dt g dt g dt gs dt yt 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 factor s(t):

U s(ϕ) ≡ GU + 3U02 M2 + ξϕ2 = P . 2 2 2 ˙ 2 MP + ξϕ + 6ϕ (ξ + ξ) At the electroweak scale s(t) ≈ 1, at inflationary scale 1 s(t) ∼ ξ ¿ 1. The one-loop anomalous dimension and β-functions of the Standard Model modified by the s-factor: µ ¶ 1 9g 2 3g 02 γ = + − 3y 2 , 16π2 4 4 t

λ ¡ 2 2 ¢ βλ = 2 18s λ + λA(t) − 4γλ, 16π µ ¶ y 2 ³ s ´ β = t − g 02 − 8g 2 + 1 + y 2 − γy , yt 16π2 3 s 2 t t 39 − s g 3 β = − , g 12 16π2 81 + s g 03 β 0 = , g 12 16π2 7g 3 β = − s , gs 16π2 6ξ β = (1 + s2)λ − 2γξ. ξ 16π2 Inflationary stage versus post-inflationary running

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 terms linear in ∆t ≡ t − tend = ln(ϕ/ϕend). This approximation will be justified in most of the Higgs mass range compatible with the CMB data. µ ¶ A λ(t) = λ 1 − 4γ ∆t + end ∆t , end end 16π2 ³ ´ ξ(t) = ξend 1 − 2γend∆t .

Here λend, γend, ξend are determined at tend and

Aend = A(tend) is a value of the running anomalous scaling at the end of inflation. The renormalization group improved potential

µ ¶ M2 2 V λ(t) Vˆ = P ' M4 2 U2 P 4ξ2(t) µ 2 ¶ 4 λend 2MP AI(tend) φ = MP 2 1 − 2 + 2 ln . 4ξend ξendφ 16π φend

Our ”old” formalism can be directly applied to determine the parameters of the CMB. They are mainly determined by the anomalous scaling AI, this quantity should be taken at tend.

We integrate the renormalization group equations from the top quark mass scale µ = Mt = 171Gev. The initial condition ξ(0) is not known. It should be determined from the CMB normalization condition for the amplitude of the power spectrum, which yields

µ ¶2 λin −9 xin exp xin 2 ' 0.5 × 10 ξin exp xin − 1 at the moment of the first horizon crossing for N = 60 which we call the “beginning” of inflation tin. This moment can be determined from the relation

MP 1 4N 1 exp xin − 1 tin = ln + ln + ln . Mt 2 3ξin 2 xin The end of inflation:

MP 1 4 tend = ln + ln . Mt 2 3ξend The duration of inflation in units of inflaton field e-foldings tin − tend = ln(ϕin/ϕend) is very short relative to the post-inflationary evolution tend ∼ 35,

1 1 ξin 1 exp xin − 1 1 tin − tend = ln N + ln + ln ' ln N ∼ 2. 2 2 ξend 2 xin 2 The coefficient A which has a big negative values at the electroweak scale becomes rather small at the inflationary scale - asymptotic freedom. Numerical analysis

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

We present λ(t) for five values of the Higgs mass and the value of top quark mass Mt = 171 GeV. The highest Higgs mass MH = 185 GeV, the lowest one corresponds to the critical (instability bound) value

c MH ' 134.27 GeV. M =184.49 2.0 H

1.5 L t H Λ

1.0 MH=180 Running 0.5 MH=170

MH=160 0.0 MH=134.27 0 10 20 30 40 50 60

t=ln H j MtL

Running λ(t) for five values of the Higgs mass above the instability threshold.. 60

40

MH=134.271 20 M =140 L H t H

I 0 A

-20

-40 MH=180

-60 10 20 30 40 50 60

t=ln H j MtL

Running anomalous scaling for the critical Higgs mass and for two masses in the stability domain. The position of the instability bound is qualitatively important for the behavior of the CMB parameters. This bound depends on the initial data for weak and strong couplings and, on the top quark mass Mt which is known with less precision. 0.970 classical 0.965 s n 0.960 Mt=169

0.955 Mt=171

0.950 Mt=173 Spectral index

0.945

0.940 130 140 150 160 170 180 190

Higgs mass MH HGeVL

The spectral index ns as a function of the Higgs mass MH for three values of the top quark mass Mt . The observational bound on the spectral index

0.94 < ns < 0.99

For Mt = 171 Gev the range of CMB compatible Higgs mass

135.62 Gev . MH . 184.49 Gev

The upper bound on ns does not generate restrictions on MH . The lower CMB bound gives both the upper and lower restrictions on MH . In the stability range of MH the anomalous scaling runs from big negative values A(0) < −20 at the electroweak scale to small negative values.

This makes the CMB data compatible with the generally accepted Higgs mass range. The knowledge of the anomalous scaling flow allows one to obtain AIend and find the parameters of the CMB power spectrum as functions of the Higgs mass. Running of ξ(t)

120 000

100 000

L 80 000 t H Ξ 60 000

MH=184.49 Running 40 000 MH=170

MH=160 20 000 MH=140 M =134.27 0 H 0 10 20 30 40 50 60

t=ln H j MtL The comparison with the results by Bezrukov and Shaposhnikov

0.97 0.968 0.966 ns 0.964 0.962

130 140 150 160 170 180 190 mH The comparison with the results by De Simone, Hertzberg and Wilczek

0.990

0.985 m

t = m 169 s

t n = 0.980 m 171 GeV t =

173 index GeV

0.975 GeV Spectral

0.970

0.965 122 124 126 128 130 132 134 136

Higgs mass mh HGeVL Conclusions and discussion

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

135 GeV . MH . 185 GeV,

I The lower CMB bound ns (k0) > 0.94. gives both the upper and lower restrictions on MH . 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 coupling ξ À 1 the effect of the Standard Model particle phenomenology on the parameters of inflation is completely encoded in one quantity – the anomalous scaling AI. I The RG running raises a large negative value of A(0) at the electroweak scale to a small negative value at the inflation scale. I This mechanism can be regarded as asymptotic freedom, 2 because AI/64π determines the strength of quantum corrections in inflationary dynamics. I The source of this asymptotic 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 propagators due to a strong non-minimal mixing in the kinetic term of the graviton-inflaton sector. Some open questions

I The correct definition of the damping s-factor for the scalar field propagators. I The problem of gauge dependence of the effective potential. I The Jordan frame versus Einstein frame. I The Cartesian coordinates for a scalar field multiplet versus spherical ones. I Possible applications to quantum cosmology. Ultimately,

it will be strongly anticipated discovery of the Higgs particle at LHC and the more precise determination of the primordial spectral index ns by the Planck satelllite that might decide the fate of this model.