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The Standard Model Higgs Boson As the Inflaton

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The Standard Model Higgs Boson As the Inflaton The StandardModel Higgsboson as the inflaton Fedor Bezrukov a,b, Mikhail Shaposhnikov a a Institut de Th´eorie des Ph´enom`enes Physiques, Ecole´ Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland b Institute for Nuclear Research of Russian Academy of Sciences, Prospect 60-letiya Oktyabrya 7a, Moscow 117312, Russia Abstract We argue that the Higgs boson of the Standard Model can lead to inflation and produce cosmological perturbations in accordance with observations. An essential requirement is the non-minimal coupling of the Higgs scalar field to gravity; no new particle besides already present in the electroweak theory is required. Key words: Inflation, Higgs field, Standard Model, Variable Planck mass, Non-minimal coupling PACS: 98.80.Cq, 14.80.Bn 1. Introduction tic coupling constant λ 10−13 [7]. This value of the mass is close to the GUT∼ scale, which is often con- The fact that our universe is almost flat, homoge- sidered as an argument in favour of existence of new neous and isotropic is often considered as a strong physics between the electroweak and Planck scales. indication that the Standard Model (SM) of elemen- The aim of the present Letter is to demonstrate that tary particles is not complete. Indeed, these puzzles, the SM itself can give rise to inflation. The spectral together with the problem of generation of (almost) index and the amplitude of tensor perturbations can be scale invariant spectrum of perturbations, necessary for predicted and be used to distinguish this possibility from structure formation, are most elegantly solved by in- other models for inflation; these parameters for the SM flation [1, 2, 3, 4, 5, 6]. The majority of present mod- fall within the 1σ confidence contours of the WMAP-3 els of inflation require an introduction of an additional observations [8]. arXiv:0710.3755v2 [hep-th] 9 Jan 2008 scalar—the “inflaton”. This hypothetical particle may To explain our main idea, consider Lagrangian of the appear in a natural or not so natural way in different SM non-minimally coupled to gravity, extensions of the SM, involving Grand Unified The- 2 M † ories (GUTs), supersymmetry, string theory, extra di- Ltot = LSM R ξH HR , (1) mensions, etc. Inflaton properties are constrained by the − 2 − observations of fluctuations of the Cosmic Microwave where LSM is the SM part, M is some mass parameter, Background (CMB) and the matter distribution in the R is the scalar curvature, H is the Higgsfield, and ξ is an universe. Though the mass and the interaction of the in- unknown constant to be fixed later. 1 The third term in flaton with matter fields are not fixed, the well known (1) is in fact required by the renormalization properties considerations prefer a heavy scalar field with a mass of the scalar field in a curved space-time background 1013 GeV and extremely small self-interacting quar- [9]. If ξ =0, the coupling of the Higgs field to gravity ∼ is said to be “minimal”. Then M can be identified with Email addresses: [email protected] (Fedor Planck scale MP related to the Newton’s constant as Bezrukov), [email protected] (Mikhail Shaposhnikov). 1 In our notations the conformal coupling is ξ = −1/6. Preprint submitted to Elsevier 4 December 2007 −1/2 18 MP = (8πGN ) = 2.4 10 GeV. This model cussed later in Section 3. Then the Lagrangian has the has “good” particle physics phenomenology× but gives form: “bad” inflation since the self-coupling of the Higgs field 2 2 4 M + ξh is too large and matter fluctuations are many orders of SJ = d x√ g R magnitude larger than those observed. Another extreme − − 2 Z ( (2) µ is to put M to zero and consider the “induced” gravity ∂µh∂ h λ 2 2 2 [10, 11, 12, 13, 14], in which the electroweak symmetry + h v . 2 − 4 − ) breaking generates the Planck mass [15, 16, 17]. This 17 happens if √ξ 1/(√GN MW ) 10 , where MW This Lagrangian has been studied in detail in many pa- 100 GeV is the∼ electroweak scale.∼ This model may give∼ pers on inflation [14, 19, 20, 24], we will reproduce here “good” inflation [12, 13, 14, 18, 19, 20] even if the the main results of [14, 19]. To simplify the formulae, scalar self-coupling is of the order of one, but most we will consider only ξ in the region 1 √ξ ≪ 1017, ≪ probably fails to describe particle physics experiments. in which M MP with very good accuracy. Indeed, the Higgs field in this case almost completely It is possible≃ to get rid of the non-minimal coupling decouples from other fields of the SM 2 [15, 16, 17], to gravity by making the conformal transformation from which corresponds formally to the infinite Higgs mass the Jordan frame to the Einstein frame m . This is in conflict with the precision tests of the 2 H ξh electroweak theory which tell that m must be below gˆ =Ω2g , Ω2 =1+ . (3) H µν µν M 2 285 GeV [21] or even 200GeV [22] if less conservative P point of view is taken. This transformation leads to a non-minimal kinetic term These arguments indicate that there may exist some for the Higgs field. So, it is convenient to make the intermediate choice of M and ξ which is “good” for change to the new scalar field χ with particle physics and for inflation at the same time. In- 2 2 2 2 deed, if the parameter ξ is sufficiently small, √ξ ≪ dχ Ω +6ξ h /MP 17 = 4 . (4) 10 , we are very far from the regime of induced grav- dh r Ω ity and the low energy limit of the theory (1) is just Finally, the action in the Einstein frame is the SM with the usual Higgs boson. At the same time, if ξ is sufficiently large, ξ 1, the scalar field be- 2 µ 4 MP ˆ ∂µχ∂ χ haviour, relevant for chaotic inflation≫ scenario [7], dras- SE = d x gˆ R + U(χ) , (5) − ( − 2 2 − ) tically changes, and successful inflation becomes pos- Z p sible. We should note, that models of chaotic inflation where Rˆ is calculated using the metric gˆµν and the with both nonzero M and ξ were considered in litera- potential is ture [12, 14, 19, 20, 23, 24, 25], but in the context of 1 λ 2 either GUT or with an additional inflaton having noth- U(χ)= h(χ)2 v2 . (6) ing to do with the Higgs field of the Standard Model. Ω(χ)4 4 − The Letter is organised as follows. We start from dis- For small field values h χ and Ω2 1, so the poten- cussion of inflation in the model, and use the slow-roll tial for the field χ is the same≃ as that for≃ the initial Higgs approximation to find the perturbation spectra parame- field. However, for large values of h M /√ξ (or ters. Then we will argue in Section 3 that quantum cor- P χ √6M ) the situation changes a lot.≫ In this limit rections are unlikely to spoil the classical analysis we ≫ P used in Section 2. We conclude in Section 4. M χ h P exp . (7) ≃ √ξ √6M P 2. Inflation and CMB fluctuations This means that the potential for the Higgs field is ex- ponentially flat and has the form Let us consider the scalar sector of the Standard 2 λM 4 2χ − Model, coupled to gravity in a non-minimal way. We U(χ)= P 1+exp . (8) 4ξ2 −√6M will use the unitary gauge H = h/√2 and neglect all P gauge interactions for the time being, they will be dis- The full effective potential in the Einstein frame is pre- sented in Fig. 1. It is the flatness of the potential at 2 χ M which makes the successful (chaotic) infla- This can be seen most easily by rewriting the Lagrangian (1), ≫ P given in the Jordan frame, to the Einstein frame, see also below. tion possible. 2 U(χ) 1.0 WMAP 50 60 λM4/ξ2/4 0.8 λφ4 mφ2 2 0.6 SM+ξh R 0.002 HZ r 0.4 λ v4/4 4 2 λM /ξ /16 0.2 0 0 v 0 0.0 χ χ χ 0 end COBE 0.90 0.95 1.00 1.05 ns Fig. 1. Effective potential in the Einstein frame. Fig. 2. The allowed WMAP region for inflationary parameters (r, n). The green boxes are our predictions supposing 50 and 60 e– 3 Analysis of the inflation in the Einstein frame can foldings of inflation. Black and white dots are predictions of usual be performed in standard way using the slow-roll ap- chaotic inflation with λφ4 and m2φ2 potentials, HZ is the Har- proximation. The slow roll parameters (in notations of rison-Zeldovich spectrum. [28]) can be expressed analytically as functions of the field h(χ) using (4) and (6) (in the limit of h2 λ NCOBE √ mH 2 2 ≫ ξ 2 49000 λ = 49000 . (13) MP /ξ v ), 3 0.027 √ ≫ ≃ r ≃ 2v 2 M 2 dU/dχ 4M 4 Note, that if one could deduce ξ from some fundamen- ǫ = P P , (9) 2 U ≃ 3ξ2h4 tal theory this relation would provide a connection be- 2 2 2 tween the Higgs mass and the amplitude of primordial 2 d U/dχ 4MP η = MP , (10) perturbations. The spectral index n =1 6ǫ +2η cal- U ≃− 3ξh2 − 3 3 4 culated for N = 60 (corresponding to the scale k = 2 4 (d U/dχ )dU/dχ 16MP 0.002/Mpc) is n 1 8(4N + 9)/(4N + 3)2 0.97.
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