Higgs Boson in RG Running Inflationary Cosmology

Higgs Boson in RG running Inflationary Cosmology 1,2, 1, Yi-Fu Cai ∗ and Damien A. Easson † 1Department of Physics, Arizona State University, Tempe, AZ 85287 2Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada An intriguing hypothesis is that gravity may be non-perturbatively renormalizable via the notion of asymptotic safety. We show that the Higgs sector of the SM minimally coupled to asymptotically safe gravity can generate the observed near scale-invariant spectrum of the Cosmic Microwave Back- ground through the curvaton mechanism. The resulting primordial power spectrum places an upper bound on the Higgs mass, which for finely tuned values of the curvaton parameters, is compatible with the recently released Large Hadron Collider data. PACS numbers: 98.80.Cq I. INTRODUCTION play an important role in the early inflationary universe if the gravitational theory is asymptotically safe. Although at present there is no explicit proof of the asymptotical Weinberg has suggested that the effective description safety, it deserves to ask whether the gravity theories un- of a quantum gravitational theory may be nonperturba- der asymptotical safety are testable in any experiment. tively renormalizable through the notion of asymptotic The current study is exactly based on a specific trunca- safety (AS) [1, 2]. In such a scenario the renormalization tion of the gravity action and certain form of RG equa- group (RG) flows approach a fixed point in the ultravio- tions according to the conjecture of asymptotical safety. let (UV) limit, and a finite dimensional critical surface of In the frame of AS gravity, the gravitational constant trajectories evolves to this point at short distance scales G and cosmological constant Λ are running along with [3, 4]. Such a fixed point was found in the Einstein- the energy scale, and thus vary throughout the cosmo- Hilbert truncation [5], and the scenario was studied ex- logical evolution. It has been argued that if there are tensively in the literature [6–11] (for recent reviews see no intermediate energy scales between the SM and AS [12, 13]). scales, the mass of the Higgs boson is predicted to be Inflationary cosmology is the most promising candi- mH = 126 GeV with only several GeV uncertainty [36]. date theory for describing the early universe [14–16]. The We find a suitable inflationary solution can be obtained paradigm solves the homogeneity, flatness, horizon and in a cosmological system which contains a Higgs boson unwanted relic problems. It also predicts a nearly scale- and AS gravity, along the lines of [37]. Different from the invariant primordial power spectrum, in agreement with idea of [34], in our model the inflationary solution can be the data from modern cosmological experiments [17]. obtained purely in the frame of AS gravity without the However, the model generally requires an as of yet un- help of the Higgs boson, but the Higgs field plays the role observed scalar field. The Higgs boson is a scalar field of seeding primordial power spectrum which explains the predicted by the SM of particle physics and exciting ev- CMB observation. In this model, the gravitational con- idence for its existence has recently been released from stant and cosmological constant vary along with the run- the Large Hadron Collider (LHC) experiment [18–20]. In ning of the cutoff scale and thus contribute extra term the past, it was hoped that the Higgs field might play a to the energy stress tensor. To take into account of the dual role as the inflaton, but the corresponding energy extra term, the model is found to be corresponding to a arXiv:1202.1285v2 [hep-th] 7 Jan 2013 scale is much lower than what inflationary cosmology typ- f(R) model [37] (see also [38, 39] for generalized discus- ically requires. Consequently, a model of Higgs inflation sions). As a consequence, there are effectively two scalar was proposed in which the Higgs field is non-minimally degrees of freedom, one being the adiabatic mode and coupled to Einstein gravity [21–27] (see earlier attempt the other being an iso-curvature mode. in [28]). It was soon realized that such non-minimal cou- pling leads to an energy scale for unitarity violation which We find the corresponding perturbation theory leads is expected to be larger than the inflationary scale; oth- to both the primordial power spectrum for the curva- erwise, the effective field description would fail (for ex- ture perturbation and the entropy perturbation. When ample, see [29–35] for detailed discussions). Therefore, the cutoff scale runs lower than a critical value, inflation difficult challenges exist for models which attempt to use abruptly ends and the Higgs field can give rise to a re- the Higgs field to drive inflation. heating phase. During this phase, the fluctuations seeded In this paper, we propose that the Higgs boson may by the Higgs field can be converted into the curvature perturbation through the curvaton mechanism [40] (see also [41–43]). We derive a relation between the spectral index of the primordial power spectrum and the Higgs ∗Electronic address: [email protected] mass. We confront this relation with the latest cosmolog- †Electronic address: [email protected] ical observations and collider experiment data, and find 2 they are consistent under a group of canonical values of of Ref. [37], the Ricci scalar and the cutoff scale are able curvaton parameters. to be identified as In this paper, we will work with the reduced Planck ξΛ 4GξΛ mass, Mp = 1/√8πGN , where GN is the gravitational R = 2(ΛIR )+ . (8) constant in the IR limit, and adopt the mostly-plus met- − ξGGN ξG ric sign convention ( , +, +, +). − Then we can reformulate the original theory as an f(R) model as follows, II. THE MODEL S = d4x√ g f(R)+ SM , (9) Z − L Consider the SM of particle physics minimally coupled to gravity with 2 [GN (R 2ΛIR)+2Z] R 2Λ f(R)= − , (10) S = d4x√ g − + SM . (1) 128πG2 Z Z − 16πG L N where we have introduced Z = ξ /ξ . In this AS gravity frame, the gravitational constant G Λ G and the cosmological constant Λ vary along the cutoff scale p. The running behaviors are approximately de- III. INFLATIONARY COSMOLOGY scribed by 1 1 2 A. Background Dynamics G(p)− G− + ξGp , (2) ≃ N 2 Λ(p) ΛIR + ξ p , (3) ≃ Λ We now turn our attention to early universe inflationary solutions. This system is most easily studied by mak- where G and Λ are the values of gravitational con- N IR ing a conformal transformation, stant and cosmological constant in the IR limit. The co- efficients ξG and ξΛ are determined by the physics near G g˜ =Ω2g , Ω2 = N , (11) the UV fixed point of RG flows in AS gravity. µν µν G The scalar sector of the SM contains the Higgs boson. We use the unitary gauge for the Higgs boson H = h where Ω2 is the conformal factor with a new scalar field √2 and neglect all gauge interactions for the time being. In √6Mp GN this case, the Lagrangian of the Higgs field is given by, φ ln , (12) ≡− 2 G SM 1 µ being introduced. ∂µh∂ h V (h) , (4) L ⊇−2 − The original system, therefore, is equivalently de- scribed in terms of two scalar fields minimally coupled where V (h) is the potential of the Higgs field, which is to Einstein gravity without RG running, typically in form of λ (h2 v2)2. 4 − Varying the Lagrangian with respect to the metric, one R˜ 1 e2b(φ) derives the generalized Einstein equation, = ( ˜ φ)2 ( ˜ h)2 V˜ (φ, h) , (13) L 16πGN − 2 ∇ − 2 ∇ − R SM AS φ ˜ Rµν gµν +Λgµν =8πG(Tµν + Tµν ) . (5) where the factor b(φ) and V (φ, h) = U(φ) + − 2 ≡ √6Mp e4bV (h), with Here the RG running of G can effectively contribute to the stress energy tensor through 4 ξΛ 2b(φ) 2b(φ) U(φ) 8πM 1 e + GN ΛIR e . (14) ≃ p ξ − AS 1 G Tµν = ( µ ν gµν )(8πG)− , (6) ∇ ∇ − Note that, the form of φ’s potential is derived from the where we have introduced the covariant derivative µ RG running of the AS gravity [37]. The last term of U(φ) µν ∇ and the operator g µ ν . The Higgs field h is proportional to G Λ . Substituting the observed val- ≡ − ∇ ∇ N IR obeys the Klein-Gordon equation. Additionally, the run- ues of GN and ΛIR, we find this term is insignificant ning of cutoff scale is controlled by the Bianchi identity, throughout the past cosmological evolution. Hence, we which requires, make the approximation µG 4 ξΛ 2b(φ) (R 2Λ)∇ +2 µΛ=0 . (7) U(φ) 8πMp 1 e , (15) − G ∇ ≃ ξG h − i Consequently, the dynamics of this cosmological system which is a sufficiently flat inflationary potential in the are completely determined. Explicitly, from the analysis regime where b(φ) 1. ≪− 3 We denote the frame proceeding the conformal trans- C. Graceful exit formation as the Einstein frame (despite the non- canonical form of the h kinetic term). Substitution Eventually, the slow roll conditions are violated, when of the flat Friedmann-Robertson-Walker (FRW) metric, φ reaches φf . Consequently, φ enters a period of fast ds2 = dt2 +a2(t)d~x2, leads to the Friedmann equations: − roll, and finally approaches φ = 0 at which point the AS gravity reduces to traditional Einstein gravity. We ˙2 2b ˙ 2 2 1 φ e h ˜ suggest two possible reheating mechanism. One is that H = 2 ( + + V ) and (16) 3Mp 2 2 the inflaton φ decays to radiation, driving the universe 1 to a phase of thermal expansion directly; the other pos- H˙ = (φ˙2 + e2bh˙ 2) , (17) sibility is that the universe reheats only after the energy 2M 2 − p scale drops sufficiently so that the SM Higgs boson is responsible for the reheating process.

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