Cosmological Constraints on the Higgs Boson Mass
aa r X i v : . [ a s t r o - ph . C O ] S e p COSMOLOGICAL CONSTRAINTS ON THE HIGGS BOSONMASS
L.A. Popa, A. Caramete
Institutul de S¸tiint¸e Spat¸iale Bucure¸sti-M˘agurele, Ro-077125 Romˆania [email protected]
ABSTRACT
For a robust interpretation of upcoming observations from PLANCK andLHC experiments it is imperative to understand how the inflationary dynamicsof a non-minimally coupled Higgs scalar field with gravity may affect the deter-mination of the inflationary observables. We make a full proper analysis of theWMAP7+SN+BAO dataset in the context of the non-minimally coupled Higgsinflation field with gravity.For the central value of the top quark pole mass m T = 171 . . ≤ m H ≤ n S and tensor-to-scalar ratio R when compared with a tensor with similar constraintsto form the standard inflation with a minimally coupled scalar field.We also show that an accurate reconstruction of the Higgs potential in terms ofinflationary observables requires an improved accuracy of other parameters of theStandard Model of particle physics such as the top quark mass and the effectiveQCD coupling constant. Subject headings: cosmology: cosmic microwave background, cosmological pa-rameters, early universe, inflation, observations
1. Introduction
The primary goal of particle cosmology is to obtain a concordant description of theearly evolution of the universe, establishing a testable link between cosmology and particlephysics, consistent with both unified field theory and astrophysical and cosmological mea-surements. On the ground, the Large Hadron Collider (LHC) at CERN is investigating the 2 –elementary particle collisions in the TeV energy range, seeking to validate a large number oftheoretical predictions of the Standard Model (SM) of particle physics and beyond. In thesky, the PLANCK Surveyor is actively taking precise measurements of the Cosmic MicrowaveBackground (CMB) temperature and polarization anisotropies.Inflation is the most simple and robust theory capable of explaining astrophysical andcosmological observations, at the same time providing self-consistent primordial initial con-ditions (Starobinsky 1979; Guth 1981; Sato 1981; Albercht 1982; Linde 1982; Linde 1983)and mechanisms for the quantum generation of scalar (curvature) and tensor (gravita-tional waves) perturbations (Mukanov 1981; Hawking 1982; Guth 1982; Starobinsky 1982;Bardeen 1983; Abbot 1984). In the simplest class of inflationary models, inflation is drivenby a single scalar field φ (or inflaton) with some potential V ( φ ) minimally coupled to theEinstein gravity. The perturbations are predicted to be adiabatic, nearly scale-invariant andGaussian distributed, resulting in an effectively flat universe.The WMAP cosmic microwave background (CMB) measurements alone (Dunkley et al. 2009;Larson et al. 2010) or complemented with other cosmological datasets (Komatsu et al. 2009;Komatsu et al. 2010) support the standard inflationary predictions of a nearly flat universewith adiabatic initial density perturbations. In particular, the detected anti-correlations be-tween temperature and E-mode polarization anisotropy on degree scales (Nolta et al. 2009)provide strong evidence for correlation on length scales beyond the Hubble radius.Alternatively, one can look to the inflationary dynamics based on models beyond theStandard Model (SM) of particle physics. The hybrid inflation models involving supersym-metric (SUSY) TeV energy scales (Dvali et al. 1994) and minimal supergravity (SUGRA)(Linde & Riotto 1997) provide natural connection between cosmology and particle physics(Cervantes-Cota & Dehnen; S¸eno˘guz & Shafi 2005). The realization of these inflationaryscenarios introduces new physics between the electroweak energy scale and the Planck scale,leading to distinct predictions of the main inflationary parameters, such as the spectral index n S of scalar perturbations and the tensor-to-scalar ratio R (Rehman et. al 2008; 2009; 2010).However, a number of recent papers (Bezrukov & Shaposhnikov 2008; Barvinsky et al. 2008;Bezrukov et al. 2009; Bezrukov et al. 2009; De Simone et al. 2009; Bezrukov et al. 2009)reported the possibility that the SM of particle physics with an additional non-minimallycoupled term of the Higgs field to the gravitational Ricci scalar can give rise to inflationwithout the need for additional degrees of freedom to the SM. This scenario is based onthe observation that the problem of the very small value of Higgs quadratic coupling re-quired by the CMB anisotropy data can be solved if the Higgs inflaton has a large couplingwith gravity (Futamase & Maeda 1989; Fakir & Unruh 1990; Komatsu & Futamase 1999;Tsujikawa & Gumjudpai 2004; Barvinsky & Kamenshchik 1994).The resultant Higgs inflaton effective potential in the inflationary domain is effectively flat 3 –and can result in a successful inflation for values of the non-minimal coupling constant ξ ∼ − , allowing for cosmological values for the Higgs boson mass in a window inwhich the electroweak vacuum is stable and therefore sensitive to the field fluctuations duringthe early stages of the universe (Espinosa et al. 2008).Limits of the validity of Higgs-type inflation have recently been debated by severalauthors. Specifically, Barb´on & Espinosa (2009) argued that the large coupling of Higgsinflaton to the Ricci scalar makes this model invalid beyond the ultraviolet cutoff scaleΛ ξ ≃ M P /ξ (here M P = 2 . × GeV is the reduced Planck mass) which is below the Higgsfield expectation value at N e -foldings during inflation, h ≃ √ N M P / √ ξ . As consequence, atthe ultraviolet cutoff scale Λ ξ at least one of the cross-sections of different scattering processeshits the unitarity bound (Burgess et al. 2009). The fact that the quantum corrections dueto the strong coupling to gravity makes the perturbative analysis to break down at energyscales above Λ ξ was interpreted as a signature of a new physics, implying higher dimensionaloperators at energies above Λ ξ . However, the theory can still be considered valid above Λ ξ if one finds some ultraviolet completion or if a very high degree of fine tuning is required,keeping in this way the unwanted contributions of higher dimensional operators small to zero(Bezrukov & Shaposhnikov 2009; De Simone et al. 2009).Recent papers (Lerner & McDonald 2010a; Lerner & McDonald 2010b; Burgess et al. 2010;Hertzberg 2010) revisit the arguments against Higgs-type inflation addressing the issue of itsnaturalness with respect to perturbativity and unitarity violation in the Jordan and Einsteinframes. It is shown that the apparent breakdown of this theory in the Jordan frame doesnot imply new physics, but a failure of the perturbation theory in the Jordan frame as acalculational method. These works demonstrate that for inflation based on a single scalarfield with large non-minimal coupling, the quantum corrections at high energy scales aresmall, making the perturbative analysis valid. As consequence, for these models there is nobreakdown of unitarity at the energy scale Λ ξ . In particular, when the single-field Higgsinflation model is analyzed in the Einstein frame there is no breakdown of the theory atenergy scales ˆ h ≥ Λ ξ , where ˆ h is the canonically normalized Higgs scalar field in the Einsteinframe. However, the inclusion of two or more scalar fields non-minimally coupled with gravity(in particular, the 3 Goldstone bosons of the Higgs doublet) causes unitarity violation in theEinstein frame at Λ ξ , making the theory unnatural (Hertzberg 2010).The present cosmological constraints on the Higgs mass are based on mapping be-tween the Renormalization Group (RG) flow equations and the spectral index of the cur-vature perturbations parameterized in terms of the number of e -foldings until the endof inflation, emerging from the analysis of CMB data combined with astrophysical dis-tance measurements. For a robust interpretation of upcoming observations from PLANCK(Mandolesi et al.2010) and LHC (Bayatian et al. 2007) experiments it is imperative to un- 4 –derstand how the inflationary dynamics of a non-minimally coupled Higgs scalar field mayaffect the degeneracy of the inflationary observables.The aim of this paper is to make a full proper analysis of the WMAP 7-year CMB mea-surements complemented with astrophysical distance measurements (Komatsu et al. 2010;Larson et al. 2010) in the context of the non-minimally coupled Higgs inflaton field withgravity. The paper is structured as follows. In Section 2 we compute the power spectra ofscalar and tensor density perturbations generated during inflation driven by a single scalarfield non-minimally coupled to gravity. In Section 3 we derive the Higgs field equationsand compute the RG improved Higgs field potential and in Section 4 we present our mainresults. In Section 5 we draw our conclusions. Throughout the paper a is the cosmologicalscale factor ( a = 1 today), κ ≡ πM − pl where M pl ≃ . × GeV is the present valueof the Planck mass, overdots denotes the time derivatives and ,ϕ ≡ ∂/∂ϕ .
2. COSMOLOGICAL PERTURBATIONS DRIVEN BY ANON-MINIMALLY COUPLED SCALAR FIELD
In this section we compute the power spectra of scalar and tensor density perturbationgenerated during inflation driven by a single scalar field non-minimally coupled to gravityvia the Ricci scalar (Fakir & Unruh 1990; Hwang & Noh 1996; Komatsu & Futamase 1998;Komatsu & Futamase 1999; Hwang & Noh 2001; Tsujikawa & Gumjudpai 2004) . The gen-eral action for these models in the Jordan frame is given by (Futamase & Maeda 1989): S J ≡ Z d x √− g (cid:20) U ( ϕ ) R − G ( ϕ )( ∇ ϕ ) − V ( ϕ ) (cid:21) , (1)where U ( ϕ ) is a general coefficient of the Ricci scalar, R , giving rise to the non-minimalcoupling, G ( ϕ ) is the general coefficient of kinetic energy and V ( ϕ ) is the general potential.The generalized U ( ϕ ) R gravity theory in Equation (1) includes diverse cases of coupling.For the generally coupled scalar field U = ( γ + κ ξϕ ), G ( ϕ ) = 1 and γ and ξ are constants.The non-minimally coupled scalar field is the case with γ = 1 while the conformal coupledscalar field is the case with γ = 1 and ξ = 1 / g µ,ν = Ω g µ,ν , Ω = 2 κ U ( ϕ ) , (2)where the quantities in the Einstein frame are marked by caret. 5 –The kinetic energy in the Einstein frame can be made canonical with respect to thenew scalar field ˆ ϕ , defined through the scalar field propagator suppression factor s ( ˆ ϕ ) as(De Simone et al. 2009; Barvinsky et al. 2009): s ( ˆ ϕ ) − = (cid:18) d ˆ ϕ d ϕ (cid:19) = 12 κ G ( ϕ ) U ( ϕ ) + 3 U ( ϕ ) ,ϕ U ( ϕ ) . (3)Thus the non-minimal coupling to the gravitational field introduces a modification to theHiggs field propagator by the factor s ( ˆ ϕ ), acting as back reaction of the gravitational field.The scalar potential ˆ V ( ˆ ϕ ) in the Einstein frame is given by:ˆ V ( ˆ ϕ ) = 14 κ V ( ϕ ) U ( ϕ ) , (4)leading to the following canonical form of the action in the Einstein frame: S E ≡ Z d x √− g (cid:20) κ R −
12 ( ∇ ˆ ϕ ) − V ( ˆ ϕ ) (cid:21) . (5) When evaluating the field equations we assume that the background space-time can bewritten in the form of a flat (k=0) Robertson-Walker line element:d s = g µ,ν d x µ d x ν = − d t + a ( t )d x (6)= Ω( x ) (cid:0) − dˆ t + ˆ a (ˆ t )d x (cid:1) , where t is the cosmic time and a is the cosmological scale factor. From the above equationwe obtain: dˆ a = √ Ω d a , dˆ t = √ Ω d t . (7)Now the Friedmann equation in the Einstein frame can be written as (Komatsu & Futamase 1998;Tsujikawa & Gumjudpai 2004): ˆ H = κ "(cid:18) d ˆ ϕ dˆ t (cid:19) + ˆ V ( ˆ ϕ ) , (8)where: ˆ H ≡ a dˆ a dˆ t = 1 √ Ω (cid:20) H + 12Ω dΩd t (cid:21) , (9)d ˆ ϕ dˆ t = (cid:18) d ˆ ϕ d ϕ (cid:19) (cid:18) d t dˆ t (cid:19) ˙ ϕ (10)Equations (9) and (10) are enough to compute the background field evolution in the Einsteinframe if the field equations in the Jordan frame are known (see the next section). 6 – Neglecting the contribution of the decaying modes, the scale dependence of the ampli-tudes of scalar (S) and tensor (T) perturbations in the Einstein frame are fully governed bythe mode equation (Mukanov 1981):d u k dˆ t + (cid:18) k − z d z dˆ t (cid:19) u k = 0 , (11)where k is the comoving wave number of the mode function u k . For the case of scalarperturbations we have (Hwang 1996; Hwang & Noh 1996; Hwang & Noh 2001):1 z S d z S dˆ t = (ˆ a ˆ H ) " (1 + ˆ δ S )(2 + ˆ δ S + ˆ ǫ ) + ˙ˆ δ S ˆ a ˆ H , (12)where: z S = ˆ a q ˆ Q S , ˆ Q s = (cid:18) d ˆ ϕ/ dˆ t ˆ H (cid:19) . (13)and slow-roll parameters ˆ ǫ and ˆ δ S are given by (Stewart & Lyth 1993):ˆ ǫ = − ˙ˆ H ˆ H , ˆ δ S = ˙ˆ Q S H ˆ Q S . (14)In the case of tensor perturbations Equation (12) has the same form with the followingreplacements: z S → z T = ˆ a q ˆ Q T , ˆ Q S → ˆ Q T = 1 , ˆ δ S → ˆ δ T = ˙ˆ Q T H ˆ Q T = 0 . (15)The power spectra of scalar and tensor perturbations are given by (Copeland et al. 1994): P S ( k ) = k π (cid:18) Q S (cid:19) | u k | a P T ( k ) = 16 k πm pl | u k | a , (16)and the spectral index of the scalar perturbations n S is obtained as usual as: n s − d ln P S ( k ) /d ln k .
3. HIGGS BOSON AS INFLATON
Higgs boson as inflaton adds non-minimal coupling to gravity (Barvinsky & Kamenshchik 1994;Bezrukov & Shaposhnikov 2008; Barvinsky et al. 2008; De Simone et al. 2009; Bezrukov et al. 2009; 7 – φ / m T ) ξ / ξ (0) λ / λ (0) y t /y t (0) S λ / λ ( ) ; ξ / ξ ( ) ; y t / y t ( ) ; S −11 V E ( φ ) / M l ψ = κ √ ξ φ m H =132.8 GeVm H =160.0 GeVm T =171.3 GeV Fig. 1.— Left panel: The running of the coupling constants normalized to their initial valuesfor m H = 132 . λ (0) ≃ .
14 (blue), ξ (0) = 1 . × (magenta), m H = 160 (dashed lines) with λ (0) = 0 .
21 (blue), ξ (0) = 2 . × (magenta) and m T = 171 . y t (0) = 0 .
91 (continuous red line). The green curves show the running of the Higgsfield propagator suppression factor s ( t ). The right-hand gray region indicates the slow-rollinflationary regime. Right panel: The Einstein frame renormalization group improved po-tential as a function of the Higgs field for m H = 132 . m H = 160(dashed line) and m T op = 171 . A S = 2 . × − at the Hubble radius crossing k ∗ =0.002 Mpc − and the vacuumexpectation value v =246.22 GeV. 8 –Bezrukov & Shaposhnikov 2009).Taking the Higgs field potential V ( ϕ ) of the Landau-Ginzburg type (by assuming that thespontaneous symmetry breaking arises through a condensate), the Jordan-frame effectiveaction has the same form as given in Equation (1) with (see e.g. Futamase & Maeda 1989,Fakir & Unruh 1990, Makino & Sasaki 1991): U ( ϕ ) = 1 + κ ξϕ κ , V ( ϕ ) = λ (cid:0) ϕ − v (cid:1) , G ( ϕ ) = 1 , (17)where: v = ( √ G F ) − / =246.22 GeV is the vacuum expectation value of the Higgs fieldthat sets the electroweak scale, λ is the quadratic coupling constat of the Higgs boson witha mass m H = √ λ v and ξ is the non-minimal coupling constant. The Jordan-frame fieldequations from the above action are given by (Komatsu & Futamase 1999; Kaiser 1995): H = κ κ ξφ ) (cid:20) V ( ϕ ) + 12 ˙ ϕ − ξHϕ ˙ ϕ (cid:21) , (18)¨ ϕ + 3 H ˙ ϕ + (cid:18) κ ξϕ (1 + 6 ξ )1 + κ ξϕ (1 + 6 ξ ) (cid:19) ˙ ϕ ϕ = κ ξϕV ( ϕ ) − (1 + κ ξφ ) V , ϕ ( ϕ )1 + κ ξϕ (1 + 6 ξ ) , (19)which in the slow-roll approximation ( | ˙ ϕ/ϕ | ≪ H and | ˙ ϕ | ≪ V ( ϕ ) ) can be written as: H ≃ κ κ ξφ ) V ( ϕ ) , (20)3 H ˙ ϕ ≃ κ ξϕV ( ϕ ) − (1 + κ ξφ ) V , ϕ ( ϕ )1 + κ ξϕ (1 + 6 ξ ) . (21)The quantum corrections due to the interaction effects of the SM particles with Higgs bosonthrough quantum loops modify the action coefficients U ( ϕ ), V ( ϕ ) and G ( ϕ ) from theirclassical expression given in Equations (1) and (17), taking the renormalization group (RG)improved forms U q ( t ), V q ( t ), G q ( t ) defined as (Barvinsky et al. 2008; De Simone et al. 2009;Clark et al. 2009; Lerner & McDonald 2009): U q ( t ) = 12 κ (cid:0) κ ξ ( t ) G q ( t ) ϕ ( t ) (cid:1) , (22) V q ( t ) = λ ( t )4 G q ( t ) (cid:0) ϕ ( t ) − v ( t ) (cid:1) , (23) G q ( t ) = e − γ ( t ) / (1+ γ ( t )) , (24)where γ ( t ) is the Higgs field anomalous dimension given in the Appendix.The scaling variable t = ln( ϕ/m T ) in the above equations normalizes the Higgs field and allthe running couplings to the top quark mass scale m T . 9 –As the energy scale of inflation is many order of magnitude above the electroweak scale( ϕ ( t ) >> v ), in the following we will approximate the Higgs potential by V ( ϕ ) ≃ λϕ / s ( t ) − = (cid:18) d ˆ ϕ ( t )d ϕ ( t ) (cid:19) = 12 κ κ ξ ( t ) ϕ ( t )(1 + 6 ξ ( t ))(1 + κ ξ ( t ) ϕ ( t )) , (25)ˆ V ( t ) = 116 κ λ ( t ) ϕ ( t )(1 + κ ξ ( t ) ϕ ( t )) . (26)The amplitude of scalar density perturbations at the Hubble radius crossing k ∗ is then givenby: A S = ˆ V π M pl ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ∗ , ǫ = 12 M pl ˆ V ,ϕ ˆ V ! . (27)We compute the various t -dependent running constants, the Higgs field propagator suppres-sion factor and the Higgs field anomalous dimension by integrating the RG β -functions ascompiled in the Appendix. The runnings of SU (2) × S (1) gauge couplings g ′ , g , the SU (3)strong coupling g s , the top Yukawa coupling y t and the Higgs quadratic coupling λ are com-puted by using two-loop quantum corrections while the running of non-minimal couplingconstant ξ is computed by using one-loop quantum corrections.One should note the importance of the quantum corrections due to non-minimal coupling.The quantum corrections to the classical kinetic sector G ( ϕ ) = 1 arise from the Higgs fieldanomalous dimension γ ( t ) occurring with a factor of 1 /ξ which in the inflationary regime( ξ ∼ ) has a negligible small contribution. In the case of a classical gravity sector U ( ϕ ) = (1 + κ ξϕ ) / κ , the conformal transformation (2) introduces a one-loop β -functionfor ξ with a term proportional to λ due to Higgs running in a loop which has a small con-tribution during inflation due to the suppression of the Higgs field propagator, while thecontribution of the remaining terms cancel to good approximation (De Simone et al. 2009).Although small, the one-loop quantum corrections due to the non-minimal coupling are notnegligible but enough for the purpose of this analysis.For each case, the t -dependent running constants are obtained as: Y ( t ) = Z tt =0 β Y ( t ′ )1 + γ ( t ′ ) d t ′ , Y = { g, g ′ , g s , y t , λ, ξ } , (28)At t = 0, which corresponds to the top quark mass scale m T , the Higgs quadratic coupling λ (0) and the top Yukawa coupling y t (0) are determined by the pole masses and the vacuum 10 –expectation value v : λ (0) = m H v [1 + ∆ H ( m H )] , y t (0) = √ m T v [1 + ∆ T ( m T )] , (29)where ∆ H ( m H ) and ∆ T ( m T ) are the corrections to Higgs and top quark mass respectively,computed following the scheme from the Appendix of Espinosa et al. (2008).The gauge coupling constants at m T scale are (Barvinsky et al. 2009): g (0) = 0 . g ′ (0) = 0 . g s (0) = 1 . ξ (0) isdetermined so that at the beginning of the slow-roll inflation t ini the non-minimally couplingconstant ξ ( t ini ) is such that the calculated value of the amplitude of density perturbationsgiven in Equation (27) agrees with the measured value of A S .Figure 1 presents the running of the coupling constants and of the Higgs field propagatorsuppression factor obtained for two different values of the Higgs boson mass. In both caseswe also show the Einstein frame renormalization group improved potential as a function ofthe Higgs field ψ = κ √ ξϕ ( t ).
4. Results4.1. The CMB Angular Power Spectra
We obtain the CMB temperature anisotropy and polarization power spectra by inte-grating the coupled Equations (9), (10) and (11) together with Equations (20) and (21)with respect to the conformal time imposing that the electroweak vacuum expectation value v =246.22 GeV is the true minimum of the Higgs potential at any energy scale ( λ ( t ) > × − −
5] Mpc − needed by the CAMB Boltzmanncode (Lewis et al. 2000) to numerically derive the CMB angular power spectra and a Hubbleradius crossing scale k ∗ = 0 . − . The value of the Higgs scalar field ϕ ∗ at this scaleis related to the quantum scale of inflation ϕ I and to the duration of inflation expressed inunits of e -folding number N through (Barvinsky et al. 2008): ϕ ∗ ϕ I = e x − , ϕ I = 64 π M pl ξ A I , x ≡ N A I π , (30) A I = 38 λ (cid:0) g + ( g + g ′ ) − y t (cid:1) − λ , (31)where the inflationary anomalous scaling parameter A I (Barvinsky & Kamenshchik 1994;Barvinsky et al. 2009) involves a special combination of quantum corrected coupling con-stants. These relations determine the value of the scaling parameter t ∗ = ln( ϕ ∗ /m T ) atHubble radius crossing k ∗ . As the inflationary observables are evaluated at the epoch of 11 – −3 −2 −1 multipole l l ( l + ) C l / π ( µ K ) C TT C TE C EE Fig. 2.— The renormalization group improved CMB temperature and polarization angularpower spectra (continuous lines) compared with the same power spectra obtained at thetree-level (dashed-lines) for m H =145 GeV (black lines) and m H =160 GeV (red lines). Inboth cases we take the top quark pole mass m T = 171 . A S = 2 . × − at Hubble radius crossing k ∗ =0.002 Mpc − and the vacuumexpectation value v =246.22 GeV. 12 –horizon-crossing quantified by the number of e -foldings N before the end of the inflation atwhich our present Hubble scale equalled the Hubble scale during inflation, the uncertaintiesin the determination of N translates into theoretical errors in determination of the infla-tionary observables (Kinney et al. 2004; Kinney & Riotto 2006). Assuming that the ratioof the entropy per comoving interval today to that after reheating is negligible, the mainuncertainty in the determination of N is given by the uncertainty in the determination of thereheating temperature after inflation. Recent studies of the reheating after inflation drivenby SM Higgs field non-minimally coupled with gravity estimates the reheating temperaturein the range (Garcia-Bellido et al. 2009; Bezrukov et al. 2009):3 . × GeV < T r < (cid:18) λ . (cid:19) / . × GeV , which translates into a negligible variation of the number of e -foldings with the Higgs mass(∆ N ∼ . N = k ∗ /aH = 59 e -foldings in viewof WMAP7+SN+BAO normalization at k ∗ (Komatsu et al. 2010; Larson et al. 2010).For each wavenumber k in the above range our code integrates the β -functions of the t -dependent running constant couplings in the observational inflationary window imposingthat k grows monotonically to the wavenumber k ∗ , at the same time eliminating those mod-els violating the condition for inflation 0 ≤ ǫ H ≡ − ˙ H/H ≤ m H =145 GeV and m H =160GeV. These plots clearly show that the CMB anisotropies are sensitive to the quantum ra-diative corrections of the SM coupling constants. We use MCMC technique to reconstruct the Higgs field potential and to derive con-straints on the inflationary observables and the Higgs mass from the following datasets.The WMAP 7-year data (Komatsu et al. 2010; Larson et al. 2010) complemented with ge-ometric probes from the Type Ia supernovae (SN) distance-redshift relation and the baryonacoustic oscillations (BAO). The SN distance-redshift relation has been studied in detailin the recent unified analysis of the published heterogeneous SN data sets - the UnionCompilation08 (Kowalski et al. 2008; Riess et al. 2009). The BAO in the distribution ofgalaxies are extracted from Two Degree Field Galaxy Redshidt Survey (2DFGRS) the SloanDigital Sky Surveys Data Release 7 (Percival et al. 2010). The CMB, SN and BAO data(WMAP7+SN+BAO) are combined by multiplying the likelihoods. We use these mea-surements especially because we are testing models deviating from the standard Friedmann 13 –
150 16000.51 L / L m a x
150 1600.9690.970.9710.9720.973 n S L / L m a x
150 16033.053.13.15 l n [ A S × ] L / L m a x
150 1603.43.63.84 x 10 −3 m H R −3 n S −3 ln [A S × ] 3 3.5 4x 10 −3 R Fig. 3.— The results of the fit of inflationary model with a non-minimal coupled Higgsscalar field to the WMAP7+SN+BAO dataset for top quark pole mass values: 168 GeV(green), 171.3 GeV (blue) and 173 GeV (magenta). The top plot in each column shows theprobability distribution of different parameters while the other plots show their joint 68%and 95% confidence intervals. All parameters are computed at the Hubble crossing scale k ∗ =0.002Mpc − . 14 –expansion. These datasets properly enables us to account for any shift of the CMB angulardiameter distance and of the expansion rate of the Universe.The likelihood probabilities are evaluated by using the public packages CosmoMC and
CAMB (Lewis & Briddle 2002; Lewis et al. 2000) modified to include the formalism forinflation driven by non-minimally coupled Higgs scalar field as described in the previoussections. Our fiducial model is the ΛCDM standard cosmological model described by thefollowing set of parameters receiving uniform priors: (cid:8) Ω b h , Ω c h , θ s , τ , A S , m H , m T (cid:9) , where: Ω b h is the physical baryon density, Ω c h is the physical dark matter density, θ s is theratio of the sound horizon distance to the angular diameter distance, τ is the reionization op-tical depth, A S is the amplitude of scalar density perturbations, m H is the Higgs boson polemass and m T is the top quark pole mass. For comparison we use the MCMC technique toreconstruct the standard inflation field potential and to derive constraints on the inflationaryobservables from the fit to WMAP7+SN+BAO dataset of the standard inflation model withminimally coupled scalar field. For this case we use the same set of input parameters withuniform priors as in the case of non-minimally coupled Higgs scalar field inflation, exceptfor Higgs boson and top quark pole masses. The details of this computation can be foundin Popa et al. (2009).For each inflation model we run 64 Monte Carlo Markov chains, imposing for each case theGelman & Rubin convergence criterion (Gelman & Rubin 1992). Figure 3 presents the con-straints on the Higgs boson mass m H , the spectral index of the scalar density perturbations n S , the amplitude of the scalar density perturbations A S and the ratio of tensor-to-scalaramplitudes R , as obtained from the fit to the WMAP7+SN+BAO dataset of the inflationmodel with non-minimally coupled Higgs scalar field for three different top quark pole massvalues. We find that n S , A S and R are dependent of the Standard Model parameters, inparticular on the Higgs quadratic coupling and Yukawa coupling. One should recall that inthe standard inflation these parameters are independent on the parameters of the StandardModel.The running of Higgs quadratic coupling λ is increased for a heavier Higgs, also receiv-ing contributions from gauge couplings { g, g ′ , g s } and top Yukawa coupling y t . In the in-flationary regime, the contribution from y t is increased as the top quark mass is variedtoward higher mass values through its experimental allowed range: 168 GeV - 173 GeV(Amsler et al. 2008). As a consequence, since we fixed the non-minimal coupling constant ξ such that the amplitude of the scalar density perturbations A S ∼ λ/ξ is at the observedvalue, A S increases for a heavier Higgs boson and a higher top quark mass value, leadingto the suppression of the spectral index of scalar density perturbations n s . Moreover, thejoint confidence regions of the scalar spectral index n s and of the ratio of tensor-to-scalar 15 –amplitudes R are anti-correlated. This can be attributed to a larger contribution of the ten-sor modes to the primordial density perturbations when Higgs boson and top quark massesare increased. Table 1 presents the mean values and the errors (68% CL) of the parametersfrom the posterior distributions obtained from the fit of the standard inflation model andthe inflation model with non-minimally coupled Higgs scalar field with m T =171.3 GeV and v =246.22 GeV to WMAP7+SN+BAO dataset. We find for Higgs boson pole mass the fol-lowing dependence on m T and α s ( m Z ) normalized in units of one standard deviations fromtheir experimental central values: m H ≃ (155 . ± . ± δ ) GeV + 3 . (cid:18) m T − . . (cid:19) − . (cid:18) α s ( m Z ) − . . (cid:19) (68% CL) , (32)where we included the overall theoretical uncertainty δ ≃ m H , n S and R as obtained from the fit of inflationarymodel with non-minimally coupled Higgs scalar field to the WMAP7+SN+BAO dataset fordifferent top quark pole mass values. Figure 4 explicitly demonstrates that the cosmologicalmeasurements not only probe the graviton-inflaton sector of the SM but also the variationof the scale of inflation due to the SM heavy particles coupled to inflation.
5. CONCLUSIONS
A number of papers have discussed bounds on the Higgs boson mass coming from de-manding stability or metastability of the lifetime of the universe (Espinosa et al. 2008). Fur-ther, by demanding that Higgs drive inflation, depending on the top quark mass and the com-putation of the RG improved effective potential, it was found that a heavier Higgs boson witha mass within the absolute stability bounds is required (Bezrukov & Shaposhnikov 2008;Barvinsky et al. 2008; Bezrukov et al. 2009; Bezrukov et al. 2009; De Simone et al. 2009;Bezrukov et al. 2009). However, the present cosmological constraints on the Higgs bosonmass are based on mapping between the RG flow and the scalar spectral index of of curva-ture perturbations.For a robust interpretation of upcoming observations from PLANCK (Mandolesi et al.2010)and LHC (Bayatian et al. 2007) experiments it is imperative to understand how the infla-tionary dynamics of a non-minimally coupled Higgs scalar field with gravity may affect the α s = g s / π is the effective QCD coupling constant
16 –
150 155 160 1653.853.93.9544.05 x 10 m H (GeV) V / ( G e V ) −3 R0.968 0.969 0.97 0.971 0.972 0.9733.853.93.9544.05 x 10 n S Fig. 4.— The dependence of the reconstructed Higgs field potential (the joint 68% and95% confidence intervals) on m H , n S and R as obtained from the fit of inflationary modelwith non-minimally coupled Higgs scalar field to the WMAP7+SN+BAO dataset for topquark pole mass values: 168 GeV (green), 171.3 GeV (blue) and 173 GeV (magenta). Allparameters are computed at the Hubble crossing scale k ∗ =0.002Mpc − . 17 –determination of the inflationary observables. The aim of this paper is to make a full properanalysis of the WMAP 7-year CMB measurements (Komatsu et al. 2010; Larson et al. 2010)complemented with geometric probes from the Type Ia supernovae (SN) distance-redshiftrelation (Kowalski et al. 2008; Riess et al. 2009) and the baryon acoustic oscillations (BAO)in the distribution of galaxies from Two Degree Field Galaxy Redshidt Survey (2DFGRS)and the Sloan Digital Sky Surveys Data Release 7 (Percival et al. 2010), in the context ofthe non-minimally coupled Higgs inflaton with gravity.We compute the full RG improved effective potential including two-loop beta functions for SU (2) × S (1) gauge couplings g ′ , g , the SU (3) strong coupling g s , the top Yukawa coupling y t and the Higgs quadratic coupling λ and one-loop beta functions for non-minimal couplingconstant ξ and vacuum expectation value v . We also include the curvature in RG flow equa-tions through Higgs field propagator suppression function s ( t ) and the Higgs field anomalousdimension γ ( t ).The initial conditions for λ and y t are properly obtained through the pole mass matchingscheme while the inflationary anomalous scale parameter A I relates the initial value of theHiggs inflation field to the quantum scale of inflation and the number of e -foldings.We use MCMC technique to reconstruct the Higgs field potential and to derive constraintson the inflationary observables and the Higgs mass from WMAP7+SN+BAO dataset. Forthe central value of the top quark pole mass m T = 171 . . ≤ m H ≤ . , (33)where we take into account the overall theoretical error δ = ± α s ( m Z ) = 0 . n S and tensor-to-scalarratio R , when compared with the similar constraints from the standard inflation with min-imally coupled scalar field. In particular, one should note the smallness of tensor-to-scalarratio ( R ∼ − ) that is challenging the future polarization experiments.We conclude that in order to obtain an accurate reconstruction of the Higgs potential interms of inflationary observables it is imperative to improve the accuracy of other parame-ters of the SM as the top quark mass and the effective QCD coupling constant.For example, it is expected that in the near future LHC will improve the determination ofthe current value of top quark mass to ∆ m T ≃ . m T ≃ . A S ∼ λ , using Equation (27) with R = 16 ǫ and fixing all param-eters at their observed values, it follows that the expected improved determination of thetop quark mass leads to an improved accuracy in the determination of the Higgs potential 18 –of about 3%. Acknowledgments
The authors acknowledge the referee the useful comments.// This work was partiallysupported by CNCSIS Contract 539/2009 and by ESA/PECS Contract C98051.
6. APPENDIX
In this appendix we collect the SM renormalization group β -functions (Ford et al. 1992),including the Higgs field propagator suppression factor s ( t ) given in Equation (25), at therenormalization energy scale t = ln( ϕ/m t ) beyond the top quark mass m t .The two-loop β -functions for gauge couplings g i = { g ′ , g, g s } are (Espinosa et al. 2008): β g i = kg i b i + k g i " X j =1 B ij g j − s ( t ) d ti y t , (34)where k = 1 / π and b = ((40 + s ( t )) / , − (20 − s ( t )) / , − , B = /
18 9 / / / / / / − ,d t = (17 / , / , . (35)For the top Yukawa coupling y t , the two-loop β -function is given by (De Simone et al. 2009): β y t = k y t (cid:20) − g − g ′ − g s + 92 s ( t ) y t (cid:21) + k y t (cid:20) − g − g g ′ + 1187216 g ′ + 9 g g s + 199 g ′ g s − g s + (cid:18) g + 13116 g ′ + 36 g s (cid:19) s ( t ) y t + 6 (cid:0) − s ( t ) y t − s ( t ) y t λ + s ( t ) λ (cid:1) (cid:21) . (36)The two-loop β -function for the Higgs quadratic coupling λ is (De Simone et al. 2009): β λ = k (cid:20) s λ − y t + 38 (cid:16) g + (cid:0) g + g ′ (cid:1) (cid:17) + (cid:0) − g − g ′ + 12 y t (cid:1) λ (cid:21) + k (cid:20) (cid:0) g − g g ′ − g g ′ − g ′ (cid:1) + 30 s ( t ) y t − y t (cid:18) g ′ g s + 3 s ( t ) λ (cid:19)
19 –Table 1: The mean values from the posterior distributions of the parameters obtained fromthe fit of the standard inflation model and Higgs inflation model with m T =171.3 GeV and v =246.22 GeV to WMAP7+SN+BAO dataset. The errors are quoted at 68% CL. Allparameters are computed at the Hubble radius crossing k ∗ =0.002 Mpc − .Model Standard Inflation Higgs InflationParameter100Ω b h ± ± c h ± ± τ ± ± θ s ± ± A S ] 3.157 ± ± n S ± ± < ± m H (GeV) - 155.372 ± λ - 0.216 ± ξ × − - 3.147 ± λ (cid:18) − g + 394 g g ′ + 62924 s ( t ) g ′ + 108 s ( t ) g λ + 36 s ( t ) g ′ λ − s ( t ) λ (cid:19) + y t (cid:18) − g + 212 g g ′ − g ′ + λ (cid:18) g + 856 g ′ + 80 g s − s ( t ) λ (cid:19)(cid:19) (cid:21) . (37)The one-loop β -function for non-minimal coupling ξ is given by (Bezrukov & Shaposhnikov 2009;Clark et al. 2009; Lerner & McDonald 2009): β ξ = k (cid:18) ξ + 16 (cid:19) (cid:18) s ( t )) λ + 6 y t − g ′ − g (cid:19) . (38)The reference Bezrukov & Shaposhnikov (2009) also gives the one-loop β -function for thevacuum expectation value v in the form: β v = k (cid:18) g ′ + 3 g − y t (cid:19) v . (39)Finally, the two-loop Higgs field anomalous dimension γ is given by (De Simone et al. 2009): γ = − k (cid:20) g g ′ − y t (cid:21) − k (cid:20) g − g g ′ − s ( t ) g ′ (cid:21) + k (cid:20) − (cid:18) g + 8524 g ′ + 20 g s (cid:19) y t + 274 s ( t ) y t − s ( t ) λ (cid:21) , (40) REFERENCES
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