Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017 The Higgs boson and cosmology rsta.royalsocietypublishing.org Mikhail Shaposhnikov Institut de Théorie des Phénomènes Physiques, École Polytechnique Review Fédérale de Lausanne, 1015 Lausanne, Switzerland Cite this article: Shaposhnikov M. 2015 The I will discuss how the Higgs field of the Standard Model may have played an important role in Higgs boson and cosmology. Phil.Trans.R. cosmology, leading to the homogeneity, isotropy and Soc. A 373: 20140038. flatness of the Universe; producing the quantum http://dx.doi.org/10.1098/rsta.2014.0038 fluctuations that seed structure formation; triggering the radiation-dominated era of the hot Big Bang; and contributing to the processes of baryogenesis and dark One contribution of 12 to a Discussion Meeting matter production. Issue ‘Before, behind and beyond the discovery of the Higgs boson’. 1. Introduction Subject Areas: A Higgs boson-like particle with mass 126 GeV has particle physics, relativity, high-energy physics recently been discovered at CERN [1,2], and thus we now have a theory of the strong, weak, electromagnetic and gravitational interactions that may be a self-consistent Keywords: effective field theory all the way up to the Planck Higgs boson, inflation, dark matter, ∼ × 18 scale E MP 2.44 10 GeV; see a recent discussion in baryogenesis [3–7]. Of course, this does not mean that the Standard Model (SM) is a correct theory of Nature up to these Author for correspondence: energies, as this would be an enormous extrapolation of known physics into a region that is not accessible to Mikhail Shaposhnikov present experiments. Nevertheless, since the assumption e-mail: [email protected] that the SM is valid up to the Planck scale is the most conservative option available to us, it has more predictive power than any other approach. The Higgs boson is a very special particle in the SM. It provides a mechanism for including weakly interacting massive vector bosons in the SM, and for ‘giving’ masses to quarks and leptons. As will be discussed in this paper, the Higgs field may also have had an important role in cosmology: it could have made the Universe flat, homogeneous and isotropic, it could have produced the fluctuations that led to structure formation and it could also have enabled the radiation-dominated epoch of the hot Big Bang to occur [8–10]. Moreover, in the modest extension of the SM by three relatively light Majorana fermions—heavy neutral leptons (HNLs)—the Higgs field is important for baryogenesis, leading to the charge asymmetric Universe, and for dark matter 2014 The Author(s) Published by the Royal Society. All rights reserved. Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017 U(c) 2 4 2 rsta.royalsocietypublishing.org lM /x /4 ......................................................... Standard Model l v4/4 lM4/x2/16 0 v Phil.Trans.R.Soc.A 0 c Figure 1. Effective potential for a canonically normalized scalar field χ (related to the Higgs field in a way that is well understood) in the Einstein frame. For large h (or χ) the potential is flat. production [11]. In addition, active neutrino masses and mixing are induced via Yukawa 373 couplings of both HNLs and left-handed neutrinos to the Higgs field. : 20140038 2. Higgs field and gravity In order to embed the SM in the cosmological framework, one has to fix the coupling of the Higgs field to gravity. In addition to the replacement of the Minkowski metric ημν by the generic curved space metric gμν , one should add to the Lagrangian a non-minimal coupling of the Higgs field to gravity (e.g. [12]): M2 S = d4x −g − P R − ξH†H R. (2.1) G 2 Here R is the scalar curvature, g is the determinant of the metric, H is the Higgs field and ξ is a new dimensionless coupling constant of the SM. As is the case with other couplings of the SM, the value of ξ cannot be fixed from within the model, but instead can be determined by specific experiments (cosmological observations in our case). ξ To elucidate√ the role of the non-minimal coupling , let us consider large Higgs fields > / ξ 2 = † / h MP ,(h H H 2), which may have existed in the early Universe. In this regime, the Higgs field not only gives masses ∝ h to fermions and vector bosons, but also determines the gravity interaction strength, which is simply the inverse coefficient in front of the scalar curvature R: eff = 2 + ξ 2 ∝ MP MP h h. In the limit of large Higgs fields, physical observables do not depend on / eff h, as in all dimensionless ratios the magnitude of h cancels out; MW MP is one such example. In particular, the physical effective potential (obtained by transforming the theory from the so-called Jordan frame to the Einstein frame) does not depend on the Higgs field, as depicted in figure 1. It has been shown [13] that the form of the potential is not changed by perturbative higher order corrections, provided the mass of the Higgs boson obeys the requirement (see also [14]): ξ M > M − 0.1 log ± 1 GeV. (2.2) H crit 1000 Here yt(μt) − 0.9361 αs(M ) − 0.1184 M = 129.1 + × 2.0 − Z × 0.5 GeV, (2.3) crit 0.0058 0.0007 where yt(μt) is the top Yukawa coupling in the MS scheme taken at μt = 173.2 GeV, and αs(MZ) μ = is the strong coupling at the scale MZ. The theoretical uncertainty in Mcrit is very small– approximately 70 MeV (see [6] and the discussion in [4,7]). The comparison of Mcrit with experiment for ξ ∼ 1 is presented in figure 2. There is 1–2 s.d. tension between the experimental Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017 Higgs mass M = 125.3 ± 0.6 GeV H 3 0.121 rsta.royalsocietypublishing.org ......................................................... ) z 0.120 M ( s a 0.119 0.118 strong coupling 0.117 Phil.Trans.R.Soc.A 0.116 170 171 172 176175174173 pole top mass Mt (GeV) Figure 2. The shaded regions account for 1 and 2 s.d. experimental uncertainties in αs (αs = 0.1184 ± 0.0007) and the pole top quark mass Mt (Tevatron: Mt = 173.2 ± 0.51 ± 0.71 GeV), and also include theoretical errors in extraction of yt from 373 experiment. The thick straight line (blue in the online version) marks the relation between αs and the pole top mass following : 20140038 from equation (2.3)ifMH is identified with Mcrit. The enclosing shaded regions correspond to 1 and 2 s.d. experimental errors in the Higgs mass. Small ellipses (red in the online version) correspond to the accuracy achievable at the e+e− collider [15]. (Online version in colour.) values of the top and Higgs masses and the bound (2.2), with the main uncertainty coming from + − Mt; it is therefore imperative to obtain more precise measurements at the future e e collider. In what follows I will assume that (2.2) is satisfied. 3. Higgs boson, cosmological inflation and the hot Big Bang It is well known that a number of important cosmological problems, such as the flatness, isotropy and homogeneity of the Universe, can be solved simultaneously by the accelerated expansion of the Universe in the distant past. This epoch of inflation is expected to be driven by some scalar field called the inflaton. The Higgs field is the only scalar field included within the framework of the SM, and in [8] it was shown that the Higgs field is a good inflaton candidate. The evolution of the Universe with the Higgs field playing the role of the inflaton (Higgs inflation) proceeds as follows [9,10]. At the first stage, as is√ the case in any chaotic inflation > / ξ scenario [16], the value of the Higgs field is large (h MP ), and it rolls slowly towards the minimum of the potential in figure 1. The potential energy of the Higgs field leads to the exponential expansion of the Universe, which then becomes flat, homogeneous and isotropic. The small-scale quantum fluctuations of the Higgs field are inflated and seed structure formation, thus leading to the creation of galaxies and clusters√ of galaxies. / ξ After the Higgs field reaches the value h MP , the slow roll ends, and the Higgs field starts to oscillate. The exponential expansion of the Universe becomes a power law, corresponding to matter domination. The Higgs field oscillations lead to the creation of the particles of the SM that couple most strongly to H, namely to intermediate vector bosons W and Z and the top quark. Eventually, W and Z thermalize through decays and inverse decays into fermions of the SM. Owing to all these processes the decay of the scalar field is completed sometime before the /ξ amplitude of the Higgs field reaches the value h MP . As a result, the Universe is heated up to ∼ 14 the temperature Treh 10 GeV [9,10]. This is the start of the hot Big Bang stage of the Universe evolution, when the Universe is dominated by radiation. The cosmological predictions of Higgs inflation can be compared with observations performed by the Planck satellite. The Higgs-inflaton potential depends on one unknown parameter, ξ.Itcan be fixed by the amplitude of the cosmic microwave background temperature fluctuations δT/T Downloaded from http://rsta.royalsocietypublishing.org/ on December 4, 2017 (a) 0.25 4 Planck + WP rsta.royalsocietypublishing.org ......................................................... ) Planck + WP + highL 0.20 Planck + WP + BAO 0.002 r predictions of 0.15 inflationary models: 0.10 Higgs inflation 0.05 tensor-to-scalar ratio ( tensor-to-scalar Phil.Trans.R.Soc.A 0 (b) 0.25 Planck + WP 373 ) Planck + WP + highL 0.20 Planck + WP + BAO : 20140038 0.002 convex r natural inflation concave 0.15 power-law inflation low-scale SSB SUSY R2 inflation 0.10 V µ f 2/3 V µ f 0.05 V µ f 2 tensor-to-scalar ratio ( tensor-to-scalar 3 V µ f N* =50 0 0.94 0.96 0.98 1.00 N* =60 primordial tilt (ns) Figure3.