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Higgs Boson Cosmology Moss IG. Higgs boson cosmology. Contemporary Physics 2015, 56(4), 468-476. Copyright: This is an Accepted Manuscript of an article published by Taylor & Francis in Contemporary Physics on 24 July 2015, available online: http://dx.doi.org/10.1080/00107514.2015.1058543. DOI link to article: http://dx.doi.org/10.1080/00107514.2015.1058543 Date deposited: 20/01/2016 Embargo release date: 24 July 2016 This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License Newcastle University ePrints - eprint.ncl.ac.uk July 21, 2015 10:28 Contemporary Physics paper To appear in Contemporary Physics Vol. 00, No. 00, Month 20XX, 1–15 Higgs boson cosmology Ian G. Moss School of Mathematics and Statistics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, U.K. (Received 00 Month 20XX; final version received 00 Month 20XX) The discovery of the Standard Model Higgs boson opens up a range of speculative cosmological scenarios, from the formation of structure in the early universe immediately after the big bang, to relics from the electroweak phase transition one nanosecond after the big bang, on to the end of the present-day universe through vacuum decay. Higgs physics is wide-ranging, and gives an impetus to go beyond the Standard Models of particle physics and cosmology to explore the physics of ultra-high energies and quantum gravity. Keywords: Higgs boson, cosmology, inflation, symmetry breaking. Acknowledgements This work was supported by the STFC under Consolidated Grant ST/J000426/1. 1. Introduction The discovery of the Higgs boson [1, 2] has completed the list of fermions (quarks and leptons), vector bosons (photons, gluons, W and Z) and the single scalar boson that make up the Standard Model of particle physics [3]. The Higgs has given us the first glimpse of a particle associated with a fundamental scalar field. This has a special significance to cosmologists, because so many of our modern theories of the early universe depend on the unique features of scalar field physics. We know that universe was once incredibly hot. As the universe cooled, it most likely went through a sequence of phase transitions. The Standard Model Higgs field has a special association with an electroweak phase transition, which took place at a temperature equivalent to an energy scale of 163 GeV around one nanosecond after the big bang. If we are fortunate, phase transitions like this can leave behind signals or relics which give us information about physics beyond the Standard Model. There is a great deal of interest in a phase transition which happened far earlier, at a time around 10−35 s after the big bang. According to the inflationary scenario, the universe was dominated then by the vacuum energy of a scalar field, causing it to undergo a period of exponential expansion [4–6]. Originally, the idea was that this field would be the Higgs field in a Grand Unified Theory uniting the strong, weak and electromagnetic forces. It is easier for the exponential expansion to get going when the energy scale of the phase transition is very high, and this fits in well with the idea of Grand Unification at around 1015 GeV. Inflationary models lead to a natural explanation for the origin of the large scale structure of the universe today. We measure this large scale structure primarily through observations of cosmic mi- crowave background (CMB) radiation and through galaxy surveys. The CMB gives us a snapshot July 21, 2015 10:28 Contemporary Physics paper of the universe from the time when it first became transparent, around 370; 000 yr after the big bang. Galaxy surveys (together with distortions of the CMB) probe the large scale structure from just after the origin of the CMB up until the present day. In inflationary models, small quantum or thermal fluctuations from the era of inflation are frozen into the geometry of spacetime. These fluctuations are the cause of the intensity fluctuations in the cosmic microwave background. Dif- ferent models of inflation, with different dependence of the vacuum energy on the scalar field, give different predictions for the cosmic microwave background fluctuations and allow us to test models of inflation. The large vacuum energy needed for inflation can arise in models where large fields correspond to large vacuum energies, provided that the universe started out with a large-magnitude scalar field. This is achieved in the chaotic inflationary scenarios, where the field starts out randomly distributed throughout space [7]. By chance, some small region has a large magnitude field and this region inflates. Eventually, with enough inflation, this region grows to encompass our entire observable universe. Once we get into the setting of these chaotic scenarios, then even a field normally associated with energy scales well below 1015 GeV, like the Standard Model Higgs field, can drive inflation. This is the main topic covered in section 3. Finally, we might enquire whether all the phase transitions are in the distant past, or whether we live in a metastable state, or ‘false vacuum’, which could decay into a new vacuum state [8]. For the observed Higgs mass around 125 GeV, the vacuum structure of the Higgs field in the Standard Model becomes rather sensitive to couplings to other massive particles. One possible configuration is a metastable state with the Higgs field having a value around 246 GeV and a true vacuum with the Higgs field close to the Planck scale 2:4 × 1018GeV. This would, of course, be very susceptible to the effects of new physics beyond the Standard Model. Decay of our ‘false’ vacuum state would be catastrophic, since it would change the masses and interactions of all the elementary particles. The fact that our vacuum has survived 13:8 Gyr since the big bang is evidence that the decay rate must be negligible. Vacuum decay is a quantum tunnelling process which is exponentially suppressed, so very long lifetimes for metastable states are not unusual. On the other hand, everyday phase transitions are often triggered by some kind of seed, a speck of ice or dust in a cloud for example causing water to condense into droplets. In the last section we shall report on recent results which consider whether a tiny black hole could act as a nucleation seed and trigger the decay of our vacuum state. The theory of elementary particles is greatly simplified by using a special set of units in which 9 ~ = c = 1. The fundamental unit is then the GeV = 10 eV = 0:1602 J (S:I:). In these units masses are given in GeV, distances in GeV−1 = 0:1973 fm (S:I:). The gravitational constant is replaced by −1=2 18 −9 the reduced Planck mass, Mp = (8πG) = 2:435 × 10 GeV = 4:3415 × 10 kg (S:I:). 2. The Higgs field We start with an overview of some of the essential features of Higgs field theory in the part of the Standard Model which unites weak and electromagnetic forces: the Weinberg-Salam model[9, 10]. This model has an SU(2) × U(1) local symmetry group. The unified Higgs field consists of two complex scalar fields arranged into a doublet H(x; t). The Higgs field is a scalar under spatial rotations, but the components mix under the SU(2) × U(1) symmetry group. The energy density of a stationary Higgs field configuration defines the Higgs potential V (H), 2 y y 2 V (H) = V0 − µ H H + λ(H H) : (1) Adding the constant V0 has no effect on the particle physics, but it becomes important for cosmology because all forms of energy affect the expansion of the universe. 2 July 21, 2015 10:28 Contemporary Physics paper Figure 1. The famous ‘Mexican hat’ shape for the Higgs potential. The potential is plotted vertically and solid lines indicate constant (φ, α) parameters for the Higgs field, leaving two angles β and γ which are not shown. The Higgs vacuum corresponds to a field lying somewhere along the brim of the hat. The potential there has to be very close to zero, otherwise it would produce a large cosmological constant. In exactly the same way in which we might represent a vector by its magnitude and direction, we can represent the Higgs field by a magnitude φ and rotation angles α, β and γ, 1 0 cos α + i sin α cos β ie−iγ sin α sin β H = p U ; U = : (2) 2 φ ieiγ sin α sin β cos α − i sin α cos β The potential has the ‘Mexican hat’ shape shown in figure 1, with the minimum of the potential at φ = v = µ2/λ. There is no dependence of the potential on the angles α, β and γ. The value of the potential at the minimum, where the Higgs field sits today, is tightly constrained by cosmology. The potential acts like a cosmological constant term in the gravitational field equa- tions and leads to exponential expansion. Whilst the current observational evidence is consistent with the universe undergoing exponential expansion, the scale is tiny today compared to any energy scale relevant to the Higgs boson. Taking into account the zero potential at the minimum, the Higgs potential can be expressed succinctly as 1 2 V (φ) = λ φ2 − v2 : (3) 4 In quantum theory, the Higgs field becomes a quantum operator with the vacuum expectation value set by the minimum of the potential, hφi = v. The Higgs boson corresponds to an excitation of the 3 July 21, 2015 10:28 Contemporary Physics paper vacuum state with mass p 00 1=2 mH = V (v) = λ v; (4) which the Large Hadron Collider has determined to be 126 GeV.
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