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Metastability of the False Vacuum in a Higgs-Seesaw Model of Dark Energy

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Metastability of the False Vacuum in a Higgs-Seesaw Model of Dark Energy PHYSICAL REVIEW D 89, 085023 (2014) Metastability of the false vacuum in a Higgs-seesaw model of dark energy † Lawrence M. Krauss1,2,* and Andrew J. Long1, 1Department of Physics and School of Earth and Space Exploration Arizona State University, Tempe, Arizona 85827-1404, USA 2Research School of Astronomy and Astrophysics, Mt. Stromlo Observatory, Australian National University, Canberra 2611, Australia (Received 24 October 2013; published 9 April 2014) In a recently proposed Higgs-seesaw model the observed scale of dark energy results from a metastable false vacuum energy associated with mixing of the standard model Higgs particle and a scalar associated with new physics at the scale of unification or the Planck scale. Here we address the issue of how to ensure metastability of this state over cosmological time. We consider new tree-level operators, the presence of a thermal bath of hidden sector particles, and quantum corrections to the effective potential. We find that in the thermal scenario many additional light degrees of freedom are typically required unless coupling constants are somewhat fine-tuned. However quantum corrections arising from as few as one additional light scalar field can provide the requisite support. We also briefly consider implications of late-time vacuum decay for the perdurance of observed structures in the universe in this model. DOI: 10.1103/PhysRevD.89.085023 PACS numbers: 12.60.Fr, 95.36.+x, 98.80.-k I. INTRODUCTION vacuum state for cosmological times, and what the impli- cations might be for the future when it decays to its true Understanding the nature of dark energy, with an ðobsÞ 4 ground state. inferred magnitude of approximately ρ ¼ 28 mev DE The organization of this paper is as follows. In Sec. II we [1–3], remains the deepest open problem in particle physics review the Higgs-seesaw model of dark energy, and in and cosmology. Observations suggest that this source particular, we estimate the lifetime of the false vacuum in has an equation of state w ¼ −1, consistent with either a this model. In Sec. III we explore three variants of the fundamental cosmological constant or false vacuum energy minimal model that extend the lifetime of the false vacuum associated with a metastable scalar field configuration. In to cosmological time scales. Since the false vacuum is only either case, quantum effects would suggest that this energy, metastable, it will eventually decay, and we consider the ρ , will depend sensitively on unknown UV physics, and DE implications of this decay in Sec. IV. We conclude it is therefore very difficult to imagine how the observed in Sec. V. small energy scale could naturally arise [4]. In particular, ρ ¼ Λ4 Λ (i) why not DE where is the UV cutoff of the ρ ¼ 0 effective field theory, (ii) why not a natural value DE , II. REVIEW OF THE HIGGS-SEESAW MODEL which could result from some symmetry constraint? The model of Ref. [5] extends the SM by introducing a The answers to these fundamental questions will most complex scalar field σ , which is a singlet under the SM likely require an understanding of a full quantum theory HS gauge groups and charged under its own global axial of gravity. Assuming they are resolvable, and that the Φ ¼ symmetry. Denoting the SM Higgs doublet as SM ultimate vacuum energy is indeed zero, one can proceed to þ ðϕ ; ϕ ÞT, the scalar sector Lagrangian is written as consider whether plausible physics, based on known SM SM energy scales in particle theory, might produce at least a L ¼j∂ Φ j2 þj∂ σ j2 − ðΦ σ Þ (1) temporary residual vacuum energy consistent with current μ SM μ HS V SM; HS observations. Recently in Ref. [5] it was proposed that a Higgs portal, mixing electroweak and grand unification where theory (GUT) MGUT scalars, might naturally produce the observed magnitude of the energy density of dark energy ðΦ σ Þ¼Ω þ μ2jΦ j2 þ λjΦ j4 due to the false vacuum energy associated with an other- V HS; HS CC SM SM þ λ jΦ j2jσ j2 þ λ jσ j4 (2) wise new massless scalar field that is a singlet under the SM mix SM HS HS HS : gauge group. The questions we examine here are whether it is possible to ensure that this field remains in its false The biquadratic term is sometimes referred to as the Higgs portal operator [6,7]. If this operator arises by virtue of *[email protected] GUT-scale physics, as argued in Ref. [5], then its value † [email protected] should naturally be extremely small in magnitude, 1550-7998=2014=89(8)=085023(7) 085023-1 © 2014 American Physical Society LAWRENCE M. KRAUSS AND ANDREW J. LONG PHYSICAL REVIEW D 89, 085023 (2014) 2 −2 4 ð Þ M M λv λ λ jλ nat j ≈ W ≃ 6.5 × 10−29 GUT : (3) UðsÞ¼ Ω − þ mix v2s2 þ HS ð1 − ϵ2Þs4: (8) mix 2 1016 CC 4 4 4 MGUT GeV σ If it is assumed that the scalar potential vanishes in the Note the absence of a mass term for the field HS, which is ð Þ¼0 assumed to only acquire a mass after electroweak sym- true vacuum, i.e. U vs , then the bare cosmological metry breaking, presumably enforced by some symmetry constant must be tuned to be principle such as conformal symmetry or supersymmetry. λv4 1 λv4 These symmetries cannot be exact, but as long as the scale Ω ¼ ≈ ½1 þ ϵ2 þ ðϵ4Þ (9) CC 4 2 4 O : of symmetry breaking (communicated by mixing to the 1 − ϵ hidden sector) is sufficiently small these assumptions can The effective cosmological constant today will then be be justified. smaller than Ω as a consequence of symmetry breaking For the purposes of studying the vacuum structure it is CC pffiffiffi phase transitions. If the scalar fields have not reached their convenient to take ϕþ ¼ 0, ϕ ¼ h= 2, and σ ¼ pffiffiffi SM SM HS true vacuum state but are instead suspended in the false s= 2 where hðxÞ and sðxÞ are real scalar fields. Then vacuum, then the vacuum energy density, ρ ≡ Uð0Þ,is the scalar potential becomes DE given by 1 λ λ λ 4 2 2 4 2 2 4 mix 2 2 HS 4 λv ϵ λ v Uðh; sÞ¼Ω þ μ h þ h þ h s þ s ; (4) ρ ¼ ≈ mix (10) CC 2 4 4 4 DE 4 2 16λ : 1 − ϵ HS Ω ρ where CC is a bare cosmological constant which must be As the notation suggests, DE should be identified with the λ ¼ 2 ð2 2Þ tuned to cancel UV contributions from the scalar field energy density of dark energy. Taking MH= v with μ2 ¼ −λ 2 sector. The tachyonic mass v induces electroweak v ≃ 246 GeV and MH ≃ 126 GeV gives symmetry breaking and causes the Higgs field to acquire a ! vacuum expectation value (VEV) hhi¼v, which in turn λ 2 1 2 2 ρ ≃ 0 97 4 mix (11) induces a mass μ ¼ λ v =2 for the field s.Ifλ < 0 DE . mev ð Þ : HS mix mix λ nat λ then this mass is tachyonic, and the true vacuum state of the mix HS theory is displaced to This value is comparable to the observed energy density, ρðobsÞ ≈ 28 4 DE mev [3]. In this way, the Higgs-seesaw model h i ≡ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiv ≈ ½1 þ ðϵ2Þ (5) h true vh 2 v O ; naturally predicts the correct magnitude for the energy 1 − ϵ density of dark energy density from the electroweak and GUT scales. For the discussion in the following sections, it rffiffiffiffiffiffiffiffiffiffiffiffi λ 1=4 ϵ will be useful here to rewrite Eq. (11) as hsi ≡ v ¼ v true s λ 1 − ϵ2 ! ! HS λ 2 ρðobsÞ λ 1=4pffiffiffi λ ≃ 0.035 mix DE (12) ≈ v ϵ½1 þ Oðϵ2Þ; (6) HS λðnatÞ ρ λ mix DE HS λ pffiffiffiffiffiffiffiffiffiffiffiffi and to note that HS remains perturbatively small ϵ ≡−λ 4λλ h i ¼ ðnatÞ where mix= HS. We will use h false v and for λ ≤ λ . h i ¼ 0 mix mix s false to denote the tachyonic false vacuum state. The success of the Higgs-seesaw model hinges upon the λ For typical values of the coupling mix, see Eq. (3), the assumption that the universe is trapped in the false vacuum. σ μ2 ⋘μ2 mass scales of the HS field are extremely small: HS The lifetime of the false vacuum can be estimated by ⋘ ∼ and vs vh v. In this limit it is a good approximation to dimensional analysis using the tachyonic mass scale, integrate out the Higgs field and work with an effective μ2 ¼ λ v2=2. Taking the same numerical values as σ HS mix field theory for the HS field alone. The field equation above, this time scale is ∂U=∂h ¼ 0 has the solution jμ j−1 ≃ ð1.4 mevÞ−1 ≈ 0.47 n sec : (13) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HS λ 1=2 ¯ð Þ¼ 1 þ ϵ HS 2 (7) h s v λ s ; Therefore, in the absence of any support, the false vacuum would have decayed in the very early universe. This observation motivates the present work, in which we will which interpolates between the false and true vacua, as explore scenarios that can provide support to the tachyonic one can easily verify. The scalar potential in the effective false vacuum, following a classification scheme outlined theory, UðsÞ ≡ Uðh¯ðsÞ;sÞ,isgivenby in Ref. [8] 085023-2 METASTABILITY OF THE FALSE VACUUM IN A HIGGS- … PHYSICAL REVIEW D 89, 085023 (2014) III.
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