Classical scale invariance in the inert doublet model

Published for SISSA by Springer Received: July 24, 2015 Accepted: August 12, 2015 Published: September 4, 2015 Alexis D. Plascencia Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham DH1 3LE, U.K. E-mail: Abstract: The inert doublet model (IDM) is a minimal extension of the Standard Model (SM) that can account for the dark matter in the universe. Naturalness arguments motivate us to study whether the model can be embedded into a theory with dynamically generated scales. In this work we study a classically scale invariant version of the IDM with a minimal hidden sector, which has a U(1)CW gauge symmetry and a complex scalar Φ. The mass scale is generated in the hidden sector via the Coleman-Weinberg (CW) mechanism and communicated to the two Higgs doublets via portal couplings. Since the CW scalar remains light, acquires a vacuum expectation value and mixes with the SM Higgs boson, the phenomenology of this construction can be modified with respect to the traditional IDM. We analyze the impact of adding this CW scalar and the Z 0 gauge boson on the calculation of the dark matter relic density and on the spin-independent nucleon cross section for direct detection experiments. Finally, by studying the RG equations we find regions in parameter space which remain valid all the way up to the Planck scale. Keywords: Higgs Physics, Beyond Standard Model, Cosmology of Theories beyond the SM ArXiv ePrint: 1507.04996 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP09(2015)026 JHEP09(2015)026 Classical scale invariance in the inert doublet model Contents 1 2 CSI in the IDM and its dark matter phenomenology 2.1 Dark matter relic density 2.2 Constraints from direct detection 3 5 7 3 Renormalization group (RG) analysis 8 4 Conclusions 12 A Other useful relations 13 1 Introduction The Lagrangian in the Standard Model (SM) contains only one scale parameter, the negative Higgs mass squared term, −µ2SM , which is quite small compared to the Planck scale; therefore, it seems reasonable to expect that it can be generated from the dynamics of the underlying theory. The concept of classical scale invariance (CSI) states that there should be no mass scales in the Lagrangian at a classical level and all the mass scales must be generated by the dynamics of the theory. In this framework it then becomes difficult to generate vastly different scales in the theory. These ideas have attracted a lot of attention recently [1–24].In our work we will follow the approach taken by the authors of ref. [3, 12], where the only mass scale in the Standard Model is generated via the ColemanWeinberg (CW) mechanism [1] in a hidden sector and then transmitted to the Standard Model through a Higgs portal interaction. One may ask if this approach of classical scale invariance implemented through a Higgs portal has implications for other extensions of the Standard Model. In this paper we investigate how the dynamical generation of electroweak symmetry scale through the Coleman-Weinberg mechanism in the hidden sector can be achieved in a model with a non-minimal Higgs sector, focusing in particular in a minimal realization of the two Higgs doublet model (2HDM) [25], which is the inert doublet model (IDM) [26, 27]. The latter was first introduced in ref. [26], where the authors give different possibilities to achieve EWSB in the 2HDM. The IDM has become particularly attractive because it provides a natural candidate for cosmologically stable dark matter [27, 28]; namely, the lightest inert neutral scalar. The IDM is a minimal extension of the SM that introduces a second complex doublet H2 and a discrete Z2 symmetry such that H1 → H1 , H2 → −H2 , –1– JHEP09(2015)026 1 Introduction where H1 stands for the Standard Model Higgs doublet and all the fields in the SM are even under this Z2 symmetry, meaning that H2 has no tree-level couplings to the SM fermions. The potential in this model is given by VIDM = µ21 |H1 |2 + µ22 |H2 |2 + λ1 |H1 |4 + λ2 |H2 |4 + λ3 |H1 |2 |H2 |2 + λ4 |H1† H2 |2 (1.1) the inert doublet consists of a neutral CP-even scalar H, a neutral CP-odd scalar A and a pair of charged scalars H ± . Imposing the requirement of an exact Z2 symmetry means that the inert H2 does not acquire a vacuum expectation value (vev), so the lightest particle in the inert doublet is stable and if it is one of the neutral scalars it can be studied as a dark matter candidate. For the rest of this work we consider MH < MA , MH + , and hence we take H to be the dark matter candidate, similar results apply if one takes A to be the lightest. The vevs for the doublets then read v hH1 i = √ , hH2 i = 0, (1.2) 2 where v = 246 GeV, and the mass of the SM Higgs boson is given by the usual relation Mh2 = −2µ21 = 2λ1 v 2 which we fix to 125 GeV. The masses of the two neutral scalars, H and A, and the charged, H ± , are given by 1 2 MH = µ22 + (λ3 + λ4 + λ5 )v 2 , (1.3) 2 1 MA2 = µ22 + (λ3 + λ4 − λ5 )v 2 , (1.4) 2 1 2 2 MH λ3 v 2 . (1.5) ± = µ2 + 2 We define the mass splittings ∆MA = MA −MH and ∆MH ± = MH ± −MH , where the mass splitting between A and H is determined by λ5 and since we consider MH < MA we take λ5 to be negative. It is convenient to work with the coupling λ 3 + λ4 + λ5 , 2 which determines the interaction between inert scalars and the SM Higgs boson. This paper is structured as follows, in section 2 we start by showing how the CW mechanism can be applied to the inert doublet model with the addition of a hidden sector and then perform a scan on the free parameters of the theory. In section 2.1 we measure the impact of introducing this hidden sector on the calculation of the relic density and in section 2.2 we calculate the spin-independent nucleon cross section and compare with current and future limits from direct detection experiments. In section 3 we perform the RG analysis on the model and show that some points satisfy vacuum stability, perturbativity, and unitarity up to the Planck scale. We close in section 4 with the conclusions. λL ≡ –2– JHEP09(2015)026 1 + λ5 [(H1† H2 )2 + (H2† H1 )2 ], 2 expanding the two doublets in their components we have ! ! G+ H+ H1 = , H2 = , √1 (v + h + iG) √1 (H + iA) 2 2 2 CSI in the IDM and its dark matter phenomenology One possibility to account for the dark matter in the universe in CSI models with a hidden sector is to extend the U(1)CW to a larger group, e.g. it has been shown that for SU(2)CW the vector bosons can account for a portion of dark matter and a scalar gauge singlet can be introduced to account for the rest of dark matter [21]. In this paper we adhere to the minimal case of having a U(1)CW symmetry and a single complex scalar Φ in the hidden sector and in order to account for dark matter we extend the SM by adding an SU(2)L vevless doublet. Since the second doublet in the IDM does not acquire a vev we will apply a similar mechanism as in ref. [12]. In this case we introd (...truncated)