]>NUPHB13216S0550-3213(14)00336-810.1016/j.nuclphysb.2014.11.001The AuthorsHigh Energy Physics – PhenomenologyFig. 1Left panel: the color coding shows the value of frel in the (mS,λHS) plane. Above the solid black line μS2<0, and modification to the electroweak transition are possible. The constraint from the perturbativity of couplings, LHC invisible Higgs width and direct dark matter searches have been imposed as explained in the text. Right panel: results from dark matter direct searches. The color coding shows how the parameter space is constrained further if the sensitivity of the direct search experiments rises. For the black points σSIeff/σLUX≤0.1. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)Fig. 2The contours, from top to bottom, show the values of λHS and mS required to produce relic density frel=1,0.1,0.01, respectively, via the freeze-in mechanism. The slope of the curves at constant ΩSh2 is 1/2, corresponding to the scaling λHS∼mS−1/2; see Eq. (3.15).Fig. 3Dark matter density as a function of the dark matter mass mψ and the Yukawa coupling gS. The inset shows the distribution of values of v(Tc)/Tc corresponding to the points in the plot. The shaded lower portion of the histogram bars correspond to points which yield frel>0.5. The constraints from the perturbativity of couplings and LHC invisible Higgs width have been taken into account. Also, we show only the points which give 0.01<frel≤1, v(Tc)/Tc>1 and Tc>40 GeV. All shown points are also compatible with the LUX constraints. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)Fig. 4Left panel: the same data points as in Fig. 3 in the (mψ,mH) plane. The color code now shows the magnitude of relative cross section σSIeff/σLUX. Right panel: the data points as a function of frel and σSIeff/σLUX. The color coding corresponds to the magnitude of the coupling gS. In both panels the constraints from the perturbativity of couplings and LHC invisible Higgs width have been taken into account. Also, we show only the points which give 0.01<frel≤1, v(Tc)/Tc>1 and Tc>40 GeV and which are not excluded by the LUX data. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)Fig. 5The relic density frel=1 produced via freeze-in mechanism as a function of dark matter mass mψ and Yukawa coupling gS. For all points frel=1, v(Tc)/Tc>1 and Tc>40 GeV.Fig. 6The figures show constraints from vacuum stability and perturbativity of the couplings. The contours in the lower right corner show the regions where the Higgs self-coupling becomes negative, while the contours in the upper right corner show the regions where one or more of the couplings become large. Finally, the horizontal contours correspond to λS becoming negative. In the left panel gs(mt)=0.4, and the contours show the dependence on λS(mt), while the right panel shows the dependence on gs(mt) at λS(mt)=0.Fig. 7Left panel: the value of the trilinear Higgs coupling, λhhh, in the model as a function of mH and cosβ. The points correspond to the ones shown in Fig. 3. Right panel: the value of the trilinear Higgs coupling as a function of the deviation Δ from the optimal values of the fit to the S and T parameters. The value Δ=1 corresponds to the 2σ limit.Strong phase transition, dark matter and vacuum stability from simple hidden sectorsTommiAlanneactommi.alanne@jyu.fiKimmoTuominenbc⁎kimmo.i.tuominen@helsinki.fiVilleVaskonenacville.vaskonen@jyu.fiaDepartment of Physics, University of Jyväskylä, P.O. Box 35 (YFL), FI-40014 University of Jyväskylä, FinlandDepartment of PhysicsUniversity of JyväskyläP.O. Box 35 (YFL)University of JyväskyläFI-40014FinlandbDepartment of Physics, University of Helsinki, P.O. Box 64, FI-00014 University of Helsinki, FinlandDepartment of PhysicsUniversity of HelsinkiP.O. Box 64University of HelsinkiFI-00014FinlandcHelsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, FinlandHelsinki Institute of PhysicsP.O. Box 64University of HelsinkiFI-00014Finland⁎Corresponding author.Editor: Hong-Jian HeAbstractMotivated by the possibility to explain dark matter abundance and strong electroweak phase transition, we consider simple extensions of the Standard Model containing singlet fields coupled with the Standard Model via a scalar portal. Concretely, we consider a basic portal model consisting of a singlet scalar with Z2 symmetry and a model containing a singlet fermion connected with the Standard Model fields via a singlet scalar portal. We perform a Monte Carlo analysis of the parameter space of each model, and we find that in both cases the dark matter abundance can be produced either via freeze-out or freeze-in mechanisms, but only in the latter model one can obtain also a strong electroweak phase transition required by the successful electroweak baryogenesis. We impose the direct search limits and consider systematically the possibility that the model produces only a subdominant portion of the dark matter abundance. We also study the renormalization group evolution of the couplings of the model to determine if the scalar sector of the model remains stable and perturbative up to high scales. With explicit examples of benchmark values of the couplings at weak scale, we show that this is possible. Models of this type are further motivated by the possibility that the excursions of the Higgs field at the end of inflation are large and could directly probe the instability region of the Standard Model.1IntroductionExtensions of the Standard Model (SM) of elementary particle interactions have been put under severe tests [1–4] after the discovery of the Higgs boson with mass mh=126 GeV at the ATLAS and CMS experiments in the CERN Large Hadron Collider [5,6]. The requirement of a light Higgs scalar boson and no other obviously accessible states at the energy scales probed so far presents a challenge for traditional model paradigms like supersymmetry and technicolor which predict an extended spectrum beyond the SM. On the other hand, the cosmological observations on the dark matter abundance and matter–antimatter asymmetry clearly require, in the elementary particle physics context, the existence of new degrees of freedom not present in the SM.One possible framework to address these aspects is to take the SM according to the current collider data and extend it with singlet fields communicating with the SM fields only through the scalar or vector portals. The singlet sector can consist of a single scalar [7,8], more complex scalar multiplets [9,10], fermions [11–13] or vectors [14,15]. The resulting spectrum can contain particles stable over the timescales of the age of the universe and contribute to the observed abundance of the dark matter. Moreover, the extended scalar potential can modify the properties of the phase transitions in the early universe with respect to the results obtained in the SM. A strong first-order electroweak transition is a prerequisite for successful electroweak baryogenesis [16], and it is well known that the electroweak phase transition in the SM is not of first order but a smooth crossover [17,18]. If the electroweak sector of the SM were fully perturbative, a first-order phase transition would arise from a cubic term generated in the Higgs effective one-loop potential by the thermal effects of fields coupled to the Higgs. However, addition of a singlet scalar can sufficiently modify the picture already by tree level effects due to the presence of T-independent dimensional parameters appearing in the scalar potential and lead to a strong first-order transition [19,20]. Consequently, the ratio v(Tc)/Tc which controls the sphaleron erasure of the baryon asymmetry can be large and lead to successful electroweak baryogenesis.The phenomenologically interesting scenario would, thus, be the one where the strong electroweak phase transition is accompanied by an explanation of the dark matter relic density by a weakly coupled massive particle (WIMP). As a concrete model paradigm in this paper, we consider simple scalar portals between the hidden sector and the visible one. In addition to the Standard Model Higgs field, the scalar sector contains a real singlet scalar S.As a simple limiting case, we consider a model where the singlet sector is solely constituted by S with a discrete Z2 symmetry [7,21–23]. Then S can also act as a dark matter candidate provided that the global minimum of the potential at zero temperature does not spontaneously break this Z2 symmetry. However, in this model it is not possible to simultaneously explain the strong electroweak phase transition and the observed dark matter relic abundance. Therefore, as a second example we consider a model where the scalar S is not assumed to have any discrete symmetries, but the singlet sector also contains a Dirac fermion [11,12,24]. We find that in this model it is possible to realize simultaneously strong electroweak phase transition and the observed dark matter relic abundance.To establish our results, we perform a Monte Carlo analysis of the parameter space to search for viable models. We impose the constraints from LHC data and precision electroweak measurements, and from the dark matter direct searches [25,26]. To improve the earlier work on models of this type [22,12], we also require that the couplings remain perturbative up to scales of O(TeV). As a special case, we consider the possibility that the model could remain perturbative up to the Planck scale. This happens only on very specific couplings and allows to single out specific benchmark models where the dark matter constraints can be satisfied and a strong electroweak transition induced. Furthermore, since it is possible that multiple components contribute to the observed dark matter abundance [27–29], we consider systematically the cases of subdominant dark matter and both freeze-out and freeze-in mechanisms to generate the dark matter abundance. In particular, we show that if the dark matter candidate is a singlet fermion, then its abundance can be generated either via the freeze-out or the freeze-in mechanism, and in both cases the model will also lead to a strong electroweak phase transition.The paper is organized as follows. In Section 2, we introduce the model and the basic constraints. Then in Section 3, we discuss the numerical results for the model with only a Z2-symmetric real singlet scalar and for the model with a real singlet scalar and a singlet Dirac fermion. In Section 4, we present our conclusions.2Model and constraintsWe consider an extended scalar sector described by the potential(2.1)V(H,S)=μH2H†H+λH(H†H)2+12μS2S2+μ33S3+λS4S4+μHS(H†H)S+λHS2(H†H)S2, which provides typical scalar portals between the hidden singlet sector and the Standard Model Higgs. In Eq. (2.1), the field H is the usual Standard Model Higgs doublet and S is a real singlet scalar. The Higgs doublet is written in terms of the components as(2.2)H=(ϕ+12(v+ϕr0+iϕi0)), where the superscript refers to the electroweak charge of the component and subscripts denote the real and imaginary parts. Note that the most general potential would have a linear term, μ13S, but this can always be removed by shifting the field S. The stability of the potential requires that(2.3)λH>0,λS>0,λHS>−2λHλS.Depending on the values of the parameters, the vacuum structure can be rich [20]. We require that the extremum which leads to correct pattern of the electroweak symmetry breaking is the global minimum at zero temperature.To study the consequences for the electroweak phase transition, we need to include the finite-temperature corrections. Since there are multiple vacua induced already at the tree level, it is sufficient to consider the finite-temperature corrections to the leading terms. In other words, we consider temperature dependent coefficients(2.4)μ1(T)3=c1T2,μS(T)2=μS2+cST2,μH(T)2=μH2+cHT2, where(2.5)c1=112(μ3+2μHS),cS=112(2λHS+3λS),cH=148(9g2+3g′2+12yt2+24λH+2λHS), in Eq. (2.1). We have assumed that the term linear in S, i.e. the one with coefficient μ1 in the potential, has been shifted away at tree level at zero temperature and therefore arises only for T≠0. Starting from T=0, we monitor the evolutions of the electroweak symmetric and electroweak broken minima. If the symmetric minimum at some temperature Tc gets deeper than the asymmetric minimum, the phase transition is possible. For these cases we determine the critical temperature, Tc, and v(Tc)/Tc to identify the parameter space domains where a strong electroweak transition can be realized.For the dark matter candidate, there are several possibilities which can be built around this scalar sector. We will consider two. The simplest, and already much studied, case is to assume the singlet scalar to have a discrete symmetry, like Z2, which renders it stable. This symmetry must then be imposed on the general potential of Eq. (2.1). The second alternative that we consider is to assume, in addition to the scalar, the existence of a singlet fermion, which then due to a conserved fermion number becomes a stable dark matter candidate. Then no symmetry restrictions need to be imposed on the scalar potential, leaving a richer possibility of patterns for the phase transition between the electroweak symmetric and broken vacua. The details and numerical result of these two model examples will be exposed more thoroughly in the next section. Here, we now discuss the general formulation of the computation of the dark matter abundance and direct detection limits.To compute the relic abundance of dark matter, ΩDM, we apply the standard freeze-out formalism.11Of course, dark matter abundance need not be due to a thermal relic. We will briefly discuss the possibility of producing the abundance via out-of-equilibrium freeze-in scenario in the models we consider when presenting the numerical results in Section 3. The number density n of the thermal relic can be computed from the Lee–Weinberg equation [30],(2.6)∂f(x)∂x=〈vσ〉m3x2H(f2(x)−feq2(x)), written in terms of scaled variables f(x)=n(x)/sE and x=sE1/3/m. Here sE is the entropy density at temperature T, m is the mass of the dark matter candidate and H is the Hubble parameter. For the averaged cross sections, we use the integral expression [31](2.7)〈vσ〉=18m4TK22(m/T)∫4m2∞dss(s−4m2)K1(s/T)σtot(s), where Ki(y) are the modified Bessel functions of the second kind and s is the usual Mandelstam invariant. Given the cross sections, we can solve the Lee–Weinberg equation for f(0) which gives the present ratio of the number density of the dark matter candidate to its entropy density. The fractional density parameter, ΩDM, can be computed from(2.8)ΩDM≃1.09⋅106mf(0). Since there is no reason to expect that all of the dark matter abundance originates from a single source, we define the fraction(2.9)frel=ΩDMh2/(Ωh2)c, where (Ωh2)c=0.12 from Planck [32].Also, there are stringent constraints for this type of models arising both from the LHC data and direct searches for dark matter. If the singlet scalar is light enough, 2mS≤mh, then an obvious constraint on new light degrees of freedom from LHC data is the invisible width of the Higgs boson. According to present LHC data [33–38], the 2σ limit for the branching fraction to invisible channels is Brinv≤0.28. On the other hand, after the symmetry breaking the scalar mass eigenstates are(2.10)h0=ϕr0cosβ+Ssinβ,H0=−ϕr0sinβ+Scosβ, and the mixing of the Higgs and the singlet scalar affects the couplings of the scalar to fermions and gauge bosons. These modifications can be constrained by the current collider data by fitting the mixing cosβ to the signal strength results. Moreover, compatibility with the electroweak precision measurements using the S and T parameters [39] needs to be checked. Finally, considering the direct searches for dark matter, the scalar portal interactions in Eq. (2.1) contribute to the spin-independent scattering cross sections on nuclei for which the LUX experiment [26] provides currently the most stringent constraints.In the cases we will consider, the WIMP couples to the nucleus via the scalars h0 and H0 with strength depending on the mixing patter of the scalars and whether the WIMP is a scalar or a fermion. In both cases the Higgs–nucleon coupling is fNmN/v, where mN=0.946 GeV, and we neglect the small differences between neutrons and protons. The effective Higgs–nucleon coupling,(2.11)fN≡1mN∑q〈N|mqq¯q|N〉, describes the normalized total quark scalar current within the nucleon. The quark currents of the nucleon have been a subject of an intensive lattice research supplemented by efforts applying chiral perturbation theory methods and pion nucleon scattering. Consequently, fN is fairly well determined currently. Following [23] we use fN=0.345±0.016, where the uncertainty in fN induces at most 20% error in the spin-independent direct detection limits.The spin-independent cross section for a WIMP scattering on nuclei is computed by considering the t-channel exchange of h0 and H0 in the limit t→0. The matrix element for this process generally contains the factors(2.12)(g˜hcosβmh2−g˜HsinβmH2)fNmNv, where the coupling between the scalars h0 and H0 and the WIMP candidate is denoted by g˜h and g˜H, respectively. In the explicit models we consider, we determine g˜h and g˜H and use the above formula when evaluating the spin-independent scattering cross section per nucleon, σSI0. Since we consider the possibility that our WIMP candidate forms only a fraction of the total dark matter abundance, we need to take this into account when comparing with the direct searches. The direct search constraints on σSI0 are given by the experimental collaborations under the assumption that frel=1. To apply the constraints under the assumption of subdominant WIMPs, we define an effective cross section(2.13)σSIeff=frelσSI0.With these preliminaries we now turn to the numerical results of two explicit models based on the scalar sector described by the potential (2.1).3Numerical resultsWe will now discuss two different concrete cases. First, assuming Z2 symmetry for the singlet scalar simplifies the potential and allows the singlet scalar to act as dark matter. However, it is impossible to saturate even a modest fraction of the observed dark matter abundance and simultaneously induce a strong first-order electroweak phase transition. Second, we release the requirement of Z2 symmetry on the singlet scalar but assume the existence of a singlet fermion. We find that in this case the scalar can induce a strong first-order electroweak transition, and the fermion can act as a dark matter candidate.3.1Scalar dark matter: Z2-symmetric caseA simple benchmark model, which has been considered extensively in literature earlier [7,21–23], is provided by imposing a Z2 symmetry on the potential in Eq. (2.1).Consider the Lagrangian(3.1)L=Lkin+LYuk−V(H,S)|Z2, where Lkin contains the kinetic terms for the scalars, appropriately gauged under the SM charges, and for all the SM gauge and matter fields. The term LYuk contains the usual Yukawa couplings between the SM matter fields and the Higgs field H. The potential V(H,S)|Z2 is obtained from Eq. (2.1) by setting μHS=μ3=0. Due to the Z2 symmetry, the singlet scalar in this model can act as a dark matter candidate. Depending on the model parameters, either the neutral component of H or the singlet can have a nonzero vacuum expectation value (vev). Since we assume the singlet to be a dark matter candidate, the T=0 vacuum has to be Z2 symmetric to ensure the stability of the dark matter candidate. This means that at zero temperature the vev, w, of S must be zero, i.e. the global minimum of the potential must be at (v,w)=(246 GeV,0). This requires, in addition to the bounds from vacuum stability in Eq. (2.3), that(3.2)μS2>−v2λHλS. In the notation of Eq. (2.10), the mass eigenstates are directly h0=ϕr and H0=S, i.e. cosβ=1. We trade the mass parameter μS2 with the physical mass of the singlet, mS2=μS2+v2λHS/2. Then the above constraint and the requirement of the stability can be combined into a bound(3.3)−2λHλS<λHS<2mS2v2+2λHλS.The known values of the electroweak scale, v=246 GeV, and the Higgs mass, mh=126 GeV, fix the model parameters λH=0.131 and μH2=−v2λH. The remaining parameters, mS2, λS and λHS are free but subject to the constraints listed above. We scan the parameter space by performing a simple Monte Carlo analysis generating a random distribution of points with(3.4)0<λS≤π,5GeV≤mS≤650 GeV.As a further constraint, we require perturbativity of the couplings reasonably far out in energy. The renormalization group (RG) equations are solved at one loop, and points which remain perturbative up to scales μ∼10 TeV are accepted. The RG equations for the gauge couplings and the Yukawa coupling of the top quark are as in the Standard Model. The β functions of the couplings are defined as(3.5)βg=dgdlnμ, and in the Z2-symmetric case they read(3.6)16π2βλH=24λH2+12λHS2−3(3g2+g′2−4yt2)λH+38(3g4+2g2g′2+g′4)−6yt4,(3.7)16π2βλHS=4λHS2+(12λH+6λS)λHS−3(32g2+12g′2−2yt2)λHS,(3.8)16π2βλS=18λS2+2λHS2.To determine how much of the dark matter relic abundance the singlet scalar S can provide for, we carry out the standard freeze-out calculation. There are three annihilation channels, SS→h0h0, VV and f¯f, where V denotes the electroweak gauge bosons, V=W,Z. The annihilation cross sections to these three distinct final states are(3.9)σhh=vh32πsvS|λHS+3mh2λHSs−mh2+imhΓh−4v2λHS2s−2mh2|2,σVV=vV4πsvSMV4λHS2|s−mh2+imhΓh|2(3+s(s−4MV2)4MV4)δV,σff=vfXf16πsvSmf2(s−4mf2)λHS2|s−mh2+imhΓh|2. Here vX=1−4mX2/s, δW,Z=1,1/2 and for QCD colour singlet fermions Xf=1 while for quarks(3.10)Xf=3(1+(32lnmq2s+94)4αs3π), where αs=0.12 is the strong coupling constant.22For the importance of this QCD correction, see Appendix A of [23].In the range mh/2<mS<mh, we factorize the annihilation to fermion and gauge boson final states to the SS→h0 fusion part times the virtual Higgs decay using the full width of the Higgs [40], which also takes the 4-body final states into account.To see if this model can provide for the strong electroweak transition, v(Tc)/Tc>1, we consider the finite-temperature corrections to the quadratic terms; the linear term in S does not arise in this Z2-symmetric case. At very high temperatures, the potential has a unique minimum at (S,H)=(0,0). As the temperature decreases, the potential generates two minima: one at nonzero S and the other at nonzero value of the neutral component of H. At some intermediate temperature these two become degenerate, and the possibility for a strong electroweak phase transition arises.To take into account the constraints from LHC, we consider the decay width of the Higgs to two singlets, h0→SS,(3.11)Γh0→SS=λHS2v232πmh1−4mS2mh2. The Higgs total decay width to the visible Standard Model channels is Γh=4.07 MeV for mh=126 GeV [40], and this implies a bound for the Higgs portal coupling.The basic result of the model is shown in the left panel of Fig. 1. The color coding shows the relative dark matter relic abundance. Above the solid black curve, λHS=2mS2/v2, μS2 is negative and modifications to the electroweak transition are possible. The figure illustrates that the parameter space of the model where (a reasonable fraction of) the relic density can be explained does not overlap with the parameter space where a strong first-order phase transition may arise.The points in the left panel of Fig. 1 have also been constrained to be compatible with the direct search results from the LUX experiment [26], which provides currently the best constraints for this type of model where only spin-independent cross section between the WIMP and ordinary matter arises. In the model we consider here, the effective couplings g˜h and g˜H in Eq. (2.12) are(3.12)g˜h=λHS2v,g˜H=0. Using these, we evaluate the cross section for elastic WIMP nucleon scattering and obtain(3.13)σSI0=14πλHS2μN2fN2mN2mh4mS2, where μN2 is the reduced mass of the WIMP–nucleon pair. To illustrate the distance of our results from the LUX bound, we show in the right panel of Fig. 1 the same points as in the left panel, with the color coding now showing the spin-independent cross section in relative to the LUX constraint.Of course the dark matter candidate need not be produced by the freeze-out mechanism as we have assumed so far. Since it is impossible to explain both the strong electroweak phase transition and the dark matter abundance with the scalar sector considered here, let us leave the phase transition for a while and focus only on dark matter. Given the fact that no trace of WIMPs has been observed in the direct searches, we consider briefly the possibility of freezing in the relic abundance [41]. Assuming that the scalar is light, it can be produced from the thermal bath of Higgs bosons. The number density of S is described by the Boltzmann equation formally similar to the one governing the density in the freeze-out case. Analogously to the well known approximate result in the freeze-out case, the dark matter abundance in this case is given by(3.14)ΩSh2≃1.09⋅1027gsgρmSΓh0→SSmh2, where gs,ρ denotes the effective degrees of freedom for the entropy and energy density, respectively. The essential feature of this mechanism is that the coupling required for the production of sufficient relic abundance is superweak; one obtains a parametric estimate(3.15)λHS≃10−12(ΩSh20.12)1/2(GeVmS)1/2.The result on the relic density in the case of freeze-in is shown in Fig. 2 as a function of the scalar mass, mS, and the portal coupling, λHS. The contours show the values of frel, defined in Eq. (2.9), as indicated explicitly in the figure.We do not consider the phenomenology of this freeze-in scenario in more detail here, but rather turn to a model where a strong electroweak phase transition can be obtained simultaneously with the observed dark matter relic density. Moreover, the relic density can be produced either via thermal freeze-out or out-of-equilibrium freeze-in scenarios.3.2Fermion dark matterIn this section we consider a model where the SM matter content is extended with the singlet sector containing a real scalar and a fermion [12,24,42]. The scalar sector is given by the full potential in Eq. (2.1), as we are not assuming Z2 symmetry for the singlet scalar. Rather, in this model the dark matter candidate is the singlet fermion, which enters through the Lagrangian(3.16)LDM=ψ¯(i∂̸−m)ψ+gSSψ¯ψ. We assume that the singlet fermion carries an exact global fermion number symmetry or that its mixing with SM neutrinos is otherwise forbidden. The main motivation for this model over the simple scalar case considered in previous section is that since the properties of the electroweak phase transition are affected by the scalar sector, and the dark matter relic density is determined by the properties of the singlet fermion, one can simultaneously explain both. As we will discuss in the following, the dark matter relic density can arise either via freeze-out or freeze-in mechanisms.At T=0, the potential (2.1) has several local extrema, which either break or conserve electroweak symmetry. There are altogether three symmetric extrema (v=0), two of which are minima. We denote one of these by S=w0. Then the two other extrema are at S=0 and S=−w0−μ3/λS. Since we do not assume Z2 symmetry for the singlet scalar, the electroweak broken minimum is at (v=246 GeV,w) where the vev of the singlet scalar, w, is not necessarily zero.We trade the parameters μH2, μS2 and μHS with v,w and w0 using the extremization conditions(3.17)μH2=−2λHv4+λHSv2w2+4w2(w−w0)(μ3+λS(w+w0))2v2,μS2=−w0(μ3+λSw0),μHS=−w(λHSv2+2(w−w0)(μ3+λS(w+w0)))v2. In the electroweak symmetric minimum, the requirement that the eigenvalues of the Hessian matrix have to be positive gives(3.18)w0(μ3+2λSw0)≥0,λHSv2+4w(μ3+λS(w+w0))≥2λHv4(w−w0)2.Moreover, we trade the parameters w, λH and m with physical masses mh=126 GeV, mH and mψ so that finally the free parameters are mψ,mH,λS,λHS,w0,μ3 and gS. We perform Monte Carlo scan of the parameter space with(3.19)0<λS<π,−2λHλS<λHS<π,−π<gS<π,−4000GeV<μ3<4000GeV,−2000GeV<w0<2000GeV,0GeV<mψ<800GeV,200GeV<mH<1400GeV. In addition to the constraints listed above, we check that the electroweak broken minimum is the deepest one at T=0 and require perturbativity up to scales μ∼1.5 TeV. The RG equations for the dimensionless couplings of the model are as follows. The β function of the Yukawa coupling, gS, is(3.20)16π2βgS=5gS3, while the β functions of the quartic couplings are(3.21)16π2βλH=24λH2+12λHS2−3(3g2+g′2−4yt2)λH+38(3g4+2g2g′2+g′4)−6yt4,(3.22)16π2βλHS=4λHS2+(12λH+6λS)λHS−3(32g2+12g′2−2yt2−43gS2)λHS,(3.23)16π2βλS=18λS2+2λHS2+8λSgS2−8gS4. Note that these reproduce the results for the Z2 symmetric scalar case considered in the previous section in the limit gS→0.Since we are not assuming Z2 symmetry for the singlet S, fields S and ϕr0 are not mass eigenstates. Rather the mass eigenstates are of the form (2.10), and the mixing pattern is constrained by the LHC data. As already discussed in Section 2, since the lighter mass eigenstate is identified with the mh=126 GeV boson observed at the LHC, the presence of the singlet component constrains the allowed values of the mixing angle. Due to the simple mixing pattern, all couplings of the lighter mass eigenstate are suppressed by cosβ relative to the couplings of the SM Higgs boson. We perform a global fit to the current data taking signal strengths from the ATLAS, CMS and Tevatron experiments [33–38]. We then allow only parameters which are within 2σ limit, cosβ>0.85, of the best fit value, cosβ=0.95. We also impose constraints from the precision electroweak measurements on the oblique corrections, i.e. S and T parameters, using formulae given in [43]. For the experimental input we use S=0.04±0.09 and T=0.07±0.08 with correlation of 0.88 from [44], and accept only points which are within the 2σ ellipsis around the central value quoted above. The constraint from the invisible width is similar as in the case of Z2 symmetric scalar. The essential difference is that now there are two channels contributing to the invisible width, h0→H0H0 and h0→ψψ,(3.24)Γh0→H0H0=λhHH232πmh1−4mH2mh2,Γh0→ψψ=gh2mh8π(1−4mψ2mh2)32, where gh=gSsinβ and the coupling λhHH is given by Eq. (A.3).To calculate the relic abundance of the singlet fermion, we again apply first the usual freeze-out formalism. The annihilation channels are similar to those of the Z2-symmetric scalar case, as the annihilation to SM fields can again proceed only via the scalar portal interactions. The cross sections are given by the fairly complicated and unilluminating expressions collected in Appendix A.The thermal corrections are given by Eqs. (2.4) and (2.5), with the additional contributions gS2/12 and −gSm/6 to cS and c1, respectively, arising from the coupling between the singlet fermion and scalar.33Note that the correction to c1 is proportional only to the m term in Eq. (3.16), and not to the part of the mass term arising from the interaction with S. In particular, if the fermion mass would be entirely due to the condensation of S, this contribution to c1 would not arise at all. Using the temperature dependent effective potential we again monitor the evolution of the electroweak symmetric and broken minima as T is increased from T=0. The critical temperature, Tc, is obtained from the condition that the symmetric minimum becomes deeper than the broken one. To check the strength of the phase transition, we evaluate v(Tc)/Tc.The results on the dark matter abundance are shown in Fig. 3 as a function of the mass of the WIMP candidate, mψ, and its coupling with the singlet scalar, gS. We see that a sizable fraction of the dark matter relic abundance can be realized with practically any mass larger than mh, but the viable values of gS are confined within narrow interval, gS≃O(0.1). On the other hand, the model now easily produces strong electroweak phase transition. This is shown in the inset of the left panel of Fig. 3, where the shaded lower (green) parts of the histogram correspond to points which yield frel>0.5.All the points shown in Fig. 3 are compatible with the direct search constraint from LUX experiment. To compare with the direct searches, we again evaluate the spin-independent WIMP–nucleon cross section. Now, the interaction between the WIMP and nucleons contains the factors(3.25)g˜h=gSsinβ,g˜H=gScosβ, which leads to(3.26)σSI0=μN2fN2mN2πv2gS2sin2βcos2β(1mh2−1mH2)2.A detailed account of the direct search limits from LUX experiment are shown in Fig. 4. The left panel shows the points from Fig. 3 in the (mψ,mH) plane, with the color coding now corresponding to the magnitude of the relative cross section σSIeff/σLUX. We see that the points furthest below the LUX bound are mostly concentrated around the region mH∼2mψ. The right panel of Fig. 4 shows the distribution of the results with respect to the values of frel and σSIeff/σLUX, with the color coding now corresponding to the values of gS. Here we see the result already observed in Fig. 3 that the large values of the relic density are confined to values of gS∼O(0.1) and that, in addition, the compatibility with the LUX bound is controlled by the mixing pattern in the scalar sector.To conclude this phenomenological analysis, let us again consider the dark matter relic density in the case of freeze in. Now the dark matter number density is produced from h0 and H0, which both contain the component of the singlet S which couples to the dark matter candidate ψ. In Fig. 5 we show the values of mass mψ and coupling gS for which Ωψh2=0.12 can be obtained. Of course, with such weak coupling the model in this part of the parameter space remains unconstrained by the dark matter direct search experiments. At all points shown in Fig. 5, also a strong electroweak phase transition can be realized. This is possible since the relic density is determined by the coupling gS while the properties of the phase transition are controlled by the couplings in the scalar sector. Similarly to the scalar case considered earlier, also here the characteristic feature of the mechanism is superweak coupling, Ωψh2∼10−12GeV/mψ, and the region where the mechanism can be applied is bounded by mψ=mH/2.3.3StabilityOne of the intriguing properties of the Standard Model with a light Higgs, mh=126 GeV, is that it is very near of being stable up to the Planck scale. In the SM the Higgs self-coupling becomes negative at one-loop level around energies μ≃108 GeV, but only so slightly that the theory remains in the region of metastability.44A higher-order analysis shows that this metastability scale is actually a little bit higher in the SM, around μ≃1011 GeV [45,46], but a one-loop analysis is sufficient for our purposes here. It is well known that extra singlets affect this situation, see e.g. [47]. For example, in our model we find that setting λS(mt)=gS(mt)=0, the Higgs self-coupling remains positive for λHS(mt)≳0.4 all the way up to the Planck scale. Currently there is some motivation for such stabilization from the measurement of the polarization of the cosmic microwave background by the BICEP-2 experiment [48]. The observation seems to be consistent with the tensor–scalar ratio r=0.16−0.05+0.06 which, if due to gravitational waves produced during inflation, sets the scale of the energy density to be very large, of the order of (1016GeV)4, during inflation. This makes the computations of metastability with respect to quantum tunneling futile, since such large energy densities allow the Higgs field to classically roll into the instability; see e.g. [49–51]. Possible solutions arising from modifying the inflationary dynamics were recently investigated in [52]. Of course another obvious solution arises if the beyond SM degrees of freedom force the scalar couplings to guarantee positivity of the potential at large field values up to the Planck scale. Let us, therefore, investigate the possibilities further in the model we have studied here.The results are shown in Fig. 6. The shaded regions in the lower right corner correspond to parameter domains where the Higgs self-coupling becomes negative and signal the vacuum instability. The shaded regions in the upper right corner show the parameter domains where some of the couplings become non-perturbative. Finally, the horizontal lines show the values where the coupling λS is driven negative. For simplicity, the results of Fig. 6 depict the Z2-symmetric case for which the Higgs self-coupling has the SM value at the electroweak scale, λH(mt)=0.128. The left panel of Fig. 6 shows the dependence on λS(mt) at value of gS(mt)=0.4. The figure shows, in particular, how sensitive the perturbativity of the model is on the value of λS: if perturbativity is to be required all the way to the Planck scale, we must have λS≲0.2. The right panel of Fig. 6 illustrates similarly the dependence on gS(mt): increasing gS(mt) above ca. 0.5 forces an instability on λS unless λHS becomes large, which in turn forces the theory into a non-perturbative domain.These results lead to a narrow range of couplings which correspond to a model which remains perturbative and stable up to the Planck scale. These ranges are λS(mt)≲0.2, gS(mt)≲0.6 and 0.35≲λHS≲0.55.A further effect on the stability is obtained by relaxing the Z2 symmetry of the potential. Although allowing trilinear terms in the scalar potential does not affect the β functions of the quartic couplings, it does change the vacuum structure thereby mixing the singlet and the doublet scalars at electroweak scale. This in turn changes the value of the coupling λH at the electroweak scale when compared with the SM value. We do a random scan over the parameter space to illustrate that within our model it is possible to obtain a considerable amount of the dark matter and simultaneously to have a strong electroweak phase transition with a stable potential all the way up to the Planck scale. As a result, we find a benchmark point with(3.27)λH(mt)=0.197,λS(mt)=0.053,λHS(mt)=0.376,gS(mt)=0.247, for which frel=0.94 and v(Tc)/Tc=1.3.Finally, let us discuss possible implications for the LHC. The above benchmark point serves as a concrete example. At this point the singlet fermion mass is mψ=679 GeV and the heavy scalar mass is mH=371 GeV. Both of these are above mh/2 so the hidden sector will remain invisible with zero branching fraction. However, even if the new degrees of freedom are not expected to manifest at LHC, the model predicts deviations with respect to SM. As already mentioned above, the mixing of the singlet and doublet scalars affects the coupling λH, and this is directly connected with the scalar potential. In the SM the Higgs self-coupling has value λSM(mt)=0.128, while for example for the parameter space point in Eq. (3.27) we have λH(mt)=0.197=1.5λSM(mt).To search for deviations from the SM all the Higgs couplings should be measured and compared with the predicted SM values. The cubic coupling of three physical Higgs bosons, λhhh, could be measured at LHC in the production of two Higgs bosons, see e.g. [53]. For the benchmark point, Eq. (3.27), we have λhhh/λhhhSM=2.00. Generally, the dependence of λhhh/λhhhSM on mS and singlet-doublet mixing cosβ within the allowed 2σ range in the model is shown in the left panel of Fig. 7. We find that sizable deviations from the SM value are indeed expected. The growth of the deviation can be traced to the dimensionful couplings, in particular to μ3, whose values can be large in comparison to the electroweak scale, v. On the other hand, the large intrinsic scales will also affect the S and T parameters through the mixing of singlet and doublet scalars. This is illustrated by the right panel of Fig. 7, which shows the trilinear Higgs coupling with respect to the deviation, Δ, from the best fit value of the oblique parameters S and T, the value Δ=1 corresponding to the 2σ limit. All our data points are naturally within this constraint by construction.4ConclusionsIn this paper we have considered simple singlet extensions of the SM motivated by the possibility to explain cosmological paradigms of dark matter and electroweak baryogenesis. For the latter we have focused on the possibility to realize a strong electroweak phase transition which is a necessary general condition for any particular realization of creating the matter–antimatter asymmetry.We considered two concrete realizations. First we studied a model where the dark matter is a singlet scalar protected by a discrete Z2 symmetry in the zero temperature vacuum. Second, we considered a model where the dark matter candidate is a singlet fermion coupled with the SM via a singlet scalar. In this case the scalar potential can be arranged to lead to strong electroweak phase transition while the singlet fermion saturates the observed dark matter abundance. In this case, the dark matter abundance can be produced either via freeze-out or freeze-in mechanisms.Between these two models, the main difference which is relevant for the phenomenological aspects of dark matter and strong first-order phase transition is that in the Z2-symmetric-scalar case both aspects arise from the same field, whereas in the fermion case the dark matter abundance is due to the fermion while the modifications to the electroweak transition are entirely due to the portal scalar. One can also imagine alternative realizations based on similar dichotomy. For example, one could consider two different scalar fields. Imposing a discrete symmetry on only one of them would make the field a plausible origin of dark matter similar to the Z2-symmetric case we considered here, while the other field would provide the necessary modifications to strengthen the electroweak phase transition.We also studied the implications of the RG equations on the stability and perturbativity of the model. We found that due to the singlet scalar, the Higgs self-coupling can remain positive at all energies below the Planck scale. With suitable values of the portal coupling, λHS, the self-coupling of the singlet scalar, λS, and the scalar Yukawa coupling, gS, the entire model was found to remain perturbative and stable all the way up to the Planck scale.A possible experimental access to the singlet sector is provided via the couplings of the Higgs boson, in particular the trilinear coupling of three physical Higgs bosons. If measured at LHC, and deviations from the corresponding value in the SM are found, this can provide, if no new states are directly observed at the LHC experiments, important clues about a possible hidden sector akin to the models studied in this paper.AcknowledgementsThis work has been financially supported by Finnish Cultural Foundation and by the Academy of Finland project 267842.Appendix ACross sectionsHere we give the formulae for the computation of the annihilation cross section for the model considered in Section 3.2. To make the equations more concise, it is useful to define the couplings(A.1)λhhh=−6λHvcos3β−3λHSvsin2βcosβ−2sin3β(μ3+3λSw)−3sinβcos2β(μH+λHSw),(A.2)λhhH=−2vsinβcos2β(λHS−3λH)+λHSvsin3β+sinβsin(2β)(−μ3+μH−3λSw+λHSw)−cos3β(μH+λHSw),(A.3)λhHH=vsinβsin(2β)(λHS−3λH)−λHSvcos3β−2sinβcos2β(μ3−μH+3λSw−λHSw)−sin3β(μH+λHSw),(A.4)λHHH=6vsin2βcosβ(λHS−3λH)−3λHSvcos3β−6sinβcos2β(μ3−μH+3λSw−λHSw)−3sin3β(μH+λHSw), and(A.5)gh=gSsinβ,gH=gScosβ,Yh=mfvcosβ,YH=−mfvsinβ,ghZ=v(g2+g′2)2cosβ,ghW=vg22cosβ,gHZ=−v(g2+g′2)2sinβ,gHW=−vg22sinβ.The squared amplitude, averaged over the initial states and summed over the final states, for the dark matter, ψ, annihilating to two scalars, hi,hj=h0,H0, is given by(A.6)|T(ψψ¯→hihj)|2=(1−δij2)(2(s−4mψ2)|∑k=h,Hgkλijks−mk2+imkΓk|2+2gi2gj2(tu−mψ2(t+u)+mψ4−mi2mj2)(1t−mψ2−1u−mψ2)2+4mψ(t−u)(1t−mψ2−1u−mψ2)∑k=h,Hgigjgkλijk(s−mk2)|s−mk2+imkΓk|2). Similarly the squared amplitudes, averaged over the initial states and summed over the final states, to fermion and gauge boson final states are(A.7)|T(ψψ¯→ff)|2=s2vψ2vf2Xf|∑k=h,HgkYks−mk2+imkΓk|2, and(A.8)|T(ψψ¯→VV)|2=2svψ2(3−sMV2+s24MV4)|∑k=h,HgkgkVs−mk2+imkΓk|2δV. In the expression for the fermion channel the factor Xf=1 for leptons in the final state, while for quarks(A.9)Xf=3(1+(32lnmq2s+94)4αs3π), where αs=0.12 is the strong coupling constant. Similarly as in the scalar case [23], also for fermion dark matter the region where this correction is important is ruled out by the invisible width of the Higgs.Then, the cross section for the process ψψ¯→ij reads(A.10)σ(ψψ→ij)=116πs2vψ2∫t−(ij)t+(ij)dt|T(ψψ→ij)|2, where(A.11)t±(ij)=mψ2+12(mi2+mj2−s)+12(1−4mψ2s)(s2+(mi2−mj2)2−2s(mi2+mj2)).References[1]D.CarmiA.FalkowskiE.KuflikT.VolanskyInterpreting LHC Higgs results from natural new physics perspectiveJ. 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