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2HDM with Z_2 symmetry in light of new LHC data

Properties of the Z_2-symmetric Two Higgs Doublet Models (2HDM) are discussed and confronted with new LHC data for a 125 GeV Higgs particle. The particle discovered at LHC in 2012 has properties expected for it in the Standard Model (SM), with a possible enhancement in the two-photon channel. SM-like Higgs scenarios can be realized in the Two Higgs Doublet Models with Z_2 (D) symmetry within the normal Mixed Model (with scalar sector as in MSSM) and the Inert Doublet Model (IDM), where a good Dark Matter (DM) candidate is present. Here we discuss both of the models.


Introduction
In the Two Higgs Doublet Models (2HDM) there are two doublets of SU (2), with the weak hypercharge Y=1. They give masses to W , Z (leading at the tree-level to ρ =1) and in principle also to the photon. Fermion masses are generated via Yukawa interactions, for which various models are considered: Model I, II, III, IV, X, Y,... [1]. Five scalars appear in these models, two charged H + and H − and three neutral ones. If CP is conserved there are two CP-even h, H and one CP-odd A particle. In the model with CP violation three neutral particles h 1 , h 2 , h 3 with undefined CP parity appear.

D-symmetric 2HDM
Study of the symmetry properties of the Lagrangian as well as of the vacuum states is crucial for understanding a real content of the theory. Here we assume the Z 2 symmetry of the potential φ S → φ S , φ D → −φ D , which we call below the D symmetry. 1 In the Mixed Model both doublets have non-zero vacuum expectation values (vevs) and are involved in the mass generation. There are five Higgs bosons and sum rules hold for the relative couplings of the neutral Higgs particles h, H, A: e.g.
Model II is assumed for the Yukawa interactions: doublet φ S couples to the down-type quarks and charged leptons, while φ D couples to the up-type quarks. D symmetry is spontaneously violated. In this model SM-like scenarios are possible for both h and H, with χ h/H V V = 1. In contrast, in the Inert Doublet Model (IDM), only one doublet (φ S ) is involved in the mass generation and there is only one SM-like Higgs boson h. The second doublet is inert (it has vev=0) and contains four scalars. Yukawa interactions are as in Model I, so the D symmetry is exact here and the neutral scalar H (or A) may play a role of the Dark Matter (DM).
The D-symmetric potential has the following form: with all parameters real. We take λ 5 < 0 without loss of generality [2]. Such potential has various possible extrema (vacua), with the following vacuum expectation values: Neutral vacua are realized for u = 0. The charged vacuum with u = 0 corresponds to breaking of U(1) QED symmetry and the appearance of the massive photon. The list of possible extrema, which can be realized as local or global minima (i.e. vacua) is given in table 1. The EW s case corresponds to the EW symmetry, i.e. lack of spontaneous breaking of EW symmetry. Existence of a stable vacuum requires [3]: Various vacua can be realized for various values of vev, which can be represented in the phase diagram (µ 1 , µ 2 ), where There are three regimes of parameter R which correspond to very different phase patterns shown in fig. 1.
In principle a model for today's Universe could be based either on the Mixed or the Inert vacuum. From theory side we assume that the considered vacua are stable and parameters of V are constrained by the perturbative unitarity: |Λ i | < 8π [4][5][6], where Λ i are the eigenvalues of the high-energy scattering matrix of the scalar sector. This leads to the upper limits on λ 1,2 In any case a condition for the existence of the particular vacuum has to be fulfilled. Existence of the Mixed vacuum is equivalent to having positive scalars' masses squared (condition for the minimum) while for the Inert vacuum this is not enough since a coexistence of Inert and Inertlike minima is possible (see fig. 1). For Inert to be the global minimum a following condition has to be fulfilled in addition [2]:

Experimental constraints
We consider both Mixed Model and IDM, taking into account also the following experimental constraints: Electroweak Precision Tests (EWPT) Values of S and T parameters are demanded to lie within 2σ ellipses in the S, T plane, with the following central values [7]: S = 0.03 ± 0.09, T = 0.07 ± 0.08, with correlation equal to 87%.
LEP We apply a model-independent limit: M H ± > 70 GeV from the direct LEP measurements. For Mixed Model the lower limit for mass of H ± is 360 GeV from b → sγ NLO analysis [8].
For the IDM we use the LEPI and LEPII bounds on the scalar masses [9,10]: of h equal to 125 GeV, an important limit on tan β = v D /v S is obtained, namely 0.18 < tan β < 5.59 (independently of sin(β − α)!). Notice, that it is possible to have SM-like H, however h needs then to have a very suppressed coupling to gauge boson, see eg. [11][12][13].
Even when all direct decay widths of h are as in SM, the loop decay widths γγ, Zγ for h may still be modified due to H ± contribution, or/and the negative sign of χ ff h coupling [14]. The H ± loop contribution leads to a 10 % (5%) suppression with respect to the SM for γγ (Zγ). The change of sign of the tth coupling has a strong influence on the decay widths γγ and Zγ by changing the destructive interference between the t and W contributions to the loop couplings present in the SM into the constructive one. The enhancement with respect to SM up to 2.3 (1.2) is possible for Γ γγ (Γ Zγ ). For the hgg coupling change of the relative sign of the b and t contributions leads to a enhancement up to 1.25.

Results for IDM with a 125 GeV Higgs
The Universe is described by the IDM if the vacuum state is given by I 1 . IDM predicts the existence of four dark scalars H, A, H ± and the SM-like Higgs particle h (we assume its mass equal to 125 GeV). λ 345 is related to a triple and quartic coupling between SM-like Higgs h and DM candidate H. λ 2 gives the quartic DM self-couplings, while λ 3 describes the Higgs particle interaction with charged scalars.
This estimation of Ω DM h 2 may be used to constrain the λ 345 coupling depending on the chosen values of masses of H and other scalars [18,20]. Ω DM h 2 does not limit the λ 2 parameter, although indirect constraints come from its relation to λ 345 parameter through the vacuum stability constraints and existence of I 1 vacuum [25,26].
Limits for M H and λ 345 coming from Ω DM h 2 are presented in fig. 2. For masses M H 72 GeV the allowed region (dark gray) of λ 345 is symmetric around zero with small values of λ 345 excluded (light gray region) due to a nonefficient DM annhilation. As the mass increases, the region of proper relic density shifts towards the negative values of λ 345 , which is due to opening of the annihilation channels into the gauge bosons final state. As it will be shown below, in the IDM R γγ > 1 is possible for λ 3 < 0. If we consider H as a DM candidate then λ 345 < 0 for R γγ > 1, meaning that it is possible to fulfill the LHC and relic density constraints for IDM.

R γγ
Here we concentrate on the two-photon decay rate of the Higgs boson, which in the IDM reads: A deviation from the value of R γγ = 1 may be caused in IDM by two factors. Firstly, the partial decay width Γ(h → γγ) IDM is modified due to the existence of the charged scalar loop [28][29][30][31][32]: where M SM is the contribution from the SM and δM IDM = The charged scalar loop can interfere either constructively or destructively with the SM contribution. Secondly, the total decay width Γ IDM (h) can be increased with respect to the SM case due to the existence of the invisible decays: h → HH and h → AA.
Performing a random scan of the parameter space, we found the regions where R γγ >1, with the maximal value of R γγ around 3.4. Fig. 3a shows values of R γγ allowed by the theoretical and experimental constraints as a function of M H . It can be seen that enhanced values of R γγ are not possible for M H < M h /2. It means that if the invisible channels are open, the total decay width is so big, that it suppresses other effects. (b) Region allowed by the constraints in the (m 2 22 , M H ± ) plane. The curves correspond to the fixed values of Rγγ (for the invisible channels closed). Figure 3: Results on R γγ for IDM, points with R γγ < 1 (R γγ > 1) are displayed in dark green/gray (light green/gray). Fig. 3b shows the region allowed by the theoretical and experimental constraints in the (m 2 22 , M H ± ) plane together with the curves corresponding to constant values of R γγ (calculated for the case with invisible decay channels closed). It can be seen that the enhancement is possible only for constrained m 2 22 region, namely: m 2 22 < −9.8 · 10 3 GeV 2 , which is equivalent to the bound λ 3 < 0 (in agreement with Ref. [32]). On the contrary, R γγ > 1 can be achieved for any value of M H ± . However, if bigger value of R γγ is demanded, then allowed values of M H ± are constrained. For example, for R γγ > 1. 6. Evolution of the Universe in 2HDM through different vacua in the past We consider 2HDM with an explicit D symmetry and assume that today the IDM describes reality. In the simplest approximation λ i terms in the potential are fixed and only mass terms vary with temperature: Here g, g are EW gauge couplings, while g t , g b are the SM Yukawa couplings) [2].
As the Universe cools down, the quadratic coefficients vary with T and the ground state of the potential V may change. Various types of evolution of the Universe from EWs phase into I 1 phase can be realized: in one (EWs → I 1 ), two (EWs → I 2 → I 1 ) or three (EWs → I 2 → M → I 1 ) steps. In general, in T 2 approximation phase transitions are of the 2nd order. The only exception is the transition between two degenerate minima I 2 and I 1 in the sequence with two phase transitions. This scenario can be realized only if R > 1. Notice, that if the Universe undergoes a series of phase transitions dark matter may appear later during the evolution, as it exists only in the Inert phase.
If R < 0 there is only one type of sequence that corresponds to the restoration of EW symmetry in the past ( fig. 4a): EWs → I 1 . This sequence can be realized when R γγ > 1, which is suggested by the recent LHC data. In the other scenarios for R < 0 the initial state of the Universe is the one with broken EW symmetry ( fig. 4b). The restoration of EW symmetry may be temporary (scenario Y), in other cases the EW symmetric state never existed.
For a certain parameter range there is a possibility of having a charged vacuum in the past ( fig.  4c). This scenario can be realized only if today we have IDM with the charged DM particle [2]. Current model independent bounds require that charged DM is heavier than 100 TeV [33]. This can be achieved without breaking the perturbative unitarity conditions with large m 2 22 . However, the sequence Z + requires that c 2 < 0 and |c 2 |/c 1 > |m 2 22 |/m 2 11 (10 5 ÷ 10 6 ). This contradicts the requirement c 1 > −c 2 based on the positivity condition [2]. This means that during the evolution Universe cannot pass through the U (1) QED breaking phase.

Summary
The 2HDM is an excellent laboratory of the beyond SM physics. Although the discovery of the 125 GeV Higgs particle is in agreement with the Standard Model prediction, many extensions of the SM have some build-in SM-like scenarios. Here we consider two very different explicit Z 2symmetric extensions with two SU(2) doublets -the very common Mixed Model and the Inert Doublet Model. Both can correspond to the SM tree-level decay modes, at the same time both may lead to the enhancement in the two-photon decay channel for the 125 GeV Higgs boson.