{'fraction_non_alphanumeric': 0.038222372423115915, 'fraction_numerical': 0.020074349442379184, 'mean_word_length': 4.109825591434985, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Natural Two-Higgs-Doublet Model

We discuss phenomenological consequences of requiring the cancellation of one-loop quadratic divergences in corrections to scalar masses within Two-Higgs-Doublet Models (2HDM). It is shown that the Inert Doublet Model (IDM) can not accommodate the requirement of the cancellation. A general 2HDM is also analyzed taking into account existing experimental constraints. Allowed regions in the parameter space, permitting the above cancellation, are determined. A degeneracy between masses of scalar bosons is observed for tanbeta \gsim 40. The possibility for CP violation in the scalar potential is discussed and regions of tanbeta-M_{H^\pm} with substantial amount of CP violation are determined. In order to provide a source for dark matter in a minimal manner, a scalar gauge singlet is introduced and discussed. The model allows to ameliorate the little hierarchy problem by lifting the minimal scalar Higgs boson mass and by suppressing the one-loop quadratic corrections to scalar masses. The cutoff originating from the naturality arguments is therefore lifted to \gsim 15 TeV depending on the mass of the lightest scalar.


I. INTRODUCTION
The goal of this work is to extend the Standard Model (SM) such that there would be no quadratic divergences to scalar masses up to the leading order at the two-loop level of the perturbation expansion. The quadratic divergences were first studied within the SM by Veltman [1], who showed that applying dimensional reduction [2] one gets the following quadratically divergent one-loop correction to the Higgs boson (h) mass where Λ is a UV cutoff and v ≃ 246 GeV denotes the vacuum expectation value of the scalar doublet. The issue of quadratic divergences was then investigated further adopting other regularization schemes (e.g. point splitting [3]) and also in [4] without reference to any regularization scheme. Since precision measurements require a light Higgs boson the correction (1) exceeds the mass itself even for small values of Λ, e.g. for m h = 130 GeV we obtain δ (SM) m 2 h ≃ m 2 h already for Λ ≃ 580 GeV. On the other hand, if we assume that the scale of new physics is widely separated from the electro-weak scale, then constraints that emerge from analysis of operators of dimension 6 require Λ ∼ > a few TeV. The lesson from this observation is that whatever is beyond the SM physics, some amount of fine tuning is necessary; either we tune to lift the cutoff above Λ ≃ 580 GeV, or we tune when precision observables measured at LEP are fitted. 1 Tuning both in corrections to the Higgs mass and in LEP physics is, of course, also a viable alternative which we are going to explore below. So, we will look for new physics in the TeV range which will allow to lift the cutoff implied by quadratic corrections to m 2 h to the multi-TeV range and which will be consistent with all the experimental constraints-both require some amount of tuning. Within the SM the requirement δ (SM) m 2 h = 0 implies m h ≃ 310 GeV. However, as is very well know, the present data favor a light SM Higgs boson -according to the PDG [5], after including all the available experimental data and taking into account theoretical uncertainties, the 99% CL upper limit for the Higgs mass reads: m h ≤ 194 GeV. Therefore, within the SM the one-loop condition δ (SM) m 2 h = 0 requires an unrealistic value of the Higgs boson mass. Examining closer the experimental constraints one finds also the following tension which emerges in the process of fitting all the available data to the SM (see [6] for a recent review). Hadronic asymmetry measurements (A c FB , A b FB , Q FB ) favour a heavy Higgs boson, with m h ∼ 500 GeV, while leptonic asymmetries (A LR , A l FB ) together with non-asymmetry precision measurements (m W , Γ Z , . . . ) favour a Higgs mass smaller by one order of magnitude. If (A c FB , A b FB , Q FB ) are omitted from the fit one obtains m h ∼ 50 GeV with an upper limit m h < 105 GeV at the 95% CL [6]. Moreover there is the LEP lower limit on the Higgs mass, m h > 114.4 GeV [7]. The fit which combines all the data is therefore of low quality. That observation suggests a modification of the SM which would allow for a heavy Higgs boson with a mass at least above the LEP limit. For that the SM prediction for the oblique parameters S and T must be modified by the extension of the SM that we are seeking.
Here we are going to construct a model which would both soften the little hierarchy problem by suppressing δ (SM) m 2 h and which would allow to lift the central value for the Higgs mass up to a value which is well above the LEP limit (presumably it would imply a better fit of the precision observables). We would like to point out also that increasing the Higgs boson mass would ameliorate the little hierarchy problem even if δ (SM) m 2 h was not suppressed (since then larger cutoff would lead to the correction of the order of the mass itself).
There are only two ways to suppress δ (SM) m 2 h /m 2 h : one can either modify the SM such that (i) larger SM-like Higgs boson mass is allowed; or (ii) extra radiative corrections to δ (SM) m 2 h emerge that partially cancel (1). The best-studied example of the second approach is provided by supersymmetric theories for which δm 2 h ≪ m 2 h up to the GUT scale, however the suppression of δ (SM) m 2 h could also be achieved through very modest means, e.g. by introducing just extra real scalar singlets to the SM [8] (although more tuning than in the supersymmetric case is necessary). The first strategy was followed in [9] within the so-called inert doublet model 2 (IDM). There, a second Higgs doublet was introduced and an exact 2 symmetry was imposed to provide a dark-matter candidate. As shown in [9] a large (400 − 600 GeV) SM-like Higgs boson mass was allowed by the addition of an extra Higgs doublet (the inert doublet). It was demonstrated that the extra contributions to the oblique parameters originating from the inert doublet (with physical fields H ± , A and S) can cancel large effects of a heavy SM Higgs, such that m h ∼ 400 − 600 GeV is allowed.
Here we propose a model which does both, i.e. suppression of δ (SM) m 2 h by contributions from some extra states (which implies reduced δ (SM) m 2 h /m 2 h ) and which modifies the results of the global fit such that much heavier SM Higgs boson is allowed (that also helps to decrease δ (SM) m 2 h /m 2 h and in addition it could eliminate the tension caused by the high LEP lower bound in the presence of a low central value from precision tests). A different approach to this problem has been proposed in [12].
Another well-known problem of the SM is the strength of CP violation (CPV) which is too weak to make the electroweak baryogenesis viable [13]. It is also worth noting a too slow phase transition (within the SM) which is another difficulty for realistic baryogenesis [13].
In light of the above remarks it seems very natural to consider simple extensions of the SM scalar sector, as for instance the Two-Higgs-Doublet Model (2HDM). Our intention is to bring the reader's attention to a region of parameter space that not only is consistent with standard theoretical requirements (positivity and unitarity) and satisfies all the relevant experimental constraints, but also offers a simple pragmatic option to reduce the size of the quadratic corrections to scalar two-point Green's functions (so in other words to scalar masses). It has been noticed a long time ago [14] that within the 2HDM one can cancel quadratically divergent corrections to two-point Green's functions for scalar particles. Some phenomenological consequences of the cancellation were discussed already in [15]. It is well known [16] that within 2HDM extensions the oblique parameters S, T and U can be modified such that SM contributions growing with m h (∝ ln m h ) could be canceled by other terms (originating from extra scalars present in the 2HDM), so that the lightest Higgs boson could be relatively heavy, see [9]. Within the 2HDM the electroweak phase transition could also be made fast enough [17] to make electroweak baryogenesis viable. The 2HDM provides also new sources of CP violation in interactions of neutral scalars. Therefore, here we will discuss 2HDMs which do not suffer from quadratic divergences in scalar two-point Green's functions, the tempered Two-Higgs-Doublet Model, seeking a model which also allows for CP violation in the scalar potential. Since we have argued above that the heavy SM-like Higgs boson would be more consistent with experimental data, we will investigate how much CP violation in 2HDMs is allowed after lifting the Higgs mass well above the LEP lower limit and by imposing the conditions needed to cancel quadratic divergences (so as to ameliorate the little hierarchy problem). We will not address here the issue of the electroweak phase transition.
In a recent publication [18], motivated by similar arguments, we have considered a version of the IDM with CP violation introduced by replacing the SM-like Higgs doublet by a pair of doublets. There, a candidate for dark matter (DM) was provided by the lightest neutral component of the inert doublet (as in the original IDM). In the model considered here, CP violation again originates from the 2HDM, however in order to accommodate a DM candidate in a minimal manner (instead of introducing the inert doublet as in [18]) we extend the model by a real singlet. 3 In fact, it is intriguing to note that the singlet is even more inert than the original inert doublet since it interacts only with the Higgs doublets and with right-handed neutrinos, having no gauge interactions.
The paper is organized as follows. In Sec. (II) we investigate theoretical and phenomenological consequences of the cancellation conditions within the general 2HDM. In order to accommodate a DM candidate we introduce an extra real scalar gauge singlet, that is discussed in Sec. III. Section IV contains our summary.

II. NON-INERT TWO-HIGGS-DOUBLET MODEL
A very appealing possibility would be to combine the IDM with the idea of canceling the one-loop quadratic divergences. However, as we have shown in [20], that is impossible because ot the vacuum stability conditions in the IDM are inconsistent with the requirement of cancellation of quadratic divergences. Since our intention is to build a 2HDM, which has no quadratic divergences at least at the one-loop level therefore, in the following we will consider a general (non-inert) 2HDM hoping for both a successful implementation of the cancellation condition and for a new source of CP violation. The price to pay will be the loss of a DM candidate. We return to that issue in Sec. III.

A. One-loop quadratic divergences
The cancellation of one-loop quadratic divergences for the scalar two-point Green's functions at zero external momenta (G i , i = 1, 2) implies [14] in the case of 2HDM type II: where v 2 ≡ v 2 1 + v 2 2 , tan β ≡ v 2 /v 1 and we adopt the notation: s β ≡ sin β and c β ≡ cos β. We note that when tan β is large, the two quark contributions can be comparable. In the type II model the mixed, φ 1 − φ 2 , Green's function is not quadratically divergent.
In the general CP-violating case, the quartic couplings λ i can be expressed in terms of the mass parameters and elements of the rotation matrix needed for diagonalization of the scalar masses (see, for example, Eqs. (3.1)-(3.5) of [21]): Re cos α i and s i = sin α i refer to the neutral-Higgs-sector rotation matrix R, the latter parametrized in terms of the angles α 1 , α 2 and α 3 according to the convention of [22].
It will be useful to adopt the following relation (emerging from the diagonalization of the neutral Higgs mass matrix [23]) between M 2 1 , M 2 2 and M 2 3 : Substitution of (12) into (6)-(11) allows to express the quartic couplings through the mixing angles together with . Then, inserting the appropriate quartic couplings into the conditions for cancellation of the quadratic divergences (4)-(5) we obtain two linear equations for M 2 1 and M 2 2 with coefficients depending on the mixing angles α i (as well as on M 2 H ± and µ 2 ). Therefore, for a given choice of α i 's, the squared neutral-Higgs masses M 2 1 , M 2 2 and M 2 3 can be determined from the cancellation conditions (4)-(5) in terms of tan β, µ 2 and M 2 H ± . Scalar masses resulting from a scan over α i , M H ± and tan β are shown in Fig. 1 for µ = 200 GeV and µ = 500 GeV. The charged Higgs boson mass was varied between 300 GeV and 700 GeV. Since only large tan β will turn out to be allowed we have chosen to display plots with 40 ≤ tan β ≤ 50 in order to illustrate a specific property of the scalar spectrum that is visible at large tan β. Under the scan, the M 2 i were calculated along the lines described above. The only extra constraints (the cancellation of quadratic divergences was, of course, guaranteed implicitly by the construction) imposed were M 2 i > 0 and M 1 ≤ M 2 ≤ M 3 . A striking degeneracy of the neutral-Higgs masses is observed for the case of large tan β. This degeneracy can be understood by expanding M 2 i for large tan β. The cancellation conditions, Eqs. (4)-(5) can then be expressed as follows: where 4 First, it is useful to notice that (14) implies Therefore, for tan β ≫ 1, we expect to have M 2 1 ≃ M 2 2 unless |R 12 R 22 | ≪ 1 or −2M 2 H ± + µ 2 is very large. 5 Secondly, solving (13)- (14) one finds that to leading order (for large tan β) Note that cancellations between the M 2 H ± and µ 2 terms are possible. This is why the degeneracy survives even for µ as large as µ = 500 GeV, see Fig. 1. On the other hand, expanding (12) for tan β ≫ 1 one obtains: Therefore (invoking unitarity of R) it is seen that the degeneracy Finally we can conclude that for large tan β one obtains M 1 ≃ M 2 ≃ M 3 ≃ µ 2 + 4m 2 b , this explains the approximate degeneracy observed in Fig. 1. 6

B. Two-loop leading quadratic divergences
The generic form of the quadratically divergent contributions to scalar two-point Green's functions at zero external momenta reads [4] where n corresponds to (n + 1)-loop contribution, λ stands for relevant coupling constants,μ is the renormalization scale and f (i) n (λ) is a calculable (order by order) function (polynomial) of the couplings. It should be noticed that at the (n + 1)-loop level there exist also sub-leading contributions that contain terms ∝ Λ 2 (ln Λ) m with m < n, so for instance at the two-loop level the leading contribution is ∝ Λ 2 ln Λ while there are also sub-leading terms ∝ Λ 2 . The coefficients of the leading terms, f (i) n (λ), can be determined recursively adopting a nice algorithm noticed by Einhorn and Jones [4]: where the sum runs over coupling constants that contribute to the coefficient f (i) n . Hereafter we will limit ourselves to the leading two-loop contributions. Therefore, to calculate f and one-loop beta functions are needed. As beta functions for the 2HDM are known [24] the cancellation condition for quadratic divergences up to the leading two-loop order can easily be determined: with In what follows, adopting (6)-(11) we will be solving the conditions (20) for the scalar masses M 2 i for a given set of α i 's, tan β, µ 2 and M 2 H ± . For the renormalization scale we will adopt v, soμ = v. 7 Then those masses together with the corresponding coupling constants, will be adopted to find predictions of the model for various observables which can be confronted with experiments.

C. Positivity and unitarity constraints
The requirements of positivity for the 2HDM model potential are well known [10]: One could also require the above conditions to be satisfied up the unification scale Λ ∼ 10 15 GeV. That approach, resulting in much stronger constraints in terms of allowed scalar masses and tan β was followed in Ref. [11]. Here, as we consider UV completion appearing at the scale of a few TeV we do not follow that line of reasoning.

D. Experimental constraints
We impose the following experimental constraints: • The oblique parameters T and S • The muon anomalous magnetic moment • Electron electric dipole moment For more details concerning the implementation of the experimental constraints, see refs. [18,21,30]. Subject to all these constraints, we find allowed solutions of (20). The recent paper [26] also contains an exhaustive analysis of experimental constraints on the 2HDM type II. The lower limit on the charged Higgs-boson mass adopted here, M ∓ ≥ 300 GeV (basically determined by the b → sγ constraint) agrees roughly with the 95% CL lower limit, 316 GeV, obtained in [26] irrespectively of tan β.

E. Allowed regions
Imposing the above conditions we find allowed regions in the tan β-M H ± plane as illustrated by the red domains in the tan β-M H ± plane, see Figs. 2 and 3 for fixed values of µ. The allowed regions were obtained scanning over the mixing angles α i and solving the two-loop cancellation conditions (20). Imposing also unitarity in the Higgs-Higgs-scattering sector [27][28][29] (yellow regions), the allowed regions are only slightly reduced. Requiring that also experimental constraints listed in the Sec. II D are satisfied one obtains green regions shown in the figure.
For parameters that are consistent with unitarity, positivity, experimental constraints and the two-loop cancellation conditions (20), we show in Figs. 4-5 scalar masses resulting from a scan over α i , M H ± and tan β. Those plots could be compared with Fig. 1. One should however remember that in the two-loop case also unitarity, positivity and experimental constraints are taken into account. Note that in Figs. 2 and 3, consistent solutions are obtained only for tan β ∼ > 15 while at the one-loop level, also a small low-tan β region was allowed after imposing all the constraints. That small low-tan β region is disallowed after the two-loop corrections are imposed, see [20] for the one-loop result. As we have noticed for the one-loop spectrum, large tan β implies similar scalar masses. This is indeed what is being observed in Figs. 4-5 also for the two-loop case. The allowed solutions "peak" around M H ± ∼ µ with 20 ∼ < tan β ∼ < 50. For µ = 200 and 600 GeV there are hardly any solutions for Λ = 2.5 TeV and no solutions were found for Λ = 6.5 TeV.

F. CP violation
Here we are going to discuss the possibility of having CP violation in the scalar potential (2), subject to the twoloop cancellation of quadratic divergences (20). In order to parametrize the magnitude of CP violation we adopt the rephasing invariants introduced by Lavoura and Silva [31] (see also [32]). We shall here use the basis-invariant formulation of these invariants J 1 , J 2 and J 3 as proposed by Gunion and Haber [33]. As is proven there (theorem #4) the Higgs sector is CP-conserving if and only if all J i are real. In the basis adopted here the invariants read [18]: It is seen that there is no CP violation when Im λ 5 = 0, see [18] for more details.
As we have noted earlier, tan β above ∼ 40 implies approximate degeneracy of scalar masses. That could be catastrophic for CP violation since it is well known that the exact degeneracy M 1 = M 2 = M 3 results in vanishing invariants Im J i and no CP violation (exact degeneracy implies Im λ 5 = 0). Using the one-loop conditions (4)-(5) one immediately finds that In fact the above result shows even more than we have anticipated. If tan β is large then Im J 1 is suppressed not only by Im λ 5 ≃ 0 (as caused by M 1 ≃ M 2 ≃ M 3 ) but also by the factor (m 2 b /v 2 ), as implied by the cancellation conditions (4)- (5). The same suppression factor appears for Im J 3 . The case of Im J 2 is more involved, however when m 2 b /v 2 is neglected all the invariants (25)- (27) have the same simple asymptotic behavior for large tan β. It is also worth noticing that tan β = m t /m b (≃ 38) implies λ 1 = λ 2 , which in turn leads to exact vanishing of Im J 1 and Im J 3 . Qualitatively those conclusions survive at the two-loop level. For a quantitative illustration we plot in Figs. 6-7 maximal values of the invariants in the tan β-M H ± plane with all the necessary constraints imposed, seeking regions which still allow for substantial CP violation. At high values of tan β these invariants are of the order of 10 −3 , in qualitative agreement with the discussion above. Note that the SM corresponding invariant ImQ = (V ud V cb V ⋆ ub V ⋆ cd ) ≃ 2 × 10 −5 sin δ KM [13], for V ij and δ KM being elements of the CKM matrix and CPviolating phase, respectively. Therefore the model considered here offers at least two orders of magnitude enhancement comparing to the SM. and M H ± ∈ (300, 700) GeV, for µ = 300, 400, 500 GeV. Red: Positivity is satisfied; yellow: positivity and unitarity both satisfied; green: also experimental constraints satisfied at the 95% C.L., as specified in the text.

G. Stability and the determination of the cutoff
It should be emphasized here that the conditions (20) eliminate the quadratic divergences only up to the leading two-loop corrections. Even though the sub-leading two-loop and higher effects are suppressed by powers of coupling constants and powers of 1/(16π 2 ), nevertheless since the ln Λ term is growing, there exists always Λ large enough, that the hierarchy problem reappears: loop corrections to masses are again of the order of the masses itself. In fact that observation allows to determine the value of the cutoff up to which higher order corrections do not reintroduce the hierarchy problem (see [25] for the analogous strategy within the SM). In general, quadratic corrections to scalar masses have the form of (18 ) where v is chosen as a renormalization scale. The following naïve estimation of f (i) n is sufficient: where the relevant coupling constants were conservatively assumed to be of the order of 4π. 8 Here we choose as the cutoff the maximal value of Λ such that the higher order corrections do not exceed the mass of the lightest scalar, M 1 : Then, e.g. for M 1 = 200 (500) GeV the cutoff is at least at Λ ∼ 2.5 (6.3) TeV. Of course, larger M 1 would imply higher Λ.
Having the cutoff determined, we should address the issue of higher-loop corrections to the equations (20) that ensure vanishing of the quadratic corrections up to leading two-loop effects. As is seen from (30), the generic form of the condition for vanishing quadratic divergence is the following where λ stands for a typical coupling constant. The last two terms shown above (sub-leading two-and leading threeloop effects) have been neglected in the present analysis. It is then easy to see that even for the cutoff as large as Λ = 6.5 TeV using a very conservative (large) value for the typical coupling, λ = 4π, the precision of the adopted approximation is of the order of 12%. Note that whenever λ < 4π or Λ < 6.5 TeV, the adopted approximations work better.

III. 2 DOUBLET + 1 SINGLET HIGGS MODEL: THE CASE FOR DARK MATTER
In this scenario we combine CP violation present in the non-inert 2HDM (allowing for softly broken 2 symmetry) with a real scalar ϕ which is a gauge singlet. The singlet provides a natural DM candidate (see [34], [35] and [8]). In this case the scalar potential is the following Note that the term ∝ ϕ 2 φ † 1 φ 2 is forbidden as it breaks the 2 symmetry in a hard way. Since ϕ is supposed to be the DM candidate, in order to ensure its stability we have imposed an extra discrete symmetry ′ 2 such that ϕ → −ϕ while other fields are neutral. The symmetry excludes terms odd in ϕ. The potential should be arranged such that the symmetry remains unbroken, so that ϕ = 0. For that it is sufficient to require Then it is easy to see that if the standard 2HDM stability conditions (23)- (24) are fulfilled then the potential (34) is also positive definite. For the mass of the singlet we obtain: Since ϕ is a gauge singlet, therefore in the presence of right-handed neutrinos which are also gauge singlets the following Yukawa interaction is allowed [8]: Note that for a number of right-handed neutrino flavours greater than 1, the Yukawa matrix Y ϕ is in general (depending on the quantum numbers of ν R under ′ 2 , see [8]) non-vanishing. In this model the conditions for cancellation of quadratic divergences are slightly modified: The last condition above guarantees vanishing quadratic divergence in corrections to the ϕ mass. Since for the positivity we assumed λ ϕ , η 1 , η 2 > 0, it is clear from the above equation that the presence of the Yukawa coupling Y ϕ is mandatory to extend the condition for cancellation of quadratic divergences to the singlet field as well. It should also be mentioned that the presence of the singlet does influence the two-loop corrections to the quadratic divergences, those effects are neglected as being small (∝ η i ).
As we have already mentioned the extra singlet ϕ provides a candidate for the DM. To estimate its present abundance we consider the dominant annihilation channels for ϕ. The Lagrangian describing relevant cubic and quartic scalar interactions reads where A detailed study of the DM within this model (with extended conditions for the cancellation of quadratic divergences (37)) will be presented elsewhere [36]. However here we would like to show that it is indeed natural to expect the right DM abundance in the presence of the singlet. For an illustration we will assume that the DM annihilation cross section is of the order of the contributions from the lightest neutral Higgs boson H 1 of mass M 1 .
For an estimate of the DM abundance, we will consider two ϕϕ annihilation mechanisms. First we assume that ϕϕ annihilate to γγ, qq, l + l − , W + W − and ZZ through s-channel H 1 exchange. Then, following [38] we obtain 9 in the non-relativistic approximation the following result for the thermally averaged annihilation cross section where Γ H1 (2m ϕ ) stands for the decay width of H 1 calculated at M 1 = 2m ϕ (in the following numerical calculations we will use the SM width for the estimate). Now we have to add the contribution from the H 1 H 1 final state. There are two contributions: due to s-channel Higgs exchange, and due to the four-point coupling. We find in the non-relativistic approximation where the quartic ϕ 2 H 1 H 1 coupling λ 11 is defined by Eq. (40) and the trilinear H 1 H 1 H 1 coupling normalized to v is denoted byλ 111 [39]. For an order-of-magnitude estimate of the DM abundance, we will use hereλ 111 = 3M 2 1 /v 2 (this choice, together with (44), reproduces results which would be obtained for the SM Higgs doublet φ SM coupled to the singlet through the term ηϕ 2 |φ SM | 2 ) and parameterize κ 1 and λ 11 through one variable η as follows: Then, following [38] for cold relics one has to solve the following equation to determine the freeze-out temperature from x f = m ϕ /T f : where σv ≡ σv 1 + σv 2 and g ⋆ counts relativistic degrees of freedom at annihilation and m Pl denotes the Planck mass. It turns out that in the range of parameters we are interested in, x f ∼ O (25), so that this is indeed the case of cold dark matter, it also implies that g ⋆ ≃ 10 − 100. Then the present density of ϕ's is given by Ω ϕ h 2 = 1.07 · 10 9 x f g 1/2 In Fig. 8 we show the 3-σ allowed band in log(η) vs. m ϕ , as constrained by the observed DM abundance Ω DM h 2 = 0.106 ± 0.008 [37]. For m h = 100 and 200 GeV we observe consequences of resonant behavior at m ϕ = m h /2. The thresholds seen at m ϕ ≃ 25 and 80 GeV are caused by the rapid change in g ⋆ as a function of temperature and by the opening of the W + W − channel for the decay of a Higgs boson of mass m h = 2m W , respectively. One can conclude that the singlet could indeed provide a realistic candidate for DM: for any m ϕ between ∼ 1 GeV and ∼ 500 GeV there exists an allowed value η for which Ω ϕ h 2 agrees with the experimental data. Note that if we had found only solutions with η ∼ > 1 and light ϕ (m ϕ ∼ < v) then this scenario would be jeopardized since the minimization condition requires m 2 ϕ > η 1 v 2 1 + η 2 v 2 2 (as µ 2 ϕ > 0). As seen from Fig. 8, this is not the case. Most of the allowed region corresponds indeed to η ∼ < 10 −1 if the singlet mass is not too low.

IV. SUMMARY
The goal of this work was to build a minimal realistic model which would allow for softening the little hierarchy problem through suppression of the quadratic divergences in scalar boson mass corrections and through lifting the mass of the lightest Higgs boson. That could be accomplished within Two-Higgs-Doublet Models. Phenomenological consequences of requiring no quadratic divergences in corrections to scalar masses within the 2HDM were discussed. The 2HDM type II was analyzed taking into account existing experimental constraints. Allowed regions in the parameter space were determined. An interesting scalar mass degeneracy was observed for tan β ∼ > 40. The issue of possible CP violation in the scalar potential was addressed and regions of tan β − M H ± with substantial strength of CP violation were identified. In order to accommodate a possibility for dark matter a scalar gauge singlet was added to the model. Requirements necessary for correct present abundance of dark matter were estimated.
The model we considered here allows to soften the little hierarchy problem by lifting the minimal scalar Higgs boson mass and by suppressing the one-loop quadratic corrections to scalar masses. The cutoff implied by the naturality arguments is lifted from ∼ 600 GeV in the SM up to at least ∼ > 2.5 TeV, depending on the mass of the lightest scalar.