Just so Higgs boson

Just so Higgs boson F. Bazzocchi IFIC - Instituto de Fı́sica Corpuscular University of Valencia, Apartado de Correos 22085 E-46071 Valencia, Spain M. Fabbrichesi and P. Ullio arXiv:hep-ph/0612280v1 21 Dec 2006 INFN, Sezione di Trieste and Scuola Internazionale Superiore di Studi Avanzati via Beirut 4, I-34014 Trieste, Italy (Dated: February 2, 2008) We discuss a minimal extension to the standard model in which there are two Higgs bosons and, in addition to the usual fermion content, two fermion doublets and one fermion singlet. The little hierachy problem is solved by the vanishing of the one-loop corrections to the quadratic terms of the scalar potential. The electro-weak ground state is therefore stable for values of the cut off up to 10 TeV. The Higgs boson mass can take values significantly larger than the current LEP bounds and still be consistent with electro-weak precision measurements. PACS numbers: 11.30.Qc, 12.60.Fr, 14.80.Cp, 95.35.+d I. MOTIVATIONS There is some tension between the value of the electroweak (EW) vacuum and the scale at which we expect new physics to become manifest according to EW precision measurements [1]. If we take the latter scale around 10 TeV as the cutoff of our effective theory, some degree of fine tuning is necessary in the scalar potential in order to guarantee the vacuum stability against radiative corrections. This little hierachy problem—and before it the more general (and more serious) problem of the large hierarchy between the EW vacuum and the GUT and Planck scale—has been used as a clue to the development of models in which the scalar sector of the standard model is enlarged to provide better stability, as, for instance, in supersymmetry, technicolor and little-Higgs models. Here we discuss a different approach in which no new symmetry is introduced to cancel loop corrections and instead the parameters of the lagrangian are such as to make the one-loop corrections vanish and thus ensure the stability of the effective potential for the scalar particles up to the energy scale at which two-loop effects begin to be sizable, namely 10 TeV. Clearly, by its very nature, such a procedure can only be applied to the little hierarchy problem and not to the more general GUT or Planck scale hierarchy problem. It is a limited solution to a little (hierarchy) problem, a problem that—contrary to those arising in much larger hierarchies—may well be contingent to the choice of the lagrangian parameters. Given the simplicity of the idea behind this approach, it is not surprising that it was suggested early on (by Veltman [2]) in the following terms: the quadratically divergent one-loop correction to the Higgs boson mass mh ,  3Λ2  2 2mW + m2Z + m2h − 4m2t , 2 16π 2 vW (1) can be made to vanish, or at least made small enough, if mh happens to be around 316 GeV at the tree level. The remaining contributions—not included in (1)—are proportional to the light quark masses and therefore negligible. Such a cancellation does not originate from any dynamics and it is the accidental result of the values of the physical parameters of the theory. The absence of this quadratically divergent term in the two-point function of the scalar bosons makes possible to increase the cut-off for the theory to a higher value with respect to the standard model (SM) where the renormalization of the Higgs boson mass and the given value of the expectation value vW impose a cut off of around 1 TeV to avoid unnaturally precise cancellations among terms. We now know that a value of mh = 316 GeV is a little over 3σ with respect to current precision measurements of EW data [1]. This however does not mean that a scenario in which the Higgs boson mass is chosen just so to make the cancellation à la Veltman is ruled out. It only means that we must either enlarge the SM with new particles propagating below 1 TeV and then redo the EW data fit [4] or introduce new physics at a higher scale, the effect of which is to correct the precision observables and make room for the shifted value of the Higgs boson mass (as described in the framework of the effective EW lagrangian in [3]). Bearing this in mind, we introduce the minimal extension to the SM in which • quadratically divergent contributions cancel at one-loop à la Veltman; • it is consistent with the EW precision data. The model, as we shall see, is quite simple and provides an explicit example of an extension of the SM in which the mass of the Higgs boson can assume significantly larger values with respect to the current lower bound without having the EW precision measurements violated. In so doing, it introduces a characteristic spectrum of states beyond the SM that can be investigated at the LHC. The model is natural in the sense that the EW vacuum is stable against a cut off of the order of 10 TeV for a large choice of parameters. It is just so because the physical parameters are chosen by hand in order to satisfy the constraints. These parameters are however numbers of order unity and not extravagantly small or large; moreover, they can be chosen among many possible values so that no unique determination is required, as it would be in the original Veltman’s condition where the only free parameter is the Higgs boson mass, or—what amounts to the same thing—the quartic coupling λ. Because among the additional states required there is a stable neutral (exotic) fermion, we also discuss to what extent this state can be considered a candidate for dark matter. II. THE MODEL: HOW THE HIGGS GOT ITS MASS We consider a model in which there are two scalar EW doublets, h1 and h2 , the lightest scalar component of which is going to be identified with the SM Higgs boson and two Weyl fermion doublets, ψ1 and ψ2 . In addition, we also introduce a Weyl fermion ψ3 which is a SU (2)W × U (1)Y singlet. Let us briefly discuss to what extent this choice of new states is the minimal extension which cancels the quadratic divergence. In the model with only two Higgs bosons it is possible to reduce—or indeed cancel—the contribution of the top quark to the quadratic divergence but not that of the gauge bosons and the cutoff cannot be raised up to 10 TeV. We comment on this class of models in sec. VII below. The mass of a single fermion doublet (with two singlets) is necessarily proportional to the scalar field vacuum expectation and cannot be varied independently of the Veltman condition (if we want to choose naturally the Yukawa of the new fermions). Two doublet fermions are also necessary in order to be anomaly free. The singlet fermion is necessary to lift the fermion degeneracy and couple the fermions to the scalar fields. The states of the model are similar to those of a SUSY minimal extension of the scalar sector of the SM into a Wess-Zumino chiral model in which the singlet boson has been integrated out. However, a model with softly broken supersymmetry cannot be the model we are discussing because the supersymmetry, if present at any scale, would make the quadratically divergence zero. The lagrangian for the scalar bosons is given by X Dµ hi Dµ hi + V [h1 , h2 ] (2) Lh = i=1,2 with the potential h i V [h1 , h2 ] = λ1 (h†1 h1 )2 + λ2 (h†2 h2 )2 + λ3 (h†1 h1 )(h†2 h2 ) + λ4 (h̃†2 h1 )(h̃†1 h2 ) + λ5 (h̃†2 h1 )2 + H.c. + µ21 (h†1 h1 ) + µ22 (h†2 h2 ) , (3) where h̃2 = iσ2 h∗2 . The potential in eq. (3) is the most general for the two Higgs doublets once we impose a parity symmetry T1 according to which the two doublets h1 and h2 are, respectively even and odd. In this way, the quadratic and quartic mixing terms are forbidden, which makes the discussion simpler. The lagrangian of eq. (3) can be studied to find the ground state that triggers the electroweak symmetry breaking. It is    √  0 v2 / 2 √ hh1 i = and hh2 i = . (4) v1 / 2 0 The requirement of matching the EW vacuum vw to this vacuum state constains one parameter of the model. The mass eigenstates of the scalar particles can thus be derived. The masses are q 2 2 m2h,H = λ1 v12 + λ2 v22 ± (λ1 v12 − λ2 v22 ) + (λ3 + λ4 + λ5 ) v12 v22 m2A = 2 −λ5 vw , (5) for the three neutral scalar bosons (two of which, h and H, are scalars and one, A, a pseudoscalar), 2 , (6) m2H + = − (λ4 + λ5 ) vw
2 for the charged boson H + after using the constraint vw = v12 + v22 in eqs. (5)–(6). By introducing the mixing angle α and β to rotate the scalar boson gauge states into the mass eigenstates, we write:     1 v2 + sin α h + cos α H + cos β A sin β H + 1 √ √ and h2 = . (7) h1 = cos β H − 2 v1 + cos α h − sin α H + sin β A 2 As usual in 2 Higgs doublet model tan β = v2 /v1 and tan 2α = λ3 v1 v2 . (λ2 v22 − λ1 v12 ) (8) The exotic fermion content of the model is given by two SU (2)W doublets:  +  0  ψ2 ψ1 Ψ1 = , Ψ = 2 ψ10 ψ2− and one SU (2)W singlet ψ3 ; we can also define the Majorana 4-components fermions current eigenstates as  0  + ψi ψ1 + ψ̃i0 = χ̃ = . i ψ̄i0 ψ̄2− The SM fermions are even under the T1 parity symmetry and therefore can have Yukawa interactions only with the scalar doublet h1 . The exotic doublet fermions Ψi are odd under this parity symmetry while the singlet ψ3 is even. We also introduce an additional parity T2 under which the Higgs bosons are even while all the exotic fermions are odd (SM particles are always even under both parities). In this way the exotic fermions do not mix with the SM fermions and may have Yukawa terms only with the scalar doublet h2 . The lagrangian for the exotic fermions is simply given by the kinetic and the Yukawa terms, that is Lψ = Lkin + Lψ m, (9) where Lkin is given by ˆ 3 Lkin = Ψ̄1 D̂Ψ1 + Ψ̄2 D̂Ψ2 + ψ¯3 ∂ψ and  − Lm = µ̃ ǫij Ψ1i Ψ2j + µ̂ ψ3 ψ3 +   k4 ˜ + √ ψ3 Ψ2i h2 j + H.c. . 2  † k1 ˜ √ ψ3 h2 Ψ1 + H.c. 2 From Lm we see that the charged Dirac fermion χ̃+ has mass mχ+ = µ̃ while once we insert eq. (4) into eq. (10) the Majorana mass matrix for the neutral states is given by √   0 −µ̃ k1 v2 /√2 M 0 =  −µ̃√ (10) 0 √ k4 v2 / 2  µ̂ . k1 v2 / 2 k4 v2 / 2 0 This matrix is diagonalized by a neutralino mixing matrix V which satisfies V T M 0 V ∗ = Mdiag . From the 20 components mass eigenvectors of eq. (10) χi=1,3 , we define the 4-components neutral fermions that will be our neutralinos  0  0  0 χ2 χ3 χ1 0 0 0 χ̃3 = , χ̃1 = χ̃2 = γ5 χ̄02 χ̄03 χ̄01 where in eq. (11) the definition of χ̃02 takes into account that the corresponding eigenvalue of the Majorana mass matrix is negative. From Lkin of eq. (10) using the mass eigenstates defined in eq. (11) we obtain the interaction terms of the new fermions with the gauge bosons i g ¯+ g h X ¯+ χ̃ γµ (V1 i ǫi PL − V2∗i PR )χ̃i 0 W + µ + H.c. + χ̃ γµ (PL + PR )χ̃+ W3µ L = √ 2 2 i=1,3
i g X h¯0 χ̃i γµ ǫi ǫj (Vi†1 V1 j − Vi†2 V2 j )PL − PR (ViT1 V1∗j − ViT2 V2∗j ) χ˜j 0 W3µ − 2 i,j=1,3 g′ ¯ + χ̃ γµ (PL + PR )χ̃+ B µ 2 i
g′ X h ¯ 0 + χ̃i γµ ǫi ǫj (Vi†1 V1 j − Vi†2 V2 j )PL − PR (ViT1 V1∗j − ViT2 V2∗j ) χ˜j 0 B µ , 2 i,j=1,3 + (11) where the factor ǫi = (−1)i−1 keeps into account the signs of the eigenvalues of the Majorana mass matrix of eq. (10). This lagrangian is necessary in order to compute the one-loop radiative corrections to the scalar potential. III. VELTMAN CONDITION REDUX As stated in the introduction, we want to stabilize the potential given by eq. (3) at one-loop level, that is we want that the one-loop δµ2i quadratically divergent contributions to µ2i be zero. As in the SM the quadratically divergent contributions arise by loops of gauge bosons, scalars and fermions. We therefore find two Veltman conditions by imposing δµ21 = 0 and δµ22 = 0 , (12) that is 9 2 3 ′2 g + g + 2(3λ1 + λ3 + λ4 ) − 12λ2t = 0 4 4 9 2 3 ′2 g + g + 2(3λ2 + λ3 + λ4 ) − (k12 + k42 ) = 0 . 4 4 (13) In eq. (13) g, g ′ are the electroweak gauge couplings, λi the parameter of the scalar potential of eq. (3) , λt the top Yukawa defined as λt = vw /v1 since the SM fermions couple only to the scalar doublet h1 and k1,4 are the Yukawa coupling of eq. (10). The contributions of the lighter SM fermions to eq. (13) have been neglected. Notice that if we did not have the parity symmetry T1 and the fermions would have interacted with both h1 and h2 we would have generated a divergent mixed contribution that could have been canceled only by a bare term. In writing eq. (13) we have taken a common cutoff Λ for the divergent loops of different states. The possibility that there exist different cutoffs for the different contributions does not change our result because a change of order O(1) in the Λs only means a similar change of order O(1) in the parameters of the model λi and ki . Once these two conditions are satisfied the scalar potential is stable at one-loop order and so is its vacuum state. We interpret these conditions as two constraints on the 10 free parameters of the model. IV. THE EW PARAMETERS S, T AND U For our purposes, the consistence of the model against EW precision measurements can be checked by means of oblique corrections. These corrections can be classified [5] by means of three parameters: αS = −4e2 [Π′33 (0) − Π′3B (0)] e2 αT = 2 2 2 [Π11 (0) − Π33 (0)] sW cW m Z αU = 4e2 [Π′11 (0) − Π′33 (0)] , (14) where the functions Πnm (q 2 = 0) represent the vacuum polarizations of the gauge vactors in the various directions of isospin space. Other corrections functions—like the functions Y and W of ref. [6]—are not relevant here because mainly sensitive to physics in which there are new vector bosons. EW precision measurements severely constrain the possible values of the three parameters S, T and U. In the SM, the data allow [1], for a Higgs boson mass of mh = 117 GeV, S = −0.13 ± 0.10 T = −0.17 ± 0.12 U = 0.22 ± 0.13 (15) These constraints must be rescaled for the different values of the Higgs boson mass. If we want the model to be consistent with the EW precision measurements within, for instance, one sigma we have three further constraints on the parameters of the model—5 of which still remain free at this point. A mass of the Higgs boson larger than the reference value will make the parameter T smaller, the size of the correction going like the ln m2h /m2Z . This can be compensated by the fermion contribution which can give a ∆T > 0 of size ∆m2 ln m2 where ∆m2 is the isospin splitting of the fermion masses. The parameter S is changed by the larger Higgs mass with a ∆S > 0, a change that is in general difficult to compensate. In our model a negative contribution to S comes about because of the fermion with both Dirac and Majorana masses which gives a negative contribution proportional to ln(mχ+ /mχ0i ), and mχ+ − mχ0i is the isospin mass splitting between the chargino and the neutralino i. Let us consider their contributions to T and S separately in a simplified model which helps in visualizing better how scalar and fermion contributions compensate one other in order to accomodate the EW experimental values. For what concerns the fermions, suppose to be in the simple case in which mχ+ ∼ mχ0i . The fermion contribution to the T parameter can be written as f f + TLR , T f = TLL (16) f takes into account the one loop contributions that arise from the vacuum polarizations of the gauge bosons where TLL f LR RL RR of the kind ΠLL 11,33 and Π11,33 , TLR the ones that arise from Π11,33 and Π11,33 . The latters are not present in the SM case. Keeping only the leading contributions, we have f TLL = − f TLR = − 2 c2W s2W m2Z π X Uijef f i,j 2 h c2W s2W m2Z π i,j 2 i h m2χ+ (mχ+ − mχi )2 i Uief f − m2χ+ + log 2 2 mZ − m2χ0 + 2 X Uijef f nX i nX (mχ0i − mχ0j )2 i 2 Uief f mχ+ mχ0i log i mχ0j mχ0i log m2χ0 o i m2Z log m2χ+ m2χ0 o i m2Z m2Z , (17) ef f where Ui,ij are effective couplings related to the neutralino mixing matrix V and are in general different for the LL f contribution and for the LR one. For example Uief f in TLL is given by Σk = Vik† Vki . Notice that when the third f neutralino decouples, the TLL contribution goes to zero when mχ+ = mχ01,2 and the isospin symmetry is restored. The f same happens also for TLR . In a manner similar to the T parameter, the S parameter receives a fermion contribution that can be splitted in f f S f = SLL + SLR , (18) with f SLL # " m2χ0 m2χ+ 1 ef f ef f i = (Uij − 1) − 2 log 2 + 2Uij log 2 3π mZ mZ f SLR = 2 ef f (yRχ+ − yR ), χ0 3π ij (19) ef f where yRχ+ = 1/2 follows by the definition of χ+ and where we have defined yR = 2Σk=1,2 T3k yRk f (Vki , Vkj ) 0 χ ij where T3k and yRk are the isospin and the hypercharge of the Majorana singlets defined in eq. (9) and f (Vki , Vkj ) is 0.4 0 m Χ3 =200 GeV Sf 0.3 Tf m Χ3 =200 GeV -0.02 0.2 -0.04 -0.06 0.1 -0.08 0 0 5 10 15 20 0 5 10 15 20 Hm Χ+ -m Χ1 L Hm Χ2 -m Χ1 L 0.4 0 Α=А4 -0.05 -0.1 -0.15 -0.2 -0.25 0.3 TEWPT Ss Ts FIG. 1: Fermion contributions to the parameters T and S as a function of their mass splitting. The plots are made for one representative value of the χ03 mass. 0.2 mA =400 GeV 0.1 mA =150 GeV SEWPT 0 200 300  400 500 600 0.5 1 mh mH HGeVL 1.5 2 2.5  mh mH  mA 3 3.5 4 FIG. 2: Scalar contributions to the parameters T and S as a function of their masses. The red horizontal lines show the central value of the current EW bounds. Notation and values of the parameters are explained in the text. a combination of different entries of the neutralino mixing matrix V . Notice that we recover the contribution of two f f SM-like doublets when the hypercharges difference in SLR gives 1/2 and Uij in SLL is equal to 1[5]. The previous expressions can be further simplified if we consider the fermion mass matrix of eq. (10) in the limit in which µ̂ is much larger than µ̃, k1,4 v2 . In this limit the neutral fermion mixing matrix is approximately given by r   mχ0 −mχ0 2 1 √1 √1 − − 2m 0   2 2   r χ3 3mχ0 −mχ0 mχ0 −mχ0  mχ0 +mχ0 mχ0  2 2 1  2 3 √1 √1 − − V =  (20) mχ0 2mχ0  . 2 2mχ0 2 2mχ0 1 3   r 1 r 1  mχ0 +mχ0 mχ0 −mχ0 mχ0 −mχ0 mχ0 −mχ0  3 1 1 3 2 1 2 1 1 m 0 2m 0 m 0 2m 0 χ χ 1 3 χ 1 χ 3 If it holds also that µ̃ larger than k1,4 v2 , mχ+ ≃ mχ01,2 , T f and S f can be easily expressed in terms of the mass splitting, see fig. 1. Notice that eq. (20) is valid for mχ01,2 < mχ03 < m2χ0 /∆mχ012 . For this reason we have plotted T f 1 and S f in fig. 1 corresponding to only one value of mχ03 , since in the range allowed differences are minimal. For what concerns the scalar sector, consider the case in which mH + ≃ mA . In this limit, T s and S s assume a simple form. We have Ts = − m2H m2h 3 2 2 + sin α log ), (cos α log 16πc2W mZ mZ (21) where α is the mixing angle in the neutral scalar sector and Ss = m2H + 1 m2 m2 m2 (log 2h + log H − 2 log + log A ). 2 2 12π mZ mZ mZ m2Z (22) We can compare the contribution of the scalar sector of our model with respect to the SM one. In the SM we have ThSM = − 3 mh log 2 8πcW mZ ShSM = 1 m2 log 2h . 12π mZ (23) EW precision measurements indicate that at 2σ mh ≤ 185 GeV [1] and therefore the introduction of the fermions in our model is justified if T s and S s exceeds the contributions ThSM and ShSM corresponding to mh ≃ 185 GeV. This is shown in fig. 2. For fixed mh and mH , and a given fermion spectrum that accomodates the T parameter, the fermion contribution to S is fixed and therefore the only freedom left is in the values of mA and mH + . In the case in which mH + ≃ mA , their total contribution to S has the same sign of the fermion one, therefore we expect that mA cannot in general be to heavy. This is verified in the numerical analysis. V. A DARK MATTER CANDIDATE? The lightest neutral exotic fermion state in the model is similar to the neutralinos in a minimal supersymmetric extension of the SM (NMSSM) in which the composition is dominated by Higgsinos. It is stable because the lagrangian does not contain couplings between the SM and the exotic fermions—or, alternatively, you can think of the lagrangian as written with a underlying conserved parity. We compute by means of the program DARKSUSY [7] its relic abundance ΩDM h2 . To do this we need the lagrangian written on the exotic fermion mass eigenstates: X X ¯ 0i (ViT3 V1 j ǫj PL + V † V1∗j ǫi PR )χ̃0j ¯ 0i χ0i + χ̃ mi χ̃ − 2Lm = i3 i,j=1,2,3 i=1,2,3 [ X (k1 U2Rn + k2 U1Rn )Hn ] n=1,2 + X ¯ 0i (ViT3 V2 j ǫj PL + V † V2∗j ǫi PR )χ̃0j [ χ̃ i3 + (k4 U2Rn + k3 U1Rn )Hn ] n=1,2 i,j=1,2,3 X X ¯ 0i χ̃ [(−i ViT3 V1 j ǫj PL + i Vi†3 V1∗j ǫi PR )(k1 cos β + k2 sin β) i,j=1,2,3 + (i ViT3 V2 j ǫj PL − i Vi†3 V2∗j ǫi PR )(k4 cos β + k3 sin β)]χ̃0j A ¯ 0i χ̃+ (ViT3 PL + V † ǫi PR )((k2 − k3 )U1C2 + (k4 − k1 ) cos β)H − + H.c. , + χ̃ i3 where Hn=1,2 = h, H are the two neutral scalars, A the pseudoscalar, H − the charged scalar, β the mixing angle defined in eq. (7) and U R is the mixing matrix related to the real neutral components of the two doublets h1 and h2 given by   cos α − sin α R , (24) U = sin α cos α where α is the mixing angle defined in eq. (8). The analysis shows that the relic abundance is always at least one order of magnitude too small than the presently favorite abundance of dark matter in the Universe. This seems to be due to the lack of cancellations among different diagrams introduced by the arbitrariness in the Yukawa couplings that makes pair annihilation rates too large. Therefore, the lightest neutral exotic fermion can at most be a marginal component of dark matter. VI. THE MODEL SOLVED The model has eleven parameters, 10 of which are in principle free once the ground state has been identified with vW . If we enforce the Veltman conditions—and thus make the one-loop quadratically divergent corrections vanish—we are left with eight parameters. These can be treaded for the masses of the 4 scalar and 4 fermion states. These can be varied and for each choice of them the S, T and U parameters computed and compared against the EW constraints. We vary the dimensionless parameters within one order of magnitude. In particular, we keep the λi and the κi between 1 and 4π (after which the the perturbative analysis may break down). Mass parameters µ and µ̃ are varied between 100 and 300 GeV. We find that for a large choice of the five remaining parameters the model is consistent with the EW precision measurements. For these choices, masses as large as 450 GeV are possible for the lightest neutral scalar Higgs boson. 300 250 mΧ 200 150 100 50 150 200 250 300 mh 350 400 450 500 FIG. 3: Distribution of values for the masses mχ vs. mh for values of the parameters within 1σ of EW precision masurements. 800 700 mA 600 500 400 300 200 100 150 200 250 300 mh 350 400 450 500 FIG. 4: Distribution of values for the masses mA vs. mh for values of the parameters within 1σ of EW precision masurements. As its mass increases those of the neutral pseudoscalar tends to favor lighter values so that there are solutions in which the lightest Higgs boson is the pseudoscalar. The lightest neutral fermion mass tends to increase together with the mass of the Higgs boson. Figure 3 shows some of the possible values we obtain for the Higgs boson and lightest neutral fermion masses for values of the parameters which satisfy within 1σ the EW precision masurements. Figure 4 shows the distribution of the masses for the scalar and pseudoscalar states under the same conditions. Our result may help in dispelling excessive surprise in not seeing a bantamweight Higgs boson with mh just above the current LEP bound of 117 GeV and should encourage searches at the LHC for a Higgs boson substantially heavier than the current LEP bound—what we can call a welterweight at mh around 300 GeV or even a cruiserweight at 500 GeV. Such a scenario has been pointed out recently in [8] and [9] in the framework of the little Higgs models[10] and in [11, 13] in a two-Higgs extension of the SM. VII. MODELS WITH TWO HIGGS BOSONS AND NO EXTRA FERMIONS Different possibilities of realizing a minimal extention of the scalar sector of the SM could have a natural cut-off Λ around few TeV while being compatible with EW precision measurements have been discussed in the last year. The authors of [11, 12, 13] have analyzed different realizations of the 2 Higgs doublets model (2HDM) and have TABLE I: Representative values (among those used in the plots) of the eight parameters of the model, and mass spectrum of the most relevant states: scalar and pseudo-scalar bosons and lightest fermion, that satisfy the bounds from EW precision measerements. µ (GeV) µ̃ (GeV) 173 287 138 128 276 438 266 381 239 180 k1 1.4 1.6 2.9 3.8 3.8 λ1 8.5 6.3 7.7 5.3 4.8 λ2 4.6 6.4 5.0 12.5 11.4 λ3 2.8 2.3 8.1 1.7 7.3 λ4 −8.5 −10.6 −7.1 −7.0 −11.9 λ5 mh (GeV) mA (GeV) mχ (GeV) −5.6 146 600 131 −2.9 210 417 96 −8.5 304 715 223 −3.5 450 460 212 −2.1 470 360 190 parametrized the fine tuning parameter in terms of the dependence of the mass of the light Higgs boson on the cut–off Λ. In the Barbieri-Hall (BH) model [11] both doublets acquire a VEV, but the small mixing angle between them makes the light scalar coupling to the top quark quite small and Λ becames proportional to the mass of the heavy neutral scalar. The mass of the heavy neutral scalar is then bounded by the requirement of satisfying the EW precision measurements and this allows Λ to reach more or less 2 TeV when the light Higgs boson has a mass mh = 115 GeV. The twin doublets model [12] is a particular version of the 2HDM in which only one doublet couples to the SM fermions. The symmetry of the model makes possible to improve the bound found in the BH model and to reach a cut-off between 3 and 4 TeV. Finally, the inert doublet model (IDM) [13] proposes a different picture. Instead of trying to justify through naturalness the existence of a light Higgs boson and a cut-off of few TeV, it describes the possibility of having a heavy Higgs while being still compatible with EW precision measurements. The cut-off of the model turns out to be of few TeV (a value that would be natural even in the SM context if the Higgs were heavy). The new feature of the IDM is that the model may be compatible with the EW precision measurements even in the presence of a heavy Higgs boson. This is realized thanks to the contribution to t he EW parameters that arises from the heavy new scalars. In general, in the different realizations of the 2HDM the T parameter receives a SM-like contribution and a contribution that arises from loops involving the new scalars. These contributions are approximately given by [14] 3 m2 m2 (cos2 (α − β) log 2h + sin2 (α − β) log H ) 2 16π cos θW mZ m2Z 1 = (cos2 α(mH + − mh )2 + sin2 α(mH + − mH )2 + (mH + − mA )2 4πs2W m2W Ta = − Tb − cos2 α(mA − mh )2 − sin2 α(mA − mH )2 ) , (25) where tan β = v2 /v1 with vi the vev of hi and α the mixing angle between the two neutral scalars. If both the doublets acquire a VEV (BH, twin and the just-so models) Tb is negligible because mA − mH + cannot be too large (for natural choice of the λi parameter of the potential). On the contrary, in the IDM Tb may not be negligible and can balance the contribution to Ta arising from a heavy Higgs boson; in this way, the model predicts a heavy Higgs boson and a cut-off around 3 TeV. In conclusion, in all the version of 2HDM the cut-off can be around 5 TeV but not much higher. Our approach is different with respect to the models that present improved naturalness. The cancellation of the Veltam condition fixes our cut-off at 10 TeV and the requirements to be compatible with the EW precision measurements and to cover the most general neutral scalar spectrum forces us to include at least a new fermion doublet. Acknowledgments This work is partially supported by MIUR and the RTN European Program MRTN-CT-2004-503369. F. B. is supported by a MEC postdoctoral grant. [1] S. 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