September 12, 2012 UMD-PP-012-021 Natural Islands for a 125 GeV Higgs in the scale-invariant NMSSM 2 1 Kaustubh Agashea, Yanou Cuia, Roberto Franceschinia 0 2 aMaryland Center for Fundamental Physics, Department of Physics, University of Maryland, p e College Park, MD 20742, U.S.A. S 0 1 Abstract ] h p - We study whether a 125 GeV standard model-like Higgs boson can be accommodated within the p e scale-invariant NMSSM in a way that is natural in all respects, i.e., not only is the stop mass and h [ hence its loop contribution to Higgs mass of natural size, but we do not allow significant tuning 1 of NMSSM parameters as well. We pursue as much as possible an analytic approach which gives v clearinsightsonvariouswaystoaccommodatesuchaHiggsmass, whileconductingcomplementary 5 1 numerical analyses. We consider both scenarios with singlet-like state being heavier and lighter 1 2 than SM-like Higgs. With A-terms being small, we find for the NMSSM to be perturbative up to . 9 GUT scale, it is not possible to get 125 GeV Higgs mass, which is true even if we tune parameters 0 2 of NMSSM. If we allow some of the couplings to become non-perturbative below the GUT scale, 1 then the non-tuned option implies that the singlet self-coupling, κ, is larger than the singlet-Higgs : v coupling, λ, which itself is order 1. This leads to a Landau pole for these couplings close to i X the weak scale, in particular below ∼ 104 TeV. In both the perturbative and non-perturbative r a NMSSM, allowing large A ,A gives “more room” to accommodate a 125 GeV Higgs, but a tuning λ κ of these A-terms may be needed. In our analysis we also conduct a careful study of the constraints on the parameter space from requiring global stability of the desired vacuum fitting a 125 GeV Higgs, which is complementary to existing literature. In particular, as the singlet-higgs coupling λ increases, vacuum stability becomes more serious of an issue. Contents 1 Introduction 2 2 The Model 6 3 Getting SM-like Higgs of mass 125 GeV 8 3.1 Perturbativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Constraints from unrealistic minima . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Push-up Scenario 17 5 Pull-down Scenario 26 5.1 λ (cid:39) 0.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 λ > 0.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Conclusions 30 1 Introduction Very recently, the ATLAS and CMS collaborations at the large hadron collider (LHC) have discovered a resonance with a mass of around 125 GeV which is consistent with a standard model (SM) Higgs boson [1]. Here, we assume it is a SM-like Higgs. In the SM, the quartic coupling for the Higgs and hence its mass is a free parameter so that the above observation can be easily “accommodated”. However, the SM is plagued by the Planck- weak hierarchy problem for which weak-scale supersymmetry (SUSY) is perhaps the most studied solution. Inturn,intheminimalsupersymmetricSM(MSSM),theHiggstree-levelquarticcoupling is given by the electroweak (EW) gauge coupling so that there is actually a prediction for the Higgs mass. The flip side is that the Higgs mass then has a tree-level upper bound of m , which is well Z below the above observation of 125 GeV. It is well-known that the contribution of the superpartner of top quark (stop) at the loop-level can raise the Higgs mass in the MSSM, but reaching 125 GeV this way requires stops to be as heavy as ∼ 5−10 TeV or tuned, large A [2, 3, 4]. The heaviness t of the stop results in reintroduction of fine-tuning of the EW scale. This issue is embodied in the the relation m2 tan2β −m2 m2 = −|µ|2− Hu Hd , (1) Z tan2β−1 that is one of the minimization constraints of the MSSM potential. Here the soft mass m2 is Hu largely determined by the stop mass via RGE, resulting in |m2 | (cid:38) 0.1m2 [5]. Given the above Hu t˜ stop mass, such large value of m2 needs to be canceled by other contributions at the right-hand Hu side of eq. (1), in particular, either against the supersymmetric mass for Higgs doublets (µ-term) 2 or the other Higgs soft mass (m2 ) [6, 7, 8]. At any rate the need for a cancellation of the large H d m2 signals that a new “little hierarchy” needs to be explained in the MSSM. This little hierarchy Hu problem in MSSM can be quantified with a fine-tuning measure [9]: dlogm2 ∆ = max Z , (2) θi dlogθi with ∆ (cid:46) 5, i.e., less than 20% tuning, conventionally taken as typical for a natural theory. The MSSM also has another drawback, partly connected with the issue of having a cancellation in eq. (1) involving µ. The issue is about why is the µ-term close to the weak scale as required phenomenologically: there is lower bound on it of ∼ 100 GeV from chargino mass, whereas much larger values would require tuning in eq. (1). An attractive solution to this µ-problem is the scale- invariantNMSSM(forareviewsee[10]). Here,anexplicitµ-termisforbiddeninthesuperpotential, and instead is generated by the vacuum expectation value (VEV) of an added SM gauge singlet, S, which is coupled to Higgs doublets, H and H , via the superpotential term u d λSH H . (3) u d In turn, this VEV for the singlet is driven by soft SUSY-breaking mass terms for singlet. As there arenoexplicitmassscalesinthesuperpotential, thismodelcanbereferredtoasthescale-invariant NMSSM. This form of superpotential can be ensured by a Z -symmetry, which can be extended 3 to the soft SUSY breaking terms as well. Thus it is often also referred to as the Z invariant 3 NMSSM 1. As a bonus of the NMSSM, we get an extra tree-level quartic coupling for the Higgs doublets from the same singlet-Higgs coupling λ which solves the µ-problem. This effect in raising the Higgs mass relative to its MSSM prediction is well known and has been investigated at length in the literature (see for instance [13, 14, 15]). Here we re-visit this issue in the light of the discovery of a SM-like Higgs boson at 125 GeV. Variations of the NMSSM may allow dimensionful terms in the superpotential. In this work, we focus on the version of the NMSSM with no mass scales in superpotential so as to keep its other merit of solving the µ-problem. We assume that the stop contribution to Higgs mass is small, as follows from natural m ,A . This means that we aim at using tree-level contributions to get close t˜ t to 125 GeV, about 110 GeV or more, and use the stop contribution to gain just the “last” 15 GeV or less in mass 2. 1The original Z NMSSM typically suffers from the difficulty of simultaneously solving domain wall problem and 3 tadpole problem which destabilizes electroweak scale [11]. A simple way to resolve this issue is to assume a low- scale inflation such that the domain walls are significantly diluted [10]. Alternatively, a suitable R-symmetry can be imposed to constrain the form of explicit Z -breaking terms which may reconcile the tension of solving domain wall 3 and tadpole problems [12]. 2The estimate of 110 GeV lower limit will be detailed in the following and roughly corresponds to small t˜ −t˜ L R mixing and the contribution of a stop with mass of 500 GeV which is a general upper bound from naturalness according to eq. (2) . 3 Our goal is to see if it involves any tuning of parameters of the NMSSM to reach the required tree-level value for Higgs mass. Such a suspicion is rather motivated. For example, it is well known that in a large region of the NMSSM parameter space, the lightest CP even state, s , is identified 1 as the SM-like Higgs. The mass of s has an upper bound at tree-level 1 m2 ≤ m2 cos22β+λ2v2sin22β. (4) s1 Z Ifweprefertopreservetheconventionalunificationofthegaugecouplings, whichisamajorsuccess of MSSM, λ is constrained to remain perturbative up to the GUT scale. This means that the value of λ at the weak scale which enters the above bound cannot be much larger than 0.6−0.7 [16]. In turn, even for such a maximal value of λ, the tree-level upper bound for the Higgs mass is very close to 110 GeV. Given that Higgs mass of 110 GeV or more at tree-level is our goal, it is clear that we must actually saturate the upper bound in this case. However, the upper bound is not a generic value of the Higgs mass in the NMSSM, i.e., we expect that the model parameters must be arranged in a specific manner to fit the data in this case. Here we shall investigate in detail what are the conditions that define the locus of maximal Higgs mass in this region of the parameter space of the NMSSM. A key quantity to identify the regions that lead to the largest Higgs boson mass (when it is the lightest CP-even state) is the mixing of the doublet-like states with the singlet-like states. Indeed it has been known that the extra quartic coupling of the NMSSM for the SM-like Higgs is not the “end of the story” since mixing with the singlet modifies the mass of the SM-like state [13] and in general pushes it away from the value of the right-hand side of eq. (4). When the singlet-like scalar is heavier, as discussed so far, its mixing with the doublet-like state results in a “pull-down” of the mass of the latter. The crucial point is that, even though the singlet-like scalar is heavier than the SM-like Higgs, it turns out that the shift of the mass of the SM Higgs-like state due to mixing with singlet state can still be significant, since this effect does not quite decouple even if singlet VEV is large 3. As mentioned already, if the singlet-Higgs coupling λ is to remain perturbative up to GUT scale, then it turns out that the maximal tree-level SM-like Higgs mass – before negative mixing effect – can only be barely around or lower than our target of 110 GeV. Thus one has to work hard to minimize the negative mixing effect on the SM-like Higgs mass in order to obtain 125 GeV mass at loop-level. There is actually a different possibility, namely, that the singlet-like scalar is actually lighter than the SM-like Higgs. In this case, their mixing provides a “push-up” effect, which raises the SM-likeHiggsmass. Thus,thetree-levelmass,beforethemixingistakenintoaccount,canbebelow 110 GeV, lessening the tension with perturbativity mentioned above for the other case. However, we face a different potential worry: the state lighter than 125 GeV necessarily has a component of Higgs doublet so that it couples to the Z boson and a bound from LEP2 applies. Thus, in this 3Although we might actually require (cid:104)S(cid:105)∼v in order to have a natural µ-term, λ(cid:104)S(cid:105). EW 4 case, one has to work hard to make sure that the singlet-like scalar is not too light or too strongly coupled to the Z boson. As we proceed with our above analysis, we distinguish two cases: the “perturbative” NMSSM that we mentioned above, which requires all the couplings do not hit a Landau pole till GUT scale, and the non-perturbative NMSSM, where such singularity of the couplings can appear below the GUT scale. The motivation is clear: the former case can manifestly maintain the successful gauge coupling unification of the MSSM. Another useful “axis” of our exploration of the parameter space is the size of soft trilinear SUSY breaking terms for Higgs doublets and singlet, A and A . We κ λ consider both cases where they are small and cases where they are large. This division clearly is relevant for some SUSY breaking mediation schemes that predict the A-terms to be small, for example, minimal gauge mediation. Finally, as far as possible, we try to come up with analytical insights into these various issues. The issue of the Higgs mass in the NMSSM has been studied recently, also in connection with the experimental finding of a 125 GeV SM-like Higgs boson. However our study is significantly different in a number of aspects with respect to the current literature. For example in Ref. [3] an explicit mass terms for the singlet and Higgs doublets are allowed. Other studies such as [17, 18] are mostly numerical and they lack a clear division into the above cases that we highlight. Our results can be summarized as follows, beginning with the case with negligible A-terms. For the perturbative case, we demonstrate that even if we allow tuning of parameters of NMSSM it is not possible to reach 125 GeV mass: for the sub-case here of pull-down, this limitation is due to a combination of the facts that the singlet-Higgs coupling λ is small to begin with and that the reduction in Higgs mass due to the mixing cannot be made small. On the other hand, for push-up, thefailuretoreach125GeVHiggsmassisduetothelightsinglet-likestatetypicallybeingruledout by LEP2. In the non-perturbative NMSSM, the singlet-Higgs coupling λ and singlet self-coupling κ can be larger. Thus in this case the maximum tree level value itself can be well above 110 GeV. Typically, one then uses the singlet-Higgs mixing to get down from the above bound to 110 GeV at tree-level. In this way we completely avoid the tuning associated with requiring the pull-down mixing effect to be near vanishing that plagued the perturbative case above. Looking closer at the non-perturbative case it turns out that, if one wishes to avoid any kind of tuning of NMSSM parameters, then we end up in a specific region, namely, singlet-Higgs coupling, λ, is of order one andsingletself-coupling, κ, islarger. SuchachoiceimpliesaLandaupoleforthesecouplingswhich is rather close to weak scale. Finally, with A-terms, for both perturbative and non-perturbative cases, there is more freedom, for example, it is possible to minimize the size of mixing in order to reach 125 GeV even in the perturbative case, but in this case it still requires tuning of A against λ µ. One of the constraints in addition to m (cid:39) 125 GeV that we explicitly take into account is the h stability of the vacuum. Although this is a complicated issue to deal with, it is essential for the 5 phenomenological viability of the model: the parameter choice that accommodates the right Higgs massshouldcorrespondtoacosmologicallystablevacuum. Thisissue,however,iseitheroverlooked or only investigated in a partial way in most existing literature trying to explain m (cid:39) 125 GeV h within NMSSM. In our work we try to do a more careful study in this regard, both analytically and numerically. Here is the outline of the rest of the paper. We begin with reviewing the model and giving relevant formulae for our analysis. This is followed by a general discussion of how to get to Higgs mass of 125 GeV, including relevant issues such as perturbativity and vacuum stability. After this setting of the stage, we discuss in detail the viability of the model in the push-up scenario first and then the pull-down case, before concluding. 2 The Model The scale-invariant NMSSM is a very well-known extension of the MSSM which was originally proposedtoovercometheµproblemandcanalsoservetoliftthetree-levelHiggsmass. Nonetheless, in this section, for the sake of completeness and to fix our notation, we give some details of the model relevant for our discussion. For a complete overview on the NMSSM we refer to [10]. The superpotential of the model is κ W = λSH H + S3, (5) NMSSM u d 3 where S is a singlet chiral superfield and H = (cid:0)H+,H0(cid:1) and H = (cid:0)H0,H−(cid:1) are the MSSM Higgs u u u d d d doublets. The soft-SUSY breaking Lagrangian is 1 −L = m2 |H |2+m2 |H |2+m2|S|2+λA H H S + κA S3. (6) soft Hu u Hd d S λ u d 3 κ Hence the complete scalar potential in this sector is: (cid:88) V = −L + |F |2+ scalar soft i Hu,Hd,S g12+g22 (cid:0)|H0|2+|H+|2−|H0|2−|H−|2(cid:1)2+ g22|H+H0∗+H0H−∗|2, (7) 8 u u d d 2 u d u d where the last line is the MSSM D-term, g ≈ 0.65 and g ≈ 0.35 denote the SU(2) and U(1) 2 1 L Y gauge couplings, and F = ∂W/∂φ are the F-terms. i i We expand the neutral scalar fields around the vacuum expectation values (VEVs) as follows (see, for example, [10] ) : H0 = v + √1 (cid:104)(cid:0)h0 +iG0(cid:1)sinβ+(cid:0)H0+iA0(cid:1)cosβ(cid:105), (8) u u v v v 2 H0 = v + √1 (cid:104)(cid:0)h0 −iG0(cid:1)cosβ −(cid:0)H0−iA0(cid:1)sinβ(cid:105), (9) d d v v v 2 S = s+ √1 (cid:0)h0+iA0(cid:1) , (10) s s 2 6 (cid:113) where v = v2 +v2 ≈ 174 GeV, tanβ = v /v and G0 is the neutral would-be Nambu-Goldstone u d u d boson. The extremization conditions of the scalar potential are v (cid:104)m2 +µ2+λ2v2+ g12+g22 (cid:0)v2 −v2(cid:1)(cid:105)−v µB = 0, u Hu d 4 u d d v (cid:104)m2 +µ2+λ2v2 + g12+g22 (cid:0)v2−v2(cid:1)(cid:105)−v µB = 0, d Hd u 4 d u u s(cid:0)m2 +κA s+2κ2s2+λ2v2−2κλv v (cid:1)−λv v A = 0, (11) S κ u d u d λ where µ = λs and B = A +κs. We can use the above three equations in order to replace the λ parameters m2 , m2 and m2 by the three VEVs. Hu Hd S The CP-even mass squared matrix in the above basis of h0, H0, h0 is v v s   λ2v2sin22β+m2 cos22β 2rv2cot2β 2λ2sv−2v2R Z M2 =  · −2v2r+2λκs2+Aλs −2Rvcot2β  , (12)  sin2β  · · κs(4κs+A )+ v2A λsin2β κ 2s λ where r ≡ (cid:16)λ2 − m2Z(cid:17)sin22β, and R = 1λ(κs+ 1A )sin2β. 2 2v2 v 2 λ The basis used in eq. (12) has the advantage that it shows clearly what state couples linearly to the SM gauge bosons, such that it can be produced at LEP2 in association with a Z boson. In fact h0 is rotated in the same way as v, so that it is exactly the linear combination of H0 and H0 which v u d is responsible for the masses of the W and Z gauge bosons. Thus only h0 has tree-level couplings v to WW/ZZ, while H0 and h0 do not. This implies that only the component of h0 in each CP-even v s v mass eigenstate is relevant for the LEP2 limit. Similarly, to the extent that ATLAS and CMS collaborations at the LHC have observed a SM- like Higgs at 125 GeV, in particular with SM-like coupling to WW/ZZ, this state (denoted by h) should be predominantly h0, with small admixture of H0, h0. v v s The CP-odd scalar mass squared matrix in the above basis of A0 and A0 is given by v s (cid:32) (cid:33) 2λs(A +κs) λ λv(A −2κs) M2 = sin2β λ . (13) A λv2(A +4κs)sin2β · λ −3κA s 2s κ The Lagrangian is such that we get a light CP-odd scalar in the two approximate symmetry limits: the U(1) symmetric limit when the A-terms are small; and the Peccei-Quinn symmetric limit R when κ and A are small. The light CP-odd scalars then correspond to pseudo-Nambu-Goldstone κ bosons from spontaneous breaking of these symmetries. If their mass is below about 10 GeV, there can be a constraint coming from Υ-decay into these CP-odd scalars. In general, the CP-odd states can also be produced in association with the CP-even scalars at LEP2. Finally, there are physical charged Higgs bosons whose mass is given by 2µB (cid:18)g2 (cid:19) m2 = +v2 2 −λ2 . (14) H± sin2β 2 7 Although it is not directly relevant for our analysis, we refer the reader to [10] for the mass matrix and the properties of the neutralinos and charginos. Without loss of generality we take λ > 0 while κ and µ can have either signs. Meanwhile, in the examples shown in the following sections we take k > 0 and µ > 0, as we do not expect the other choices of signs to result in any qualitative change of our findings. 3 Getting SM-like Higgs of mass 125 GeV Asmentionedabove,the125GeVSM-likeHiggsdiscoveredattheLHCislikelytobepredominantly h0, with a small admixture of H0, h0. Neglecting this mixing, its mass squared is given by M2 v v s 11 in eq. (12): the m2 piece is the same as in the MSSM and the λ2 piece results from the extra Z quartic of the NMSSM. However, as mentioned in the introduction, the mixing of h0 with H0 and v v h0 modifies the mass from the above value, lowering (raising) if h0 mixes with heavier (lighter) s v states. In fact the SM-like Higgs mass can be written as: m2 = m2 cos22β +δm2 + h Z hloop λ2v2sin22β+δm2 . (15) hmix We further discuss below, in turn, each term in the above formula. The first line is the MSSM value, including the stop mixing effect for which we use the approximate formula from eqs. (69-71) of [19]: δm2 = 3m¯4t (cid:104)ln Mt˜+ Xt + ln Mmtt˜ (cid:0)3 m2/v2−16g2(cid:1)(cid:18)X +2ln Mt˜(cid:19)(cid:105) (16) hloop 2π2v2 m 4 32π2 t s t m t t where the “stop-mixing” parameter X is given by t 2(A −µ/tanβ)2(cid:104) (A −µ/tanβ)2(cid:105) t t X = 1− , (17) t M2 12M2 t˜ t˜ m¯ = m /(1+4α /(3π))istheMStopquarkmass,m beingthepolemass,g istheQCDcoupling t t s t s and M2 ≡ m m is the geometric average of the two stop mass eigenvalues. t˜ t˜1 t˜2 Wedonotassumead-hoclargemixinginthestopsectorwhichtendstoincreasetheHiggsmass. We also assume 500 GeV as a rough upper limit of stop mass M as it is favored by naturalness of t˜ weak scale 4. So we take typical values µ = 200 GeV, tanβ = 1.5, M (cid:46) 500 GeV and A = 0 in t˜ t eq. (16) and find the stop loop contribution to m can be up to 15 GeV. Therefore our goal here is h to achieve a tree-level mass for the SM-like Higgs of 110 GeV or more, up to 125 GeV. Of course if one would allow a heavier, hence less natural stop our analysis would be overly restrictive. 4Asemphasizedinreference[3],forλ∼2,even1.5TeVstopmasscanbenatural. However,forsuchlargevalues ofλ,thetree-levelboundisalreadywellabove125GeVandsoalargeloopcontributiontoHiggsmassfromaheavy stop is not really needed anyway. 8 Moving onto the second line in eq. (15), the first term here is the contribution characteristic of the NMSSM coupling λ. Notice that for a moderate or large λ, in the NMSSM, m is maximized h at tanβ ≈ 1 in contrast to the case in the MSSM, where m is maximized at tanβ → ∞. Thus, h we choose small tanβ in our analysis. As it will play an important role in what follows, we have introduced the notation δm2 in the second line here to encode the effect from the mixing of h0 hmix v with H0 and h0. v s For the discussion here, and for our analytic study in Sections 4 and 5, we shall neglect the h0−H0 mixing and only deal with the h0−h0 mixing part of δm2 , reducing the 3-by-3 mixing v v v s hmix problem into a simpler two states mixing problem. The justifications of this approximation are as follows. Typically we have to choose parameters so that the h0 − H0 and h0 − H0 mixings are v v s v small effects. The reason is that H0 must be somewhat heavy, because H0 is accompanied by a v v charged Higgs of similar mass which is constrained by b → sγ [23]. As H0 gets heavier its effect v on m decouples since the h0 −H0 mass mixing term does not scale with H0 mass, as shown in h v v v eq.(12). FurthermoreintheNMSSMsmalltanβ iswellmotivatedandtheh0−H0 mixingactually v v vanishes at tanβ = 1. In contrast, the effect of the h0 −h0 mixing on SM-like Higgs mass cannot v s be decoupled. The main reason is that both the h0 −h0 mass term and h0 mass scale with the v s s VEV for S. Similar to the h0 −H0 mixing we neglect the h0 −H0 mixing since it affects the mass of the v v s v doublet like state only at higher order and it also vanishes at tanβ = 1. We stress that we neglect h0−H0 and h0−H0 mixing only to give analytical arguments, of course, the full mixing is included v v s v in our numerical analysis. As anticipated in the introduction, there are two possibilities for the mass of the SM-like Higgs boson as far as the effect of the mixing effect in eq. (15) is concerned: • The “pull-down” case where the singlet-like is heavier than SM-like Higgs, such that the mixing reduces SM-like Higgs mass. In this case, (i) M2 < M2 and (ii) M2 > (110 GeV)2 such that after the pull-down mixing 11 33 11 effect, we end up with a SM-like Higgs tree-level mass of 110 GeV. Obviously, eq. (12) then suggests that λ (cid:38) 0.7. • The “push-up” case where the singlet is lighter that the SM-like state, so that the mixing increases the SM-like Higgs mass. By definition this implies that (i) M2 > M2 and (ii) M2 < (110 GeV)2 since, after the 11 33 11 effect of the mixing is included, we want to get to 110 GeV at tree-level. Therefore, based on eq. (12), we must have λ (cid:46) 0.7. Elaborating on the above discussion we are in position to outline the following argument. If the model has to be perturbative up to the GUT scale, then in the pull-down scenario, the region of 9 tanΒ(cid:61)1.5 130 125 (cid:68)V 120 e G (cid:64) 115 x a m mh 110 105 Non(cid:45)Perturbative 100 0.60 0.62 0.64 0.66 0.68 0.70 Λ Figure 1: The green solid line is the maximal mass of the SM-like Higgs boson for tanβ = 1.5 from eq. (15) in the “pull-down” scenario. The two horizontal blue dashed lines are the reference values 125 GeV, that is about the observed mass of the SM-like boson, and 110 GeV, that is the minimal value of the tree-level mass to attain 125 GeV after the inclusion of radiative corrections from the stop, taking M (cid:46) 500 GeV. The light (dark) purple shaded regions mark the values of λ that, t˜ √ according to the one-loop RGE, get larger than 2π, our definition for non-perturbative, below the GUT scale for κ = 0.1 (κ = 0) at the weak scale. To allow larger values of λ to be compatible with perturbative unification we have introduced extra matter in 4 copies of 5+¯5 of SU(5) at 5 TeV. the NMSSM parameter space that can give a SM-like Higgs mass as large as 125 GeV is not at all a generic one. For the push-up case we shall indicate how we get a similar conclusion, though for a different reason. If the coupling λ must remain perturbative up to the GUT scale, it cannot exceed a generously estimated upper-bound at about 0.7 5. Even plugging in such large value of λ in eq. (15) one finds that, in the pull-down scenario, effect of the mixing must be minimal to keep the physical Higgs mass above 110 GeV at the tree-level. This fact was already mentioned in the introduction, but now we demonstrate it in Figure 1 and it suggests that the parameters of the model must be arranged in a specific pattern such to have δm2 (cid:39) 0. (18) hmix Thisconspiracyofparametersseemsratherunnatural,orcanbetakenasasuggestionforapeculiar relation that must come from a UV construction, and therefore we introduce a measure of “tuning” to quantify to what extent we are requiring cancellations among apparently unrelated parameters of the model. In a fashion similar to what is done for the tuning of the EW scale (or Z boson mass) as given in eq. (2), we introduce the tuning measure for the Higgs mass and expand the expression using 5Infactλistypicallyconstrainedtobebeloworaround0.6,asingeneraltheneedforanon-vanishingκreduces the maximal value of λ compatible with perturbativity up to the GUT scale. 10