A Fake Split Supersymmetry Model for the 126 GeV Higgs
AA Fake Split-Supersymmetry Modelfor the 126 GeV Higgs
Karim Benakli ♣ , Luc Darm´e ♥ , Mark D. Goodsell ♦ and Pietro Slavich ♠
1– Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7589, LPTHE, F-75005, Paris, France2– CNRS, UMR 7589, LPTHE, F-75005, Paris, France
Abstract
We consider a scenario where supersymmetry is broken at a high energy scale, outof reach of the LHC, but leaves a few fermionic states at the TeV scale. The particlecontent of the low-energy effective theory is similar to that of Split Supersymmetry.However, the gauginos and higgsinos are replaced by fermions carrying the same quan-tum numbers but having different couplings, which we call fake gauginos and fakehiggsinos. We study the prediction for the light-Higgs mass in this Fake Split-SUSYModel (FSSM). We find that, in contrast to Split or High-Scale Supersymmetry, a126 GeV Higgs boson is easily obtained even for arbitrarily high values of the super-symmetry scale M S . For M S (cid:38) GeV, the Higgs mass is almost independent ofthe supersymmetry scale and the stop mixing parameter, while the observed value isachieved for tan β between 1 . . ♣ [email protected] ♥ [email protected] ♦ [email protected] ♠ [email protected] a r X i v : . [ h e p - ph ] J un ontents M S , the FSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Gauge coupling unification . . . . . . . . . . . . . . . . . . . . . 82.2.2 Mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The LHC experiments have completed the discovery of all of the particles predicted bythe Standard Model (SM). The uncovering of the last building block, the Higgs boson[1, 2], opens the way for a more precise experimental investigation of the electroweaksector. Of particular interest is understanding the possible role of supersymmetry.Supersymmetry (SUSY) can find diverse motivations. From a lower-energy pointof view, (i) it eases the problem of the hierarchy of the gauge-symmetry-breakingscale versus the Planck scale; (ii) it provides candidates for dark matter; (iii) it allowsunification of gauge couplings and even predicts it within the Minimal SupersymmetricStandard Model (MSSM). On the other hand, supersymmetry can be motivated as anessential ingredient of the ultraviolet (UV) theory, having String Theory in mind. In thelatter framework, there is no obvious reason to expect supersymmetry to be brokenat a particular scale, which is usually requested to be much below the fundamentalone. The original motivations of low-energy supersymmetry might then be questioned.In fact, the Split-Supersymmetry model [3–5] abandons (i) among the motivations ofsupersymmetry, while retaining (ii) and (iii). The idea of Split SUSY is to consideran MSSM content with a split spectrum. All scalars but the lightest Higgs are takento be very massive, well above any energy accessible at near-future colliders, while thegauginos and the higgsinos remain light, with masses protected by an approximate R -symmetry.It is important to emphasise that, even if supersymmetry is broken at an arbitrarilyhigh scale M S , its presence still has implications at low energy for the Higgs mass. ndeed, in supersymmetric models the value of the Higgs quartic coupling is fixed oncethe model content and superpotential couplings are given. This provides a boundarycondition at M S for the renormalisation group (RG) evolution down to the weak scaleto predict the value of the Higgs mass. As a result, in the Split-SUSY model theprediction for the light-Higgs mass can be in agreement with the measured value ofabout 126 GeV only for values of M S not exceeding about 10 GeV (see for examplerefs [6, 7]).In this paper we consider a new scenario. In the spirit of Split SUSY, we assumethat fine-tuning is responsible for the presence of a light Higgs. A first difference isthat our UV model is not the MSSM but it is extended by additional states in theadjoint representation of the SM gauge group, as in ref. [8]. Such a field contenthas been discussed in the so-called Split Extended SUSY [9, 10] (see also [11, 12] forrelated work), where it was assumed that the additional gaugino-like and higgsino-likestates arise as partners of the SM gauge bosons under an extended supersymmetry, anddifferent hierarchies between the Dirac and Majorana masses have been considered inref. [13]. Furthermore, in related work a similar scenario to ours was recently presentedin [14].A fundamental difference between our scenario and the usual Split SUSY or theclosely related models mentioned above is that R -symmetry is strongly broken and doesnot protect the gauginos from obtaining masses comparable to the scalar ones [14]. Inthe simplest realisation presented here, in order to keep the extra states light, we endowthem with charges under a new U (1) F symmetry. An N = 2 supersymmetry origin ofthe new states [9, 10] raises then the difficulty of embedding U (1) F in an R -symmetryand will not be discussed here.For M S lower than the MSSM GUT scale M GUT ≈ × GeV, the achievement ofunification requires additional superfields which restore convergence of the three SMgauge couplings. This set of extra states can be chosen to be the ones that are requiredfor unification of gauge couplings in Dirac gaugino unified models [15], and can safelybe assumed to appear only above M S , not affecting the discussion in this work: theproperties of our model are fixed at M S and, as we shall establish, any corrections thatwe cannot determine are tiny.Below the supersymmetry scale M S , the field content of the model is the same as inthe usual Split SUSY, but the gauginos are replaced by very weakly coupled fermions inthe adjoint representation that we call “fake gauginos”, and the higgsinos are replacedby weakly coupled fermion doublets that we call “fake higgsinos”. At the TeV scalethe model looks like Split SUSY with fake gauginos and higgsinos, hence the name ofFake Split-SUSY Model (FSSM). As we will show, a remarkable consequence of thedifferent couplings of the fake gauginos and higgsinos to the Higgs boson, compared tothe usual gauginos and higgsinos, is that a prediction for the Higgs mass compatiblewith the observed value can be obtained for arbitrarily high values of M S .The plan of the paper is as follows. In section 2, we describe the field contentof the Fake Split-SUSY Model and a possible realisation using a broken additional U (1) F symmetry. The latter will be at the origin of the desired hierarchy betweendifferent couplings and mass parameters. We explain how the effective field theory ofthe FSSM compares with the usual Split SUSY. Section 3 briefly discusses the colliderand cosmological constraints. Section 4 presents the predictions of the model for the iggs mass. The assumptions, inputs and approximations used in the computationare described in section 4.1, while numerical results are presented in section 4.2. Wealso provide a comparison with the cases of Split SUSY and High-Scale SUSY, showingthe improvement for fitting the experimental value of the Higgs mass for arbitrarilyhigh values of the supersymmetry scale M S . Our main results, and open questionsrequiring further investigation, are summarised in the conclusions. Finally, the two-loop renormalisation group equations (RGEs) for the mass parameters of Split SUSYare given in an appendix. At the high SUSY scale M S , we extend the MSSM by additional chiral superfields anda U (1) F symmetry. There are three sets of additional states: Fake gauginos (henceforth, F-gauginos) are fermions χ Σ in the adjoint repre-sentation of each gauge group, which sit in a chiral multiplet Σ having scalarcomponent Σ. These consist of: a singlet S = S + √ θχ S + . . . ; an SU (2) triplet T = (cid:80) a T a σ a /
2, where T a = T a + √ θχ aT + . . . and σ a are the three Paulimatrices; an SU (3) octet O = (cid:80) a O a λ a /
2, where O a = O a + √ θχ aO + . . . and λ a are the eight Gell-Mann matrices.2. Higgs-like SU (2) W doublets H (cid:48) u and H (cid:48) d (henceforth, F-Higgs doublets) withfermions appearing as fake higgsinos (henceforth, F-higgsinos).3. Two pairs of vector-like electron superfields (i.e. two pairs of superfields withcharges ± U (1) Y ) with a supersymmetric mass M S . For M S (cid:46) M GUT these fields restore the possibility of gauge coupling unification, because theyequalise the shifts in the one-loop beta functions at M S of all of the gauge groupsrelative to the MSSM [15].In contrast to the usual Split-SUSY case – and also in contrast to the usual Diracgaugino case – we do not preserve an R -symmetry. This means that the gauginos havemasses at M S , moreover the higgsino mass is not protected, thus a µ term of order M S will be generated for the higgsinos.However, we introduce an approximate U (1) F symmetry under which all the adjointsuperfields and the F-Higgs fields H (cid:48) u and H (cid:48) d have the same charge. The breakingof this symmetry is determined by a small parameter ε which may correspond to theexpectation value of some charged field divided by the fundamental mass scale of thetheory (at which Yukawa couplings are generated); this reasoning is familiar fromflavour models. We can write the superpotential of the Higgs sector of the theory In the following, bold-face symbols denote superfields. chematically as W = µ H u · H d + Y u U c Q · H u − Y d D c Q · H d − Y e E c L · H d + ε (cid:16) ˆ µ (cid:48) d H u · H (cid:48) d + ˆ µ (cid:48) u H (cid:48) u · H d + ˆ Y (cid:48) u U c Q · H (cid:48) u − ˆ Y (cid:48) d D c Q · H (cid:48) d − ˆ Y (cid:48) e E c L · H (cid:48) d (cid:17) + ε (cid:16) ˆ λ S S H u · H d + 2 ˆ λ T H d · T H u (cid:17) + ε (cid:16) ˆ λ (cid:48) Sd S H u · H (cid:48) d + ˆ λ (cid:48) Su S H (cid:48) u · H d + 2 ˆ λ (cid:48) T u H d · T H (cid:48) u + 2 ˆ λ (cid:48) T d H (cid:48) d · T H u (cid:17) + ε ˆ µ (cid:48)(cid:48) H (cid:48) u · H (cid:48) d + ε (cid:20)
12 ˆ M S S + ˆ M T Tr( TT ) + ˆ M O Tr( OO ) (cid:21) , (2.1)where we have neglected irrelevantly small terms of higher order in ε . Even if chosen tovanish in the supersymmetric theory, some parameters in eq. (2.1), such as the bilinear µ terms, obtain contributions when supersymmetry is broken. In order to keep trackof the order of suppression, we have explicitly extracted the parametric dependence on ε due to the U (1) F charges, such that all the mass parameters are of O ( M S ), and allthe dimensionless couplings are either of order one or suppressed by loop factors.Note that the “fake states” can appear as partners of the MSSM gauge bosonsunder an extended N = 2 supersymmetry that is explicitly broken at the UV scaleto N = 1. The imprint of N = 2 is the extension of the states in the gauge sectorinto gauge vector multiplets and Higgs hyper-multiplets which give rise to the fakegauginos and higgsinos when broken down to N = 1. The quarks and leptons of theMSSM should be identified with purely N = 1 states. The difficulty of such a scenarioresides in making only parts of the N = 2 multiplets charged under U (1) F . It is thentempting to identify the U (1) F as part of the original R -symmetry. We will not pursuethe discussion of such possibility here.We will now review the spectrum of states resulting from eq. (2.1). The Higgs softterms, and thence the Higgs mass matrix, can be written as a matrix in terms of thefour-vector v H ≡ ( H u , H d ∗ , H (cid:48) u , H (cid:48) ∗ d ) − M S L soft ⊃ v † H O (1) O (1) O ( ε ) O ( ε ) O (1) O (1) O ( ε ) O ( ε ) O ( ε ) O ( ε ) O (1) O ( ε ) O ( ε ) O ( ε ) O ( ε ) O (1) v H . (2.2)In the spirit of the Split-SUSY scenario, the weak scale is tuned to have its correctvalue, and the SM-like Higgs boson is a linear combination of the original Higgs andF-Higgs doublets: H u ≈ sin β H + . . . , H d ≈ cos β iσ H ∗ + . . . , (2.3) H (cid:48) u ≈ ε H + . . . , H (cid:48) d ≈ ε iσ H ∗ + . . . , (2.4)where β is a mixing angle and the ellipses stand for terms of higher order in ε . Dueto the suppression of the mixing between the eigenstates by the U (1) F symmetry, thispattern is ensured. Note that, if we wanted to simplify the model, we could impose We use a (cid:100) hat to denote the suppressed terms. n additional unbroken symmetry under which the F-Higgs fields transform and arevector-like – for example, lepton number. In this way we would remove the mixingbetween the Higgs and F-Higgs fields. This is unimportant in what follows, since weare only interested in the light fields that remain.Eqs (2.3) and (2.4) show that the SM-like Higgs boson is, to leading order in ε , alinear combination of the fields H u and H d . Thus, the Yukawa couplings are unaffectedcompared to the usual Split-SUSY scenario. The original higgsinos are rendered heavy,while the light fermionic eigenstates consist of ˜ H (cid:48) u and ˜ H (cid:48) d , with mass µ of O ( ε M S )and an O ( ε ) mixing with the original higgsinos.Since we are not preserving an R -symmetry, the original gaugino degrees of free-dom will obtain masses of O ( M S ), and we will also generate A -terms of the sameorder (although there may be some hierarchy between them if supersymmetry break-ing is gauge-mediated). On the other hand, since the adjoint fields transform under a(broken) U (1) F symmetry, Dirac mass terms for the gauginos and also masses for theadjoint fermions are generated by supersymmetry breaking, but they are suppressedby one and two powers of ε , respectively. We can write the masses for the gauginos λ and the adjoint fermions χ as − ∆ L gauginos = M S (cid:20) λλ + O ( ε ) λχ + O ( ε ) χχ + h . c . (cid:21) , (2.5)giving a gaugino/F-gaugino mass matrix M / ∼ M S (cid:18) O ( ε ) O ( ε ) O ( ε ) (cid:19) . (2.6)This leaves a heavy eigenstate of O ( M S ) and a light one of O ( ε M S ), where the lighteigenstate is to leading order χ + O ( ε ) λ .We will assume that the Dirac masses are generated by D-terms of similar orderto the R -symmetry-violating F-terms. This means that B -type mass terms for theadjoint scalars are generated of O ( ε M S ) too. However, the usual supersymmetry-breaking masses for the adjoint scalars S, T, O will not be suppressed, and thereforewill be at the scale M S : − ∆ L adjoint scalars = M S (cid:20) | Σ | + O ( ε )( 12 Σ + 12 Σ ∗ ) (cid:21) . (2.7)This is straightforward to see in the case of gravity mediation, and in the case of gaugemediation we see that the triplet/octet adjoint scalars acquire these masses – as thesfermions do – at two loops (while in this case the singlet scalar would have a massat an intermediate scale, but couplings to all light fields suppressed). This resolves ina very straightforward way the problem, typical of Dirac gaugino models, of havingtachyonic adjoints [16–18]. M S , the FSSM Below the supersymmetry scale M S , we can integrate out all of the heavy states andfind that the particle content of the theory appears exactly the same as in Split SUSY: his is why we call the scenario Fake Split SUSY. Above the electroweak scale, we haveF-Binos ˜ B (cid:48) , F-Winos ˜ W (cid:48) and F-gluinos ˜ g (cid:48) with (Majorana) masses m ˜ B (cid:48) , m ˜ W (cid:48) and m ˜ g (cid:48) ,respectively, and F-higgsinos ˜ H (cid:48) u,d with a Dirac mass µ .We can also determine the effective renormalisable couplings. The F-gauginos andF-higgsinos have their usual couplings to the gauge fields. The F-gluinos have onlygauge interactions, whereas there are in principle renormalisable interactions betweenthe Higgs, F-higgsinos and F-electroweakinos. The allowed interactions take the form L eff ⊃ − H † √ g u σ a ˜ W (cid:48) a + ˜ g u ˜ B (cid:48) ) ˜ H (cid:48) u − H T iσ √ − ˜ g d σ a ˜ W (cid:48) a + ˜ g d ˜ B (cid:48) ) ˜ H (cid:48) d . (2.8)Since the gauge couplings of all the particles are the same as in the usual Split-SUSYcase, the allowed couplings take the same form. However, the values differ greatly. Thecouplings in eq. (2.8) descend from the gauge current terms, given by L gauge current ⊃ − H † u √ g σ a λ a + g (cid:48) λ Y ) ˜ H u − H † d √ g σ a λ a − g (cid:48) λ Y ) ˜ H d − H (cid:48) † u √ g σ a λ a + g (cid:48) λ Y ) ˜ H (cid:48) u − H (cid:48) † d √ g σ a λ a − g (cid:48) λ Y ) ˜ H (cid:48) d , (2.9)where λ , λ Y are the gauginos of SU (2) and hypercharge in the high-energy theory,but there are also terms of the same form from the superpotential terms ε ˆ λ S,T , ε ˆ λ (cid:48) Su,d , ε ˆ λ (cid:48) T u,d involving the fields χ S and χ T . When we integrate out the heavy fields, wethen see that in our model the couplings are doubly suppressed:˜ g u ∼ ˜ g d ∼ ˜ g u ∼ ˜ g d ∼ ε . (2.10)We recall that, in the usual Split-SUSY case, we would have instead ˜ g u = g sin β ,˜ g d = g cos β , ˜ g u = g (cid:48) sin β and ˜ g d = g (cid:48) cos β , where β is the angle that rotates theHiggs doublets H u and H d into one light, SM-like doublet and a heavy one.The remaining renormalisable coupling in the theory is the Higgs quartic coupling λ , which at tree level is determined by supersymmetry to be λ = 14 (cid:0) g + g (cid:48) (cid:1) cos β + O ( ε ) . (2.11)The tree-level corrections at O ( ε ) come from the superpotential couplings ˆ λ S andˆ λ T , and from the O ( ε ) mixing between the Higgs and F-Higgs fields. Additional O (1)contributions to this relation could arise if the SUSY model above M S included new,substantial superpotential (or D-term) interactions involving the SM-like Higgs, butthis is not the case for the model described in section 2.1. There are, however, smallloop-level corrections to eq. (2.11), which we will discuss in section 4.The O ( ε ) corrections to the ˜ g (1 , u,d ) and λ couplings are not determined fromthe low-energy theory and are thus unknown. However, in this study we focus onmodels where the set of F-gauginos and F-higgsinos lies in the TeV mass range, whichcorresponds to values of ε of the order of ε ∼ (cid:114) TeV M S , (2.12) hich gives a ε ranging between 10 − to 10 − when M S goes from the highest GUTscale of 10 GeV down to 100 TeV, the lowest scale considered here. With such valuesof ε , we have verified that we can safely neglect the contribution of ˜ g (1 , u,d ) to therunning of the Higgs quartic coupling, and that the shift in the Higgs mass due to thetree-level corrections to λ is less than 2 GeV for M S >
100 TeV, falling to a negligiblysmall amount for M S > One of the main features of the MSSM that is preserved by the split limit is theunification of gauge couplings. At the one-loop level, the running of gauge couplingsin our model is the same as in Split SUSY because the Yukawa couplings only enter attwo-loop level. However, we have verified that gauge-coupling unification is maintainedat two loops in our model.
From the discussion above we can then read off the mass matrices after electroweaksymmetry breaking. In the basis ( ˜ B (cid:48) , ˜ W (cid:48) , ˜ H (cid:48) d , ˜ H (cid:48) u ) the neutralino mass matrix is M χ = m ˜ B (cid:48) ε M Z ε M Z m ˜ W (cid:48) ε M Z ε M Z ε M Z ε M Z − µε M Z ε M Z − µ . (2.13)We see that there is a mixing suppressed by ε = TeV M S . For example, if the F-higgsino isthe lightest eigenstate, it will be approximately Dirac with a splitting of the eigenvaluesof order ε M Z /µ ∼ (cid:16) TeV M S (cid:17) M Z .We then write the chargino mass matrix involving the ˜ H (cid:48) + , ˜ H (cid:48)− and the chargedF-gauginos ˜ W (cid:48) + and ˜ W (cid:48)− . The mass terms for the charginos can be expressed in theform − ( v − ) T M χ ± v + + h . c . , (2.14)where we have adopted the basis v + = ( ˜ W (cid:48) + , ˜ H (cid:48) + u ), v − = ( ˜ W (cid:48)− , ˜ H (cid:48)− d ). This gives M χ ± = (cid:18) m ˜ W (cid:48) ε M W ε M W µ (cid:19) . (2.15)Again we have very little mixing.Clearly, the mixing coefficients of order ε in the mass matrices are dependent onquantities in the high-energy theory that we cannot determine. However, because theyare so small, they have essentially no bearing on the mass spectrum of the theory(although they will be relevant for the lifetimes). From now on, given the smallness of ε , we shall not keep explicit track of the numerical coefficients infront of it, thus we will use ε n as a shorthand for O ( ε n ). Comments on cosmology and colliders
The signatures of Fake Split SUSY concern the phenomenology of the F-higgsinosand F-gauginos, and thus share many features with the usual Split-SUSY case. Theydiffer quantitatively in that the lifetimes are parametrically enhanced: the decay ofheavy neutralinos and the F-gluino to the lightest neutralino must all proceed ei-ther via ε -suppressed mixing terms or via sfermion interactions, and, since the F-higgsinos/gauginos only couple to sfermions via mixing, each vertex is therefore sup-pressed by a factor of ε or ε . Hence the lifetimes are enhanced by a factor of ε − [14,19];in particular the F-gluino lifetime is τ ˜ g (cid:48) (cid:39) ε × (cid:18) M S GeV (cid:19) × (cid:18) m ˜ g (cid:48) (cid:19) ∼ sec × (cid:18) M S GeV (cid:19) × (cid:18) m ˜ g (cid:48) (cid:19) , (3.1)where on the second line we used m ˜ g (cid:48) = ε M S . The constraints from colliders thendepend upon whether the gluino decays inside or outside the detector; the latter willoccur for M S (cid:38) . M S (cid:46) GeV.While the bound is no longer necessarily exact, because the relationship between themass and lifetime is different in our case, it still approximately applies. If, on theother hand, the gluino decays well after the end of BBN such that it deposits verylittle energy at BBN times, then other constraints become relevant: it can distort theCMB spectrum and/or produce photons visible in the diffuse gamma-ray background.Finally, when the gluino becomes stable compared to the age of the universe, in ourcase corresponding to M S (cid:38) GeV, very strong constraints from heavy-isotopesearches become important, as we shall briefly discuss below.
One way to attempt to allow the gluino to decay is to have a gravitino LSP. In minimallycoupled supergravity, the gravitino has mass F √ M P , where F is the order parameter ofsupersymmetry breaking. If supersymmetry breaking is mediated at tree level to thescalars, the supersymmetry scale could be as high as √ F (we could even have somefactors of π if we allow for a strongly coupled SUSY-breaking sector, but that will notsubstantially affect what follows) and so we could potentially have a gravitino lighter han the gluino if M S (cid:46) × GeV × (cid:16) m ˜ g (cid:48) (cid:17) / . (3.2)In this case, the F-gluino can decay to a gravitino and either a gluon or quarks, po-tentially avoiding the above problems. However, this relies on the couplings to thegoldstino; since we have added Dirac and fake-gaugino masses, these are no longer thesame as in the usual Split-SUSY case, and a detailed discussion will be given else-where [25]. The effective goldstino couplings are the Wilson coefficients C ˜ Gi of ref. [19],and in our model we find C ˜ Gi = − ε g s √ , i = 1 ... C ˜ G = − ε m ˜ g (cid:48) √ . (3.3)For i = 1 ... g (cid:48) → ˜ G + X ) (cid:39) ε m g (cid:48) πF (3.4)and hence, for M S ∼ √ F (the maximal value), m ˜ g (cid:48) = ε M S , we find the F-gluinolifetime to be τ ˜ g (cid:48) (cid:39)
600 sec × (cid:18) M S GeV (cid:19) × (cid:18) m ˜ g (cid:48) (cid:19) . (3.5)Hence this cannot be useful to evade the cosmological bounds: the gravitino couplingsare simply too weak. For F-gluinos stable on the lifetime of the universe, in our case corresponding to M S (cid:38) GeV, remnant F-gluinos could form bound states with nuclei, which would bedetectable as exotic forms of hydrogen. The relic density is very roughly approximatedby Ω ˜ g h ∼ (cid:16) m ˜ g (cid:48)
10 TeV (cid:17) , (3.6)although this assumes that the annihiliations freeze out before the QCD phase tran-sition and are thus not enhanced by non-perturbative effects; for heavy F-gluinos thisseems reasonable, but in principle the relic density could be reduced by up to three or-ders of magnitude. However, the constraints from heavy-isotope searches are so severeas to render this moot: the ratio of heavy isotopes to normal hydrogen X/H shouldbe less than 10 − for masses up to 1 . − for masses up to10 TeV [27], whereas we find XH ∼ − (cid:16) m ˜ g (cid:48) TeV (cid:17) . (3.7)If the F-gluino is stable, then we must either: . Dilute the relic abundance of F-gluinos with a late period of reheating.2. Imagine that the reheating temperature after inflation is low enough, or thatthere are several periods of reheating that dilute away unwanted relics before thefinal one.In both cases, we must ensure that gluinos are not produced during the reheatingprocess itself, which may prove difficult to arrange: even if the late-decaying particledecays only to SM fields, if it is sufficiently massive then high-energy gluons may beamong the first decay products, which could subsequently produce F-gluinos whichwould not be able to annihilate or decay away.The safest solution would be for a decaying scalar to have a mass near or belowtwice the F-gluino mass. Then we must make sure that the decays where the productsinclude only one F-gluino – and, because of the residual R-parity, one neutralino –are sufficiently suppressed, assuming that the neutralino is somewhat lighter thanthe F-gluino. However, such processes are suppressed by a factor of ε , which shouldsufficiently reduce the branching fraction of decays by a factor of 10 if M S (cid:38) GeV.Such a scenario would possibly still have difficulty producing sufficient dilution if theuniverse is thermal before the final reheating: suppose that the final reheating occurswhen the universe is at a temperature T decay and reheats the universe to a temperature T R , then the dilution is of order (cid:16) T R T decay (cid:17) . However, if we require the universe toundergo BBN only once, then both temperatures are bounded: T decay > T BBN ∼ MeV, but also T R (cid:46) m ˜ g (cid:48) /
50 to ensure that the freeze-in production of F-gluinos isnot too large. Then the amount of dilution achieved is only of order 10 for 2 TeVF-gluinos, insufficient to evade bounds from heavy-isotope searches.We conclude that for a high M S (cid:38) the most plausible cosmological scenariois option (2) above: a final reheating temperature T R (cid:46) m ˜ g (cid:48) /
50 which occurs eitherdirectly at the end of inflation or after at least one additional period of low-temperatureentropy injection.
Even though the F-gluino may be stable on the lifetime of the universe, the heavy neu-tralinos are not (although they may decay on BBN timescales in the case of extremelyhigh M S ): they can decay to the lightest neutralino and a Higgs boson via their ε -suppressed Yukawa couplings, so not involving any heavy mass scale. This suppressiondoes however render the F-bino effectively inert in the early universe once the heavyneutralinos have decoupled; the F-bino would be produced essentially by freeze-infrom decays and annihilations of the heavier neutralinos – which have usual weak-scalecross sections and so could potentially thermalise. Moreover, the charginos will still de-cay rapidly via unsuppressed weak interactions to their corresponding neutralino; themass splitting between charginos and neutralinos is produced by loops with electroweakgauge bosons and is of the order of a few hundred MeV. If we imagine a modulus inscenario (2) above that reheats the universe having mass less than twice that of the F-gluino, but greater than 2 m ˜ W (cid:48) or 2 µ , or where m ˜ W (cid:48) / , µ/ < T R (cid:46) m ˜ g (cid:48) /
50, we couldpotentially have a neutralino dark matter candidate, but the detailed investigation ofthis possibility is left for future work. Fitting the Higgs mass
Our procedure for the determination of the Higgs-boson mass is based on the onedescribed in ref. [6] for the regular Split-SUSY case. We impose boundary conditionson the MS-renormalised parameters of the FSSM, some of them at the high scale M S ,where we match our effective theory with the (extended) MSSM, and some othersat the low scale M Z , where we match the effective theory with the SM. We thenuse RG evolution iteratively to obtain all the effective-theory parameters at the weakscale, where we finally compute the radiatively corrected Higgs mass. However, inthis analysis we improved several aspects of the earlier calculation, by including thetwo-loop contributions to the boundary condition for the top Yukawa coupling, thetwo-loop contributions to the RG equations for the Split-SUSY parameters, as well assome two- and three-loop corrections to the Higgs-boson mass.At the high scale M S , the boundary condition on the quartic coupling of the light,SM-like Higgs doublet is determined by supersymmetry: λ ( M S ) = 14 (cid:20) g ( M S ) + 35 g ( M S ) (cid:21) cos β + O ( ε ) , (4.1)where g and g are the electroweak gauge couplings of the FSSM in the SU (5) nor-malisation (i.e. g = g and g = (cid:112) / g (cid:48) ), β is the mixing angle entering eq. (2.3),and the additional terms of O ( ε ), which we neglect, arise from suppressed superpo-tential couplings and from the mixing of the two MSSM-like Higgs doublets with theadditional F-Higgs doublets. In contrast with the Split-SUSY case, a large µ -termand A -terms are no longer forbidden by R -symmetry (as the latter is broken at thescale M S ), and the threshold corrections proportional to powers of | A t − µ cot β | /M S can in principle alter the boundary condition in eq. (4.1). For very large values of M S , the top Yukawa coupling that controls these corrections is suppressed, and theireffect on the Higgs mass is negligible. For lower values of M S , on the other hand, theeffect becomes sizable, and it can shift the Higgs mass by up to 6 GeV when M S ∼ GeV [7]. This allows us to obtain the desired Higgs mass for a lower value of tan β forfixed M S , or a lower M S for a given value of tan β . As our main purpose in this workis to study the possibility of pushing M S to its highest values, in the following we shalltake the stop-mixing parameter to be vanishing, and we will neglect all of the one-loopcorrections described in refs [6, 7].As mentioned in section 2.2, the effective Higgs–higgsino–gaugino couplings ˜ g u , ˜ g d ,˜ g (cid:48) u and ˜ g (cid:48) d are of O ( ε ), and we set them to zero at the matching scale M S . The RGevolution down to the weak scale does not generate non-zero values for those cou-plings, therefore, in contrast with the case of the regular Split SUSY, the F-higgsinosand F-gauginos have negligible mixing upon electroweak symmetry breaking, and theydo not participate in the one-loop corrections to the Higgs-boson mass. Indeed, theelectroweak F-gauginos and the F-higgsinos affect our calculation of the Higgs massonly indirectly, through their effect on the RG evolution and on the weak-scale bound-ary conditions for the electroweak gauge couplings, and we find that the precise valuesof their masses have very little impact on the prediction for the Higgs mass. On the ther hand, the choice of the F-gluino mass is more important due to its effect on theboundary conditions for the strong and top Yukawa couplings.To fix the soft SUSY-breaking F-gaugino masses, we take as input the physicalF-gluino mass M ˜ g (cid:48) , and convert it to the MS parameter m ˜ g (cid:48) evaluated at the scale M ˜ g (cid:48) according to the one-loop relation m ˜ g (cid:48) ( M ˜ g (cid:48) ) = M ˜ g (cid:48) g π , (4.2)where g is the strong gauge coupling of the FSSM. We then evolve m ˜ g (cid:48) up to thescale M S , where, for simplicity , we impose on the other two F-gaugino masses theGUT-inspired relations m ˜ B (cid:48) ( M S ) = (cid:20) g ( M S ) g ( M S ) (cid:21) m ˜ g (cid:48) ( M S ) , m ˜ W (cid:48) ( M S ) = (cid:20) g ( M S ) g ( M S ) (cid:21) m ˜ g (cid:48) ( M S ) . (4.3)We can then evolve all of the F-gaugino masses down to the weak scale. For whatconcerns the F-higgsino mass µ , we take it directly as an MS input parameter evaluatedat the scale M Z .The gauge and third-family Yukawa couplings, as well as the vacuum expectationvalue v of the SM-like Higgs (normalised as v ≈
174 GeV), are extracted from thefollowing set of SM inputs [28, 29]: the strong gauge coupling α s ( M Z ) = 0 . α ( M Z ) = 1 / . Z -boson mass M Z = 91 . G F = 1 . × − GeV − ;the physical top and tau masses M t = 173 . ± . M τ = 1 .
777 GeV; and therunning bottom mass m b ( m b ) = 4 .
18 GeV. We use the one-loop formulae given in theappendix A of ref. [6] to convert all the SM inputs into MS running parameters of theFSSM evaluated at the scale M Z . However, in view of the sensitivity of λ to the precisevalue of the top Yukawa coupling g t , we include the two-loop QCD contribution to therelation between the physical top mass M t and its MS counterpart m t . In particular,we use: m t ( M Z ) = M t g (4 π ) C + g (4 π ) (cid:16) C SM + C ˜ g (cid:48) (cid:17) + Σ t ( m t ) EW , (4.4)where g is computed at the scale M Z using eq. (A.1) of ref. [6], Σ t ( m t ) EW denotes theterms in the one-loop top self energy that do not involve the strong interaction, and C = 163 − M t M Z , (4.5) C SM = 282118 + 163 ζ (1 + ln 4) − ζ − M t M Z + 22 ln M t M Z , (4.6) C ˜ g (cid:48) = 899 + 4 ln m g (cid:48) M Z (cid:32)
133 + ln m g (cid:48) M Z − M t M Z (cid:33) . (4.7) Although the patterns of neutralino and chargino masses are important for collider searches, in ourmodel they have negligible impact on the Higgs mass and so the exact relation is not important. he boundary condition for the top Yukawa coupling of the FSSM is then given by g t ( M Z ) = m t ( M Z ) /v ( M Z ). The two-loop SM contribution C SM in eq. (4.6) is fromref. [30], while to obtain the two-loop F-gluino contribution C ˜ g (cid:48) in eq. (4.7) we adaptedthe results of ref. [31] to the case of a heavy Majorana fermion in the adjoint repre-sentation of SU (3). For an F-gluino mass of a few TeV, the inclusion of C ˜ g (cid:48) in theboundary condition for g t becomes crucial, as it changes the prediction for the Higgsmass by several GeV. Alternatively, one could decouple the F-gluino contribution fromthe RG evolution of the couplings below the scale M ˜ g (cid:48) , include only the SM contri-butions in the boundary conditions for g t and g at the scale M Z , and include thenon-logarithmic part of C ˜ g (cid:48) as a threshold correction to g t at the scale M ˜ g (cid:48) . We havechecked that the predictions for the Higgs mass obtained with the two procedures arein very good agreement with each other.To improve our determination of the quartic coupling λ at the weak scale, weuse two-loop renormalisation-group equations (RGEs) to evolve the couplings of theeffective theory between the scales M S and M Z . Results for the two-loop RGEs ofSplit SUSY have been presented earlier in refs [7, 32, 33]. Since there are discrepanciesbetween the existing calculations, we used the public codes SARAH [34] and
PyR@TE [35]to obtain independent results for the RGEs of Split SUSY in the MS scheme. Takinginto account the different conventions, we agree with the RGE for λ presented inref. [32], and with all the RGEs for the dimensionless couplings presented in section3.1 of ref. [33]. However, we disagree with ref. [33] in some of the RGEs for the massparameters (our results for the latter are collected in the appendix). Concerning theRGEs for the dimensionless couplings presented in ref. [7], we find some discrepancies in two-loop terms proportional to g and g .At the end of our iterative procedure, we evolve all the parameters to a commonweak scale Q W , and obtain the physical squared mass for the Higgs boson as M H = λ ( Q W ) √ G F (cid:104) − δ (cid:96) ( Q W ) (cid:105) + g t v π (cid:20) g (3 (cid:96) t + (cid:96) t ) − g t (cid:18) (cid:96) t − (cid:96) t + 2 + π (cid:19)(cid:21) + g g t v π ln m g (cid:48) Q W , (4.8)where (cid:96) t = ln( m t /Q W ). The one-loop correction δ (cid:96) ( Q W ), which must be computedin terms of MS parameters, is given in eqs (15a)–(15f) of ref. [36], while the two-loopcorrections proportional to g g t and to g t come from ref. [37]. We have also includedthe leading-logarithmic correction arising from three-loop diagrams involving F-gluinos,which can become relevant for large values of m ˜ g (cid:48) /Q W . This last term must of coursebe omitted if the F-gluinos are decoupled from the RGE for λ below the scale M ˜ g (cid:48) .In our numerical calculations we set Q W = M t to minimise the effect of the radiative In particular, in ref. [7] the coefficient of g in the RGEs for g t , g b , g τ , ˜ g u and ˜ g d should be changedfrom − / − /
4, while the coefficient of g in the RGEs for ˜ g u and ˜ g d should be changed from − / − /
12. In the RGE for λ , the terms proportional to g , λg and g g should be corrected in accordancewith ref. [32]. We thank A. Strumia for confirming these corrections. Higgs-mass predictions as a function of the SUSY scale M S for FSSM, High-Scale SUSY andSplit SUSY. We set M ˜ g (cid:48) = µ = 2 TeV and tan β = 1 or 40. The green-shaded region indicates a Higgs massin the range [124 , corrections involving top quarks, but we have found that our results for the physicalHiggs mass are remarkably stable with respect to variations of Q W . We find that, in the FSSM, the dependence of the physical Higgs mass on the SUSYscale M S differs markedly from the cases of regular Split SUSY or High-Scale SUSY(where all superparticle masses are set to the scale M S ). Figure 1 illustrates this dis-crepancy, showing M H as a function of M S for M ˜ g (cid:48) = µ = 2 TeV. The solid (black)curves represent the prediction of the FSSM, the dashed (red) ones represent the pre-diction of High-Scale SUSY, and the dot-dashed (blue) ones represent the predictionof regular Split SUSY (the predictions for the latter two models were obtained withappropriate modifications of the FSSM calculation described in section 4.1). For eachmodel, the lower curves were obtained with tan β = 1, resulting in the lowest possiblevalue of M H for a given M S , while the upper curves were obtained with tan β = 40.As was shown earlier in ref. [7], the Higgs mass grows monotonically with the SUSYscale M S in the Split-SUSY case, while it reaches a plateau in High-Scale SUSY. Inboth cases, the prediction for the Higgs mass falls between 124 and 127 GeV only fora relatively narrow range of M S , well below the unification scale M GUT ≈ × GeV.In the FSSM, on the other hand, the Higgs mass reaches a maximum and then startsdecreasing, remaining generally lower than in the other models. It is therefore much
Running of the Higgs quartic coupling λ in the FSSM and in the usual Split-SUSY case fortan β = 1 and 1 .
5. We set M S = 2 × GeV and M ˜ g (cid:48) = µ = 2 TeV. easier to obtain a Higgs mass close to the experimentally observed value even for largevalues of the SUSY scale. For example, as will be discussed later, when tan β ≈ . M S between 10 GeV and M GUT .This new behaviour originates in the RG evolution of λ in the FSSM, which differsfrom the case of Split SUSY. In figure 2 we show the dependence of λ on the renormal-isation scale Q in the two theories, imposing the boundary condition in eq. (4.1) at thescale M S = 2 × and setting tan β to either 1 or 1 .
5. Even though we impose thesame boundary condition in both theories, the fact that the effective Higgs–higgsino–gaugino couplings are zero in the FSSM induces a different evolution. Indeed, in SplitSUSY the contributions proportional to four powers of the Higgs–higgsino–gauginocouplings enter the one-loop part of β λ with negative sign, as do those proportionalto four powers of the top Yukawa coupling, whereas the contributions proportional tofour powers of the gauge couplings enter with positive sign. For M S (cid:38) GeV, thetop Yukawa coupling is sufficiently suppressed at the matching scale that removing theHiggs–higgsino–gaugino couplings makes β λ positive. This prompts λ to decrease withdecreasing Q , until the negative contribution of the top Yukawa coupling takes overand λ begins to increase.Figure 2 also shows that, for values of tan β sufficiently close to 1, the quarticcoupling λ can become negative during its evolution down from the scale M S , only tobecome positive again when Q approaches the weak scale. This points to an unstablevacuum, and a situation similar to the one described in ref. [38]. However, it was lready clear from figure 1 that, for tan β = 1, the FSSM prediction for the Higgs massis too low anyway. For the values of tan β large enough to induce a Higgs mass in theobserved range, the theory is stable. This is illustrated in figure 3, where we show thecontours of equal Higgs mass on the M S – tan β plane, setting M ˜ g (cid:48) = µ = 2 TeV. Thegreen-shaded region corresponds to a Higgs mass in the observed range between 124and 127 GeV, while the yellow-shaded region is where λ becomes negative during itsevolution between M S and the weak scale, and the vacuum is unstable. It can be seenthat, for M S (cid:38) GeV, a Higgs mass around 126 GeV can be comfortably obtainedfor either tan β ≈ . β ≈ .
6. The unstable region is confined to values of tan β very close to 1, and only for M S (cid:38) GeV. For lower values of M S , the top Yukawacoupling is not sufficiently suppressed at the matching scale and β λ is always negative,therefore there is no region of instability.We investigated how our results are affected by the experimental uncertainty onthe top mass. An increase (or decrease) of 1 GeV from the central value M t = 173 . M S . For larger values of M t , the observed valueof M H is obtained for tan β closer to 1, and the green regions in figure 3 approach theunstable region. The size of the unstable region is itself dependent on M t (i.e. the regionshrinks for larger M t ) but the effect is much less pronounced. Consequently, raisingthe value of the top mass may lead to instability for large M S (e.g. for M S (cid:38) GeVwhen M ˜ g (cid:48) = 2 TeV). Considering an extreme case, for M t = 175 GeV we would see asubstantial overlap of the experimentally acceptable regions with the unstable regionaround M S ≈ M GUT . On the other hand, for values of M t lower than 173 . β further away from 1, andthe vacuum is always stable for the correct Higgs mass.Finally, in figure 4 we show the contours of equal Higgs mass on the M ˜ g (cid:48) – tan β plane, setting M S = 2 × GeV and µ = 2 TeV. The colour code is the same as infigure 3. It can be seen that the region where the FSSM prediction for the Higgs massis between 124 and 127 GeV gets closer to the unstable region when the F-gluino massincreases. However, the dependence of M H on M ˜ g (cid:48) is relatively mild, and only when M ˜ g (cid:48) is in the multi-TeV region do the green and yellow regions in figure 4 overlap. Weconclude that if we insist on enforcing exact stability and setting M S ≈ × GeV,then obtaining a Higgs mass compatible with the observed value constrains the gluinomass to the few-TeV region.
We have defined a model – the FSSM – which has the same particle content at lowenergies as Split SUSY, but has a substantially different ultraviolet completion andalso low-energy phenomenology:1. We discussed in section 2.2 that the F-gaugino and F-higgsino couplings to theHiggs are suppressed by ε .2. The effective operators leading to the decay of the charginos/heavier neutralinos,which are generated by integrating out the sfermions, are also suppressed, because Contour plot of the prediction for the Higgs mass on the M S – tan β plane, for M ˜ g (cid:48) = µ = 2TeV. The yellow-shaded region indicates where λ becomes negative during its running between M Z and M S .The green-shaded region indicates a Higgs mass in the range [124 , Figure 4:
Same as figure 3 on the M ˜ g (cid:48) – tan β plane, with M S = 2 × GeV and µ = 2 TeV. he adjoint fermions χ do not have a gauge-current coupling to the sfermions. Asdiscussed in section 3, the lifetimes are enhanced by a factor ε − . This makes thegauginos/higgsinos very long-lived; we must appeal to a non-thermal history ofthe universe with a low reheating temperature to avoid unwanted relics.3. Since we no longer have an R -symmetry, the usual corrections to the Higgs quarticcoupling at the SUSY scale proportional to powers of | A t − µ cot β | /M S are inprinciple no longer negligible. However, as we discussed in section 4, in Split-SUSY scenarios those corrections are less important than in the MSSM, becausethe evolution to the large scale M S suppresses the top Yukawa coupling thatmultiplies them [6, 7].4. Finally, the main result of this paper was presented in section 4, and concerns theprecision determination of the Higgs mass in this model. Its value is substantiallydifferent than in either High-Scale or Split SUSY; in particular we can find 126GeV for any SUSY scale, with a vacuum that is always stable when the F-gluinomass is not too large.We have found that a standard-model-like Higgs boson with a mass around 126GeV can be obtained for low values of tan β . For low values of M S , the exact valueof tan β is subject to modification that we estimated when considering the presence ofadditional contributions to the quartic Higgs coupling from the unsuppressed A -terms.For larger values of M S , the latter contributions are negligible.In supersymmetric theories, the theorem of non-renormalisation of the superpoten-tial implies that supersymmetry cannot be broken by perturbative effects. It is eitherbroken at tree level or by non-perturbative effects. The former implies that the scaleof supersymmetry breaking is of the order of the fundamental (string) scale M ∗ , andunless this is taken to lie at an intermediate energy scale [39], it predicts a heavyspectrum. In studies of low-energy supersymmetry, the use of non-perturbative effectsattracted most interest because it allows the generation of the required large hierarchyof scales through dimensional transmutation. It is then interesting to investigate thefate of the former possibility when the supersymmetry scale is pushed to higher values.For Split and High-Scale SUSY, it is difficult to justify a very high O ( M GUT ) SUSYscale, since in that regime they predict the Higgs mass to be too high (unless one pushesto the limits of the theoretical and experimental uncertainties, see e.g. refs [7, 40]).Here, we have shown that the situation is different in the Fake Split SUSY Model.It is tempting to consider that while supersymmetry is broken at tree level in a secludedsector, the scale M S ∼ M GUT could be induced through radiative effects [17] from thefundamental scale M S ∼ αM ∗ , where α is a loop factor. We postpone the constructionof explicit realisations of this possibility for a future study. Acknowledgments
We thank Emilian Dudas, Jose Ramon Espinosa, Mariano Quir´os, Alessandro Strumiaand Carlos Tamarit for useful discussions. ppendix: Two-loop RGEs for Split-SUSY masses In this appendix we list the two-loop RGEs for the fermion-mass parameters of SplitSUSY in the MS scheme. Defining dm x d ln Q = β (1) m x π + β (2) m x π , ( m x = m ˜ g , m ˜ B , m ˜ W , µ ) , (A1)we obtained, using the public codes SARAH [34] and
PyR@TE [35], β (1) m ˜ g = − g m ˜ g , β (2) m ˜ g = − g m ˜ g , (A2) β (1) m ˜ B = (˜ g + ˜ g ) m ˜ B + 4˜ g ˜ g µ , (A3) β (2) m ˜ B = (cid:20) (cid:0) ˜ g + ˜ g (cid:1) −
72 ˜ g ˜ g − (cid:0) ˜ g ˜ g + ˜ g ˜ g (cid:1) − (cid:0) ˜ g ˜ g + ˜ g ˜ g (cid:1) + 518 (cid:0) ˜ g + ˜ g (cid:1) (cid:18) g + g (cid:19) − (cid:0) ˜ g + ˜ g (cid:1) (cid:0) g b + 3 g t + g τ (cid:1)(cid:21) m ˜ B + (cid:2) g ˜ g + 3˜ g ˜ g − g ˜ g ˜ g ˜ g (cid:3) m ˜ W + (cid:20) g + 24 g − ˜ g − ˜ g − g − g (cid:21) ˜ g ˜ g µ , (A4) β (1) m ˜ W = (cid:0) − g + ˜ g + ˜ g (cid:1) m ˜ W + 4˜ g ˜ g µ , (A5) β (2) m ˜ W = (cid:20) − (cid:0) ˜ g + ˜ g (cid:1) −
212 ˜ g ˜ g − (cid:0) ˜ g ˜ g + ˜ g ˜ g (cid:1) − (cid:0) ˜ g ˜ g + ˜ g ˜ g (cid:1) − g + 18 (cid:0) ˜ g + ˜ g (cid:1) (cid:18) g + 91 g (cid:19) − (cid:0) ˜ g + ˜ g (cid:1) (cid:0) g b + 3 g t + g τ (cid:1)(cid:21) m ˜ W + (cid:2) ˜ g ˜ g + ˜ g ˜ g − g ˜ g ˜ g ˜ g (cid:3) m ˜ B + (cid:20) g + 48 g − ˜ g − ˜ g − g − g (cid:21) ˜ g ˜ g µ , (A6) β (1) µ = (cid:18) − g − g + 34 ˜ g + 34 ˜ g + 14 ˜ g + 14 ˜ g (cid:19) µ + 3˜ g ˜ g m ˜ W + ˜ g ˜ g m ˜ B , (A7) β (2) µ = (cid:20) − g + 1359400 g − g g −
158 ˜ g −
158 ˜ g −
14 ˜ g −
14 ˜ g + 33160 g (cid:0) ˜ g + ˜ g + 3˜ g + 3˜ g (cid:1) + 3332 g (cid:0) ˜ g + ˜ g + 11˜ g + 11˜ g (cid:1) − (cid:0) ˜ g ˜ g + ˜ g ˜ g + ˜ g ˜ g + ˜ g ˜ g (cid:1) −
454 ˜ g ˜ g − g ˜ g +3˜ g ˜ g ˜ g ˜ g − (cid:0) ˜ g + ˜ g + 3˜ g + 3˜ g (cid:1) (cid:0) g t + 3 g b + g τ (cid:1)(cid:21) µ + (cid:20) g + 2710 g − g − g (cid:21) ˜ g ˜ g m ˜ W + (cid:20) g + 910 g − ˜ g − ˜ g (cid:21) ˜ g ˜ g m ˜ B . (A8) ote Added After the appearance of our paper in preprint, the author of ref. [33] revised his calcula-tion of the two-loop RGEs in Split SUSY. His results for the RGEs of the fermion-massparameters are now in full agreement with ours.
References [1] G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the searchfor the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys.Lett. B , 1 (2012) [arXiv:1207.7214 [hep-ex]].[2] S. Chatrchyan et al. [CMS Collaboration], “Observation of a new boson at a massof 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B , 30 (2012)[arXiv:1207.7235 [hep-ex]].[3] N. Arkani-Hamed and S. Dimopoulos, “Supersymmetric unification without lowenergy supersymmetry and signatures for fine-tuning at the LHC,” JHEP (2005) 073 [hep-th/0405159].[4] G. F. Giudice and A. Romanino, “Split supersymmetry,” Nucl. Phys. B (2004)65 [Erratum-ibid. B (2005) 65] [hep-ph/0406088].[5] N. Arkani-Hamed, S. Dimopoulos, G. F. Giudice and A. Romanino, “Aspects ofsplit supersymmetry,” Nucl. Phys. B (2005) 3 [hep-ph/0409232].[6] N. Bernal, A. Djouadi and P. Slavich, “The MSSM with heavy scalars,” JHEP (2007) 016 [arXiv:0705.1496 [hep-ph]].[7] G. F. Giudice and A. Strumia, “Probing High-Scale and Split Supersymmetrywith Higgs Mass Measurements,” Nucl. Phys. B (2012) 63 [arXiv:1108.6077[hep-ph]].[8] P. Fayet, “Massive Gluinos,” Phys. Lett. B , 417 (1978).[9] I. Antoniadis, K. Benakli, A. Delgado, M. Quiros and M. Tuckmantel, “Splittingextended supersymmetry,” Phys. Lett. B , 302 (2006) [arXiv:hep-ph/0507192];[10] I. Antoniadis, K. Benakli, A. Delgado, M. Quiros and M. Tuckmantel, “Splitextended supersymmetry from intersecting branes,” Nucl. Phys. B , 156 (2006)[arXiv:hep-th/0601003].[11] M. S. Carena, A. Megevand, M. Quiros and C. E. M. Wagner, “Electroweak baryo-genesis and new TeV fermions,” Nucl. Phys. B (2005) 319 [hep-ph/0410352].[12] J. Unwin, “R-symmetric High Scale Supersymmetry,” Phys. Rev. D (2012)095002 [arXiv:1210.4936 [hep-ph]].[13] G. Belanger, K. Benakli, M. Goodsell, C. Moura and A. Pukhov, “DarkMatter with Dirac and Majorana Gaugino Masses,” JCAP (2009) 027[arXiv:0905.1043 [hep-ph]].[14] E. Dudas, M. Goodsell, L. Heurtier and P. Tziveloglou, “Flavour models withDirac and fake gluinos,” arXiv:1312.2011 [hep-ph].
15] K. Benakli, M. D. Goodsell, F. Staub and W. Porod, “A constrained minimalDirac gaugino supersymmetric standard model”, in preparation .[16] K. Benakli and M. D. Goodsell, “Dirac Gauginos in General Gauge Mediation,”Nucl. Phys. B (2009) 185 [arXiv:0811.4409 [hep-ph]].[17] K. Benakli and M. D. Goodsell, “Dirac Gauginos, Gauge Mediation and Unifica-tion,” Nucl. Phys. B (2010) 1 [arXiv:1003.4957 [hep-ph]].[18] C. Csaki, J. Goodman, R. Pavesi and Y. Shirman, “The m D − b M Problem ofDirac Gauginos and its Solutions,” arXiv:1310.4504 [hep-ph].[19] P. Gambino, G. F. Giudice and P. Slavich, “Gluino decays in split supersymme-try,” Nucl. Phys. B (2005) 35 [hep-ph/0506214].[20] V. Khachatryan et al. [CMS Collaboration], “Search for Heavy Stable ChargedParticles in pp collisions at √ s = 7 TeV,” JHEP (2011) 024 [arXiv:1101.1645[hep-ex]].[21] G. Aad et al. [ATLAS Collaboration], “Search for stable hadronising squarks andgluinos with the ATLAS experiment at the LHC,” Phys. Lett. B (2011) 1[arXiv:1103.1984 [hep-ex]].[22] S. Chatrchyan et al. [CMS Collaboration], “Searches for long-lived charged parti-cles in pp collisions at √ s =7 and 8 TeV,” arXiv:1305.0491 [hep-ex].[23] G. Aad et al. [ATLAS Collaboration], “Search for long-lived stopped R-hadronsdecaying out-of-time with pp collisions using the ATLAS detector,” Phys. Rev. D (2013) 112003 [arXiv:1310.6584 [hep-ex]].[24] A. Arvanitaki, C. Davis, P. W. Graham, A. Pierce and J. G. Wacker, “Limitson split supersymmetry from gluino cosmology,” Phys. Rev. D (2005) 075011[hep-ph/0504210].[25] E. Dudas, M. Goodsell and P. Tziveloglou, “Goldstini and Dirac gaugino masses,” in preparation .[26] T. K. Hemmick, D. Elmore, T. Gentile, P. W. Kubik, S. L. Olsen, D. Ciampa,D. Nitz and H. Kagan et al. , “A Search for Anomalously Heavy Isotopes of Low Z Nuclei,” Phys. Rev. D (1990) 2074.[27] P. F. Smith, J. R. J. Bennett, G. J. Homer, J. D. Lewin, H. E. Walford andW. A. Smith, “A Search For Anomalous Hydrogen In Enriched D-2 O, Using ATime-of-flight Spectrometer,” Nucl. Phys. B (1982) 333.[28] J. Beringer et al. [Particle Data Group Collaboration], “Review of Particle Physics(RPP),” Phys. Rev. D (2012) 010001.[29] M. Muether [Tevatron Electroweak Working Group and CDF and D0 Collabora-tions], “Combination of CDF and DO results on the mass of the top quark usingup to 8 . f b − at the Tevatron,” arXiv:1305.3929 [hep-ex].[30] J. Fleischer, F. Jegerlehner, O. V. Tarasov and O. L. Veretin, “Two loop QCDcorrections of the massive fermion propagator,” Nucl. Phys. B (1999) 671[Erratum-ibid. B (2000) 511] [hep-ph/9803493].
31] L. V. Avdeev and M. Y. Kalmykov, “Pole masses of quarks in dimensional reduc-tion,” Nucl. Phys. B (1997) 419 [hep-ph/9701308].[32] M. Binger, “Higgs boson mass in split supersymmetry at two-loops,” Phys. Rev.D (2006) 095001 [hep-ph/0408240].[33] C. Tamarit, “Decoupling heavy sparticles in hierarchical SUSY scenarios: Two-loop Renormalization Group equations,” arXiv:1204.2292 [hep-ph].[34] F. Staub, “SARAH 4: A tool for (not only SUSY) model builders,”arXiv:1309.7223 [hep-ph].[35] F. Lyonnet, I. Schienbein, F. Staub and A. Wingerter, “PyR@TE: Renormaliza-tion Group Equations for General Gauge Theories,” arXiv:1309.7030 [hep-ph].[36] A. Sirlin and R. Zucchini, “Dependence of the Quartic Coupling H(m) on M( H )and the Possible Onset of New Physics in the Higgs Sector of the Standard Model,”Nucl. Phys. B (1986) 389.[37] G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidoriand A. Strumia, “Higgs mass and vacuum stability in the Standard Model atNNLO,” JHEP (2012) 098 [arXiv:1205.6497 [hep-ph]].[38] D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala,A. Salvio, A. Strumia, “Investigating the near-criticality of the Higgs boson,”[arXiv:1307.3536 [hep-ph]].[39] K. Benakli, “Phenomenology of low quantum gravity scale models,” Phys. Rev.D (1999) 104002 [hep-ph/9809582].[40] A. Delgado, M. Garcia and M. Quiros, “Electroweak and supersymmetry breakingfrom the Higgs discovery,” arXiv:1312.3235 [hep-ph].(1999) 104002 [hep-ph/9809582].[40] A. Delgado, M. Garcia and M. Quiros, “Electroweak and supersymmetry breakingfrom the Higgs discovery,” arXiv:1312.3235 [hep-ph].