A low Fermi scale from a simple gaugino-scalar mass relation
DDESY 13–174November 2013
A low Fermi scalefrom a simple gaugino-scalar mass relation
F. Br¨ummer a,b and W. Buchm¨uller ba Scuola Internazionale Superiore di Studi Avanzati SISSA/ISAS,Via Bonomea 265, I-34136 Trieste, Italy b Deutsches Elektronen-Synchrotron DESY,Notkestraße 85, D-22607 Hamburg, Germany
Abstract
In supersymmetric extensions of the Standard Model, the Fermi scale of electroweak sym-metry breaking is determined by the pattern of supersymmetry breaking. We present anexample, motivated by a higher-dimensional GUT model, where a particular mass relationbetween the gauginos, third-generation squarks and Higgs fields of the MSSM leads toa Fermi scale smaller than the soft mass scale. This is in agreement with the measuredHiggs boson mass. The µ parameter is generated independently of supersymmetry break-ing, however the µ problem becomes less acute due to the little hierarchy between the softmass scale and the Fermi scale as we will argue. The resulting superparticle mass spectradepend on the localization of quark and lepton fields in higher dimensions. In one case,the squarks of the first two generations as well as the gauginos and higgsinos can be inthe range of the LHC. Alternatively, only the higgsinos may be accessible at colliders. Thelightest superparticle is the gravitino. The unification of gauge couplings and the prediction of viable dark matter candidates providesa strong theoretical motivation for supersymmetric extensions of the Standard Model withTeV superparticle masses [1–3]. So far searches for heavy superparticles at the Large HadronCollider (LHC) have only led to lower bounds on scalar quark and gluino masses of about 1–2 TeV [4, 5]. On the other hand, the discovery of a 126 GeV Higgs boson [6, 7] allows, withoutor with supersymmetry, for an extrapolation of the Standard Model up to the scale of grandunification.The Higgs boson mass is consistent with the mass range predicted by the minimal su-persymmetric standard model (MSSM). However, since the Higgs mass significantly exceedsits tree-level upper bound of 91 GeV, quantum corrections are large, which generically re-quires multi-TeV scalar masses. This raises the question why the Fermi scale, the expectation1 a r X i v : . [ h e p - ph ] M a r alue of the Higgs field, (cid:104) H (cid:105) = ( √ G F ) − / = 246 GeV, is much smaller than the scale ofsupersymmetry breaking, and the required fine-tuning of seemingly unrelated parameters isoften considered as unnatural. Possible answers to this question invoke the anthropic principleand the string landscape, as in split supersymmetry [8], the focus point idea [9], or similaraccidental cancellations between non-universal gaugino and scalar masses at the grand unifi-cation scale [10]. The naturalness problem might also be solved in a non-minimal extension ofthe MSSM with additional sub-TeV degrees of freedom (for instance, the NMSSM, reviewedin [11]), or through non-decoupling effects such as in [12]. In this note we restrict ourselves to the MSSM, and attempt to answer a question which isintimately connected with the naturalness problem: Is there a well motivated and simple setof boundary conditions for the GUT-scale soft terms which favours a ‘little hierarchy’ betweenthe soft and the electroweak scale? And, what can we expect from this soft mass pattern forthe upcoming second LHC run?Our main findings can be summarized as follows. Since the µ parameter of the MSSM canbe generated independently of supersymmetry breaking, it is technically natural to choose itsmaller than the typical soft SUSY breaking parameters, say, of the order of the electroweakscale. Usually, explaining why µ should be of the order of the soft masses is a well knownchallenge (the ‘ µ problem’) in SUSY model building. Here, as we will argue, the µ problembecomes less severe once one accepts a little hierarchy. To obtain proper electroweak symmetrybreaking at large tan β , the loop-corrected up-type Higgs soft mass needs to be of the sameorder as µ at the scale where the MSSM is matched to the Standard Model, requiring anaccidental cancellation between the tree-level and radiative contributions to this parameter.We identify a simple soft mass pattern which suggests this cancellation, and which is motivatedby a six-dimensional GUT model (although we expect that there are other models that canlead to the same pattern). Within this model, we obtain an estimate for the possible rangeof the gluino mass, which will be partly probed at LHC-14. Squarks and sleptons may alsobe within reach, and by construction there are higgsino-like charginos and neutralinos withelectroweak-scale masses which can be discovered at a linear collider.Note that we are not claiming to solve the fine-tuning problem: The fine-tuning in our modelis as large as one would expect from a generic MSSM-type model without large contributionsto the lightest Higgs mass from stop mixing, i.e. at the permille level. Our model predicts therelevant soft terms only up to factors of order one, and while the predicted pattern non-triviallyallows for a little hierarchy, these unknown factors still need to be tuned in order to actuallyrealize it. We anticipate that fully understanding the origin of the cancellations involved willrequire a better understanding of the complete UV theory. Matching the MSSM to the Standard Model
The scalar potential for the MSSM Higgs fields depends on the higgsino mass µ , which is aparameter of the superpotential, and the soft supersymmetry breaking parameters m H u , m H d See [13] for a recent review of naturalness in supersymmetry in the light of the first LHC run. Bµ , V = (cid:0) m H u + | µ | (cid:1) H † u H u + (cid:0) m H d + | µ | (cid:1) H † d H d + Bµ (cid:0) H Tu iσ H d + c . c . (cid:1) + 18 (cid:0) g + g (cid:48) (cid:1) (cid:16) H † u H u − H † d H d (cid:17) + 12 g H † u H d H † d H u . (1)Our starting assumption is that the only scalar with an electroweak-scale mass is the lightestHiggs, while all others (in particular the remaining Higgs bosons) are much heavier. In thisso-called decoupling limit the Higgs vacuum expectation value (vev) is approximately alignedwith the lightest mass eigenstate. It is convenient to work with the fields H and H (cid:48) definedby H u = sin β H + cos β iσ H (cid:48)∗ , H d = cos βiσ H ∗ + sin βH (cid:48) , (2)with tan 2 β = 2 Bµm H u − m H d , (3)such that the quadratic part of the potential is diagonal in the new fields: V = m H † H + m (cid:48) H (cid:48)† H (cid:48) + 18 (cid:0) g + g (cid:48) (cid:1) (cid:16) cos 2 β (cid:16) H † H − H (cid:48)† H (cid:48) (cid:17) − sin 2 β (cid:0) H T iσ H (cid:48) + c . c . (cid:1)(cid:17) + 12 g H † H (cid:48) H (cid:48)† H , (4)where m = | µ | + m H u sin β + m H d cos β − Bµ sin 2 β , (5) m (cid:48) = | µ | + m H u cos β + m H d sin β + Bµ sin 2 β . (6)Within the MSSM the measured mass of the lightest Higgs boson requires large radiativecorrections from heavy stop squarks. Therefore we take the scale M S = ( m ˜ t m ˜ t ) / to be muchlarger than the electroweak scale, of the order of several TeV. At the scale M S the MSSM ismatched to the Standard Model with scalar potential V = m H † H + 12 λ (cid:16) H † H (cid:17) , (7)where λ | M S = 14 (cid:0) g + g (cid:48) (cid:1) cos β (cid:12)(cid:12) M S . (8)The Higgs mass parameter m encodes the prediction for the electroweak scale, v = − m /λ ,with λ being O (1). In the usual notation for the tree-level mass eigenstates, would-be Goldstone bosons, and mixing angles (seee.g. [14]) this corresponds to H = (cid:0) G + , v + ( h + iG ) / √ (cid:1) T , H (cid:48) = (cid:0) ( H + iA ) / √ , H + ∗ (cid:1) T , and α = β − π/ onditions for a little hierarchy When keeping the electroweak scale fixed, the tree-level contribution to the lightest Higgs massis maximized at large tan β (since in that limit | cos 2 β | → m Z = 91 GeV. The region of at least moderately large tan β (cid:38)
10 is therefore favouredby the large observed Higgs mass of 126 GeV, with the discrepancy accounted for by radiativecorrections. The Standard Model-like Higgs field H is then predominantly H u .By Eq. (3), using tan 2 β = 2 / (cot β − tan β ), large tan β implies Bµ (cid:28) m H d (9)in the generic case that (cid:12)(cid:12) m H u (cid:12)(cid:12) (cid:46) m H d . In the following we will take the µ parameter to begenerated independently of supersymmetry breaking. Since µ and Bµ are both governed by aPeccei-Quinn symmetry, unless Bµ is merely accidentally small due to radiative corrections,the reason underlying relation (9) is that the effective symmetry breaking scale is below thesoft mass scale, as will be discussed in more detail momentarily. In that case also µ is small: | µ | (cid:28) m H d . (10)Furthermore, since at large tan β we have m (cid:39) | µ | + m H u + m H u − m H d tan β , (11)a little hierarchy requires that m H u is small, (cid:12)(cid:12) m H u (cid:12)(cid:12) (cid:28) m H d . (12)Together with Eq. (3) this impliestan β (cid:39) m H d Bµ . (13)Relations (9), (10) and (12) are thus necessary to obtain a Fermi scale much smaller thanthe soft mass scale, assuming that tan β is at least moderately large and that µ and Bµ areconnected. We now proceed to discuss the possible origins of these conditions. Why should m H u be small? Choosing µ and Bµ small is technically natural, and this choice is radiatively stable. Bycontrast, radiative corrections to the Higgs soft masses are sizeable. In particular, as is wellknown, no symmetry protects m H u from loop corrections due to the large top Yukawa cou-pling. Condition (12) is technically unnatural, which is a manifestation of the usual fine-tuningproblem in the MSSM. It requires large cancellations between the radiative contributions andthe tree-level value of m H u . Let us discuss these in some more detail. We do not consider exceptionally small values for m H d , which could occur in exotic mediation schemes orbe induced by RG running at large y b (i.e. extremely large tan β (cid:38) m H d are given below. The last term in Eq. (11) is often neglected. However, in the case of a large matching scale M S it is generallyimportant, even for large values of tan β . M S which we cannot neglect.Turning first to the renormalization group running, the tree-level RG-improved Higgs po-tential at M S can be expressed as a function of the running Higgs mass parameters and ofthe running gauge couplings. The Higgs mass parameters at the scale M S depend on theirGUT-scale values, but also on the GUT-scale soft masses of all fields with sizeable couplingsto the Higgs sector. These are the third-generation scalars and the gauginos (with the gluinoentering because of its large coupling to the stops and sbottoms). We find for tan β = 15 m H u (cid:12)(cid:12) M S = − . . . (cid:99) M − . . . (cid:99) M (cid:99) M + 0 . (cid:99) M + 0 . (cid:99) M (cid:98) A t + 0 . (cid:99) M (cid:98) A t − . (cid:98) A t + . . . (cid:98) m H u − . (cid:98) m U − . . . (cid:98) m Q , for M S = . . TeV . (14)Here the hatted quantities on the RHS denote GUT-scale soft parameters, with (cid:98) A t normalizedto the top Yukawa coupling. We have taken the GUT scale to be fixed at M GUT = 1 . × GeV, and omitted all terms with coefficients smaller than 0 .
05. The coefficients are largelyinsensitive to tan β , as long as tan β (cid:38)
10; for instance, for tan β = 30 the coefficients ofthe (cid:99) M term are −{ .
07; 1 .
11; 1 . } and all other coefficients differ from Eq. (14) at most by0 .
01. Another source of uncertainty is the experimental uncertainty in the top mass. We havechecked that the uncertainty obtained from varying m t by 1 σ around its central value of 173 . .
01. The GUT-scale values forYukawa and gauge couplings have been obtained using the two-loop RG code
SOFTSUSY [15].With the assumptions of large tan β and of negligible stop mixing at the GUT scale, i.e. neg-ligible (cid:98) A t , the matching scale is in principle rather sharply determined by the lightest Higgsmass m h . This is because the radiative corrections to m h depend mainly on the stop masses(and on the RG-induced stop mixing parameter at the TeV scale). In practice however thereis still a large uncertainty, partly because of the uncertainty in y t , but mostly because ofthe theory uncertainty in computing m h from a given soft mass spectrum. We have chosen M S = 5 ± . SOFTSUSY , SuSpect [16] and
FeynHiggs [17] and also compatible with the three-loop analysis in [18] whichis based on the
H3M code [19].The equivalent of Eq. (14) for m H d reads m H d (cid:12)(cid:12) M S = − . . . (cid:99) M + . . . (cid:99) M + . . . (cid:98) m H d − . (cid:98) m U , for M S = . . TeV . (15)5hile the coefficients in Eq. (15) show a more pronounced tan β -dependence, the overall run-ning of m H d remains moderate in the range 10 (cid:46) tan β (cid:46)
40. In this region m H d | M S is thereforeof the order of (cid:98) m H d , which is generically of the order M S .In addition to the RG running of the tree-level parameters there are important finite correc-tions to the Higgs potential due to top-stop loops, which affect the Higgs masses (for a detaileddiscussion and references, see e.g. [20]). They amount to replacing m H u,d in Eqns. (9)–(13) by¯ m H u,d , where¯ m H u = m H u ( M S ) − t v sin β , ¯ m H d = m H d ( M S ) − t v cos β . (16)Here the tadpole terms t i are computed from the minimization conditions for the full one-loopeffective potential [20]. Using the one-loop results of [20], it turns out that the dominantcorrections to the Higgs masses are obtained in the limit where the top Yukawa coupling isthe only non-vanishing coupling, and where the stop squarks are approximately unmixed anddegenerate with mass m ˜ t = M S . In the MS scheme at the renormalization scale M S , one finds t ≈ , t v sin β ≈ y t π m t , (17)i.e., only m H u is significantly modified by the finite corrections to the Higgs potential.In this paper we are interested in the question how electroweak symmetry breaking canoccur at a scale significantly below the scale of supersymmetry breaking, i.e. how the conditions m < | m | (cid:28) M S can be realized from Eq. (11) at small | µ | . In particular, how canrelation (12) be satisfied? An important observation is that in Eq. (14) the scalar contributionsapproximately cancel for equal stop and H u masses, (cid:98) m H u = (cid:98) m Q = (cid:98) m U ≡ m . This is thebasis of the ‘focus point’ idea [9]. However, as the matching scale increases, the cancellationbetween the scalar soft mass contributions becomes less precise. The actual focussing point ofthe RG trajectories, where the m coefficient vanishes, is only obtained for M S close to theelectroweak scale.As Eq. (14) shows, the remaining positive contribution to m H u by scalar masses can becompensated by the negative contribution from gaugino masses. Assuming universal gauginomasses as suggested by unification, (cid:99) M = (cid:99) M = (cid:99) M = M / , and taking the correction Eq. (17)into account, one obtains for M S = 5 TeV¯ m H u (cid:12)(cid:12) M S = − . M / + 0 . M / (cid:98) A t − . (cid:98) A t + 0 . m , (18)subject to the uncertainties mentioned above. In the following we shall be interested in thecase | (cid:98) A t | (cid:46) M / . A cancellation between the gaugino and the scalar contribution then occursfor a particular ratio M / /m : M / = κ m , (cid:46) κ (cid:46) . (19)This can also be seen from Fig. 1, which shows ¯ m H u ( M S ) as a function of M S for differentvalues of the ratio κ = M / /m at negative, vanishing, and positive (cid:98) A t .Models predicting a relation of this type therefore show some promise for obtaining a littlehierarchy. In Section 3 we will present an example with all the required properties: A moderate6 igure 1: ¯ m H u (more precisely ¯ m H u / (cid:113) | ¯ m H u | ) as a function of the matching scale M S for various values of theparameter κ = M / /m . Top: (cid:98) A t = − M / , center: (cid:98) A t = 0, bottom: (cid:98) A t = + M / . Here tan β = 15 and M GUT = 1 . × GeV. We have indicated the range of M S preferred by the Higgs mass (which we took to be5 ± . | ¯ m H u | around the electroweak scale in yellow. µ and small Bµ . Why should µ and Bµ be small? It is well known that the higgsino mass µ plays a special role among the dimensionful pa-rameters of the MSSM. It preserves supersymmetry, but it breaks a U(1) Peccei-Quinn (PQ)symmetry under which the Higgs bilinear is charged. The soft masses and trilinear soft terms,by contrast, break supersymmetry but preserve U(1) PQ . The Higgs soft mass mixing parameter Bµ breaks both SUSY and PQ symmetry.For concreteness, assume that SUSY is broken by some singlet spurion X with (cid:104) X (cid:105) = F X θ ,and that U(1) PQ is broken supersymmetrically by some spurion Y , such that the followingterms are allowed in the Lagrangian: L = (cid:90) d θ Y p M p − (cid:18) XM (cid:19) H u H d + (cid:90) d θ XM (cid:0) | H u | + | H d | (cid:1) + h . c . (20)The K¨ahler terms in Eq. (20) can be absorbed in the superpotential terms by a field redefinition.The power p depends on the PQ charges of H u H d and of Y . Bare µ and Bµ terms µH u H d | θ and XH u H d | θ are forbidden by U(1) PQ , which also forbids the operators X † H u H d (cid:12)(cid:12) θ ¯ θ and | X | H u H d (cid:12)(cid:12) θ ¯ θ . Consequently, the effective µ parameter is µ ∼ Y p M p − , (21)and Bµ is proportional to both µ and the SUSY-breaking vev, Bµ ∼ Y p M p F X ∼ µM S , (22)where F X /M ∼ M S is the scale of the scalar and gaugino soft mass parameters.Choosing Y such that | µ | (cid:28) M S (23)is technically natural, since PQ breaking is a priori unrelated to SUSY breaking. The ‘ µ problem’ is usually formulated as the need for an explanation why the SUSY-breaking softmasses are of the same order as µ . Here this is not the case: In contrast to the common SUSYmodel building approach, we obtain µ and the SUSY breaking soft terms from two independentscales. As soon as we allow for a little hierarchy, the µ problem becomes less severe as we willargue momentarily. Indeed the most interesting parameter choice has µ maximally separatedfrom M S , to the extent that is allowed by experimental data.With the conditions (12) and (23), electroweak symmetry can be broken with all threeterms in Eq. (11) being of the order of the electroweak scale. The required fine-tuning is noworse than the fine-tuning needed in the more common case where µ is of the order of thesoft breaking terms, and cancelled against a similarly large ¯ m H u . In our case we are insteadcancelling large radiative contributions to the ¯ m H u parameter against each other.8emarkably, if the conditions (23) are satisfied with ¯ m H u sufficiently small, then the elec-troweak scale is parametrically given not by M S but by µ . This is most easily seen by setting¯ m H u = 0, m H d = ηM S , Bµ = ζ | µ | M S at the scale M S , with η and ζ of the order one (or atleast small compared to M S /µ — in the next section we will consider a model where ζ ∼ /κ ,with κ ≈ .
25 as in Eq. (19)). One then obtains m H = (cid:18) | µ | ζ | µ | M S ζ | µ | M S η M S (cid:19) , (24)leading to − m (cid:39) (cid:18) ζ η − (cid:19) | µ | . (25)For ζ > η the Higgs mass matrix Eq. (24) has a negative eigenvalue even though the diagonalentries are both positive. In fact, for ζ (cid:29) η the electroweak scale is given by a seesaw-typeformula, m (cid:39) − ( ζ | µ | M S ) η M S = − ζ η | µ | < . (26)A very similar pattern has previously been investigated in the context of gauge-mediatedsupersymmetry breaking, where the hierarchy between m H d (or equivalently M S ) and | µ | isnot due to a PQ symmetry but due to a loop factor [21]. Let us emphasize that a sufficientlylarge value of Bµ , and therefore ζ , is crucial for electroweak symmetry breaking, which takesplace irrespective of the sign of m H u .As already emphasized we have no symmetry reason for ¯ m H u = 0. In the more generalcase (cid:12)(cid:12) ¯ m H u (cid:12)(cid:12) (cid:28) M S , electroweak symmetry breaking imposes a lower bound on | µ | , | µ | > η ζ − η ¯ m H u . (27)Note that there is also a phenomenological lower bound on | µ | : Since tan β is parametricallygiven by m H d /Bµ ∼ η M S / ( ζ | µ | ), and should not exceed a value ≈
60 in order to avoid non-perturbative Yukawa couplings, the hierarchy between µ and M S cannot be too large. Thus,for fixed M S , µ is bounded from below. The most relevant bound for the model of the nextsection will however turn out to be the direct experimental lower limit | µ | (cid:38)
100 GeV fromchargino searches at LEP.At this point let us briefly return to the µ problem. If we set ¯ m H u = 0 and ignore theassociated fine-tuning for a moment, it is clear from the Higgs mass matrix Eq. (24) and fromEq. (26) that the soft mass scale may be decoupled from the scale of electroweak symmetrybreaking (which is essentially given by µ ). In a hypothetical universe with very light down-typequarks, there would also be no restriction on the ratio M S /µ ∼ tan β , so M S could in principlebe very large, and the µ problem would be circumvented. Realistically, however, this line ofreasoning is invalidated to some extent by the experimentally known bottom and top quarkmasses. The known value of m b leads to an upper bound on tan β , while the known value of m t implies that the top Yukawa coupling is large, and that a relation such as ¯ m H u = 0 willtherefore be spoiled by large loop corrections. These two arguments point towards a soft massscale M S which is not too far above the electroweak scale; the 126 GeV Higgs mass furtherfixes the ‘little hierarchy’ to amount to 1–2 decades. In summary, the µ problem is still present,but somewhat alleviated when allowing for a little hierarchy between M S and the Fermi scale(as seems to be forced upon us by LHC data).9 Supersymmetry breaking in higher-dimensional GUTs
We shall now present an explicit example which realizes the conditions for a seesaw-type patternof electroweak symmetry breaking discussed in the previous section. Consider a six-dimensional(6d) GUT model, with the third quark-lepton generation and the Higgs fields located in thebulk and the first two families localized at 4d branes or orbifold fixed points. Such a modelhas been derived as an intermediate step [22] in a compactification of the heterotic string tothe supersymmetric standard model in four dimensions [23, 24]. Supersymmetry is supposedto be broken by the F -term of a chiral superfield located at some fixed point.In the following we shall restrict our discussion to the case of strong coupling at the cutoffscale. The couplings of the supersymmetry breaking brane field to Higgs, matter and gaugefields can then be estimated by means of ‘naive dimensional analysis’ (NDA) following [30].The localization of the fields fixes the structure of the Lagrangian L d = L bulk ( W α , Φ) + (cid:88) i δ ( y − y i ) L i ( W α , Φ , φ ) , (28)where y i are the positions of the 4d branes, and W α , Φ and φ denote bulk gauge fields,bulk chiral fields, and brane chiral fields, respectively. Matching 6d and 4d theories at thecompactification scale, the gauge couplings and Planck masses are related by1 g V = 1 g , M V = M , (29)where V is the volume of the two compact dimensions. In order to define the theory one has to introduce a UV cutoff Λ. If loop corrections at thescale Λ are suppressed by (cid:15) , the Lagrangian Eq. (28) can be expressed in terms of dimensionlessfields (cid:99) W α / Λ / , (cid:98) Φ / Λ and (cid:98) φ/ Λ, W α ( x, y ) = Λ / ( (cid:15)(cid:96) ) / (cid:99) W α ( x, y )Λ / , Φ( x, y ) = (cid:18) Λ (cid:15)(cid:96) (cid:19) / (cid:98) Φ( x, y )Λ , (30) φ i ( x ) = (cid:18) Λ (cid:15)(cid:96) (cid:19) / (cid:98) φ i ( x )Λ . (31)The fields W α , Φ and φ are assumed to have canonical kinetic terms in 6d and 4d, respectively,and the rescaled fields (cid:99) W α , (cid:98) Φ and (cid:98) φ have canonical dimensions in 4d. According to NDA theLagrangian (28) now takes the form L d = Λ (cid:15)(cid:96) (cid:98) L bulk (cid:32) (cid:99) W α Λ / , (cid:98) ΦΛ , ∂ Λ (cid:33) + (cid:88) i δ ( y − y i ) Λ (cid:15)(cid:96) (cid:98) L i (cid:32) (cid:99) W α Λ / , (cid:98) ΦΛ , (cid:98) φ Λ , ∂ Λ (cid:33) , (32)where all couplings are O (1) and (cid:96) D = 2 D π D/ Γ( D/
2) is a geometrical loop factor, with (cid:96) = 128 π , (cid:96) = 16 π . (33) In the considered GUT model one has V = 2 π R R , where R and R are the radii of the orbifold. Themodel has a Wilson line in the direction of R which breaks the GUT symmetry. With R ≥ R , the mass ofthe lowest lying Kaluza-Klein state is 1 / (2 R ). Identifying this mass with the GUT scale M GUT (cid:39) × GeV,one obtains V − / (cid:39) × GeV (see [25]). (cid:15) (cid:39) X to the bulk fieldsare given by −L sb = Λ (cid:15)(cid:96) (cid:26) (cid:90) d θ Λ (cid:32) (cid:98) µ Λ (cid:98) H u Λ (cid:98) H d Λ (cid:32) (cid:98) X Λ (cid:33) + (cid:98) X Λ (cid:32) tr (cid:34) (cid:99) W α Λ / (cid:99) W α Λ / (cid:35) + (cid:98) Q Λ (cid:98) H u Λ (cid:98) U Λ (cid:33) + h . c . (cid:33) + (cid:90) d θ Λ | (cid:98) X | Λ (cid:32) | (cid:98) H u | Λ + | (cid:98) H d | Λ + | (cid:98) Q | Λ + | (cid:98) U | Λ + | (cid:98) D | Λ + | (cid:98) L | Λ + | (cid:98) E | Λ (cid:33) (cid:27) , (34)where H u , H d , W α , Q , U , D , L and E denote Higgs fields, gauge fields and third generationquark and lepton fields, respectively. H u is part of the 6d gauge multiplet, Q and U belongto the same hypermultiplet, and the cubic term Q H u U is part of the 6d gauge interactions. From the gauge kinetic term one reads off the gauge coupling g ∼ ( (cid:15)(cid:96) ) / Λ . (35)The mass parameter (cid:98) µ is an additional free parameter which can be much smaller than thecutoff scale Λ due to an accidental PQ symmetry as discussed in Section 2. From Eqs. (30)and (34) one obtains the Lagrangian for canonically normalized bulk fields, −L sb = (cid:96) (cid:96) D − (cid:26) (cid:90) d θ (cid:32) ˆ µH u H d (cid:32) (cid:98) X Λ (cid:33) + (cid:98) X Λ (tr [ W α W α ] + g Q H u U + h . c . ) (cid:33) + (cid:90) d θ | (cid:98) X | Λ (cid:0) | H u | + | H d | + | Q | + | U | + | D | + | L | + | E | (cid:1) (cid:27) . (36)Finally, the replacement Φ( x, y ) → V − / Φ( x ) yields the couplings of canonically normalizedzero modes, −L sb = (cid:96) (cid:96) V (cid:26) (cid:90) d θ (cid:32) ˆ µH u H d (cid:32) (cid:98) X Λ (cid:33) + (cid:98) X Λ (tr [ W α W α ] + g Q H u U + h . c . ) (cid:33) + (cid:90) d θ | (cid:98) X | Λ (cid:0) | H u | + | H d | + | Q | + | U | + | D | + | L | + | E | (cid:1) (cid:27) . (37)In Eq. (34) we have assumed a universal coupling of the SUSY breaking field to bulk fields.The focus point cancellation discussed in Section 2 requires approximately equal mass termsof H u , Q and U at a level of about 5%. In the considered model the equality of mass terms isguaranteed by a symmetry only for U and E , which belong to the same SU(6) hypermultipletin six dimensions. For all other fields a dynamical reason is needed. The couplings of brane and This model has two pairs of equivalent fixed points [22]. Hence, there will be at least two SUSY breakingfields, at a pair of equivalent fixed points. For the following discussion this complication is irrelevant and willbe ignored. Trilinear terms for the other matter fields are also allowed but will not be written explicitly. y t = g (cf. Eq. (37)), with the large values of tan β considered in Section 2 [25].It is conceivable that the FI terms present in the model [22] lead to approximately equal massterms, but a detailed study of the compactification dynamics is beyond the scope of this paper.Replacing now the brane field X by its SUSY breaking vacuum expectation value F X , weobtain from Eq. (37) the wanted mass parameters of the zero modes for gaugino fields, Higgsand higgsino fields and third generation scalar quark-lepton fields, L soft = − (cid:18) µ h u h d + Bµ H u H d + 12 M / tr λ a λ a + A t y t Q H u U + h . c . (cid:19) − (cid:0) m + | µ | (cid:1) (cid:0) | H u | + | H d | (cid:1) − m (cid:0) | Q | + | U | + | D | + | L | + | E | (cid:1) , (38)where y t = g , and m ∼ κ (cid:32) (cid:98) F X Λ (cid:33) , M / ∼ κ (cid:98) F X Λ ∼ κ m , A t ∼ M / (39) µ ∼ κ ˆ µ , Bµ ∼ κ ˆ µ (cid:98) F X Λ ∼ κ µm , (40)with κ = (cid:96) (cid:96) V = (cid:96) (cid:96) (cid:18) M Λ (cid:19) M V / . (41)For Λ = M and GUT-scale extra dimensions, i.e. V − / (cid:39) × GeV, this yields κ (cid:39) . M / ∼ . m . (42)Let us emphasize that this relation is not at all generic, but based on a 6d GUT picture,supersymmetry breaking by a brane field and the assumption of strong coupling at the UVcutoff which is chosen to be the 6d Planck mass. The prediction for analogous models witha different number of GUT-scale extra dimensions is not too different, however: the generalexpression for D dimensions and Λ = M D reads κ = (cid:96) D (cid:96) (cid:32) M D − V D − (cid:33) D − (43)and yields κ = 0 .
08 (0 .
09) for D = 5 ( D = 7), assuming the same compactification radius asabove. For even larger D the loop factor enhancement becomes dominant, and κ grows ratherlarge. The precise choice of the compactification scale sensitively affects the prediction for κ ,and its proper value is dependent on the model details. The following discussion applies to our D = 6 orbifold model with V − / (cid:39) × GeV.Comparing with the electroweak symmetry breaking pattern of the last chapter, we findthat in this model the κ parameter is of the correct order of magnitude to explain a small ¯ m H u ζ parameter issomewhat large at ζ ∼ /κ ∼
4; therefore if ¯ m H u were completely negligible, we would obtaina slightly too large electroweak scale from Eq. (26): v = − m λ ≈ (4 µ ) λ (44)which is not compatible with the experimental lower bound | µ | (cid:38)
100 GeV. However, a finitenegative ¯ m H u can easily cure this.An important quantity is the gravitino mass. From Eq. (39) one obtains for the scalar massparameter m (cid:39) √ (cid:96) (cid:18) M Λ (cid:19) F X M . (45)Together with m / = F X / ( √ M ) this yields m / (cid:39) √ (cid:96) (cid:18) Λ M (cid:19) m (cid:39) . (cid:18) Λ M (cid:19) m . (46)Hence, unless the cutoff significantly exceeds the 6d Planck mass, the gravitino will be thelightest superparticle. The result (46) is consistent with the analysis carried out in [28]. Let us finally consider the first and second quark-lepton generations, which are localizedat two equivalent fixed points (see [22]) that may or may not coincide with the localization ofthe supersymmetry breaking fields. In the second case one has −L (cid:48) soft = ˜ m (cid:88) i =1 , (cid:0) | Q i | + | U i | + | D i | + | L i | + | E i | (cid:1) , (47)with ˜ m = 0, correponding to the boudary conditions of gaugino mediation. In the first case,the scalar mass terms are obtained from Eq. (34), −L (cid:48) sb = Λ (cid:15)(cid:96) (cid:88) i =1 , (cid:90) d θ Λ | (cid:98) X | Λ (cid:32) | (cid:98) Q i | Λ + | (cid:98) U i | Λ + | (cid:98) D i | Λ + | (cid:98) L i | Λ + | (cid:98) E i | Λ (cid:33) . (48)Performing the transition to canonically normalized fields using (31), one finds in the case ofstrong coupling ( (cid:15) = 1),˜ m (cid:39) κ m ∼ m . (49)Hence, in this case, unlike gaugino mediation, first- and second-generation scalars will beheavier than third-generation scalars. Note that the predictions of masses obtained from naive dimensional analysis have an uncertainty of O (1).This includes the effect of a colour factor which was included in the calculations in [28] and which has beenomitted in the present discussion for simplicity. Strictly speaking this is not possible with the exact particle content of [22], since in this model there is nosuitable brane-localized singlet which could play the role of X . However a slight variation of this model mightwell contain a suitable candidate. Prospects for phenomenology and outlook
In the six-dimensional GUT model which we discussed in the previous section, the localizationof fields and the breaking of supersymmetry by a brane field determine the pattern of scalarand gaugino masses. The Higgs bosons, third generation squarks and sleptons, and gauginosare bulk fields. Their masses depend on κ = M / /m , as determined by the matching scale M S , the sign of (cid:98) A t and m (which in turn is related to the matching scale by renormalizationgroup running). In Fig. 2 the gluino mass at M S is shown as function of M S for both signs of (cid:98) A t . The resulting predicted range of gluino masses,2 TeV (cid:46) M | M S (cid:46) , (50)is a consequence of the allowed range of matching scales and the sign ambiguity of (cid:98) A t .The first two generations are brane fields. Their masses strongly depend on the localizationof the supersymmetry breaking field X . There are two possibilities:(A) The matter fields and X are localized on different branes. This implies the familiarpattern of gaugino mediation, and squarks and sleptons of the first two generations arelighter than those of the third generation. (B) The matter fields and X are localized on the same brane. According to Eq. (49), derivedin the previous section, the squarks and sleptons of the first two generations are thenheavier than those of the third generation.A further important parameter is the higgsino mass µ . If µ is generated independentlyof supersymmetry breaking, generically one would expect µ ∼ M S / tan β , as discussed inSection 2. In the model of Section 3, since Bµ is enhanced by a factor 1 /κ , we estimate µ ∼ κM S / tan β , which implies that for M S = 5 ± . β , the µ parameter should actually be close to the electroweak scale ( | µ | (cid:46)
100 GeV being excluded bychargino searches). A soft upper bound can be estimated by conservatively setting tan β = 5, κ = 1 / M S = 6 . µ (cid:46)
450 GeV.In summary, the mass spectrum we predict is characterized by heavy third-generationsquarks and sleptons, heavy extra Higgs bosons, gluino masses starting from about 2 TeV,higgsino-like charginos and neutralinos with electroweak-scale masses, and squarks and sleptonswhich are either extremely heavy (B) or generated by gaugino mediation (A). In the lattercase, standard SUSY searches for jets and missing energy, as well as searches for direct sleptonproduction, will be promising channels at LHC-14. In any case, the light higgsinos can besearched for and measured at a linear collider [32, 33].Table 1 shows a number of superpartner mass spectra. The first three columns correspondto three different values of M S in scenario (A). For a relatively low matching scale M S = 3 . M S = 5 TeV is more challenging,but squarks and gluinos may still be accessible at high integrated luminosities. The third caseof M S = 6 . Note that for the same localization of fields, but a different mechanism of supersymmetry breaking, thirdgeneration squarks and sleptons can also be lighter than those of the first two generation [31]. igure 2: The running gluino mass M | M S as a function of the matching scale M S for various values of theparameter κ = M / /m . Top to bottom curves: κ between 0 .
33 and 0 .
19 in steps of 0 .
02. The solid linescorrespond to (cid:98) A t = + M / and the dashed lines to (cid:98) A t = − M / , with the colour coding the same as in Fig. 1.Note that the relation between M S and M for fixed κ is only approximately linear. As before, tan β = 15, M GUT = 1 . × GeV. We have indicated the range of M S preferred by the Higgs mass (which we tookto be 5 ± . M as a function of M S , for the two cases (cid:98) A t = ± M / (black strips), and the minimal and maximal M which can be obtained (dashed horizontal lines),when restricting | m H u | to be of the order of the electroweak scale as in Fig. 1. One finds 1 . (cid:46) M (cid:46) . The last column of Table 1 shows a spectrum for the case that the first- and second-generation scalar masses are non-vanishing at the GUT scale and given by M soft = 30 TeV(scenario (B) above). In this case the overall soft mass scale also for the third generation andthe gluinos is higher. The reason is that we are keeping M S = 5 TeV fixed, and the first twosquark generations significantly decrease the stop masses when running down from the GUTscale through two-loop effects, up to a point where the stop mixing contribution to the lightestHiggs mass can become very significant [32, 36]. This case is not covered by our semi-analyticdiscussion in Section 2, which does not account for possible large contributions to the runningfrom the first two generations, but can nevertheless be dealt with numerically. As is evidentfrom Table 1, all states are too heavy to be seen at colliders in the foreseeable future, with thepossible exception of the higgsinos.Finally, the matching scale M S also determines the gravitino mass. From Eq. (46) oneobtains40 GeV (cid:39) m / (cid:39)
80 GeV . (51)Here we have chosen the 6d Planck mass as the cutoff scale, and we have varied m between4 TeV and 8 TeV according to Table 1.The starting point of our discussion has been the compatibility of the measured Higgs bo-son mass, and the associated large matching scale M S , with a Fermi scale significantly smallerthan M S . We have shown that for a small higgsino mass µ , not controlled by supersymmetrybreaking, and universal Higgs and stop masses at the GUT scale, a small Fermi scale arises forsuitable relations between gaugino and scalar masses. It is interesting that a simple examplecan be obtained within the context of a higher-dimensional GUT model. The matching scale,15ight 1st & 2nd generation heavy 1st & 2nd generation M S = 3 . M S = 5 TeV M S = 6 . M S = 5 TeV χ
127 109 141 185 χ
140 116 146 189 χ ±
133 112 144 187 χ
430 700 990 1100 χ , χ ±
820 1300 1900 2100 H , A , H ± g u i , ˜ d i , ˜ c i , ˜ s i × ˜ t t b b µ , ˜ e
350 560 800 3 × ˜ µ , ˜ e
610 1000 1400 3 × ˜ τ τ Table 1: Example mass spectra computed with
SOFTSUSY 3.3.10 [15] for different matching scales. The pa-rameters are tan β = 15, (cid:98) A t,b = 0, M / = (1 , . , . , .
45) TeV, m = (4 . , . , . , .
7) TeV for the thirdgeneration and the Higgs fields. For the three columns on the left, the GUT-scale scalar soft masses of the firsttwo generations vanish, whereas for the rightmost column they are ˜ m = 30 TeV. All masses in the table are inunits of GeV. together with the related gaugino-scalar mass ratio, and the value of tan β determine the su-perparticle mass spectrum. If the matching scale turns out to be lower than about 5 TeV, thisscenario will be probed by the upcoming next LHC run, with searches for gluinos, squarks andalso sleptons being promising channels. Moreover, our setup favours light higgsinos, which canbe searched for at a linear collider. The lightest superparticle is the gravitino. Acknowledgements
We thank J. Kersten, M. Ratz und M. Winkler for helpful comments. This work was supportedin part by the German Science Foundation (DFG) within the Collaborative Research Center676 “Particles, Strings and the Early Universe”. The work of FB was supported in part byERC Advanced Grant 267985 “Electroweak Symmetry Breaking, Flavour and Dark Matter”.
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