Surveying the Phenomenology of General Gauge Mediation
aa r X i v : . [ h e p - ph ] D ec SCIPP 08/13
Surveying the Phenomenology of GeneralGauge Mediation
Linda M. Carpenter University of California Santa CruzSanta Cruz [email protected]
Abstract
I explore the phenomenology, constraints and tuning for severalweakly coupled implementations of multi-parameter gauge mediationand compare to minimal gauge mediation. The low energy spectra aredistinct from that of minimal gauge mediation; a wide range of NLSPsis found and spectra are significantly compressed, thus tunings maybe generically reduced to a part in 10 to a part in 20.
Introduction
Gauge mediation is a predictive and flavor blind communication mechanismfor SUSY breaking [1]. The earliest and simplest implementation of minimalgauge mediation involved communication of SUSY breaking from the hiddensector through a single set of vector like messengers charged under standardmodel gauge groups. The messengers couple to a hidden sector field h X i = x + θ F x and acquire a supersymmetric and nonsupersymmetric mass term, W = λXφ ˜ φ → λ h x i φ ˜ φ + λF x φ ˜ φ (1.1)Defining Λ ≡ F x /x we see that one loop gaugino masses are generated M λi ∼ α i π Λ (1.2)in addition to two loop scalar masses e m = 2Λ " C (cid:16) α π (cid:17) + C (cid:16) α π (cid:17) + 53 (cid:18) Y (cid:19) (cid:16) α π (cid:17) . (1.3)Minimal gauge mediation predicts that all scalar and gaugino mass termsare related by a single mass scale. Sparticles get masses proportional topowers of their gauge couplings, therefore, there is a large mass hierarchy inthe spectrum placing sparticles charged under QCD far above the others. Alarge mass hierarchy in the SUSY spectrum is not just a prediction of gaugemediation but also results from the other leading candidates for SUSY com-munication, anomaly mediation and mSUGRA [2]. Recent bounds on quicklydecaying NLSPs place a strong limit on Λ, the mass scale of minimal gaugemediation, at 91 TeV [3]. For non-prompt NLSP decay lower bounds comefrom charged sparticle masses and indirect constraints such as the inclusivetri-lepton signal. In both cases a heavy and thus tuned spectrum results, butthis applies especially to the case of prompt NLSP decay. As experimentspush the spectrum of minimal gauge mediation higher, non-minimal modelsbecome more phenomenologically (and for many philosophically) attractive.Simple extensions to MGM have been scattered throughout the litera-ture, see for example [4] [5]. Recently [6] have proposed a simple frameworkfor counting allowed parameters in general gauge mediated schemes. Thegeneral framework allows for at most six mass parameters to determine thelow energy spectrum. There are now a variety of viable models implementing1GM in a more systematic way, however a detailed analysis of the parameterspace of non-minimal SUSY models has not been done. Though constraintson minimal frameworks like MSUGRA are well understood, in non-minimalscenarios it isn’t even clear exactly what bounds on sparticle masses are -for example see recent work on gluinos [7] or look through PDG to see howmany analyses are solely MSUGRA based[8].For a given low energy spectrum in the GGM framework, there may beseveral ways to complete the model in the hidden sector. Model building ingeneral gauge mediation may be accomplished with weakly coupled renor-malizable theories [9] [10], or in frameworks with non-perturbative dynamics[11] [12]. GGM models may be direct, with messengers participating in SUSYbreaking, or indirect. In fact the general framework of GM does not requiremessengers at all. The goal of this work is not to build specific UV comple-tions to models, or to insist on a specific fix in the Higgs sector. Rather itis to look at the low energy predictions and pick out interesting regions ofparameter space where the phenomenology is far from standard and/or thetuning is significantly less then the minimal case.That said, many implementations of GGM result in split SUSY like spec-tra, where scalars are much heavier than gauginos and tuning is quite bad.This may be the case for example if Poppitz-Trivedi type mass terms arelarge. Such terms add to scalar masses only as they preserve R symme-try and are accounted for by non-vanishing messenger supertrace [13]. Suchterms occur generically in certain classes of GGM models. This work willnot address models with large mass splitting between scalars and gauginos,rather it will attempt to focus on theories that predict light spectra whichlook different from both the MGM and split SUSY spectra. The focus herewill be limited the predictions of weakly coupled renormalizible hidden sec-tors with and R symmetry [10]. Such models avoid split spectra and rely onmultiple hidden sector spurions and different numbers of messenger multi-plets in various representations of SU(5) to change the number of low energyparameters.For example, consider the case of N sets of messengers in a 5, 5 represen-tation of SU(5). These messengers may couple to multiple scalar fields whichget vevs and F terms. The messenger couplings are λ aiℓ X a ¯ ℓ i ℓ i + λ aiq X a ¯ q i q i (1.4)where the index i counts the N messengers and the index a counts the hidden2ector fields.There are then four parameters which determine the low energy spectrum,Λ iq = λ aiq F a λ biq x b ; Λ iℓ = λ aiℓ F a λ biℓ x b . (1.5)Counting this way we see that one set of 5, 5 leads to two parameters,one set of 10, 10 yields three and so on. Scanning over the complete space ofgauge mediated parameters is complicated. However, the possible low energyphenomenologies become very nonstandard for cases with just two or threeparameters. This work will focus on such cases, with the hope that thesescans will sample much of the interesting low energy phenomenology of thefull six parameter space. What follow in section 2 is an overview of param-eter counting for weakly coupled GGM models. Section 3 is a catalogue ofthe most relevant constraints on GGM parameter space. The remaining sec-tions map out the viable parameter space for different multi-parameter GMscenarios; section 4 deals with 2 parameter GM, section 5 with 3 parameters.Section 6 concludes. In counting parameters we must of course first count the number of scales Λ i which determine the gauge mediated soft mass spectrum. There is howevera difference between the number of parameters predicted by general gaugemediation and the number of parameters involved in producing a viable spec-trum.Models within the gauge mediated framework must include additionalstructure that generates µ and B µ terms since these parameters do not fallwithin the definition of gauge mediation, i . e . these terms occur wether theSM gauge couplings are turned off or not. However, any possible completionthat generates µ may also effect Higgs soft masses. Therefore even thoughGGM makes Higgs soft mass predictions at each point in parameter space,the soft masses must be thought of as parameters to be scanned over. Contri-butions to soft masses squared may either be positive or negative, thereforewe must consider benchmark scenarios, the most conservative assuming assmall a positive contribution to Higgs soft masses as possible, as well as sce-narios assuming large negative mass squared contributions. The parametersin this sector are the b-term, µ , tan β ,and the soft masses m Hu and m Hd .3or the purposes of this paper all scans are done fixing tan β at 10, once thesoft masses are selected, µ and B µ are solved for using the conditions of theelectroweak minimum.The SUSY breaking scale also effects the spectrum and experimentalconstraints. The SUSY breaking scale controls the mass of the gravitino,the LSP in gauge mediation, and determines the lifetime of sparticles. Asthe SUSY breaking scale is varied, sparticles may decay inside or outsideof the detector drastically changing the phenomenological constraints. Inaddition, the sparticle masses run from the SUSY breaking scale altering thelow energy spectrum. Two benchmark scenarios will be considered; for thecase of prompt sparticle decay the low SUSY breaking scale is taken to be 10 GeV, for decay outside of the detector the high SUSY breaking benchmarkis chosen to be 10 GeV.The number of copies of messengers effects the spectrum but requiressome caution. In MGM the gaugino masses scale like N the number ofmessengers, and the scalar mass squareds scale like the √ N . The ratio ofscalar to gaugino masses may thus be compressed by a factor of √ N . Thishas significant consequences for searches, for example, large N may make thedifference between spectra with a bino-like LSP and a stau LSP. Howeverincreasing N may itself alter the number of GGM parameters. An examplewill illustrate the point. In the case of two parameter GM, achieved withone messenger set in a 5 of SU(5), the masses are given by the 2D subset ofequation 1, m ˜ g = α π Λ ; m ˜ w = α π Λ ; m ˜ b = α π ( 23 Λ + Λ ) ; (2.1) m s = 2 (cid:18) C ( α π ) Λ + C ( α π ) Λ + Y ( α π ) ( 23 Λ + Λ ) (cid:19) . (2.2)What happens if we want two copies of the messengers? Generally this willnot just multiply these equations by N, but will instead put us in a fourdimensional parameter space, m ˜ g = α π (Λ + Λ ′ ); m ˜ w = α π (Λ + Λ ′ ); (2.3) m ˜ b = α π ( 23 (Λ + Λ ′ ) + (Λ + Λ ′ ));4 s = 2 (cid:16) C ( α π ) (Λ + Λ ′ ) + C ( α π ) (Λ + Λ ′ )+ Y ( α π ) ( 23 (Λ + Λ ′ ) + (Λ + Λ ′ ) (cid:19) . (2.4)Where the four parameters are the two sums of scales ΣΛ i and sums of thesquare of scales ΣΛ i . This counting will be true for any number or messengermultiplets beyond two. We see that in our example there is a 2-D subspaceof this 4 dimensional space where Λ i = Λ ′ i which would multiply both thescalar mass squareds and the gaugino masses by 2.Finally the predicted Higgs mass must exceed the experimental bound.For standard signals LEP puts a lower mass bound on the Higgs of 114 GeV[14]. However the MSSM predicts a tree-level Higgs mass of at most themass of the Z. There are several ways to solve this problem. The standardsolution is to include loop corrections from very heavy stops. At one loopthe stop correction is overcounted, at two loops a stop of 500 GeV gives aHiggs mass of roughly 105 GeV [15] [17]. In order to push the Higgs abovethe mass bound much heavier stops are needed. However, loop correctionsfrom heavy stops also make large contributions to m Hu , which must the becanceled down to m Z in the electroweak potential. For this reason all modelsbecome tuned to roughly one part in δm Hu /m Z . Barring the heavy stopsolution, models may be completed with a variety of other fixes. The first isby assuming a hidden Higgs scenario, where the Higgs is in fact lighter than114 GeV, but decays in a nonstandard way so as to have escaped detection.This may happen in R parity violating scenarios [18] or in the NMSSM[19] .The second is by assuming extra operators in the Higgs sector, for examplethe dimension 5 operators proposed by Dine Seiberg and Thomas [20]. Theseoperators alter the calculation of the electroweak minimum and present oneor two extra parameters. In particular dimension 5 operators change theHiggs potential in the following way δ V + δ V = 2 ǫ H u H d ( H † u H u + H † d H d ) + ǫ ( H u H d ) + h . c . (2.5)where ǫ ≡ µ ∗ λM , ǫ = − m SUSY λM . (2.6)5nd thus add to the Higgs mass δ ǫ m h = 2 v ǫ r + 2 ǫ r sin(2 β ) + 2 ǫ r ( m A + m Z ) sin(2 β ) − ǫ r ( m A − m Z ) cos (2 β ) p ( m A − m Z ) + 4 m A m Z sin (2 β ) ! (2.7)This does not alter the condition for electroweak minimum by much but itdoes call for an intermediate scale of physics at a TeV. As per the calculationof Dine et. al. , at tanβ
10 and choices of λ/M of order 10 − , The Higgs massmay be set above LEPs bound for squark masses of 300 GeV. In this work,all scans are done assuming a value λ/M = 2x10 − . gauginos prompt decay non-prompt decaybino-like NLSP χ +
229 102.7˜ g
320 130gluino NLSP˜ g
315 270wino-like NLSP m χ + − m χ < m π m π < m χ + − m χ < χ +
206 45 † Table 1: direct mass bounds on gauginos decaying through different NLSPs, † assumes light sneutrino massesTevatron places strong inclusive bounds on charged massive stable parti-cles, or CHAMPS, which must be taken into account in configurations wherethe LSP is both long lived and charged. Bounds on CHAMPS are quite re-strictive, the upper limit on production for weakly interacting particles is 10 f b [16]. 6irect bounds on gauginos depend strongly on the NLSP, and if the SUSYbreaking scale is low or high. A recent redux of bounds is found in reference[21]. The absolute lower limit on chargino masses, which is independent ofdecay products, comes from LEP 1 and is 45 GeV. In the case of a bino-likeNLSP and sufficient mass splitting LEP places a chargino mass bound of102.7 GeV [22]. In the case that the NLSP is wino-like, the mass splittingbetween the chargino and lightest neutralino may be very small. If it is under3 GeV, the previously stated bound no longer holds. If the mass splittingis less than the pion mass then the chargino lives a very long time [23] andits existence may be ruled out by CHAMP data. The lower limit on longlived stable charginos is 206 GeV [16]. However LEP’s analyses also assumea small contribution from t channel sneutrinos, if sneutrinos are light thebounds do not apply . In the case of low SUSY breaking scale such that thechargino decays inside the detector, the mass bound is a very restrictive 229GeV [3].A decay mode independent lower mass bound on the gluino has beenset at 51 GeV [24]. In regions of parameter space that obey MSUGRA likerelations, the gluino can decay through real or virtual squarks to a bino-likeNLSP. Recent work has placed a strict lower bound on the gluino decaying tojets plus missing energy with nonunified gaugino masses at 130 GeV [7], andhas also made exclusions in the gluino-bino mass plane. Previous searchesfor gluinos had been confined to the MSUGRA scenario, where the currentPDG lower bound is quoted at 308 GeV [8]. MSUGRA assumes that thegluino decays to a bino LSP which is much lighter than the gluino itself,which would mean a substantial amount of missing energy in the event.However in non MSUGRA scenarios the amount of missing energy decreasesas the gluino mass and bino mass are moved closer together. Therefore theTevatron’s missing energy events cuts would cause gluinos to be missed fora range of masses. Recent work not only moves the MSUGRA bound upfrom 308 GeV, but it makes exclusion in the full gluino-bino mass plane.These bino vs. gluino mass constraints will be considered in the followingscans. In the case of prompt decay, if the bino is the NLSP the experimentalconstraints are stronger; there is a search for gluinos decaying to squarksplus diphotons and missing energy which placed the gluino mass bound at320 GeV [25]. This search had relatively low cuts on missing energy andthus it is not possible to avoid constraints. In the case that the gluino is theNLSP, non-prompt decay of the stable gluino is ruled out by stopped gluinosearches to at least 270 GeV for particle lifetimes under 3 hours[26]. For7rompt decay Tevatron’s monojet search rules it out to 320 GeV [27]. The trilepton signal is a distinctive SUSY signature. Tevatron has donesearches for the final state: three leptons plus missing energy [28]. In theSUSY framework this is a signal for the decays of a charged and neutralgaugino. The process is χ + → ˜ l + ν, χ → ˜ l + l where ˜ l decays to leptonand χ . Bounds on the trilepton cross section are extremely small. There isroom for half an event at two inverse f b of data. Thus if ǫσ ( bf ) L = N wesee that even for a low efficiency, for example 5 percent, there is only roomfor f b ’s of production. There are three possible outs: first, we may make thegauginos heavy to reduce the production cross section. Second we may signif-icantly decrease the branching fraction into leptons and missing energy. Forexample if the intermediate sleptons are far off shell, the decay will insteadproceed through virtual gauge bosons; in this case the branching fractionto leptons plus missing energy is lowered because of significant contributionfrom branching fraction to jets. Finally, the chargino may be made Higgsinolike, in which case the final decay will favor taus, excluded by the trileptonanalysis. The triletpton search highly constrains any model which predictsdominant decays of charginos and neutralinos which proceed through slep-tons, e.g. in regions of SUSY parameter space where the neutralino is theNLSP and the sleptons are light, which is exactly the case in minimal gaugemediation with non-prompt NLSP decay. This bound puts a new lower limiton the mass of the lightest chargino beyond the direct searches. The newbound for charginos in the MSUGRA scenario is 145 GeV [29] and is ex-pected to be 160 GeV with 12 f b − of data. One expects that with one fewerparameter, minimal gauge mediation will have a bound at least about asstrict. For GGM the bound will depend on the spectra and will be a strongconstraint for models with certain LSPs and mass hierarchies.The Higgs sector conditions of electroweak minima are well known: anymodel must reside in a region of SUSY parameter space where the elec-troweak symmetry is broken and the Higgs acquires the proper vev. Theparameters in this sector are the b-term, µ , tan β ,and the soft masses m Hu and m Hd but they are not all independent, the conditions of the electroweakminima remove two of these parameters (also recall there are two additionalinequalities which are constraints for the stability of the potential).The mass of the charged Higgs, the charginos and the stops are predicted8t each point in parameter space. These parameters are involved in theprediction of the process b → sγ . There is a tight bound on this process[30]. However, the NNLO prediction is for the first time below the measuredvalue, leaving some space for SUSY corrections [31]. Scans are conductedover all parameter space calculating Γ( b → sγ ) at each point.Finally the Higgs mass must be sufficiently increased. no EWSB m χ = 150 GeV m χ = 45 GeVm χ = 102.7 GeV m χ - m χ <
3 G e V + Λ ( GeV ) + + ++ Λ ( GeV ) m χ - m χ = G e V + Figure 1: Plot of Λ vs Λ with various constraints. Allowed parameter spaceis shaded. The arc on the bottom left is LEP’s right handed selectron bound,the dotted line indicates where the mass difference between the chargino andthe selectron is zero.Consider a two dimensional GGM parameter space achieved with a setof messengers in the 5, 5 representation. The two GM parameters are the9cales Λ and Λ and the sparticle spectrum is given by m ˜ g = α π Λ ; m ˜ w = α π Λ ; m ˜ b = α π ( 23 Λ + Λ ) (4.1) m s = 2( C ( α π ) Λ + C ( α π ) Λ + Y ( α π ) ( 23 Λ + Λ ) (4.2)In the case that Λ = Λ we are reduced to the minimal gauge mediatedrelations. We see immediately that lower Λ in relation to Λ may compressthe SUSY spectrum, while raising the ratio will make the QCD hierarchyeven bigger than the minimal scenario. In parts of the parameter spacewhere Λ is sufficiently less than Λ the gluino can be made lighter than thebino, making it the NLSP. In regions where Λ is sufficiently larger than Λ ,then m ˜ w < m ˜ b , and the NLSP is wino-like. Everywhere else in parameterspace, the NLSP is bino-like.The MGM prediction of heavy squarks relative to the sleptons does notnecessarily hold. Because the sleptons hypercharge is larger than that of uptype squarks by a factor of 3, one finds that for regions of mass parameterssuch that α α Λ < Λ the sleptons are heavier than the squarks.We now concentrate on the constraints on parameter space in the highSUSY breaking scale scenario. The first constraint is to satisfy the conditionsfor the electroweak minimum. In the case that m Hu does not get largenegative mass contributions from the µ generating mechanism, this requires m Hu to run down due to the stop coupling. This in turn sets a lower limiton squark masses and hence Λ . Extra contributions to m Hu will shift theEWSB constraint around the parameter space, increasing or decreasing theminimum allowable Λ and altering the size of the µ term as µ cancels the softHiggs mass in the minimization conditions of the electroweak potential. Forthe choice of dimension five Higgs sector operator, the Higgs mass remainsabove 110 and 114 GeV for stop masses of 310 and 360 GeV respectively.Figure 1 gives a picture of parameter space for the benchmark scenario wherethe Higgs soft masses are set at their gauge mediated soft mass predictions.We see that regions of low Λ are ruled out, including those regions wherethe gluino is the NLSP.Of the remaining constraints, the most restrictive in the region of bino-like NLSP is the trilepton search. Where the selectron is lighter than thechargino and next to lightest neutralino, the branching fraction of a gauginopair into three leptons plus missing energy is large. Interpolating from the10SUGRA analysis we see that with 12 f b ’s of data the lower limit shouldbe around 150 GeV where the sleptons are light. As the selectron mass isincreased the branching fraction drops. However if the masses of the charginoand lightest neutralino get too closer together, the resulting sleptons will betoo soft. The Tevatron’s search required 4 GeV of p T for leptons [32]. Track-ing the relaxation of the trilepton constraint due to small mass differencerequires more rigorous analysis. In Figure 1 the strip of parameter spacewith mass difference between the chargino and lightest neutralino below 10GeV is labeled. This space may be ruled out by further analysis.In addition, there is a lower bound from the model independent gluinosearch [7]. However, the ratio of gluino to squarks mass is moderate and thebino mass constraint becomes less strict as ˜ q ˜ g production begins to competewith gluino pair production, therefore the model independent gluino massbound only rules out a small slice of parameter space. In the region whereΛ is large, the chargino and bino masses are sufficiently split and the min-imum allowable chargino mass is 102.7 GeV. However, in the region of lowΛ the sneutrinos are quite light and for this reason a high chargino massconstraint does not apply. Additionally in regions of low Λ compared to Λ ,the wino-like neutralino is the NLSP. Parameter space is allowed where themass splitting of the χ + and χ is above the pion mass and where the χ + isabove 45 GeV. For mass splittings smaller than the pion mass the charginois long lived and ruled out by CHAMP searches.The regions of parameter space where the spectrum is most compressedare those with bino-like NLSP where Λ < Λ . A typical spectrum follows,sparticle mass (GeV)˜ τ ν q ul µ χ + χ g mz /δ mhu
40 TeV, Λ
65 TeV with Higgs softmasses set at the GGM prediction. mz /δ mhu is the tuning measure.11s a comparison, the minimal case satisfying all bounds requires squarksof around 700 GeV and a tuning of at least a part in 45. Then for thissimplest extension of MGM with a conservative benchmark scenario, thespectrum can be compressed by a factor of 1.5 and the tuning reduced bythe square of that, though the qualitative nature of the spectrum is similarto that of minimal gauge mediation.In principle, if models exist with large negative contributions to the Higgssoft mass squared, the minimum allowable Λ can be decreased. For example,with models that add an additional 200 GeV to the up type Higgs mass,allowed up type squark masses can be pushed below 250 GeV. Here the Higgsmass is above 110 and 114 GeV for stop masses above 260 and 312 GeV. TheTevatron’s current squark search limits the squark masses above 329 GeV,however PDG’s bound for cascade decays where the gluino mass is less thanthe squark mass is less restrictive, only 224 GeV [8]. One would expect amodel independent analysis to be slightly less restrictive than this as themissing energy of the event in the general case is less than the MSUGRAcase since the neutralino and gluino mass ratio is compressed. Note that inscenarios where the m Hu get a very large negative mass squared contributionfor any Higgs mass fix, parameter space is opened in the region gluinosNLSPs. However a stable gluino NLSP is ruled out in the given region bybounds on stopped gluinos, since gluino masses in this region are less than200 GeV [26]. Beyond that the a lower limit on the squarks may be set dueto the Higgs mass bound. In general the Higgs mass of 110 GeV may onlybe surpassed for squark masses not much below 300 GeV, resulting in anirreducible tuning of 10 percent.In the case of a promptly decaying NLSP, the parameter space is moreseverely restricted. The charginos of mass bound of 229 GeV holds [3] andrules out much of parameter space, unless the mass splitting between charginoand neutralino is small. In this case having two parameters in gauge media-tion is not sufficient to avoid tunings and restrictions on parameter space. If we take the case of messengers in a 5, 5 representation and increase thenumber of messenger multiplets, we are analyzing a 2-D subspace of 4 pa-rameter gauge mediation. Five 5, 5 messengers is more or less the maximumcompatible with unification, in this case I have chosen to map out the sce-nario with 4 messengers. Here we get a factor of √ compared to Λ the NLSP is the wino-like neutralino.If the stau decays outside the detector, the most restrictive bounds onthe parameter space come from CHAMPs. Though bounds on stable stausthemselves are not that restrictive, the overall production cross section limitfor staus is dominated by production from gluino and chargino cascades.Tevatron places a bound on weakly produced CHAMPS at 10 f b . Produc-tion from gluinos can be safely supressed for gluinos above 500 GeV. Themost restrictive bound however comes from chargino pair production, whichrequires chargino masses of 250 GeV for sufficient suppression. This beatsall other bounds and forces us into a corner of parameter space similar tothat of the case for one messenger with non-prompt NLSP decay.In the case of prompt decay, GMSB searches place a lower bound onright handed staus of 82.5 GeV in case of stau NLSP. In areas with bino-likeNLSP the 102.7 GeV chargino bound is the most restrictive. In this casesquark masses may be pushed down to about 450 GeV again with a spec-trum qualitatively like the case for minimal gauge mediation with multiplemessengers.Below in Table 3 is a typical spectrum for non-prompt decay of the NLSPsparticle mass (GeV)˜ τ ν q ul µ χ g mz /δ mhu = 40TeV Λ = 58 TeV with Higgs soft masses set to the GGM prediction.13 Three Parameters g l u i n o N L S P e x c l u d e d m χ = 45 GeV Λ (GeV) Λ (GeV) ν NLSP + noEWSB Figure 2: Plot of Λ vs Λ with various constraints. Λ E = 80 TeV and theSUSY breaking scale is highThree parameter GM may be achieved with a set of messengers in a 10,10 representation of SU(5). The relations for three parameter GM are givenby m ˜ g = α π (2Λ + Λ ); m ˜ w = α π ; m ˜ b = α π ( 83 Λ + 43 Λ + 2Λ ) (5.1) m s = 2( C ( α π ) Λ Q + C ( α π ) Λ w + Y ( α π ) Λ Y (5.2)where Λ Q = (2Λ + Λ ), Λ w = 3Λ and Λ Y = ( Λ + Λ + 2Λ )Here, with just three parameters, spectra diverge from the qualitative fea-tures of minimal gauge mediation. The gaugino masses are now independent.It is possible to make the gluino the NLSP by canceling the scales Λ and Λ
20 000 40 000 60 000 80 000020 00040 00060 00080 000 χ NLSP χ NLSP ν NLSP ν NLSP g l u i n o N L S P Λ (Gev) Λ (Gev) Λ = E Figure 3: Plot of Λ vs Λ for regions of differing NLSP. Λ E = 160 TeV andthe SUSY breaking scale is highwhile the bino may be made heavy by choosing large Λ E . The choice of largeΛ E also compresses the mass difference between sleptons and squarks. Forregions where Λ is sufficiently low compared to the other scales, the sneu-trino may be the NLSP. The analysis that follows scans the two dimensionalsubspace of 3 parameter GM fixing the scale Λ E at the relatively large valueof 80 TeV.For the case of high SUSY breaking scale, the gluino can be made to be theNLSP. However, it is too light ( <
150 GeV) to be allowed by stopped gluinoconstraints. Elsewhere in parameter space, the bino remains the NLSP, ex-cept where Λ is sufficiently smaller than the other scales such that thesneutrino is the NLSP. This occurs for low values of Λ except near the elec-troweak trough. For these large values of Λ E the U(1) mass contributionsto scalars are relatively heavy. In low Λ regions the charged fermions areheavier than the charginos. However, the sneutrinos are light in regions of15ow Λ therefore the chargino mass bounds are not strong. Again there isa small window of wino-like light neutralino. However, this window is ruledout as the NLSP in the area is not the neutralino but the sneutrino. Finally,some part of parameter space is ruled out by model independent gluino tojets plus missing energy searches. Figure 2 shows a scan of the high SUSYbreaking scale three parameter space.Constraints on promptly decaying sparticles in this 3 parameter case aresimilar. With the same choice of Λ E , gluino NLSPs are still ruled out, thistime by monojet analyses. The chargino mass bound is increased to 229 GeVfor bino-like NLSP. In the case of a sneutrino NLSP, the chargino decays inthe lepton plus missing energy channel and the chargino mass constraint ismuch weaker. Unlike the minimal and two parameter case, prompt decay inthe three parameter case is almost as unconstrained as non-prompt decay.In both cases, since U(1) contributions to scalar masses can be madevery large. The possibility exists for an extremely compressed spectrum,about a factor of 3 compression from the minimal scenario. However thecompression results from raising charged slepton masses rather than fromdecreasing squark masses, therefore tuning remains at about a part in 20.Below, Table 4 shows a typical point in parameter space for 3 parametergauge mediation in the case of non-prompt decay.sparticle mass (GeV)˜ τ ν q ul µ χ + χ g mz /δ mhu
10 TeV Λ
28 TeV with noextra contribution to Higgs soft masses.If the scale Λ E is made even larger in comparison to the other scales thephenomenology becomes even more complex. Figure 4 shows a survey ofpossible NLSPs for the high SUSY breaking scale benchmark and the choiceΛ E = 160 TeV. In this case the region over which the gluino is the NLSP16s increased. This time there is a small strip of parameter space where thegluino is still the NLSP and is more massive than the bound demanded bythe stopped gluino searches. Here the gluino may be up to around 360 GeV.In addition since Λ E is larger than Λ over much of parameter space, theregion in which the sneutrino is the NLSP increases. In the region where thegluino is the NLSP, the squarks are quite heavy. This is because while massparameters cancel for the gluino, they add in quadrature for the squarks,and squark masses end up around 1.5 TeV. This region is highly tuned andthe spectrum is quite hierarchical, however it offers the possibility of a SUSYspectrum where everything decays to multi-jets plus missing energy. In re-gions with a sneutrino NLSP, the tuning in the spectrum can remain quitereasonable. The spectrum may become exceedingly compressed with righthanded squarks and sleptons around the same mass, while certain decays -for example that of the sleptons - become very nonstandard. We have seen that GGM can produce a variety of non-standard results. Scan-ning over just 3 GGM parameters qualitative features of the spectrum maychange drastically; NLSPs may vary from wino to bino to sneutrino to stauto gluino. Several features seem to be generic over the space. First of all get-ting a gluino NLSP is difficult. For low numbers of parameters prompt decayof the NLSP is highly constrained, while parameter space is quite open as wereach the 3 parameter case. In addition, though spectra may be compressed,squarks usually remain the heaviest sparticles. An exception is the threeparameter space with very large U(1) mass parameter, where squarks andsleptons may have similar masses. In general, tuning in the multi-parameterspace may be reduced to one part in 20, less than half the tuning of theminimal case; in some instances tuning may be as low as a part in 10.Many possibilities exist for interesting analyses that could further restrictthis space. Examples include: a full trilepton analysis for the case of non-prompt decay in minimal and multiparameter space; a model independentsquark search that did not rely on gaugino mass unification; a gluino-binoexclusion analysis for the case of light squarks; analyses that explore sparticlebounds in cases of low mass splitting between the chargino and a wino-likeNLSP; and slepton and chargino mass bounds in the case of a sneutrinoNLSP. 17 cknowledgments
This work was supported in part by DOE grant number DE-FG03-92ER40689.I would like to thank Michael Dine, Howie Haber, and Andrew Blechman forhelpful discussions.
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