Travels on the squark-gluino mass plane
Joerg Jaeckel, Valentin V. Khoze, Tilman Plehn, Peter Richardson
IIPPP/11/52; DCPT/11/104
Travels on the squark-gluino mass plane
Joerg Jaeckel, Valentin V. Khoze, Tilman Plehn, and Peter Richardson Institute for Particle Physics Phenomenology,Department of Physics, Durham University, United Kingdom Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Germany (Dated: December 4, 2018)Soft supersymmetry breaking appears in the weak-scale effective action but is usually generated athigher scales. For these models the structure of the renormalization group evolution down to theelectroweak scale leaves only part of the squark-gluino and slepton-gaugino mass planes accessible.Our observations divide these physical mass planes into three wedges: the first can be reached byall models of high-scale breaking; the second can only be populated by models with a low mediationscale; in the third wedge squarks and gluinos would have to be described by an exotic theory. Allusual benchmark points reside in the first wedge, even though an LHC discovery in the third wedgewould arguably be the most exciting outcome. a r X i v : . [ h e p - ph ] S e p I. INTRODUCTION
Searches for supersymmetry are one of the most visible tasks of the LHC experiments [1, 2]. To interpretthe data they have to rely on specific SUSY models determining the mass spectrum and the decay patterns.Limiting the Higgs sector to two doublets, a good starting point for such an interpretation is the MSSM definedat the weak scale. However, for practical purposes one needs to significantly constrain its vast parameterspace. After taking into account the strong constraints for example from flavor physics [3] and electric dipolemoments [4] we are left with O (20) parameters which can be relevant for LHC searches or observations [5]. Afurther reduction of this parameter space is traditionally achieved in terms of simplest constrained realizationssuch as the CMSSM/mSUGRA [6, 7], gauge mediation [8–10] or anomaly mediation [11] (see also [12] for anoverview).These models share two important features. First, by construction they have a small or even minimalnumber of free parameters to describe all the soft supersymmetry breaking terms. Second, the soft parametersin these models are determined at a high scale arising from an underlying theory of supersymmetry breakingand mediation. At the mediation scale M the values for the soft parameters are initialized. For example, ingauge mediation, M = M mess , an effective mass of the messenger fields transmitting supersymmetry breakingto the Standard Model sector. In these models M mess is typically taken to be in the range 10 − GeV.In gravity mediation models M is set by M Planck which, with the additional assumption of grand unification,in CMSSM is traded down to M GUT . In order to make contact with the scale at which experiments operatethe soft terms have to undergo renormalization group evolution down from the mediation scale M to the weakscale [12, 13].In this paper we point out that all such high scale models automatically impose severe restrictions onsuperpartner masses at collider energies. In the strongly as well as weakly interacting sfermion-gaugino massplanes as much as half of the available parameter space becomes inaccessible. For example in the squark-gluinocase, squarks cannot become significantly lighter than gluinos. Similar relations hold between sleptons andelectroweak gauginos in all high scale models.The details of these restrictions dominantly depend on one parameter, namely the value of the mediationscale. As a result, each sfermion-gaugino mass plane can be divided into three wedge-shaped regions. Oneregion can be reached by all usual models of high-scale supersymmetry breaking. A second wedge can onlybe populated by models with a mediation scale M < M
GUT , while sfermions and gauginos in the third wedgewould have to originate from a theory which either does not have a high SUSY scale or a qualitatively differentRG evolution. Thus, from measuring gaugino and sfermion masses we can draw powerful conclusions on theway supersymmetry is realized in Nature.Conversely, when searching for supersymmetry one should make as few assumptions as possible about theway supersymmetry breaking is realized. This definitely includes its high scale origin. With the next roundof SUSY searches at the LHC being imminent, new sets of benchmark points and test models will be definedto determine, optimize and calibrate the search strategies. In order to minimize the bias of assumed specificmodels it may be useful to include also points which do not originate from high scale models and which aredistributed more democratically on the sfermion-gaugino planes accessible at collider energies. One way toobtain points not prejudiced towards high scale models is to use the MSSM defined directly at the weak scale.A manageable incarnation of this idea is the so-called phenomenological MSSM or pMSSM [14]. Alternatively,one can use the simplified model approach for constructing test models based on kinematic considerations anda selection of a small number of allowed sparticle species [15]. To some degree, squark and gluino searches bothby ATLAS [1] and CMS [2] are already following this route.This paper is organized as follows: in the next section we will show explicitly how the renormalization groupevolution from the high scale M to the weak scale restricts the accessible regions in the squark-gluino plane. InSec. III we extend our discussion to binos, winos and sleptons. In particular, we discuss the additional compli-cations arising from electroweak symmetry breaking. In Sec. IV we investigate the distribution of benchmarkpoints as well as a variety of test models. Finally, in Sec. V we summarize our findings and conclude. II. SQUARK VS GLUINO MASS
The key to understanding the coverage of the squark-gluino mass plane is the renormalization group equationfor these masses. It has been known for a long time [12, 13] that the gaugino masses strongly impact the runningof the sfermion masses to the weak scale. Starting from a high scale, they generate contributions to the softsfermion masses even if the initial soft sfermion masses vanish.To illustrate this structure, we can approximately solve the RG equations in SUSY-QCD. In the absence ofYukawa couplings we find schematically m q ( Q ) ∼ m q ( M ) + A ˜ q m q + A ˜ g M g M ˜ g ( Q ) ∼ M ˜ g ( M ) + B ˜ g M ˜ g . (II.1)Numerically, A ˜ g dominates. The running of the gluino mass does not include any squark mass terms on theright-hand side. The reason is that Majorana fermion masses are protected by the R symmetry, in analogy tothe chiral symmetry for the Dirac masses of the Standard Model fermions. This feature persists for the entireMSSM and can be exploited for example to decouple all scalars from a high-scale SUSY model while keepingall gauginos light enough to ensure gauge coupling unification, dark matter, etc [16].Moving on to the full theory, for the first two generations we neglect the Yukawa couplings [3]. If the trilinear A -terms are proportional to the Yukawa couplings, the same holds for the renormalization group contributionsfrom them. Using this, the RG equations for the first generation sfermions read [12],16 π ddt m f = − (cid:88) r C r g r | M r | + 2 Y ˜ f g S , (II.2)where M r are the gaugino masses, r = (1 , ,
3) the (bino,wino,gluino) labels, g r the gauge couplings not in theGUT normalization for U (1), and S := Tr( Y m ) = (cid:88) generations (cid:16) m Q L − m u R + m d R − m L L + m e R (cid:17) + m H u − m H d . (II.3)The Casimir invariants and hypercharge assignments for the relevant fermions are( C , C , C ) = (cid:18) Y , , (cid:19) ( Y ˜ Q L , Y ˜ u R , Y ˜ d R , Y ˜ L L , Y ˜ e R ) = (cid:18) , − , , − , (cid:19) . (II.4)The gaugino masses, couplings, and scalar masses then evolve according to16 π ddt M r = − b r g r M r π ddt g r = − b r g r π ddt S = − b g S ( b , b , b ) = ( − , − , . (II.5)Comparing Eq.(II.2) and Eq.(II.5) we indeed see that the gaugino masses contribute to the running of thesfermion masses but not vice versa.Equation (II.2) can easily be integrated, m f ( Q ) = m f ( M ) + Y ˜ f b (cid:20) α ( Q ) α ( M ) − (cid:21) S ( M ) + (cid:88) r =1 C ˜ fr b r (cid:20) − α r ( M ) α r ( Q ) (cid:21) M r ( Q ) . (II.6) Log (cid:72) M (cid:76) (cid:72) m s qu a r k (cid:144) M g l u i n o (cid:76) m i n G e V (cid:60) M (cid:60) M G U T no high scale model M gluino m s qu a r k Figure 1: Left: minimal ratio m ˜ q /M ˜ g as a function of the mediation scale. The blue curve assumes universal gauginomasses at the GUT scale, whereas for the red curve only M ˜ g ( M GUT ) is non-zero. Right: accessible regions in the m ˜ q - M ˜ g plane, assuming gaugino mass unification. Their boundaries correspond to mediation scales M = M GUT = 2 · GeVand M = 10 GeV. The thick green line shows the simplified model ATLAS exclusion with 1 .
04 fb − [18]. The dotsshow benchmark points from Refs. [1, 2, 19, 20]. Because we will mainly be interested in sfermion masses smaller than the gaugino masses, the on-shell correctionsto the gaugino masses are small, so we can identify the gaugino mass parameter with the mass M r . In addition,we can average over the light squark masses. The term proportional to the hypercharge and S then drops outand we find for the average squark mass m q ( Q ) : = 14 (cid:104) m Q L + m u R + m d R (cid:105) ( Q ) = m q ( M ) + 14 (cid:88) ˜ f =2 ˜ Q L , ˜ u R , ˜ d R (cid:88) r =1 C ˜ fr b r (cid:20) − α r ( M ) α r ( Q ) (cid:21) M r ( Q ) . (II.7)Similar averaged expressions can be obtained for the sleptons.At scales Q below M the U (1) and SU (2) running implies α ( M ) /α ( Q ) >
1, while for SU (3) this ratio is lessthan one. Thus all three terms in the r sum on the right hand side of Eq.(II.7) are positive. Assuming thatthe initial soft sfermion mass terms are non-negative, i.e. avoiding tachyonic sfermions at the high scale , weobtain minimal sfermion mass values at the low scale Q as a function of the gaugino masses.Our argument is most straightforward for the first generation squarks where electroweak symmetry breakingeffects play no role. Given a fixed gluino mass we find the lowest possible squark mass when the wino andbino masses vanish. The red curve in the left panel of Fig. 1 gives the minimal ratio of squark to gluino massaveraged over ˜ u L,R and ˜ d L,R . If instead of very light weak gaugino masses we assume gaugino unification thismass ratio slightly increases, as can be seen from the blue curve in Fig. 1.Different mediation scales, which we implicitly assume for any SUSY model, put restrictions on the achievablephysical squark masses in terms of a lower limit on m ˜ q /M ˜ g . The constraints on the mass-ratio m ˜ q /M ˜ g can beinterpreted as region boundaries on the two-dimensional squark-gluino mass plane as shown on the right panelin Fig. 1.Beyond our basic observation, we need to make a technical aside on the role of the low scale in Fig. 1.Eq.(II.7) depends on the choice of the renormalization point Q defining the physical masses observable at theLHC. This dependence is logarithmic and therefore quite weak. In the left panel of Fig. 1 we simply choose Q = 1 TeV. In the right panel we included this dependence by evaluating m ˜ q ( m ˜ q ) and M ˜ g ( M ˜ g ). Therefore, thelines separating the three regions are not entirely straight. For models with high scale tachyons see [17].
Ignoring any high-scale physics features we start from phenomenological weak-scale SUSY models populatingthe entire squark-gluino mass plane. The more we then increase the scale of mediation, the stronger theconstraints become and the smaller the area in the mass plane we can cover. Turning this argument around,the position of a low-energy supersymmetric model on the squark-gluino plane can be used to find an upperlimit on the possible mediations scales or even make a statement about the absence of such a scale.In the right panel of Fig. 1 we divide the full plane into three regions: region I (blue) can easily be reachedby all known models of SUSY-breaking, including gravity and gauge mediation. To illustrate this we have alsoindicated in Fig. 1 a set of SUSY benchmark points proposed and studied over the years in Refs. [1, 2, 19, 20].Region II (green) corresponds to SUSY models where the breaking is mediated in the window 10 GeV − M GUT .It is not accessible to gravity mediation but provides a good home for gauge mediation. Finally, if SUSY shouldbe discovered in Region III (orange) its breaking would have to be described by an exotic theory. It wouldhave to descend from a theory with no or little separation between the electroweak and the SUSY mediationscales, excluding anything similar to gauge and gravity mediation. These and other possibilities will be furtherdiscussed in Sect. V.There are different ways to study region III. One way is to start from the so-called phenomenological MSSM(pMSSM) [14] Lagrangian where all MSSM soft parameters are defined at the weak scale and no assumptionson the SUSY breaking mechanism need to be made. For studies along these lines see [5]. Alternatively we canutilize the simplified model approach [15], where one reduces the number of decay topologies and with it theparameter dependence of branching ratios to a level where only the masses of the particles appearing in theproduction and decay channels have to be tracked. The main difference between these two approaches is thatfor the weak-scale pMSSM several decay topologies can contribute to a given signature and that non-trivialbranching ratios are included in the analysis. From our point of view both approaches are well suited to avoidLHC searches based on a theory bias.One example is a simplified model with light squarks and gluinos and a massless neutralino. The 95 %confidence level exclusion contour for this model based on 1 .
04 fb − of ATLAS data from [18] is shown inFig. 1. The considerable mismatch between these exclusion contours and their CMSSM counterparts [1, 2](where they are defined) is mostly due to different neutralino mass assumptions. We can, however, easily modifythe simplified model by assuming a light rather than massless bino and locking its mass to an appropriatelyrescaled gluino mass, m ˜ χ ∼ M = α /α M . This would largely be equivalent to the high-scale motivatedmodels where they are possible, while avoiding assumptions about the scale and the precise nature of SUSYmediation mechanisms which should be results of an analysis instead of assumptions.Last but not least, it should be noted that the regions of the squark-gluino plane which lie outside the usualhigh scale motivated region are particularly interesting from a phenomenological point of view. If a gluino (orother color octet) becomes significantly heavier than the color-triplet squark it is likely that we will reconstructtwo hard decay jets, in addition to the well understood softer QCD jet radiation [22]. The observation of forexample four such hard jets would clearly point to the production of a pair of color octet particles [23]. Thereconstruction of the effective mass is also easier if we see several hard decay products, so we can correlate itwith the number of jets, to get a first global guess at the properties of the new particles [23]. Finally, longeron-shell decay chains with hard decay products are the basis of any kind of SUSY parameter analysis, whichfor example rely on the decay ˜ g → ˜ b → ˜ χ → ˜ (cid:96) → ˜ χ [5, 24]. III. SLEPTON VS BINO/WINO MASS
Similarly to the squark-gluino mass plane discussed in Sect. II, we can also project SUSY models onto theelectroweak slepton-gauginos mass planes. Again, the region attainable for models with a reasonably highmediation scale turns out to be wedge shaped.Using Eq.(II.6) we can compute ratios of the left and right handed slepton masses to the bino and winomasses. Before doing that let us address the term ∝ S which is not positive definite. In the simplest andmost commonly used models, like the CMSSM or gauge mediation with universal Higgs masses, S = 0 at themediation scale and remains so 1-loop. Thus the second term in Eq. (II.6) is absent. For more general models,including models with non-universal Higgs masses the S term is generally non-zero. One way to address thisissue would be to average over the charged sleptons similarly to what was done for squarks in the previoussection. Instead we will choose to work with left and right handed sleptons separately and make use of the fact Log (cid:72) M (cid:76) (cid:72) m s l e p t o n , R (cid:144) M ga u g i n o (cid:76) m i n Log (cid:72) M (cid:76) (cid:72) m s l e p t o n , L (cid:144) M ga u g i n o (cid:76) m i n Figure 2: Left: minimal ratios m ˜ e R /M j for the bino (blue) and wino (red) as a function of the messenger scale. Whereasthe bino curve is independent of all other masses, the wino curve assumes universal gaugino masses. Right: minimalratios m ˜ e L /M j for bino/wino (blue/red), assuming universal gaugino masses. For the yellow curve which shows m ˜ e L /M ,only the bino mass is taken to be non-zero. that the hypercharge has opposite sign for the left and the right handed species. Therefore, if the effect is tolower the sfermion to gaugino mass ratio in one case, it will unavoidably increase it in the other. Thus we willproceed with the determination of the minimal slepton to gaugino mass ratios derived from Eq. (II.6) without the S term. The caveat is that a non-zero S has the potential to lower either the right or left handed sfermionmasses but never both. Hence one of the minimal ratios cannot be lowered.We show the minimum values for all four combinations of left and right handed sleptons compared to binoand wino in Fig. 2. The corresponding regions in the two-dimensional mass planes are shown in Fig. 3. Naively,one would think that the renormalization group running should be flatter than in the case of squarks, dueto the smaller gauge couplings. However, the relative contribution of the gauginos to the sfermion masses in N o N e u t r a li n o D a r k M a tt er G e V (cid:60) M (cid:60) M G U T no high scale model M bino m s e l ec t r o n , R G e V (cid:60) M (cid:60) M G U T no high scale model M wino m s e l ec t r o n , L Figure 3: Three regions in the sfermion-gaugino mass plane, for a bino or wino and left and right handed selectrons.The color coding is the same as in in Fig. 1. We assume gaugino mass unification. The “No Neutralino Dark Matter”diagonal indicates where selectrons are lighter than the lightest neutralino. We also display benchmark points presentedin [1, 2, 19, 20]. In the left panel the dots indicate χ and selectron masses whereas in the right panel they correspondto χ ± and left handed selectron masses. m selectron, L (cid:62) m selectron, R m selectron, L (cid:60) m selectron, R M bino M w i n o Figure 4: Left and right handed selectron masses for a non-chiral input at the messenger scale, as well as S = 0. In theblue (dark green) region the left (right) handed selectrons are heavier for M > GeV. In the light green region theright handed selectrons can be heavier for sufficiently large M . Eq.(II.6) is proportional to the relative change in the gauge coupling divided by the beta function coefficientwhich is of the same order of magnitude for all three gauge groups.The reason for the significant difference between the ratios for the left and right handed selectrons is thechiral nature of the electroweak interactions and the gauge structure of the gauginos. Even if the initialsupersymmetry breaking for the sfermions is chirality blind, left and right handed sfermions will be split duringthe renormalization group evolution. The typical example for such a mediation is gravity. In contrast, gaugemediation does have a chiral structure already at the messenger scale.The main difference between the squark-gluino case and the slepton-gaugino results shown in Fig. 3 is thetranslation of the Lagrangian parameters into the masses of the physical states. While for the gluino we onlyhave to take into account a moderately small correction to the on-shell mass scheme, the weak gauginos aregenerally mixed.What we can study at the Lagrangian parameter level, however, are the different slepton masses. FromFig. 2 we already know that renormalization scale evolution separates left and right handed selectrons. Foruniversal gaugino masses the contribution to the left handed sfermions is always bigger and therefore left handedsfermions are heavier.The two-dimensional slepton mass plane in Fig. 4 shows the ordering of the left and right handed massesas a function of the bino and wino masses assuming chiral degeneracy at the messenger scale. Gaugino massunification, as often assumed in LHC searches, implies M ∼ M . This translates into a solid prediction m ˜ e L > m ˜ e R . However, for non-universal gaugino masses [21] this can be different. If the bino is significantlyheavier than the wino the right handed sfermions could indeed be heavier. Therefore, in the same way that weshould not unnecessarily assume the squark-gluino mass hierarchy as described in the previous section, LHCsearches should not be based on the assumption that the lighter sleptons do not couple to the wino.Our discussion so far has been in terms of bino and wino components of the electroweak gauginos, but asalready noted, due to the effects of electroweak symmetry breaking the bino and wino are not the appropriatemass eigenstates. Their mass matrix is given by M ˜ χ = M − c β s w m Z s β s w m Z M c β c w m Z − s β c w m Z − c β s w m Z c β c w m Z − µs β s w m Z − s β c w m Z − µ , (III.1)where s w = sin θ w , c w = cos θ w , etc. This mass matrix is real and symmetric, so its eigenvalues are real.Accordingly, the mass matrix squared is positive definite and its smallest eigenvalue is smaller than any of itsdiagonal elementsmin m ˜ χ < min (cid:34)(cid:114) M + m Z − cos(2 θ w )2 , (cid:114) M + m Z θ w )2 , (cid:114) µ + m Z ± cos(2 β )2 (cid:35) < min (cid:34)(cid:114) M + m Z − cos(2 θ w )2 , (cid:114) M + m Z θ w )2 (cid:35) . (III.2)For M , (cid:29) m Z the smallest eigenvalue of the neutralino mass matrix is usually smaller than both M and M .Therefore, the minimum curves for m ˜ e /M in Fig. 2 also set a lower limit on the ratio of selectron/slepton tothe smallest neutralino mass.Because of the wealth of additional parameters the relevant question is if these bounds are saturated. Ina first attempt we assume gaugino mass unification, which means the bino is roughly 6 times lighter thanthe gluino. Current LHC constraints imply M ˜ g >
750 GeV, translating into M >
125 GeV, so our originalassumption M (cid:29) m Z is reasonable. For illustration purposes we also assume large µ , so we can consider thelimit m Z (cid:28) | M ± µ | , | M ± µ | and M , (cid:28) µ . In this regime the lightest neutralino is bino-like and its massis given m ˜ χ = M − m Z s w µ − M [ M + µs β ] = M (cid:20) O (cid:18) m Z µM (cid:19)(cid:21) . (III.3)In this limit the bound in the slepton-gaugino mass plane can indeed be saturated. Our expectations for theratios between neutralino and bino masses are confirmed in the test models briefly discussed in the next section.Corresponding points are shown in the left panel of Fig. 5.One might be curious to see how the SUSY benchmark points included in the squark-gluino plane in Fig. 1are distributed on the electroweak mass planes. The black dots in the left panel of Fig. 3 denote values of themass for ( m ˜ χ , m ˜ e R ) for those benchmark points. As expected, they lie in the high-scale region. The only pointlocated in the green region corresponding to messenger scales below 10 GeV is a gauge mediated point witha very low messenger scale of 80 TeV. We will continue the discussion of benchmark points and models in thenext section.In the left panel of Fig. 3 we also introduce a “No Neutralino Dark Matter” line. Below it, a bino-likeneutralino cannot be dark matter. It would decay into the lighter right handed selectron which cannot be darkmatter, as it is charged. This requirement is strongly correlated with a high mediation scale, i.e. once werequire the bino to be the dark matter candidate we automatically constrain the available parameter space tothe fraction accessible by high scale SUSY breaking. Perhaps an obvious point to note is that if dark matteris not made of neutralinos all points below the dashed line remain perfectly viable.If the lightest neutralino is bino-like in the limit of large | µ | , the second lightest neutralino is wino-like. In thiscase we can interpret dark green region in the right hand panel of Fig. 3 as the area for ˜ χ vs. m ˜ e L inaccessibleto high scale models. However, here we need to be careful with possible gaugino-Higgsino mixing effects.We can apply the same argument as for neutralinos to the chargino sector with its mass matrix M ˜ χ ± = (cid:18) X T X (cid:19) with X = (cid:18) M √ s β m W √ c β m W µ (cid:19) . (III.4)Its eigenvalues are given by (each twice), m χ ± j = 12 (cid:104) | M | + | µ | + 2 m W ∓ (cid:113) ( | M | + | µ | + 2 m W ) − | µM − m W s β | (cid:105) . (III.5)Again, we find that the smallest eigenvalue is bounded from above asmin m ˜ χ ± < min (cid:104)(cid:113) M + 2 s β m W , (cid:113) M + 2 c β m W (cid:105) < (cid:113) M + m W . (III.6)For m W (cid:28) M this smallest eigenvalue is typically below M . Therefore, the wino curves for m ˜ e /M in Fig. 2can be interpreted as lower limits on selectron to lightest chargino mass ratio as a function of the messengermass. Hence, the separation into three regions in the right hand panel of Fig. 3 can be directly interpreted interms of physical masses. Again, for illustration we have indicated the distribution of the benchmark points. N o N e u t r a li n o D a r k M a tt er G e V (cid:60) M (cid:60) M G U T no high scale model M bino m s e l ec t r o n , R G e V (cid:60) M (cid:60) M G U T no high scale model M wino m s e l ec t r o n , L Figure 5: The sfermion-gaugino mass plane, for a bino or wino and left and right selectrons and with the same colorcoding as Fig. 1. The dots represent scans over high-scale models, namely the CMSSM (black), a low-scale CMSSM(blue) and pure generalized gauge mediation (red) (see text for details).
IV. BENCHMARK AND TEST MODELS
As shown in Section II, no supersymmetric model arising from a theory with a high scale and containingMajorana gauginos can cover the squark-gluino mass plane. As a result, large regions in the squark-gluinoand slepton-gaugino mass planes are not populated by such high scale models. For example, assuming gravitymediation at M GUT , only roughly half of the parameter space corresponding to the light blue area in Fig. 1will be covered. For gauge mediation this effect is slightly more moderate as such models can enter into and(if the mediation scale is chosen suitably low) cover the green area in Fig. 1.This shortcoming becomes particularly obvious when we study benchmark points provided by theorists tohelp guide the LHC experiments. Ref. [26] lists a standard set of the benchmark compiled for and used atthe LHC [1, 2, 19, 20]. These benchmarks are shown as black dots in Fig. 1. The first and most importantrequirement on benchmark points is to represent the available parameter space. The distribution we observe inFig. 1 clearly shows that this is not the case, provided we consider the weak-scale MSSM the model the LHClooks for. All benchmark points populate the region of the squark-gluino mass plane which can be linked tohigh-scale SUSY breaking. In addition, reminiscent of the population of Scotland (or Canada), the vast majorityof benchmark points in Fig. 1 live along the southern border which saturates the m ˜ q /M ˜ g mass ratio, i.e. valuesof the squark mass where the renormalization group induced contribution shown in Eq.(II.6) dominates overthe soft breaking scalar mass. One of the underlying reasons for this squeezed distribution is that most of thebenchmark points are CMSSM points. In the CMSSM all sfermions have the same initial mass at the GUTscale characterized by the parameter m which is typically chosen to be of the order of the electroweak scale. Atthe same time the contributions arising from gauginos scale with their gauge couplings and gluino contributionsare therefore dominant.For the weakly interacting particles all but one benchmark point also lie in the upper region. Indeed byconstruction all those points lie even above the “No Neutralino Dark Matter” line. The benchmark pointsare now more spread out because the initial value of the universal CMSSM sfermion mass is comparable toelectroweak gaugino contributions. Nevertheless, they still cover only a restricted area of parameter space.To illustrate more generally ( i.e. not just based on the limited set of benchmark points) how the sfermion-gaugino mass plane is populated by high-scale models we show in Fig. 5 a large set of parameter points scanningover a variety of test models:– the CMSSM with tan β = 3 , ,
40 and A = 0;– the same initial soft parameters (tan β = 3 , , A = 0) but at lower M = 2 · , · GeV;0– pure general gauge mediation (pGGM) with M mess = 10 , , GeV, see Refs. [20, 25, 26] for details.Following our discussion in the previous section we use for the x axis coordinates the masses of the lightestneutralino m ˜ χ and the lightest chargino m ˜ χ +1 . The (black) CMSSM points indeed cover the accessible parame-ter space and saturate the minimal ratios for m ˜ e /M , . The (blue) low-scale “gravity mediation” points extendinto the intermediate M wedge though they do not approach the lower end it. The pure general gauge media-tion points marked in red extend further into this intermediate region. The same models have a qualitativelyvery similar behavior on the squark-gluino mass plane.We also note that in the left panel the pGGM points do not extend to arbitrarily high neutralino masses.This is special to this model which becomes non-perturbative for parameter values that correspond to largebino masses. V. CONCLUSIONS
Many LHC searches for supersymmetry are conveniently interpreted on the squark-gluino mass plane. Inthis note we have argued that in the MSSM all sfermion-gaugino mass planes can be divided into three wedgeshaped-regions: the first region with high squark masses is accessible to all types of SUSY models includingthose with a high mediation scale M (cid:38) M GUT . The second region with intermediate values of the sfermion togaugino mass ratio requires a mediation scale
M < M
GUT . Finally, the third region with the low sfermion togaugino mass ratios cannot be accessed by any MSSM type model with a mediation scale M ≥ GeV. Themodels in this third wedge would have to be described by an exotic SUSY theory. Discovering SUSY in thisregion would be a particularly surprising and exciting outcome.What does “exotic” mean in this context and how might such theories look? In general, any renormalizationgroup evolution of scalar masses sufficiently different from the one considered here could result in a theory livingin the third wedge. The renormalization group equations we employed are inherently MSSM equations. A non-MSSM matter content could therefore give an exotic theory. One well understood example of this are modelswith Dirac gauginos [27]. In these theories the Dirac gaugino masses simply do not determine the running thesfermion masses [28]. Of course, this is just one example of an exotic theory arising from a non-MSSM setup.One of our technical assumptions was that the sfermion masses at the mediation should not be non-tachyonic.In principle, allowing such tachyons is a way to lower the physical sfermion to gaugino mass ratios below minimalvalues for high scale models we have computed in this note. The examples of these models examined in [17] oftencontain low lying color breaking vacua and while in general models of this type are not necessarily excludedthey need to be carefully screened for dangerous instabilities.A third large class of exotic theories are models without a significant separation between the electroweak scaleand the scale at which the soft terms are generated. Practically, such models could be described by effectiveactions with soft terms defined at the collider scale, avoiding any renormalization group evolution. An evensimpler approach to generate model points would be to use various versions of simplified models [15].From an LHC perspective the striking result of our study is that these fundamentally very different structurescan be classified in terms of the standard scalar-gaugino mass planes and that their physics is essentiallydetermined by one parameter, the mediation scale of SUSY breaking.
Acknowledgments
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