Phenomenological Implications of Deflected Mirage Mediation: Comparison with Mirage Mediation

We compare the collider phenomenology of mirage mediation and deflected mirage mediation, which are two recently proposed"mixed"supersymmetry breaking scenarios motivated from string compactifications. The scenarios differ in that deflected mirage mediation includes contributions from gauge mediation in addition to the contributions from gravity mediation and anomaly mediation also present in mirage mediation. The threshold effects from gauge mediation can drastically alter the low energy spectrum from that of pure mirage mediation models, resulting in some cases in a squeezed gaugino spectrum and a gluino that is much lighter than other colored superpartners. We provide several benchmark deflected mirage mediation models and construct model lines as a function of the gauge mediation contributions, and discuss their discovery potential at the LHC.


Introduction
TeV scale softly broken supersymmetry (SUSY) (for recent reviews, see [1,2]) is one of the best-motivated candidates for physics beyond the Standard Model (SM). Theories with softly broken supersymmetry, such as the minimal supersymmetric standard model (MSSM), will be tested thoroughly at the Large Hadron Collider (LHC). The collider phenomenology of such theories depends in detail on the associated soft supersymmetry breaking terms, which in turn are largely governed by the way in which supersymmetry breaking is mediated from a hidden sector to the SM fields. As the number of possible soft supersymmetry breaking parameters is vast even in minimal extensions of the SM such as the MSSM, it is fruitful to develop models of supersymmetry breaking. Exploring the possible mediation mechanisms for supersymmetry breaking is crucially important for LHC tests of the hypothesis that supersymmetry is present at TeV energy scales.
Viable supersymmetry breaking models are based on the hidden sector paradigm, in which supersymmetry is broken in a hidden or secluded sector and communicated to the observable sector via the interactions of mediator fields. As is well known, the phenomenological implications of these models are largely insensitive to the details of the hidden sector, and instead are governed by the mediation mechanism that is responsible for the transmission of supersymmetry breaking to the MSSM fields. Most models of supersymmetry breaking involve one of the three most popular mediation mechanisms: (i) gravity mediation, (ii) gauge mediation, and (iii) (braneworld-motivated) "bulk" mediation models. Gravity-mediated terms [3] arise from couplings that vanish as the Planck mass M P → ∞; examples include the (tree-level) minimal supergravity (mSUGRA) model (also known as the constrained MSSM (CMSSM)) and modulus mediation models [4], and the (loop-suppressed) anomaly mediation models [5]. Gauge-mediated terms arise from loop diagrams involving new messenger fields with SM charges [6,7,8]. Bulk-mediated terms arise from bulk mediator fields in braneworld scenarios. Examples include gaugino mediation [9] and Z ′ mediation [10]. Certain gravity-mediated models, such as the pure anomaly mediation scenario [5] (which requires sequestering), are also bulk mediation models. Typically, one of these mediation mechanisms is assumed to dominate (see e.g. [11]), for simplicity and practicality and/or to solve a given problem of the MSSM, such as the µ problem or the flavor/CP problems.
A complementary approach is to consider models in which more than one mediation mechanism plays an important role. Such "mixed" scenarios are motivated within stringmotivated constructions, such as the Kachru-Kallosh-Linde-Trivedi (KKLT) approach to moduli stabilization [12]. A prototype KKLT-motivated example is mirage mediation [13,14], a phenomenological model in which the tree-level gravity (modulus)-mediated terms and the (loop-suppressed) anomaly-mediated terms are comparable in size, contrary to naive expectations. As a result, the soft masses unify at a scale that is typically well below the scale where they are generated, resulting in "mirage" unification. Mirage mediation has distinctive phenomenological implications compared with standard minimal supergravity models [15,16,17,18]. These features include a gaugino mass pattern that is typically more squeezed than the standard mSUGRA/CMSSM gaugino mass pattern [19] and reduced low energy fine-tuning [20], for which the details depend on the ratio of the gravity-mediated contributions to the anomaly-mediated contributions.
The recently proposed deflected mirage mediation scenario is an extension of mirage mediation in which gauge-mediated supersymmetry breaking terms are also present and competitive in size to the gravity-mediated and anomaly-mediated soft terms [21,22]. This framework is denoted as deflected mirage mediation because the mirage unification scale in the gaugino sector is shifted or "deflected" from its mirage mediation value due to threshold effects associated with the gauge mediation messengers fields. In deflected mirage mediation, the gauge-mediated terms arise from the couplings of an additional matter modulus field X and vectorlike messenger pairs with nontrivial SM quantum numbers. The ratio of the gauge-mediated and anomaly-mediated terms depends on the details of the stabilization mechanism for the mediator field X. In [22], it was found that if the stabilization mechanism is dominated by supersymmetry breaking effects, generically the gauge-mediated and anomaly-mediated contributions are comparable. Deflected mirage mediation provides a general framework in which to explore mixed supersymmetry breaking scenarios at the LHC, where well-known single mediation mechanism models can be recovered by judiciously adjusting dimensionless parameters in the theory.
In this paper, we explore the question of how the collider phenomenology of deflected mirage mediation differs from that of standard mirage mediation, following previous work on the sparticle spectrum [22,23] and dark matter constraints [24]. More precisely, we focus our attention on the effects of the gauge mediated terms by varying the ratio of gauge to anomaly mediation contributions and/or the number of messenger pairs, as compared to benchmark KKLT mirage mediation scenarios. The differences in the collider phenomenology of the deflected mirage mediation scenarios compared to the corresponding pure mirage mediation scenarios depend primarily on the deflection of the gaugino mass mirage unification scale. This depends in turn on the size of the messenger scale and whether the threshold effects are large or small. For small threshold effects, the pattern of soft masses does not differ greatly from the corresponding pure mirage mediation limit. However, if the threshold effects are large, one can have situations in which the gaugino mirage unification scale can be deflected from a high scale value to the TeV scale. In such situations, the gaugino mass spectrum is squeezed, resulting in gluinos that are typically lighter than other colored superpartners. The LHC phenomenology is thus dominated by gluino production in this case, with soft decay products due to the compressed chargino and neutralino mass spectrum.
The outline of this paper is as follows. In Section 2, we review the theoretical framework of deflected mirage mediation and enumerate the supersymmetry breaking terms of the theory. An overview of mirage unification and the properties of the resulting low energy mass spectra in both mirage mediation and deflected mirage mediation is given in Section 3. We then discuss the collider phenomenology of benchmark deflected mirage mediation models and compare it to that of pure mirage mediation benchmark scenarios in Section 4. In Section 5, we provide our conclusions and outlook.

Theoretical Framework
We begin with a brief review of mirage mediation, a phenomenological model motivated from the KKLT flux compactification approach within Type IIB string theory [12]. In this setup, the MSSM fields are confined to a stack of D branes that are localized in a higher-dimensional bulk Calabi-Yau space, and the hidden (supersymmetry breaking) sector consists of anti-branes at the tip of the warped throat geometry. The tree-level gravity mediation terms are where T is the Kähler modulus. The anomaly mediation terms are m (anomaly) soft where C is the conformal compensator of the gravity multiplet. In the KKLT scenario, there is a cancellation between the superpotential terms from the fluxes and the nonperturbative terms, leading to a supersymmetry-preserving vacuum with stabilized moduli but with a negative cosmological constant. Supersymmetry is then broken by an uplifting potential of the form (T + T ) −np , where n p = 2 in the KKLT model, due to the anti-branes at the tip of the warped throat. After cancelling the cosmological constant, the following mirage mediation relation is obtained [13,14]: in which M P = 2.4 × 10 18 GeV is the reduced Planck mass, and m 3/2 is the gravitino mass. As m 3/2 is typically ∼ 100 TeV in this class of models, the factor of ln(M P /m 3/2 ) is numerically close to 4π 2 . Hence, the tree-level gravity mediation terms are comparable to the anomaly mediation terms in mirage mediation. In deflected mirage mediation, the observable sector matter content also includes a gauge singlet X and vectorlike messenger pairs Ψ, Ψ with SM gauge charges, which are fields that are generically present in string-derived models. In general, X can acquire an F term vacuum expectation value, leading to gauge mediated terms of the form Depending on the stabilization mechanism for X, it was shown in [21,22] that in general, in deflected mirage mediation. To see this more clearly, let us begin with the effective supergravity theory of KKLTinspired models. Labeling the observable sector (MSSM) fields as Φ and taking a diagonal matter metric for simplicity, the Kähler potential at leading order is where n X and n i are the modular weights of X and Φ i , respectively. The superpotential is In Eq. (2.7), W 0 is the part of the superpotential that governs (together with the uplifting potential) the supersymmetry breaking effects [12], W 1 (X) = λ n X n describes the possible self-couplings of X, and W MSSM takes the form in which µ 0 ij are supersymmetric mass parameters, and y 0 ijk are the (unnormalized) Yukawa couplings. The gauge kinetic functions are assumed to take the form (2.9) The messengers are taken to be Ψ, Ψ are 5,5 representations of SU (5), as is standard in many models of gauge mediation. Here N will denote the number of such messenger pairs. Upon computing the soft terms using standard supergravity techniques, we obtain the observable sector soft supersymmetry breaking Lagrangian, which is of the usual form in which m 2 i are the soft scalar mass-squared parameters, M a are the gaugino masses, and A ijk are trilinear scalar interaction parameters. 1 Recalling that above the mass scale of the messengers M mess ≡ λ X , the beta functions depend on not only the MSSM fields, but also on the messenger pairs, the soft terms at the GUT scale M G and the messenger threshold effects at M mess are as follows: Gaugino Masses. The gaugino mass parameters are given by in which the threshold corrections are Here g 0 is the unified gauge coupling at M G , and the beta functions b ′ a are related to their MSSM counterparts by b ′ a = b a + N , with (b 3 , b 2 , b 1 ) = (−3, 1, 33 5 ) (in our conventions, b a < 0 for asymptotically free theories).

Trilinear terms. The trilinear terms are
where γ i is the anomalous dimension of Φ i .
Soft scalar masses. The scalar mass-squared parameters are given by where the threshold corrections are In the above, c a is the quadratic Casimir, and γ i , We now replace the F terms with the parameterization given in [21,22], as follows: in which M 0 ≡ F T /(T + T ) sets the overall scale of the soft terms. The dimensionless parameter α m is the α parameter of mirage mediation: it denotes the relative importance of anomaly mediation with respect to gravity mediation. In the KKLT case α m = 1; the case of α m = 2 (i.e, an uplifting potential with n p = 1, which has no obvious string theory realization) has also been considered in the literature, for reasons that will be clear shortly.
The dimensionless parameter α g denotes the relative importance of the gauge-mediated terms with respect to the anomaly-mediated terms. The values of α g depend on the details of the stabilization of X [21,22]. For radiative stabilization, while for self-couplings of the form W 1 = λ n X n , Here n is restricted to the values n ≥ 3 or n < 0, i.e., stabilization by higher-dimensional operators or non-perturbative effects, respectively. With the parametrization given in Eqs. (2.17)-(2.18), the soft terms at M G take the form and the threshold terms are given by The parameters of the model are the mass scales M 0 and M mess , as well as the dimensionless quantities α m , α g , the number of SU (5) messenger pairs N , the modular weights n i , tan β, and signµ (the model-dependent µ and Bµ parameters are replaced as usual by the Z boson mass, tan β, and the sign of µ). Here, we will fix the modular weights to the standard values n i = 1/2 for all SM matter fields, and n i = 1 for the MSSM Higgs doublets. With this choice, we see that in deflected mirage mediation, in addition to the usual tan β and signµ, there are the continuous parameters M 0 and M mess , the discrete parameter N , and the parameters α m and α g , which take on discrete values within a particular string framework but can be taken to be continuous parameters within a purely phenomenological approach. Hence, deflected mirage mediation models in this class have 6 parameters plus one sign. This is to be compared with the 4 continous parameters (the universal scalar mass M 0 , the universal gaugino mass M 1/2 , and the universal trilinear scalar coupling A 0 , and tan β) plus one sign (the sign of µ) in mSUGRA/CMSSM models. We will see that despite its relatively small extension of parameters compared to mSUGRA/CMSSM models, deflected mirage mediation provides a rich framework for LHC phenomenology.

(Deflected) Mirage Unification and Particle Mass Spectra
To compare deflected mirage mediation to mirage mediation, we provide here a discussion of the prototypical feature of mirage mediation, which is the phenomenon of mirage unification and its resulting profound impact on the low energy spectrum (see also [21,22,23] for previous discussions). Mirage mediation is thus named because at the one-loop order, the soft terms unify not at the unification scale M G ∼ 10 16 GeV as in the case of mSUGRA/CMSSM models, but rather at a "mirage" scale [13,14,15]: where α m is the ratio of anomaly mediation to gravity mediation terms given in Eq. (2.17).
For the KKLT value of α m = 1, the mirage unification occurs at ∼ 10 10 GeV, as shown in Fig. 3. Here we have included two-loop effects in the running, which spoil the precise unification, though the general features are maintained. The mass spectrum, also given in Fig. 3, shows that this model has a relatively heavy gluino, but has a slightly compressed spectrum with respect to corresponding mSUGRA/CMSSM models. Smaller values of α m have mirage unification scales closer to M G (with M mir = M G in the limit of α m = 0), while larger values of α m result in lower mirage unification scales. The case of α m = 2 results in a mirage unification scale at TeV energies, resulting in a highly compressed superpartner spectrum and a reduced little hierarchy problem [16,20]. In deflected mirage mediation, mirage unification is maintained in the gaugino sector (though lost in general in the scalar sector), as follows [21,22]: with the parameter ρ given by 3) show that the mirage mediation limit is obtained for N = 0 (i.e., the absence of messengers). If α g = 0, corresponding to vanishing gauge mediation contributions, the messengers still contribute to anomaly mediation, as reflected in Eq. compared to that of the pure mirage mediation case. For fixed α m , N , and M mess , the effects of a nonzero α g are straightforward to understand: for α g > 0, ρ increases and M mir is lowered, while for α g < 0, ρ decreases and M mir is correspondingly increased.
As an example of a model with gaugino mass mirage unification near the TeV scale, in Fig. 3 we show the renormalization group running of the gaugino and soft scalar masssquares, as well as the particle mass spectrum at the TeV scale, for the case of α g = 1 and M mess = 10 12 GeV. Clearly, the gaugino mass spectrum is highly squeezed, with a very light gluino (the lightest of the colored superpartners). Indeed, for N = 3 the beta-function coefficient for the soft supersymmetry breaking parameter M 3 vanishes at one loop above the messenger scale, and hence the soft mass for the gluino then runs very little between the high energy input scale and the messenger scale. Therefore, the gluino mass is much smaller for these cases than for analogous pure mirage mediation models. Furthermore, in contrast to the pure mirage mediation case with α m ≃ 2, which also has mirage unification near the TeV scale, the deflected mirage mediation scenarios with low-scale mirage unification have heavier scalars. The full superpartner spectrum is thus stretched with respect to the pure mediation scenario due to the effects of gauge mediation.
In contrast, we show in Fig. 3 a deflected mirage mediation scenario with the same parameters as that of the previous figure, except that α g = −0.5. In this case, there is a much higher gaugino mirage unification scale that is close to that of the pure mirage mediation limit, since with this choice of parameters ρ ≈ 1. As a result, the particle mass spectrum strongly resembles that of Fig. 1, with heavier colored superpartners and no strong degeneracy between the lightest chargino and neutralino. This feature depends on the messenger scale; for α g < 0, there are always pairs of α g and M mess for which ρ = 1 and thus M mir is given by the pure mirage mediation limit. Furthermore, if the messenger scale is below the mirage unification scale as given in the pure mirage mediation case by Eq. (3.1), the mirage unification behavior is maintained not only for the gauginos, but also for the scalars (though at a different scale than the pure mirage mediation limit due to the presence of the messengers). For α g > 0, we can find model points in which the gaugino mass unification scale is sub-TeV, resulting in a flipped gaugino mass spectrum in which the wino is the lightest superpartner, similar to the gaugino pattern in the pure anomaly mediation limit. An example is shown in Fig. 3, in which all parameters are the same as in Fig. 2, except that M mess = 10 5 GeV. We see, therefore, that deflected mirage mediation models roughly can be categorized according to their values of α g as follows: the case of large threshold effects (α g > 0, nonperturbative stabilization), and the case of small threshold effects (α g < 0, radiative or higher-dimensional stabilization). This can be easily understood from Eqs. (2.21)-(2.25); for α g > 0, F X /X and F C /C have the same sign, and thus the threshold effects for gauge and anomaly mediation constructively interfere, while for α g < 0, the threshold effects destructively interfere. For small thresholds, the particle mass spectra are typically similar to a corresponding pure mirage mediation spectrum, while for large thresholds, deflected mirage mediation can result in significantly different mass spectra with non-standard gaugino mass patterns characterized by a squeezed gaugino mass spectrum with relatively light gluinos and a lightest superpartner with a mixed wino-bino-Higgsino content, similar to gauge messenger models [25,26]. This is phenomenologically interesting for a variety of reasons, including fine-tuning considerations [20], LHC signatures [27,19,28,29], and dark matter signals [30,31,33,32,34,24]. However, the thresholds must not be too large, otherwise the gluino can become the lightest superpartner. Therefore, viable deflected mirage mediation models with large threshold effects have bounds on the allowed range of α g , depending on the other parameters. The examples we have shown here, while not chosen to optimize the dark matter predictions, encompass all of these possibilities.

Collider Phenomenology
The soft term expressions in Eqs. (2.21)-(2.25) have sufficient complexity to produce a wide variety of possible low-energy superpartner spectra. The analysis of Section 3 gives evidence of the wide diversity of outcomes which can arise. In this section we will turn our attention to how the addition of gauge mediation to the mirage pattern of particle masses influences the collider phenomenology of this class of models.

Comparison of Mirage and Deflected Mirage Mediation
We will begin by considering a set of benchmark points designed to illustrate some of the differences between pure mirage mediation models and deflected mirage mediation models, focusing on the case of large thresholds (α g > 0) as described in Section 3. The benchmark models were chosen such that the deflected mirage mediation models satisfy the Higgs mass bound and the dark matter relic density constraints; note that the pure mirage mediation models given here do not satisfy these constraints, but are shown for sake of comparison. The high scale input parameters which define these points are collected in Table 1, along with the physical particle masses at the TeV scale and the composition of the lightest neutralino, which is the lightest superpartner (LSP) in these models. The six model points are grouped into three pairs, with the first model in each pair being a pure mirage mediation model with the number of vector-like 5 +5 messenger fields set to N = 0, while the second model is a deflected mirage point for which we have taken N = 3. For all of the model points in Table 1, we have set tan β = 10 and have chosen modular weights n i = 1/2 for the matter fields and n i = 1 for the Higgs fields, as described previously.
The first pair are models for which there is mirage unification at the TeV scale in the gaugino sector. Point 1 is a pure mirage mediation model with α m ≃ 2, and hence the scalars are also unified at TeV energies in this case. Point 2 is a deflected mirage mediation model with α m = 1 and α g > 0, which has TeV-scale gaugino mirage unification. The unification is not exact due to two-loop effects: for Point 2 the three gaugino soft masses at the electroweak scale are M 1 = 500 GeV, M 2 = 494 GeV, and M 3 = 574 GeV, while for Point 1 they are M 1 = 929 GeV, M 2 = 929 GeV, and M 3 = 1062 GeV. For Point 1, the electroweak symmetry breaking conditions constrain the µ-parameter to a relatively small value of µ = 239 GeV. The approximate unification at the electroweak scale is then between the gluino and the heavier pair of neutralinos and heavier chargino. The lighter pair of neutralinos are mostly Higgsino-like, as indicated by the LSP composition given in Table 1. For Point 2, however, the gaugino spectrum is compressed and the gluino approximately unifies with the entire ensemble of neutral and charged gauginos.
The last two pairs show the effects of keeping α m fixed and adding nonvanishing gauge mediation contributions by including messenger fields and a nonvanishing α g . The pair represented by Points 3 and 4 is designed to show the impact of adding the effects of gauge mediation on the resulting gaugino masses. Both points have M 0 = 1 TeV and α m = 0.6, yet the two cases have very different phenomenology. The spectrum for the pure mirage mediation point is similar to typical mSUGRA/CMSSM models, with a bino-like LSP, large mass gap between the lightest and second lightest neutralinos, gluinos and squarks of roughly comparable size, and a relatively light set of sleptons. In contrast, the deflected mirage mediation point has a mixed bino/wino-like LSP, a degenerate trio of N 1 , N 2 and C 1 , and a very light gluino relative to the squarks and sleptons. The final pair, represented by Points 5 and 6, are models which again have the same values of α m = 1 and M 0 = 800 The first model in each pair is a mirage mediation model and the second is a deflected mirage mediation model with N = 3. All masses are given in GeV. Low energy physical masses are given at the scale 1 TeV.
GeV. These models have very similar spectra for the light superpartners, but very different values for the gluino and squark masses and the µ-parameter. The pure mirage mediation point (Point 5) has a predominantly bino-like LSP, the deflected mirage mediation point (Point 6) has a neutralino LSP which is a mixed bino-wino-Higgsino state.
To study the collider signatures of these points at the LHC, 50,000 events were generated for each model at √ s = 14 TeV using PYTHIA 6.4 [35]. Generated events are passed to PGS4 [36] to simulate the detector response. Events are analyzed using the PGS4 level one triggers, designed to mimic the CMS trigger tables [37]. Object-level post-trigger cuts were also imposed. We require all photons, electrons, muons and taus to have transverse   given for all cases in which Lint ≤ 100 fb −1 . We require a minimum of 100 signal events in the no-lepton and single lepton channels, and a minimum of ten signal events in the multi-lepton channels.
momentum p T ≥ 10 GeV and |η| < 2.4 and we require hadronic jets to satisfy |η| < 3. Additional post-trigger level cuts were implemented for specific analyses, as described below. The total cross section for superpartner production is given in Table 2 for each benchmark model. To a first approximation the total cross section is dependent solely on the size of the gluino mass and thus deflected mirage mediation models offer the prospect of larger LHC signals relative to comparable pure mirage mediation models. The trigger efficiency is estimated using the level one trigger table of PGS4 and represents the fraction of the 50,000 generated events that passed the trigger criteria. As we will see below, however, the actual number of events that pass post-trigger cuts will often be much smaller.
The addition of the gauge messenger sector has its most striking effect in the pair of Points 3 and 4, producing a difference in the total production cross section of almost three orders of magnitude. For Point 3, collecting 50,000 signal events will require over 1000 fb −1 of integrated luminosity, while Point 4 achieves this in just over 1 fb −1 . For smaller values of N , the gluino mass would be larger and hence the expected signal size would diminish. Triggering efficiencies are generally slightly better for models with a less compressed gaugino mass spectrum. This results in slightly harder leptonic decay products at the final stages of cascade decays of gluinos and squarks. The PGS4 default level one trigger criteria requires leptons (e ± and µ ± ) to have p T ≥ 10 GeV in the dilepton channel, p T ≥ 15 GeV for an isolated lepton produced with a tau, and p T ≥ 20 GeV for a single isolated lepton produced in association with hard jets. In addition to this trigger requirement, standard supersymmetry search algorithms involving jets, leptons and missing transverse energy generally also demand minimum p T values for leptonic objects.
To demonstrate the differences between deflected mirage mediation models and their pure mirage mediation model analogs, we will here concern ourselves with counting observables associated with traditional discovery channels for supersymmetry [38], reserving a more detailed analysis of collider signatures for the following subsection. These five signatures are collected in Table 2 for 50,000 generated events at √ s = 14 TeV. These signatures are defined as follows. All five require transverse sphericity S T ≥ 0.2 and at least 250 GeV of E T except for the trilepton signature, where only E T ≥ 200 GeV is required. Multijets here refers to events with at least four jets with the transverse momenta of the four leading jets satisfying p T ≥ (200, 150, 50, 50) GeV, respectively. For this signature we impose a veto on isolated leptons. For the single lepton, opposite-sign dilepton, same-sign dilepton and trilepton signatures we include only e ± and µ ± final states and demand at least two jets with the leading jets satisfying p T ≥ (100, 50) GeV, respectively. The drastic reduction in the multijet and the leptonic signatures for the deflected mirage mediation models seen in Table 2 is caused by the small mass gap between the LSP and either the gluino or the lightest chargino/second neutralino. This is also true of the TeV-scale mirage unification model of Point 1.
Since the total cross sections vary significantly between the benchmark models, a more relevant number for comparison of the discovery potential between model points is the amount of integrated luminosity necessary to observe a clear excess of events over the Standard Model background. For this we generated a sample of 5 fb −1 Standard Model background events, consisting of Drell-Yan, QCD dijet, tt, bb, W /Z+jets and diboson production. Scaling the weight of this sample relative to the total cross section in Table 2 we can determine when a S/ √ B = 5σ excess will be detectable at √ s = 14 TeV. The results are given in Table 3. Note that we only extrapolate the value of S/ √ B for cases where L int ≤ 100 fb −1 and we require at least 100 signal events for the multijet and single-lepton channels and at least 10 signal events for the multi-lepton channels. With the exception of Point 3, all of these benchmark points will give clear signals in the multijet channel early in the high luminosity phase of the LHC with √ s = 14 TeV. Leptonic discovery channels will generally take longer to observe. Point 5, despite its modest production cross-section of 0.6 picobarns, gives sizable signals in all leptonic channels within the first 10 fb −1 . This is largely due to the mass ordering m N 1 < m τ 1 < m N 2 , which does not appear in any of the deflected mirage mediation models considered here. While several points would produce O(1000) signal events in 1 fb −1 at √ s = 7 TeV, the reliance on multijet + E T channels and absence of strong leptonic signals suggest that these points will be challenging to discover in the first year of LHC running.

Influence of α g on LHC Phenomenology
In this subsection, we wish to study in greater detail how the size of the gauge-mediated  contribution to soft supersymmetry breaking affects the expected collider signatures at the LHC for deflected mirage mediation scenarios. To do so we construct four model "lines" in which the various parameters determining the soft supersymmetry breaking masses are fixed at specific values, while allowing the parameter α g to vary. These points represent a variety of overall mass scales and spectra. We summarize the relevant input parameters in Table 4. For each case, we have chosen to fix N = 3, n i = 1/2 for the matter representations and n i = 1 for the Higgs fields, and tan β = 10. Each line involves five discrete points with α g = {−1.0, −0.5, 0.0, 0.5, 1.0}.
From Eq. (2.21), we see that the magnitude of all three soft gaugino mass parameters will diminish as α g is varied from O(1) negative values to O(1) positive values. The effect is strongest for the gluino, as the contribution to its mass from gauge mediation is proportional to the value of g 2 3 (M mess ) at the messenger scale. As mentioned in Sec. 3, we expect some value of α g to exist above which the gluino will become the LSP, which happens for α g = 0.5 and 1.0 in model line B. However, states which are charged only under U (1) Y , such as the right-handed sleptons, are largely unaffected by the variation in α g since the threshold correction to their soft masses scales as g 4 1 (M mess ). As a result the lightest stau will have a roughly constant mass across the entire model line. For α g < 0, there can be points for which the lightest neutralino is heavier than the lightest stau, which occurred for α g = −1.0 in model line A. The other points all yielded a reasonable spectrum and proper electroweak symmetry breaking at the low-energy scale.
The collider phenomenology of these models is dictated first and foremost by the overall mass scale of the superpartners -particularly those which carry SU (3) quantum numbers. The masses of these states vary dramatically with α g . The mass of the gluino and lightest stop are listed in the first two columns of Table 5. Model lines C and D were chosen to produce very light squarks and gluinos. While such deflected mirage mediation models typically run into challenges with the dark matter and Higgs bounds, they were chosen here as examples for comparison because they are much more favorable with respect to the LHC than cases with heavier superpartners. Model lines A -C all involve a mass for the lightest Higgs state which satisfies m h ≥ 113 GeV over all α g values. For line D, we have m h ≤ 112 GeV along the model line. However, we will retain this model line as the signatures are representative of a large class of deflected mirage mediation models.   Table 4. Low-lying superpartner masses are given in units of GeV as well as the wavefunction composition of the LSP neutralino. Here mq 1 is the lightest first generation squark, mt 1 is the lighter stop, and we have defined the two mass differences As has been pointed out recently [39], once event rates are normalized to the overall mass scale of the colored superpartners the next most important factor determining the inclusive signatures for a model at the LHC is the hierarchy of low-lying superpartner masses. This is particularly true for leptonic signatures produced through the production and decay of light neutralino and chargino states. A comprehensive examination of the possible hierarchy patterns in deflected mirage mediation models in the manner of [40,41] is beyond the scope of the present study and will be presented elsewhere [42]. Here we will simply list the value of the lightest neutralino mass, the lightest slepton mass (generally a scalar tau), the gluino and lightest stop mass, and the two mass differences between the lightest neutralino and the next two lightest gaugino states which we denote as (4.1) These values are collected in Table 5. The mass differences ∆ ± and ∆ 0 decrease monotoni-   Table 4. The total cross section for production of superpartners and PGS4 level one trigger efficiency are given in the first two columns. The following four columns give the number of events in each channel per 50,000 generated events. The definitions of these signatures are modified slightly from those of Section 4.1.
cally with increasing α g values because the gaugino spectrum becomes more squeezed, with the result that eventually the wino is lighter than the bino. This implies a softening of the leptonic decay products of cascade decays involving these states. Hence, one typically encounters a point at which the on-shell decays of the chargino (or second-lightest neutralino) to a slepton become kinematically forbidden -further suppressing leptonic final state signatures. The above properties are common to many models in which anomaly-mediated supersymmetry breaking becomes important [43]. Indeed, as α g is increased many properties of the gaugino sector are more and more like the anomaly mediation limit, since the moduli/gravity mediated contributions cancel with the gauge mediated contributions in this case. As a partial illustration of this trend, we give the wave-function composition of the lightest neutralino in terms of bino, wino and Higgsino percentages in the final three columns of Table 5. While cases such as model line C will exhibit the bino-like LSP behavior characteristic of mSUGRA/CMSSM models, others such as lines B and D will allow for a much richer set of LSP properties, including cases with a "well-tempered" neutralino [32].
To analyze the signatures of these models at the LHC, 50,000 events were generated for each of the points in Table 4 at √ s = 14 TeV using PYTHIA 6.4. The decision to use a fixed number of events, as opposed to a certain fixed integrated luminosity, is based on the widely differing total cross-sections for supersymmetric particle production across these model lines. The total supersymmetric cross-sections range from an exceedingly small 1 fb for Point A2 to the much larger value of 0.46 nb for Point D5, roughly following the mass scale of the SU (3)-charged superpartners. The overall supersymmetric cross-sections and combined level one trigger efficiencies are collected in Table 6. As mentioned in the previous subsection, the drop in trigger efficiencies with increasing α g is due in part to the diminishing mass gaps between low-lying gaugino states, resulting in softer jets and leptonic decay products and fewer events entering the sample via leptonic triggers.
While the total event rate for a fixed integrated luminosity clearly distinguishes the various points in each line, we are here interested in how the introduction of a non-vanishing α g value changes the collider phenomenology from the pure mirage mediation case with α g = 0. In order to make meaningful statements, therefore, we will work with our fixed event rate samples. For simplicity and clarity of presentation, in the analysis that follows we consider the deflected mirage mediation signal only and do not include Standard Model backgrounds. All signature selection criteria begin with a cut on missing transverse energy of E T ≥ 100 GeV and a cut on transverse sphericity given by S T ≥ 0.2. Additional selection cuts are imposed as described below.
We begin with the inclusive multijet signature [44] which is favored as a supersymmetry discovery channel. In this subsection we will define this channel by the requirement of at least 4 jets, the two hardest of which satisfy p jet T ≥ 150 GeV while the third and fourth hardest must satisfy p jet T ≥ 50 GeV. We also impose a veto on leptons (e and µ) and require E T ≥ 200 GeV for this channel. The total number of such events for each model point is listed under the heading "Multijet" in Table 6. The dramatic fall in the event count for larger values of α g is in part due to the rather severe cut on missing energy. In Figure 5 we plot the distribution in E T across all events satisfying S T ≥ 0.2 for lines A and D. The PGS4 level one trigger requires E T ≥ 90 GeV for the inclusive E T trigger, producing a sharp drop in the observed event rate below this threshold. In the region α g > ∼ 0.5 the distribution is clearly shifted to smaller values, with the majority of events falling below the E T = 200 GeV cut. Similar behavior occurs for the other two model lines. The supersymmetric sample size can be increased by relaxing this constraint, but only at the expense of including more of the (already sizable) Standard Model background.
In addition to being a discovery mode, the multijet channel has also been suggested as a tool for crudely measuring the overall mass scale of the superpartners. More specifically, the peak in the distribution of the variable M eff , defined by the sum of the missing transverse energy and the transverse momenta of the four hardest jets in the event is known to track the mass of the lowest-lying colored superpartner, most often the gluino [45]. This continues to be the case in deflected mirage mediation. In Figure 6 we show the distribution in M eff as defined by of the signal for Point A5 is directly related to the decreased missing energy in this case. For the other points in the two lines the total number of events remains roughly constant, with the peak in the distribution at a value roughly given by the gluino mass. Events involving high p T jets and isolated leptons with missing transverse energy can also be excellent discovery modes for supersymmetry [46]. The final three columns of Table 6 list the number of events involving one, two and three isolated leptons (respectively) satisfying p ℓ T ≥ 20 GeV. Each of these signatures requires two or more jets satisfying p jet T ≥ 50 GeV and E T ≥ 200 GeV. The dilepton signature involves precisely two leptons of opposite sign, though they can be of any flavor (again, lepton here implies e or µ). Two features can be seen in the leptonic data of Table 6. The broader feature is the general reduction in leptonic activity as the value of α g is increased along each line. The reduction is most severe for the multi-lepton signals when the mass gap between N 0 2 or C ± 1 and the LSP drops below 50 GeV. For events with one or more lepton, the p T of the leading lepton remains relatively large. A representative example is given for the models of line C in Figure 7. The number of events with p ℓ T ≥ 100 GeV remains relatively constant (hence the roughly constant number of events in the "1 Lepton" channel), while the number of events with softer leptons drops with increasing α g . The softening of the second (or third) lepton in the event is even more dramatic, resulting in fewer multi-lepton events. For models with ∆ ± < ∼ 5 GeV, almost all signatures of the jets + leptons variety fail to pass the trigger requirements or the leptonic p T cuts we impose. The other feature involves the slight increase in multi-lepton events between Points A3 and A4 in line A and between Points C2 and C3 in line C. Both are the result of the decreasing mass differences between the lightest chargino/next-to-lightest neutralino and the LSP. For Point A3 the spoiler modes N 0 2 → N 0 1 h and C ± 1 → N 0 1 W ± dominate their respective decay tables, with the former producing mostly b-jets in the final state. For Point A4, however, both of these modes are kinematically inaccessible and the three body decay modes N 0 2 → N 0 1 ff and C ± 1 → N 0 1 f f ′ are populated. This results in a significant increase in the number of multi-lepton final states. The effect is also present in line C, where the three body decay modes turn on only for Points C4 and C5. For Point C2 the dominant spoiler mode is N 0 2 → N 0 1 h, but for Point C3 this mode is inaccessible and N 0 2 → N 0 1 Z dominates, leading to a dramatic increase in the number of opposite-sign (OS) dilepton and trilepton final states.

Effective Mass (GeV)
The opposite-sign dilepton channel is particularly important as it provides crucial information on the mass differences between low-lying gaugino eigenstates. For example, a typical strategy for measuring the mass difference between light neutralinos is to form the flavor-subtracted dilepton invariant mass for events with at least two jets satisfying p jet T ≥ 60 GeV, at least 200 GeV of E T and two opposite-sign leptons [45]. The invariant mass distribution formed from the subset involving two leptons of opposite flavor is subtracted from that involving two of the same flavor, i.e., the combination (e + e − + µ + µ − − e + µ − − e − µ + ), to reduce background. For three-body decays involving   point does not yield sufficient events to produce a meaningful measurement. However, point B1 clearly shows an edge in the invariant mass as expected. Point B2, while having a much lower number of events, does indicate the expected shape based on the value of ∆ 0 . For line D an edge in the distribution can be clearly delimited in four of the five points. For points D2-D4, the edges clearly track the decreasing mass difference between the two lightest neutralinos (4.1) as α g is increased.

Opposite-Sign Dilepton Invariant Mass
As discussed in Section 3, for large thresholds (α g > 0) the gauge mediated effects can produce a superpartner spectrum which is more compressed than the analogous case in pure mirage mediation. This can be seen in the values of the invariant mass edge in the second plot in Figure 8, and can also be seen in cascade decays involving squarks and gluinos. For example, in multijet events one can construct the following ratio where p jet i T is the transverse momentum of the i-th hardest jet in the event. For this signature we do not impose a lepton veto, but we require that there be at least one jet with p jet T ≥ 100 GeV and at least three more jets with p jet T ≥ 50 GeV. This signature was shown to be effective in models based on the mirage pattern of gaugino masses [47], and is here quite effective at capturing the increasing softness of the products of cascade decays as the value of α g is changed. The distribution in the values of this ratio is plotted in Figure 9 for model lines B and D. The smaller the value of r jet , the larger the mass difference between the initially produced gluino or squark and the lighter neutralino and chargino states. As α g → 1, the ratio tends towards r jet = 0.5 in both cases. Plots for the other model lines give similar results. Empirically we find the peak value of r jet tracks the value of α g with roughly the same values across each of our model lines. This suggest that signatures such as these may prove extremely powerful at determining the value of α g once non-universality in the gaugino sector is firmly established.

Conclusions
We have investigated the collider phenomenology of deflected mirage mediation, a stringmotivated "mixed" supersymmetry breaking scenario in which effects from gravity mediation, anomaly mediation, and gauge mediation all contribute to the MSSM soft terms. Our focus has been to compare the implications for LHC physics between deflected mirage mediation and the well-known mirage mediation framework, which includes gravity and anomaly mediation, but not gauge mediation. The procedure was to explore deflected mirage mediation models together with standard mirage mediation benchmark models, either by directly comparing models with similar gaugino mass spectra, or by investigating the effects of switching on gauge mediation starting from pure mirage mediation scenarios.
The results show that there is a broad variety of phenomenological outcomes within deflected mirage mediation, depending on the messenger scale and the size of the threshold effects from gauge mediation. One interesting class of examples have a deflected gaugino mirage unification scale at TeV energies, leading to a squeezed spectrum in which the gluino can be the lightest colored superpartner, which in turn results in LHC signals with softer jets and leptons than in standard MSSM models. The effects of gauge mediation can also have a large impact on the total superpartner production cross section, in some cases by several orders of magnitude. For the deflected mirage mediation examples studied here, the most robust discovery mode will be the multijet channel. The ratio of events with one lepton and high-p T jets to those with zero leptons is also capable of distinguishing between the two different paradigms.
For the MSSM fields, they take the form