Electroweak signatures of gauge-mediated supersymmetry breaking in multiple hidden sectors

This paper discusses electroweak collider signatures of the NMSSM with multiple-sector gauge mediation. We focus on the production of neutralinos and charginos which cascade decay into standard model particles and lighter supersymmetric particles, with special emphasis on final states with multiple photons. A search strategy for signatures with at least three photons is presented and compared with current exclusion limits based on two-photon searches. We show that in many regions of the parameter space our strategy gives stronger constraints than the existing two-photon analysis for these models.


Introduction
Supersymmetry (SUSY) is one of the main targets in the search for physics beyond the standard model (BSM) at the Large Hadron Collider (LHC) but, so far, no evidence in its favor has been found. There are many aspects of the current experimental situation that are problematic for SUSY. The first, most obvious, is the absence of direct detection of superpartners, pushing some of their masses well beyond the electroweak (EW) scale. Secondly, SUSY breaking must not introduce excessive neutral flavor-changing interactions. Thirdly, the discovery of the Higgs boson [1,2] at 125 GeV pushes the minimal supersymmetric standard model (MSSM, see [3] and references therein) into a highly fine-tuned regime requiring stop masses even larger than current direct searches or large A t -terms. Lastly, the lack of signal in direct dark matter searches has made the neutralino less appealing as a dark matter candidate. It should be kept in mind however that the severity of some of these obstacles is partly model dependent and can be mitigated by considering non-minimal models of SUSY. In our opinion this warrants for enlarging the class of models being targeted beyond the original MSSM with the usual SUSY breaking mechanisms.
As a specific example of the approach above, we consider the R-parity conserving nextto-minimal supersymmetric standard model (NMSSM, see [4,5] and references therein) coupled via gauge mediation (GM, see [6] and references therein) to multiple (n) SUSY breaking sectors. The advantages of GM are that it does not suffer from excessive neutral flavor-changing processes and that it provides a dark matter candidate, the gravitino, which is not in tension with the bounds from direct dark matter searches. The reason for concentrating on the NMSSM is on the other hand to alleviate the need for large loop contributions to the Higgs mass by allowing its tree-level value to exceed the mass of the Z boson. This is particularly useful when using GM as the SUSY-breaking mechanism since the A t -terms generated are typically very small, which would put further stress on generating the right Higgs mass within the MSSM alone. Our interest in studying multiple sectors of gauge-mediated supersymmetry breaking stems from their hitherto unexplored -1 -

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experimental signatures with multiple gauge bosons. If one leaves aside fine-tuning issues all our results apply just as well to the MSSM with multiple GM breaking sectors.
Multiple SUSY breaking sectors were first introduced in the context of gravity mediation [7,8]; see also [9][10][11][12][13][14][15][16]. They were then considered for GM first in [17]. As in ordinary GM, the phenomenology is mostly driven by the lightest observable sector particle (LOSP). In [18] the LOSP was taken to be a gaugino or a stau, in [19] the case of a higgsino LOSP was considered, in [20] the LOSP was a gaugino and the main production mode was via a slepton pair, in [21] both higgsino and gaugino LOSPs decaying to a heavy SM boson were considered and in [22] the LOSP was a gaugino and the main production mode was vector boson fusion. For related work see also [23][24][25][26].
In this work, we focus on electroweak production of charginos and neutralinos which cascade decay into SM particles and lighter SUSY particles. While the chargino sector (comprised ofχ ± 1 ,χ ± 2 ) is identical to that of the MSSM, the neutralino sector is extended by n states, one from each SUSY breaking sector. The five heaviest mass eigenstates (χ 0 1, · · · ,χ 0 5) largely coincide with the five neutralinos from the NMSSM while the n lightest mass eigenstates (G i ) have a large overlap with the pseudo-goldstinos (PGLDs, denoted byη i ) arising from the n SUSY breaking sectors. The lightestG i coincides with the nearly massless helicity ±1/2 components of the gravitino. We consider the case where all sleptons and squarks are decoupled from the spectrum, resulting in the LOSP being theχ 0 1, which is almost purely bino. The LOSP will cascade decay to the collider-stable next-to-lightest G i by the emission of one or several SM bosons. In this paper we discuss different collider signatures and perform a detailed analysis for the case of multi-photon (n γ ≥ 3) final states.
The paper is organized as follows. In section 2 we set the theoretical basis for our analysis. We describe the EW sector of the NMSSM lagrangian coupled to multiple GM sectors and present the main collider signatures of interest. In section 3 we focus on multi-photon signatures, and construct the benchmark points to be analyzed. Section 4 concerns the details of the simulation, section 5 the object definition and event selection and section 6 the background estimates. We present the results on the expected reach of the multiphoton analysis in section 7 and offer our conclusions in section 8. Details on the recast of the existing ATLAS search for the two photon signal and on the validation of our analysis are collected in the appendix.

Gauge-mediated NMSSM with multiple sectors
For definitiveness we work in the CP preserving version of the NMSSM without holomorphic linear and quadratic soft terms, characterized by a scalar potential We use the conventions of [4] and refer to that paper for definitions of the quantities involved.

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We choose the following parametrization of the Higgs vacuum expectation values (vevs) where the normalizations have been chosen in such a way that M B = i M B(i) and similarly for the other soft terms. We expand to find the 5 × n block M MIX of the mass matrix mixing the PGLDs with the NMSSM gauginos, higgsinos and singlino: (2.10) with D Y = − 1 4 g 1 v 2 cos 2β and D T 3 = + 1 4 g 2 v 2 cos 2β being the D-terms associated to the gauge groups U(1) Y and SU(2) L . Lastly, the pure PGLD contribution [17] is given by an n × n symmetric matrix obeying the constraint of having one zero eigenvalue whose eigenvector is proportional to (f 1 , f 2 . . . f n ) T , corresponding to the true goldstino that becomes the ±1/2 helicity component on the nearly massless gravitino. Such a matrix has the form (2.11) where µ ij are mass parameters characterizing the mixing between the hidden sectors induced by their coupling to the NMSSM [17].
The Lagrangian involving only NMSSM fields is left unchanged and the couplings of two or more PGLDs are subleading and therefore neglected. The couplings in (2.9) between the NMSSM fields and the PGLDs are relevant because they mediate the exotic decays of the neutralinos after rotating to the mass eigenbasis. In the gauge basis they read In the phenomenological analysis we will only be interested in the n = 3 case since the next-to-lightestG i is collider stable. This means that, in practice, the collider signature of the n = 2 case is very similar to the usual n = 1 scenario explored by LHC searches [31,32]  (e.g. two photons plus missing transverse momentum) but with a massiveG i as an "effective" gravitino. We will not consider the n ≥ 4 cases since these tend to have collider-stablẽ G i and soft photons leading to signatures that are indistinguishable from the n = 3 case.
For ease of comparison with the standard SUSY literature we still denote the mass eigenstates aligned with the NMSSM neutral fermions byχ diag.X . The scenarios phenomenologically interesting that we typically obtain in our exploration of parameter space (see section 3 for details) have, in order of increasing mass: a massless goldstinoG ≈η 1 , followed by two neutralinos that are mostly PGLDs,G ≈η 2 andG ≈η 3 , a mostly bino neutralinoχ 0 1 ≈B and finally a nearly degenerate chargino and neutralino pair mostly aligned with the corresponding winos,χ ± 1 ≈W ± andχ 0 2 ≈W 3 . We do not consider the heavier charginos and neutralinos that are mostly aligned with higgsinos and singlinos. We also decouple all sfermions and the gluinos and consider a scalar sector with all scalars heavier than the h(125).
In this configuration the main collider signature will be the production of charginos and neutralinos with their subsequent decay into a cascade of SM particles and lighter neutralinos. See figures 1-3 for some illustrative leading order (LO) Feynman diagrams. The decays of theG i are largely determined by the interactions of (2.12). We concentrate on the case where the dominant processes generate three or more photons as depicted in figures 1 and 2, compared with figures 3 which has at most two photons in the final state. We keepG in the final state due to the fact that it is typically long-lived. Displaced vertices or long-lived particle searches might be relevant for its discovery but will not be considered here. This type of signature is dominant in the case when mG 100 GeV and the M W (3) is suppressed so that the wino-like stateχ ± 1 decays dominantly toχ 0 1 (and not directly toG ).

Benchmark points with multi-photon signatures
Having constructed the model in section 2 we can now look for regions of parameter space of the NMSSM with multiple SUSY breaking sectors that generate signatures with at least three photons. Our search for benchmark points proceeds in two steps: first we determine Figure 3. Typical signal Feynman diagrams with two Z/γ. the bosonic sector and then we use the parameters found as input to finding the fermionic sector.
As far as the bosonic sector is concerned, after solving the tadpole equations, the scalar potential depends on the six parameters λ, k, A λ , A k , v s , tan β. Interestingly, it is possible to solve for the soft terms k, A k and A λ in terms of the charged and pseudoscalar masses m H ± , m a 1 and m a 2 (> m a 1 ), with the only extra requirement that m 2 a 2 + m 2 W > m 2 H ± . After fixing the values of m H ± , m a 1 and m a 2 , one only needs to scan over tan β, v s and λ obeying the reality constraint Of course, one could also solve for these remaining three parameters as functions of the three scalar masses, but the expressions and the required reality constraints would be quite unwieldy. We believe this is an optimal compromise between analytical and numerical analysis for our purposes. The scan proceeds as follows. We start by letting the above masses range between 400 GeV ≤ m a 1 ≤ 1000 GeV, 1000 GeV ≤ m H ± ≤ 1400 GeV and m H ± ≤ m a 2 ≤ 1400 GeV as well as 1.5 ≤ tan β ≤ 2.5. For each of these choices, we scan over the acceptable range of λ and v s , where λv s / √ 2 > 1500 GeV is required in order to have a heavy higgsino, and check for vacuum stability and a tree-level Higgs mass between 100 GeV and 110 GeV, so that the radiative corrections can raise it to 125 GeV. If we find a solution, we save it and move to the next point in the masses and tan β.
For each one of the bosonic benchmark points we then proceed to study the neutralino and chargino sector. The large parameter space of the fermionic sector ( , µ ij ) accommodates many different experimental signatures with only a specific subset relevant for multiphoton signatures. This subset can be defined more precisely by some physical conditions that we describe below.
One requirement is the presence of at least two promptly decayingG i with a partial decay width into photons Γ > 10 −12 GeV. A second requirement is mG < 90 GeV to avoid decays into Z bosons and mG − mG > 30 GeV and mχ0 1 − mG > 30 GeV to avoid decays with soft photons. As a third requirement, a hierarchy in the partial widths into photons is imposed to enhance a decay chain topology Γ(χ . Lastly, we demand the suppression of a direct decay of the lightest chargino into one of theG i , which would also reduce the number of photons in the final state. After imposing these conditions a total 138 different benchmark points are obtained. These are the targets of our analysis.
The requirement of a prompt decay into photons is what makes it necessary to consider the three hidden sector case instead of the two sector one. As had already been hinted by the estimates in [20], after a thorough scan we have been unable to find points in the two sector model where the last decayG →G γ is prompt. The signature of the two sector case is thus the same as that of the usual GM scenario but with the massless goldstino replaced by a massiveG in the usual decay chains:χ On the contrary, the three sector model can give rise to decay chains containing up to two prompt photons, for a maximum of four photons per process:

Simulation of multi-photon processes
The sensitivity of the ATLAS experiment to the NMSSM+GM models with multi-photon final states is estimated with simulated signal and background events in pp collisions at √ s = 13 TeV center-of-mass energy. Events are generated with the MG5_aMC@NLO program [33] using default dynamical factorization and renormalization scales and the NNPDF 3.1 NLO LUXQED parton distribution function (PDF) set with α s (µ) = 0.118 [34]. Parton level events are processed through Pythia8 [35] for showering and hadronization and through Delphes 3 [36] for fast detector simulation. The default Pythia8 and ATLAS Delphes cards are used.
Guided by the mass ranges populated by the benchmark points described in the previous section, signal events are generated in a grid scanning (mχ±    [38,39] and based on the NMSSM implementation [40]. The model includes the SM tree-level interactions and the interactions described in section 2. The decay chain syntax, which adopts the spin correlated narrow width approximation [41], is used. The simulation of the SM background is carried out analogously. The following backgrounds are simulated at LO: pp → Z/W ± + γγ, pp → Z/W ± + γγγ, pp → γγ, pp → γγγ and pp → tt + γγ. Based on these simulations, the dominant backgrounds after applying the selection criteria defined in eq. (5.2), are found to be pp → W ± + γγ and pp → γγ, where the third identified photon is fake, i.e. it originates from hadronic activity (mostly pions decaying into photons) or from misreconstructed electrons. For these processes, simulations at next-to-leading order (NLO) in QCD are performed and used in the analysis. All simulations are carried out with the parton-level cuts reported in table 1.
To avoid double counting we also perform a matching for samples with different photon multiplicities according to the prescription introduced in [42]. The observed effect of the matching is found to be negligible.

Object definition and event selection
The signal events of interest are characterized by at least three photons, large missing transverse momentum from the presence of the two long-livedG , and a large amount of activity from the visible decay products of the charginos and neutralinos. The objects used in the analysis are defined in the default ATLAS Delphes card [36]. Here we give a brief overview of their main characteristics.
Photons, γ, are reconstructed from energy deposits in the electromagnetic calorimeter (ECAL) with no matching track in the inner tracking system. Photons are required to have a p T > 30 GeV, |η| < 2.37 and an isolation I < 0.12 within the cone ∆R < 0.5. The isolation variable is defined by the scalar sum of the p T of all objects (not including the candidate) within a cone of ∆R around the candidate divided by the candidate p T ,

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Leptons, , refers to electrons and muons only, not τ leptons. Electrons are reconstructed from energy deposits in the ECAL and a matching track in the inner tracking system. Muons are reconstructed from hits in the inner tracking system and the muon spectrometer. Leptons are required to have p T > 25 GeV and |η| < 2.47, and to be isolated with I < 0.12 and I < 0.25 for electrons and muons respectively, both within the cone ∆R < 0.5.
Jets, j, are reconstructed with the Fastjet [43] package using the anti-k t algorithm [44] with a distance parameter R = 0.4. They are required to have p T > 25 GeV and |η| < 2.47. [GeV]) = (1000, 780), while keeping at least 80 MC events for the diphoton background, the following three-photon signal region SR 3γ was obtained:

Missing transverse energy, E
We note that the choice of selection criteria is partially driven by limited MC statistics and is thus not fully optimal. In particular a harder cut on H T is expected to further improve the sensitivity. In addition to the three-photon signal region SR 3γ , we also investigate the sensitivity of the two-photon signal region SR γγ W-H (referred to in this paper as SR γγ ) from [31]. This signal region requires

Background estimates
The rate of fake photons in the W ± + γγ sample is substantial, originating from a misreconstructed electron from the W ± → e ± ν decay or from the W ± → τ ± ν → e ± νν cascade decay, and is parametrized by a table of efficiencies in the Delphes card. On the other hand, the rate of fake photons in the pp → γγ sample is significantly lower. Here, fake photons mainly arise from hadronic activity, which is quite suppressed by requiring the photons to have large p T and to be isolated from other activity in the calorimeters and tracking detectors. In order to avoid having to simulate a prohibitive amount of events we -9 -

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adopt a parametrization of fake photons originating from jets, i.e. we select a jet instead of a photon and apply a reweighting of the event according to the fake rate [45]: The next most relevant background is tt + γγ (in the fully leptonic decay mode) and W ± + γγγ, each contributing with less than 5% of the W ± + γγ sample.
For the systematic uncertainties related to the modeling of fake photons, estimates are based on those found in the ATLAS search [31]. For the γγ process, where fake photons are modeled by eq. (6.1), we use a ±100% uncertainty, while a ±30% uncertainty is used for the W ± + γγ samples (and others including Z + γγ and tt + γγ) where the modeling of fake photons from electron and hadronic activities is done by Delphes.
The systematic uncertainty on the total cross sections of these backgrounds associated with the choice of scales is assessed by considering seven different values for the renormalization and factorization scales. Based on an envelope of the seven samples, the uncertainties on the γγ and W ± + γγ cross sections are estimated to be ±20% and ±7% respectively. Since radiative corrections at NNLO are large at high values of H T [46], an uncertainty of ±50% is assumed for the W ± + γγ process in the SR 3γ even though our computation at NLO gives only a ±7% scale dependence on the total cross section. For the γγ sample we use the NLO scale dependence of ±20%. For the other processes we consider a 60% uncertainty since they were estimated at LO.
The uncertainties on the background estimates are added in quadrature and are summarized in table 2. The backgrounds included in "others" are tt + γγ, W ± + γγγ, Z + γγ and Z + γγγ (the last two in the Z → + − , νν decay mode).
The effective cross sections of the background processes after parton-level cuts (see table 1), σ 0 , and after the SR 3γ selection criteria (see eq. (5.2)), σ SR 3γ , are shown in table 3, including systematic uncertainties for the latter. For a given integrated luminosity L in fb −1 , this results in an expected number of background events where only systematic uncertainties are included.

Results
We perform a typical cut and count analysis in SR 3γ and SR γγ and use the following Asimov formula to extract bounds on the signal cross section [47][48][49]: Here, S and B are the estimated number of signal and background events respectively, and δ is the relative systematic uncertainty on the background estimate. The expected upper limit at 95% confidence level on the number of signal events, S 95 exp , is defined as the -10 -

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Fake photon Radiative corrections Total  The expected upper limits from SR 3γ and SR γγ on the number of signal events, S 95 exp , and on the effective signal cross section, σ 95 exp , are given for each luminosity scenario in table 4. The upper limits on the production cross section times branching ratio of the main process (eq. (4.1)) are shown in figure 4 for the chargino-neutralino grid. To extract these bounds, the selection efficiencies 3γ and γγ are derived and used for each grid point. The γγ computation is validated by a comparison with the ATLAS analysis [31] (see appendix A for details). As can be seen, SR 3γ gives more stringent limits than SR γγ for the main process    Table 5. Average ratio of efficiencies for each process with respect to the main process (4.1) (first line) for SR γγ and SR 3γ . The uncertainties are the standard deviations of the ratios across the five grid points from which the averages have been obtained.
throughout the entire mass grid. The improvement in sensitivity is particularly significant in the low mass region reaching between one and two orders of magnitude. Therefore, if final states with three or more photons are as common as those with less than three, our strategy would ensure a better sensitivity. This is the case for the benchmark points discussed in section 3. The upper limits on the production cross section times branching ratio for the complementary grids are shown in figure 5. Also here, the SR 3γ performs better in the whole mass region considered, and is particularly effective in regions of low masses and small mass splitting.
For each of the benchmark points described in section 3, an exclusion test is carried out where the upper limit on the signal cross section from either SR γγ or SR 3γ is compared to the production cross section of the benchmark point. The results are shown in figure 6. Black circles depict benchmark points excluded by both SR γγ and SR 3γ , the few green crosses are excluded only by SR γγ , red squares are excluded only by SR 3γ and blue triangles are not excluded by either of the two signal regions. It can be noted that a large set of benchmark points are only excluded by SR 3γ and not by SR γγ . As these are realistic models, several processes contribute to any given benchmark point. Hence to obtain the upper limits we compute the cross sections for the various processes and multiply these with the corresponding selection efficiencies γγ or 3γ . In the case of the main process, the efficiency from the closest point in the chargino-neutralino grid is used for a given benchmark point. For each of the secondary processes the ratio of efficiencies with respect to the main process is calculated for the following points in the chargino-neutralino grid: (mχ0 [GeV]) = (800, 500), (900, 700), (1300, 700), (1400, 900), (1400, 1100). The averages of those ratios across the five grid points are given in table 5 and are used to scale the efficiency of the main process to obtain approximate efficiencies for the secondary processes in a given benchmark point. For γγ , the relative variation of the ratios across the five grid points is below 11% for all processes. In the case of 3γ the relative variation is below 3% for all processes with three or more photons. The variation for processes with two photons is larger, but these have a negligible contribution to the overall efficiency.

Conclusions
This paper presents studies of the NMSSM with multiple sectors of gauge-mediated supersymmetry breaking and strategies to search for such models at the Large Hadron Collider. The models considered are restricted to those with three additional sectors and the collider signatures studied involve at least three photons. The production ofχ ± 1 andχ 0 2, both decaying to the collider-stableG while emitting two photons and a W or Higgs boson is used as a benchmark process. Limits are placed on the production cross section times branching ratio in a grid spanning theχ

A Comparison with existing two-photon ATLAS search
In this appendix the predictions of the ATLAS search with a diphoton and E miss T signature in [31], referred to as the diphoton search, are compared to those in this paper in order to validate the analysis method used.
In the signal region SR γγ (eq. (5.3)) [31] predicts 2.05 +0.65 −0.63 SM background events for L = 36.1 fb −1 , resulting in an expected model-independent upper limit on the visible cross section for non-SM processes of σ 95 exp = 0.122 fb (see table 4). Using the simplified statistical model described in section 7 with z = 2 we obtain an expected upper limit σ 95 exp = 0.113 fb. In addition to comparing the cross-section limits, our selection efficiencies for SR γγ ( γγ ) are compared to those in [31] ( ATLAS ). 3 We do this by generating events for the following process pp → (χ ± 1 → W ± (χ 0 1 →G γ)) + (χ  table 6 for four different mass points. Since in [31] the bino-likeχ 0 1 decays to theG by emitting either a photon or a Z boson, where BR(χ 0 1 →Gγ) ∼ cos 2 θ W , the same is assumed for our generated signal samples. The discrepancy between the selection efficiencies is expected since the results presented in this paper are based on a simplified detector simulation, neglect the effects of pileup and use jets reconstructed with a distance parameter R = 0.6 rather than the R = 0.4 used by the ATLAS analysis. Moreover, signal samples in which theχ 0 1 decays by emitting a Z boson rather than a photon have been omitted.

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To validate our background estimate, our prediction N = 0.77 ± 0.27 for W ( ν)γγ is compared to the corresponding N = 1.08 +0.65 −0.63 from [31]. The quoted uncertainty includes both statistical and systematic effects, so the central value of the prediction differ by ∼ 40%. This is expected since NNLO corrections are large at large H T [46].
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.