Natural NMSSM with a Light Singlet Higgs and Singlino LSP

Supersymmetry (SUSY) is an attractive extension of the Standard Model (SM) of particle physics which solves the SM hierarchy problem. Motivated by the theoretical $\mu$-term problem of the Minimal Supersymmetric Model (MSSM), the Next-to MSSM (NMSSM) can also account for experimental deviations from the SM like the anomalous muon magnetic moment and the dark matter relic density. Natural SUSY, motivated by naturalness considerations, exhibits small fine tuning and a characteristic phenomenology with light higgsinos, stops and gluinos. We describe a scan in NMSSM parameter space motivated by Natural SUSY and guided by the phenomenology of an NMSSM with a slightly broken Peccei-Quinn symmetry and a lightly coupled singlet. We identify a scenario which survives experimental constraints with a light singlet Higgs and a singlino lightest SUSY particle. We then discuss how the scenario is not presently excluded by searches at the Large Hadron Collider (LHC) and which channels are promising for discovery at the LHC and International Linear Collider.


Introduction
With the discovery of the 125 GeV Higgs boson h 125 by ATLAS [1] and CMS [2] at the Large Hadron Collider (LHC), particle physics enters a new era. In the Standard Model (SM) of particle physics, the properties of the Higgs boson are determined by theory once the mass is known [3]. At present, their measurements are consistent with the SM prediction [4,5,6,7,8,9].
But the SM is not complete. Experimentally, it does not account for Dark Matter (DM), the anomalous muon magnetic moment or the strong CP problem, among other things. Theoretically, it suffers from the hierarchy problem. Supersymmetry (SUSY) solves the hierarchy problem by introducing a fermionic partner for each SM boson and a bosonic partner for each SM fermion [10]. SUSY with conserved R parity provides a natural candidate for DM, the Lightest Supersymmetric Partner (LSP), and can account for the anomalous muon magnetic moment by introducing new particles in loops.
The principle of naturalness in physics maintains that an effective physical theory approximately valid below some characteristic scale should not be very sensitive to the correct theory above that scale [11]. Applied to electroweak symmetry breaking in SUSY, this implies that the success of the effective SM Higgs theory disallows SUSY too far above the electroweak scale [12]. In particular, the characteristic mass spectrum of Natural SUSY includes light superpartners of the Higgs bosons, top quark and gluon near the electroweak scale.
The Minimal SUSY Model (MSSM) contains only the SM particles and their superpartners, together with an enlarged Higgs sector: one neutral pseudoscalar, two neutral scalars and two charged scalars which arise from the two Higgs doubletsĤ u andĤ d necessary for the Higgs mechanism in SUSY [3]. But the MSSM suffers from the so-called µ-term problem, which prevents the term µĤ uĤd in the MSSM superpotential from reaching the electroweak scale without fine tuning [10,13,14].
The Next-to MSSM (NMSSM) solves the µ-term problem by introducing a singletŜ and replacing µĤ uĤd with λŜĤ uĤd . The Z 3 invariant NMSSM superpotential is [13,14] where λ and κ are free parameters. An effective µ-term is generated as the vacuum expectation value ofŜ, µ ef f = λ Ŝ , reaching a natural scale without fine tuning [14,13]. In addition to the Higgs content of the MSSM, the NMSSM contains an additional pseudoscalar and an additional scalar so that the NMSSM Higgs sector consists of two neutral pseudoscalars (a 1 , a 2 ), three neutral scalars (h 1 , h 2 , h 3 ) and two charged scalars (H + , H − ) [15,16]. The NMSSM Higgs sector is fully determined at tree level by λ and κ, A λ and A κ (soft trilinear couplings), µ ef f and tan β (ratio of H u , H d vacuum expectation values) [15].
One notable version of the NMSSM is the Peccei-Quinn (PQ) symmetric NMSSM, characterized by κ = 0 [17,15,13,14]. The PQ symmetric NMSSM explains why there is so little CP violation in the strong sector by exhibiting an axion, the massless pseudoscalar a 1 . In the NMSSM with a slightly broken PQ symmetry, with small κ and A κ , the a 1 acquires a small mass proportional to κA κ but can still solve the strong CP problem [15,18,19,20,21].
Scenarios with a light NMSSM pseudoscalar Higgs, motivated variously by the strong CP problem, naturalness, the anomalous muon magnetic moment, the η b mass spectrum, and the similarity of the baryon density to the dark matter density, have been discussed in the literature [15,22,23,24,25,26,27,28]. In this study we assume a light NMSSM pseudoscalar a 1 with 2m τ < m a1 < 2m B . Motivated by the LEP Zbb feature near m bb ≈ 60 GeV [29], we identify this as an h 1 candidate. We further identify the h 125 as the second lightest neutral scalar h 2 of the NMSSM and note that the h 125 signal strength measurements at the LHC [6,5] place the heavier NMSSM a 2 , h 3 , H + in the effective MSSM decoupling limit.
2 Effective MSSM (λ, κ ≈ 0) We now consider the phenomenology of the NMSSM with a slightly broken PQ symmetry in which the singlet S is completely decoupled from the doublets H u and H d (λ = 0). We then consider how the phenomenology is altered when the singlet is allowed a weak coupling to the doublets (λ ≈ 0). The case λ, κ ≈ 0 is known as the effective MSSM [14].
For λ = 0, there is no mixing of the singlet with the doublets. The generic couplings in the NMSSM have been detailed in [30,14]. We adopt the notation of the latter, denoting S 2 ij (P 2 ij ) as the jth component of mass eigenstate h i (a i ), where j = 1, 2, 3 corresponds to the u doublet, the d doublet, and singlet respectively. For purely singlet h 1 and a 1 , S 13 = P 13 = 1 and all other S 1j , P 1j vanish, so the a 1 cannot decay to SM particles since their coupling is proportional to P 11 = 0 or P 12 = 0, and similarly for the h 1 . The a 1 is stable and the only allowed h 1 decay for m h1 ≈ 60 GeV is h 1 → a 1 a 1 . The singlet sector is decoupled from the SM sector.
Furthermore, for the case λ = 0, the singlet sector is decoupled from the MSSM sector. One neutralino is pure singlino whose mass, at tree level, is related to the a 1 mass by m χ = −2m 2 a1 /3A κ [17,15]. For m a1 ≈ 10 GeV and |A κ | of O(1) GeV, consistent with a slightly broken PQ symmetry, this yields m χ ≈ 60 GeV. In this study we identify the singlino as the LSP χ 1 . Denoting N 2 ij as the jth component of χ i , where j = 1, 2, 3, 4, 5 corresponds to bino, wino, u higgsino, d higgsino, and singlino respectively. Neutralinos heavier than the singlino LSP have zero singlino component, N 15 = 1 and all other N i5 vanish. No heavier neutralino can decay to the singlino since the coupling is proportional to N i5 = 0 for i > 1. The NLSP χ 2 is stable for conserved R parity.
However, when the singlet is allowed a weak coupling to the doublets (λ ≈ 0), small mixing between the singlet sector and the SM and MSSM sectors is possible.
To summarize, we assume an effective MSSM with mostly singlino LSP χ 1 and m χ1 ≈ 60 GeV. The a 1 and h 1 are mostly singlet with dominant decays to SM τ pairs and a 1 pairs, respectively. The a 1 , h 1 and χ 1 can be produced in neutralino decays. For m χ2 ≈ 70 GeV or below and λ < O(10 −2 ), the χ 2 decays outside of the effective tracking volume. For m χ3 ≈ 120 GeV or above, χ 3 → χ 1 h 1 is dominant. Finally, the h 1,2 mass sum rule yields |κ/λ| ≈ 0.176 for µ ef f = 300 GeV. These considerations, together with naturalness, inform the parameter ranges in the scan described in the next section.

Parameter Scan
The parameter scan is performed with NMSSMTools4.4.0 [32,33,34,35,36,37], probing 10 8 random points. We trade the soft trilinear parameters A λ , A κ , A t for m P , m A , X t , defined by [15,16] Here m A (m P ) is the diagonal component of the CP odd doublet (singlet) mass matrix and X t is the stop mixing parameter.
The parameters scanned are λ, κ, m A , m P , µ ef f , tan β, M 2 , X t and m Q3 . We fix the gaugino masses M 1 and M 3 with the unification constraints M 1 = 1 2 M 2 and M 3 = 3M 2 . We further assume m Q3 = m U3 . All other squark and soft trilinear parameters are fixed to 1500 GeV, and  Table 1. NMSSM parameters and their scan ranges. Additionally, κ is constrained to satisfy 0.125λ < |κ| < 0.225λ. The point h60 (κ = 0.006088 and Aκ = −1.087 GeV) is taken from points surviving the scan and is described in Section 5 the slepton mass parameters are fixed to 200 GeV. See Table 1 for scanned parameter ranges. Motivated by the PQ symmetric NMSSM, we scan small κ and A κ or equivalently, from Equation 4, small κ and small m P . The lower range bound of m P (9.9 GeV) is informed by the anomalous muon magnetic moment study [25], while the upper bound (10.5 GeV) is informed by the η b mass spectrum study [26]. Then κ is scanned in the range −0.01 < κ < 0.01 and is also required to satisfy 0.125λ < |κ| < 0.225λ since this requires m h1 ≈ 60 GeV within several GeV. We scan moderately small λ in the range 0 < λ < 0.1. Since the h 125 signal strength constraints are applied in the scan, m A is allowed to go into the effective MSSM decoupling limit m A ≫ m Z to accommodate the SM-like couplings of the h 125 .
The neutralino and chargino masses are largely determined by µ ef f , M 1 and M 2 which, from naturalness considerations are bounded above in the scan by 300 GeV [12]. At tree level, the stop masses are m 2 t1,t2 = m 2 Q3 + m 2 t ± m t X t for m Q3 = m U3 [15]. Naturalness informs the m Q3 range since light stops are compatible with small fine tuning.
The tree level Higgs mass in the MSSM is bounded by m 2 h < m 2 Z cos 2 2β, requiring a large loop correction for the h 125 . In the NMSSM the upper bound on m 2 h has an additional O(λ 2 v 2 ) term. The stop mixing parameter X t partly determines the one loop correction [12]: where the parameter mt is defined by ). The correction is strongly dependent on the top mass m t . In the scan m t = 172.5 GeV.
To allow the large correction required by the h 125 , but with small mt required by Natural SUSY, the stop mixing X t is allowed to contribute up to its maximal possible correction at X max t = √ 6mt. In the scan NMSSMTools4 calculates the Higgs mass spectrum at one-loop level including external momentum for self-energies and two-loop level excluding external momentum [38,39].

Surviving Points
The suite of constraints imposed by NMSSMTools4 while scanning includes experimental results from a wide variety of sources, including: -Anomalous muon magnetic moment ∆a µ measured by BNL E821 [40] -DM relic density Ω DMh 2 measured by Planck [41], direct DM exclusion by LUX [42] Loose constraints imposed during the scan require m h2 ≈ 125 GeV within 3 GeV and impose an upper bound on each h 125 signal strength χ 2 , calculated as in [43]. Of the 10 8 points scanned, 42 survive the constraints imposed during the scan. Constraints are tightened after the scan. The low mass Higgs sector must satisfy: 9.9 < m a1 < 10.5 GeV 50 < m h1 < 70 GeV 122 < m h2 < 128 GeV Finally, the sum of h 125 signal strength χ 2 are required to satisfy i χ 2 i < 13. Of the 42 points surviving the scan constraints, 15 points survive these final constraints.
In order to demonstrate the naturalness of the surviving points, we examine the fine tuning metric F max ≡ max a∈A ∂(log m 2 Z ∂(log a 2 ) calculated by NMSSMTools4. This metric yields the largest fine tuning over fundamental parameter set A. Surviving points have small fine tuning, 5 < F max < 10, light stops 300 < mt 1 < 400 GeV and light gluinos 500 < mg < 650 GeV. While agreement is not universal on which F max values characterize low fine tuning [19], studies have considered F max of order O(10 2 ) to be typical for the NMSSM [44] and O(10 1 ) to be low fine tuning [24,23,22,45]. A recent study seeking to establish naturalness as objective, model-independent and predictive concludes that a SUSY model with F max < 30 is natural, while one with F max < 10 is stringently natural [46].

Benchmark h 60
It has been noted that points surviving the scan represent a Natural NMSSM with slightly broken PQ symmetry. They also exhibit a light pseudoscalar Higgs with m a1 ≈ 10 GeV, a light scalar Higgs with m h1 ≈ 60 GeV, a  Table 2. Doublet and singlet components of the a1, h1 and gaugino and singlino components of the χ1, χ2,χ3 in h60.
singlino LSP DM candidate with m χ1 ≈ 60 GeV annihilating via χ 1 χ 1 → bb, and a light stop with mt 1 ≈ 350 GeV. The benchmark point h 60 satisfies the threshold criterion m χ3 > m h1 + m χ1 with the largest branching ratio for χ 3 → χ 1 h 1 of all surviving points in the scan. The lowest branching ratio for this decay in the surviving points which reach threshold is 65%, while the highest is 80%. This ensures production of a 1 from h 1 → a 1 a 1 in stop pair events witht 1 See the last column of Table 1 for the numerical values of the parameters which define h 60 . See Figure 1, generated with PySLHA [47] using the SUSY Les Houches Accord (SLHA) [48,49] file produced by NMSSMTools4, for the mass spectrum and decays in h 60 .
For components P 1j , S 1j , N ij of the a 1 , h 1 , χ 1 , χ 2 and χ 3 in h 60 see Table 2. The LSP χ 1 is dominantly singlino, while the a 1 and h 1 are dominantly singlet. The χ 1 mixing with doublinos and gauginos is small, as is the mixing of the a 1 and h 1 with the doublets. The phenomenology of a singlino LSP at the LHC has been considered in [31,50,51,52,53,54,55,56,57]. The phenomenology of light stops in the NMSSM, and how they avoid exclusion at the LHC, has been recently considered in [58].
For the numerical values of the masses and dominant branching ratios of the low mass spectrum of the h 60  benchmark, see Table 3. The a 1 and h 1 of the benchmark avoids the LHC search exclusion for straightforward reasons. Both ATLAS and CMS have searched for gluon fusion gg → a → µ + µ − but critically omit the Υ region and therefore cannot exclude m a1 ≈ 10 GeV [59,60]. ATLAS has searched for gluon fusion gg → h → aa for 2m τ < m a < 2m B but does not report limits for m h < 100 GeV [61]. CMS has searched for the same channel, but only reports limits for m h > 90 GeV with m a < 2m τ [62] or for the h 125 with 4 < m a < 8 GeV [63]. More decisively, the gluon fusion cross sections for a 1 and h 1 production in the benchmark are greatly reduced relative to the h 125 .
In the neutralino and chargino sector, both ATLAS and CMS have studied χ 2 χ + 1 production [64,65,66]. For example, the searches which assume decays to sleptons or to bosons also assume that m χ + 1 = m χ2 , motivated by models with a bino-like χ 1 and wino-like χ 2 and χ + 1 . But in h 60 the χ 1 is singlino, and manifestly m χ + 1 = m χ2 . The χ 2 χ + 1 searches which assume dominant decays to sleptons cannot exclude h 60 where ml > m χ + 1 , m χ2 . Such searches might be sensitive to χ 5 χ + 2 events, but here the cross section is reduced and the final states are more complex. Of the searches which assume dominant decays to bosons, only the W χ 1 Zχ 1 final state case applies. In this case both W and Z are very far off mass shell in h 60 , in which case it is unlikely that the exclusion can apply.
In the stop sector, both ATLAS and CMS report exclusion. For a summary of the ATLAS results, see [67,68]. For a bibliography of CMS results see [69]. No exclusion is given for the NMSSM, however, and exclusion for simplified models cannot be easily interpreted in the NMSSM context. For example, the analyses which assumet → tχ 1 with 100% branching ratio cannot exclude h 60 , for which this branching ratio is O(10 −3 ). h 60 does containt → tχ 3 with a branching ratio O(10 −1 ), but the subsequent χ 3 decay produces a much more complex final state with less missing energy than assumed by the searches. The stop pair searches which assumet → bχ + 1 with 100% branching ratio assume very specific cases of mass relationships be-  [70,71,72,73,74,75] on generated h 60 events. Event simulation of the gluino, stop and chargino/neutralino pair production is carried out with Pythia8.205 [76,77]. The SLHA file produced by NMSSM-Tools4 for h 60 is used with Pythia8, which features a dedicated NMSSM model with functionality for SLHA input.
See Table 4 for the exclusion r max , the ratio of the 95% confidence level lower limit on the h 60 signal presence to the measured 95% confidence level limit, of the analyses with maximum sensitivity to h 60 chargino/neutralino, stop and gluino pair production. Only for gluino pair production is r max > 1, indicating that both ATLAS and CMS have ruled out a gluino with mg ≈ 611 GeV in h 60 but neither has ruled out the stop and chargino/neutralino sectors of h 60 . However, since mg is determined by the gaugino mass M 3 , which can be easily increased without otherwise impacting the lower energy h 60 phenomenology, we simply assume mg ≈ 855 GeV or greater since this reduces the gluino pair production cross section by a factor of 13 relative to h 60 .
Note that r max = 0.6 for cms 1502 06031, which exhibits a 3σ excess in the low dilepton mass region [78]. If the h 60 stop mass is reduced such that the stop pair production cross section is enhanced by a factor of 1.5, then this analysis becomes sensitive to h 60 with the reduced mt ≈ 315 GeV.

Collider Signature
Since the stop is relatively light in h 60 , the cross section for pair production is large and makes cascade production of the a 1 and h 1 accessible at the LHC. Gluon fusion production of a 1 and h 1 is less promising. The reduced tth 1 (tta 1 ) coupling, which appears in the gluon fusion top loop, is of order O(10 −1 ) (O(10 −5 )) relative to the SM ttH SM coupling for a SM Higgs boson of the same mass.
In h 60 stop pair production, the cascade dominantly contains two top quarks. In gluino pair production it dominantly contains four top quarks. These are strong handles on any potential background. Some top pair tt background may be irreducible, but other backgrounds should be negligible.
We now describe a targeted study of the sensitivity to h 60 at the LHC. Signal events are generated with Pythia8 as described in Section 5. Background tt events are also generated in Pythia8. Fast detector simulation is performed with Delphe3.2.0 [71]. The Delphes3 detector card for CMS is modified to reproduce the tight electron, tight muon and b tag efficiencies reported by CMS [79,80,81]. The signal selection seeks the decay a 1 → µ + µ − in gluino and stop pair events and employs a standard selection for semileptonic top pair events, together with a selection for a 1 → µ + µ − , in which one top quark decays via t → bW → bℓν and the other via t → bW → bqq ′ . The requirements for the Run 1 analysis are these: exactly one tight electron with E T > 25 GeV and no isolation requirement missing transverse energy E miss T > 85 GeV four or more jets with E T > 20 GeV, at least two of which are b-tagged two or more tight muons with p T > 2 GeV, no isolation requirement and d 0 /σ d0 < 5 -zero net charge and 9.7 < m µ + µ − < 10.3 GeV in the leading and subleading muons The a 1 candidate is then reconstructed from the leading and subleading muons. The muon azimuthal impact parameter significance requirement d 0 /σ d0 < 5 ensures that the muons are consistent with prompt production. For the Run 2 analysis, we assume √ s = 14 TeV and dtL = 300 fb −1 . We use the Delphes3 simulation with mean pileup 50. The selection is identical to the Run 1 analysis except that the electron, jet, and muon thresholds are raised to 30 GeV, 30 GeV and 4 GeV, respectively.
After full signal selection, the SM top background is nearly negligible. Multiple jet events produced by QCD have not been simulated, but with the nominal selection this background is expected to be very small. In data, the nonpeaking h 60 events in the candidate a 1 distribution can be mistaken for QCD multijets events, however, so these are considered background in the significance calculation. In the SM top background, the candidate a 1 muons originate from τ lepton, D meson or B meson decays. In the nonpeaking h 60 events, they originate either from SM τ ,  Table 5. NLO cross sections fort1t1 and tt production at the LHC in Runs 1 ( √ s = 7, 8 TeV) and 2 ( √ s = 14 TeV). We omit thegg yields. Also shown are the expected yields for peaking events (Np), yields for nonpeaking events (Nn) and signal significances after the full signal selection described in the text. For √ s = 7 TeV, we show in parentheses yields and significance for a variation of h60 in which the χ2 decays outside of the effective tracking volume.
B or D decay or from NMSSM a 1 → τ µ τ , The proportion of peaking to nonpeaking signal events is sensitive to the details of h 60 . For example, if the slepton masses are raised above the threshold for decay from χ + 2 , then the peaking signal is enhanced and the nonpeaking signal is reduced. Similarly, if the branching ratio for χ 2 → χ 1 a 1 is raised at the expense of χ 2 → χ 1 Z ⋆ , the peaking signal is enhanced and the nonpeaking signal is reduced. Finally, if the χ 2 width is sufficiently small, its decay vertices may lie outside the effective tracking volume, making nonpeaking background from χ 2 effectively invisible.
Pythia8 is a leading order generator, but next to leading order cross sections obtained by the LHC SUSY Working Group [82,83] are used to normalize the event yields. See Figure 2 for the reconstructed a 1 mass distribution after full signal selection, where the distribution for a variation of h 60 in which the χ 2 decays outside of the tracking volume is also shown. See Table 5 for expected peaking and nonpeaking event yields and signal significances after The advantages of the International Linear Collider (ILC) for studying low mass NMSSM Higgs bosons has been noted in [15]. At the ILC, the standard Higgstrahlung production channel e + e − → Zh 1 is suppressed in the NMSSM due to the measured SM-like h 125 → ZZ ⋆ signal strength and the NMSSM coupling sum rule 3 1=1 ξ 2 ZZhi = 1 [14]. Instead, in h 60 we note the possibility of resonant production e + e − → a 1 h 1 , with cross section of several hundred picobarns at √ s = m Z . For √ s = 500 GeV, pair production of all neutralinos and all charginos is accessible with cross sections nearing a picobarn, as well as a 1 h 1 and Zh 2 production cross sections of about a hundred femtobarns.
In h 60 the Za 1 h 1 coupling is small enough to have evaded LEP searches [84,85] but large enough to be produced copiously at the ILC running on the Z pole. The ILC sensitivity to h 60 in operating scenarios described in [86] defined by beam polarization, luminosity and √ s and will be evaluated in a forthcoming companion study.

Conclusion
We have reviewed the motivation for a natural NMSSM with a slightly broken PQ symmetry and a lightly coupled singlet featuring a light singlet pseudoscalar a 1 , light singlet scalar h 1 and a light singlino LSP χ 1 DM candidate annihilating via χ 1 χ 1 → bb.
A random parameter space scan is performed subject to a full suite of experimental constraints, including the anomalous muon magnetic moment, the DM relic density and collider searches. Surviving points are characterized by low fine tuning, and abundant pseudoscalar a 1 production identifies the benchmark point h 60 . In addition this benchmark features light stops and light higgsinos, all characteristic of Natural SUSY.
The benchmark avoids the current LHC exclusion limits. For the a 1 and h 1 , this is due to the reduced gluon fusion cross sections. For other SUSY searches, this is primarily due to the search assumption that stops, neutralinos and charginos will decay directly to the LSP χ 1 with no intermediate SUSY particles in the decay chain. But in the h 60 benchmark the χ 1 is singlino and couples weakly to the rest of SUSY. Thus due to the light mass spectrum the decay chains can contain many intermediate SUSY particles, making the final states more complex with less missing energy than in the simplified search scenarios.
Finally, we report that the potentially fruitful discovery channels at the LHC for the benchmark considered are stop and gluino pair production with eithert 1 → χ + 2 b → χ 3 W b ort 1 → χ 3 t and χ 3 → χ 1 h 1 → χ 1 a 1 a 1 . We conclude with a fast simulation study that with a targeted signal selection the LHC may already be sensitive to h 60 in Run 1. We have also pointed out the possibility to observe at the ILC resonant e + e − → a 1 h 1 at √ s = m Z and pair production of all neutralinos and charginos at √ s = 500 GeV.