Natural NMSSM after LHC Run I and the Higgsino dominated dark matter scenario
We investigate the impact of the direct searches for SUSY at LHC Run I on the naturalness of the Next-to-Minimal Supersymmetric Standard Model (NMSSM). For this end, we first scan the vast parameter space of the NMSSM to get the region where the fine tuning measures and at the electroweak scale are less than about 50, then we implement by simulations the constraints of the direct searches on the parameter points in the region. Our results indicate that although the direct search experiments are effective in excluding the points, the parameter intervals for the region and also the minimum reaches of and are scarcely changed by the constraints, which implies that the fine tuning of the NMSSM does not get worse after LHC Run I. Moreover, based on the results we propose a natural NMSSM scenario where the lightest neutralino as the dark matter (DM) candidate is Higgsino-dominated. In this scenario, and may be as low as 2 without conflicting with any experimental constraints, and intriguingly can easily reach the measured DM relic density due to its significant Singlino component. We exhibit the features of the scenario which distinguish it from the other natural SUSY scenario, including the properties of its neutralino-chargino sector and scalar top quark sector. We emphasize that the scenario can be tested either through searching for signal at 14 TeV LHC or through future DM direct detection experiments.
In the supersymmetric models such as the Minimal Supersymmetric Standard Model (MSSM) MSSM-1 (); MSSM-2 () and the Next-to-Minimal Supersymmetric Standard Model (NMSSM) NMSSM-1 (); NMSSM-2 (), the boson mass is given by Baer:2012uy ()
where and represent the weak scale soft SUSY breaking masses of the Higgs fields and respectively, and are their radiative corrections, is the Higgsino mass and . As was shown in Baer:2012cf (), the corrections and can be obtained from the effective Higgs potential at loop level, and in case of a large , their largest contributions arise from the Yukawa interactions of third generation squarks, which are
In above formulae, denotes the renormalization scale in getting the effective potential, and its optimized value is usually taken as with and being the light and heavy top squarks (stop) respectively. Obviously, if the observed value of is obtained without resorting to large cancelations, each term on the right hand side of Eq. (1) should be comparable in magnitude with , and this in return can put non-trivial constraints on the magnitudes of and . Numerically speaking, we find that requiring the individual term to be less than leads to and upper bounded by about 200 GeV and 1.5 TeV respectively. In history, the scenario satisfying the bounds is dubbed as Natural SUSY (NS) Baer:2012uy ().
In the MSSM, the NS scenario is theoretically unsatisfactory due to at least three considerations. First, since the Higgsino mass is the only dimensionful parameter in the superpotential of the MSSM, its typical size should be of the order of the SUSY breaking scale. Given that the LHC searches for supersymmetric particles have pushed the masses of gluinos and first generation squarks up to above ATLAS-Multi-jets (); CMS-Multi-jets (), seems rather unnatural. Second, the relic density of the dark matter (DM) predicted in the NS scenario is hardly to coincide with its measured value. Explicitly speaking, it has been shown that if the DM is Higgsino-dominated 111Throughout this work, we denote the mass eigenstates of the neutralinos by with ranging from 1 to 4 (5) for MSSM (NMSSM), and assume an ascending mass order for the by convention. With such an assumption, the lightest neutralino as the lightest supersymmetric particle (LSP) is regarded as the DM candidate., its density is usually about one order smaller than its measured value Baer:2012uy (), alternatively if it is Bino-dominated, the correct density can be achieved only in very limited parameter regions of the MSSM Cao:2015efs (). These features make the NS scenario disfavored by DM physics. Third, the NS scenario is further exacerbated by the uncomfortably large mass of the recently discovered Higgs particle ATLAS-2012 (); CMS-2012 (): its value lies well beyond its tree-level upper bound , and consequently stops heavier than about must be present to provide a large radiative correction to the mass MSSM-Early-1 (); MSSM-Early-2 (); MSSM-Early-3 (); MSSM-Early-4 (); MSSM-Early-5 (); MSSM-Early-6 (). This requirement seems in tension with the naturalness argument of Eq. (1). In fact, all these problems point to the direction that the NS scenario should be embedded in a more complex framework. Remarkably, we note that the NMSSM is an ideal model to alleviate these problems.
The NMSSM extends the MSSM by one gauge singlet superfield , and it is the simplest SUSY extension of the Standard Model (SM) with a scale invariant superpotential (i.e. its superpotential does not contain any dimensionful parameters) NMSSM-1 (); NMSSM-2 (). In this model, the Higgsino mass is dynamically generated by the vacuum expectation value of , and given that all singlet-dominated scalars are lighter than about , its magnitude can be naturally less than . These additional singlet-dominated scalars, on the other hand, can act as the mediator or final states of the DM annihilation Cao:2015loa (), and consequently the NS scenario in the NMSSM with a Singlino-dominated DM can not only predict the correct relic density, but also explain the galactic center -ray excess Cao:2015loa (); Cao:2014efa (); Guo:2014gra (); Bi:2015qva (). Moreover, in the NMSSM the interaction can lead to a positive contribution to the squared mass of the SM-like Higgs boson, and if the boson corresponds to the next-to-lightest CP-even Higgs state, its mass can be further lifted up by the singlet-doublet Higgs mixing. These enhancements make the large radiative correction of the stops unnecessary in predicting , and thus stops can be relatively light NMSSM-Early-1 (); NMSSM-Early-2 (); NMSSM-Early-3 (); NMSSM-Early-4 (); NMSSM-Early-5 (); NMSSM-Early-6 (); NMSSM-Early-7 (); NMSSM-Early-8 ().
So far studies on the NS scenario in the NMSSM are concentrated on the assumption that is Singlino-dominated Cao:2015loa (); Cao:2014efa (); Guo:2014gra (); Bi:2015qva (); Singlino-NMSSM-1 (); Singlino-NMSSM-2 (); Singlino-NMSSM-3 (); Singlino-NMSSM-4 (); Singlino-NMSSM-5 (); Singlino-NMSSM-6 (); Singlino-NMSSM-7 (); Singlino-NMSSM-8 (); Singlino-NMSSM-9 (); Singlino-NMSSM-10 (); Singlino-NMSSM-11 (); Singlino-NMSSM-12 (); Singlino-NMSSM-13 (); Singlino-NMSSM-14 (); Singlino-NMSSM-15 (); Singlino-NMSSM-16 (); Singlino-NMSSM-17 (). In this case, the branching ratio of the golden channel in the LHC search for a moderately light stop is highly suppressed. Instead, mainly decays into the Higgsino-dominated and in following way Singlino-NMSSM-13 ()
where denotes either boson or a neutral Higgs boson. These lengthened decay chains can generate softer final particles in comparison with the golden channel, and consequently weaken the LHC bounds in the stop search. This feature is also applied to other sparticle searches, and it has been viewed as an advantage of the NMSSM in circumventing the tight constraints from the LHC searches for SUSY. In this work, we consider another realization of the NS scenario where is Higgsino-dominated. In our scenario, the Higgsinos and the Singlino are degenerated in mass at level, and consequently they mix rather strongly to form mass eigenstates with and being Higgsino-dominated and Singlino-dominated respectively. Since the role of the Singlino component in is to decrease the DM annihilation rate, may achieve the relic density measured by Planck and WMAP experiments Planck (); WMAP () without contradicting the DM direct search experiments such as LUX LUX (); LUX-1 (). The phenomenology of our scenario is somewhat similar to that of the popular NS scenario in the MSSM, which was proposed in Baer:2012uy (), but our scenario has following advantages
In the parameter regions allowed by current LHC searches for SUSY, it may have a lower fine tuning in getting the boson mass. Meanwhile, it has broad parameter regions to predict the right relic density (see our discussions in Sect. III).
The mass gaps of and from are sizable, e.g. and , and consequently the leptons from the decay chains and are usually energetic. As a result, our scenario can be tested at future LHC experiments by the process . By contrast, in the NS scenario of the MSSM the leptons are very soft and hardly detectable due to the small mass splittings: NS-SUSY-Compressed ().
Since all the light particles in our scenario, i.e. , , and , have sizable component, the main decay modes of include
and each mode corresponds to different signals. Because the signal of pair production is shared by rich final states, the LHC bounds on are usually weakened.
Moreover, we remind that the phenomenology of our scenario is different from that of the NS scenario with a Singlino-dominated DM. This can be seen for example from the decay modes of , which are presented in Eq.(2) and Eq.(3) respectively. We also remind that our scenario was scarcely discussed in literatures. In fact, within our knowledge only the work SNMSSM-Higgsino () briefly commented that may be Higgsino-dominated in the constrained NMSSM.
This work is organized as follows. In Sec. II, we briefly recapitulate the framework of the NMSSM, then we scan its parameter space by considering various constraints to get the NS scenarios in the NMSSM. Especially, we take great pains to implement the constraints from the LHC searches for SUSY by multiple packages and also by detailed Monte Carlo simulations, like what the work Natural-NMSSM-Simulation () did. After these preparation, we exhibit in Sec. III the features of the NS scenario with a Higgsino-dominated DM, including its favored spectrum and the properties of the neutralinos and stops, and subsequently in Sec. IV we take several benchmark points as examples to show the detection of our scenario in future experiments. Finally, we draw our conclusions in Sec.V. The details of our treatment on the LHC searches for SUSY are presented in the Appendix.
2 The Structure of the NMSSM and Our Scan Strategy
2.1 The Structure of the NMSSM
The NMSSM extends the MSSM by adding one gauge singlet superfield , and since it aims at solving the problem of the MSSM, a discrete symmetry under which the Higgs superfields and are charged is adopted to avoid the appearance of dimensionful parameters in its superpotential. Consequently, the superpotential of the NMSSM can be written as NMSSM-1 ()
where is the superpotential of the MSSM without the -term, and the dimensionless parameters , describe the interactions among the Higgs superfields.
The Higgs potential of the NMSSM is given by the usual F-term and D-term of the superfields as well as the soft breaking terms, which are given by
with , and representing the scalar component fields of , and respectively. Considering that the physical implication of the fields and is less clear, one usually introduces following combinations NMSSM-1 ()
where is second-order antisymmetric tensor with and , and with and denoting the vacuum expectation values of and respectively. In this representation, the redefined fields () are given by
These expressions indicate that the field corresponds to the SM Higgs doublet with and denoting the Goldston bosons eaten by and bosons respectively, and the field represents a new doublet scalar field which has no tree-level couplings to the W/Z bosons. These expressions also indicate that the Higgs sector of the NMSSM includes three CP-even mass eigenstates , and which are the mixtures of the fields , and , two CP-odd mass eigenstates and which are composed by the fields and , as well as one charged Higgs . In the following, we assume and , and call the state the SM-like Higgs boson if its dominant component comes from the field .
In the NMSSM, the squared mass of the filed is given by
where the last term on the right side is peculiar to any singlet extension of the MSSMNMSSM-1 (), and its effect is to enhance the mass of the SM-like Higgs boson in comparison with the case of the MSSM. Moreover, if the inequation holds, the mixing of the field with the field in forming the SM-like Higgs boson can further enhance the mass. In this case, is a singlet-dominate scalar, acts as the SM-like Higgs boson, and due to the enhancement effects the requirement does not necessarily need the large radiative correction of stops NMSSM-Early-1 (); NMSSM-Early-4 (). We remind that the singlet-dominated physical scalars (i.e. the mass eigenstates mainly composed by and respectively) are experimentally less constrained, and in case that they are lighter than about , naturally lies within the range from to .
In practice, it is convenient to use NMSSM-1 ()
as input parameters, where , and in Eq.(5) are traded for , and by the potential minimization conditions, and is replaced by the squared mass of the CP-odd field , which is given by
Note that represents the mass scale of the new doublet , and it is preferred by current experiments to be larger than about .
When we discuss the naturalness of the NMSSM, we consider two fine tuning quantities defined by Baer:2013gva ()
where represents the SM-like Higgs boson, denotes SUSY parameters at the weak scale, and it includes the parameters listed in Eq.(8) and top quark Yukawa coupling with the latter used to estimate the sensitivity to stop masses. Obviously, () measures the sensitivity of () to SUSY parameters at weak scale, and the larger its value becomes, the more tuning is needed to get the corresponding mass 222As was pointed out in Baer:2013gva (), if the NMSSM is considered as the low energy realization of an (unknown) overarching ultimate theory, and can be thought of as providing a lower bound on electroweak fine-tuning. Any parameter point with a low and implies that the ultimate theory may be low fine-tuned at high energy scale. By contrast, if the point correspond to large and , the underlying theory must be fine-tuned.. In our calculation, we calculate and by the formulae presented in Ulrich-Fine-Tuning () and lambd-SUSY-recent-2 () respectively.
The NMSSM predicts five neutralinos, which are the mixtures of the fields Bino , Wino , Higgsinos and Singlino . In the basis , the neutralino mass matrix is given by NMSSM-1 ()
If and , the Bino and Wino fields are decoupled from the rest fields. In this case, the remaining three light neutralinos () can be approximated by
where the elements of the rotation matrix roughly satisfy
with denoting the mass of . In the following, we are interested in the parameter region featured by , and , which is hereafter dubbed as the NS scenario with being Higgsino-dominated. From Eq.(12) and Eq.(13), one can conclude that this scenario has following characters
The lightest two neutralinos and are Higgsino-dominated, and is Singlino-dominated. Their masses should satisfy following relations: , and .
As far as is concerned, its largest component comes from field. If the splitting between and is significant, its Singlino component may also be quite large. The importance of the Singlino component is that it can dilute the couplings of with and bosons, Higgs scalars and SM fermions, and consequently the density of can coincide with the DM density measured by WMAP and Planck experiments.
The and components in should be comparable, and they are usually much larger than component of , i.e. .
As for , the relation usually holds.
2.2 Strategy in Scanning the Parameter Space of the NMSSM
In this part, we perform a comprehensive scan over the parameter space of the NMSSM by considering various experimental constraints. Especially, we take great pains to implement the constraints from the direct searches for SUSY at the LHC. After this procedure, we get the NS scenario with a Higgsino-dominated .
|Type I||Type II||Type III||Type IV|
We begin our study by making following assumptions about some unimportant SUSY parameters:
We fix all soft breaking parameters for the first two generation squarks at . Considering that the third generation squarks can affect significantly the mass of the SM-like Higgs boson, we vary freely all soft parameters in this sector except that we assume for right-handed soft breaking masses and for soft breaking trilinear coefficients.
Considering that we require the NMSSM to explain the discrepancy of the measured value of the muon anomalous magnetic moment from its SM prediction, we treat the common value for all soft breaking parameters in the slepton sector (denoted by hereafter) as a free parameter.
We fix gluino mass at , and treat the Bino mass and the Wino mass as free parameters since they affect the properties of the neutralinos.
Then we use the package NMSSMTools-4.9.0 NMSSMTools () to scan following parameter space:
where all the parameters are defined at the scale of . During the scan, we use following constraints to select physical parameter points:
All the constraints implemented in the package NMSSMTools-4.9.0, which include the -boson invisible decay, the LEP search for sparticles (i.e. the lower bounds on various sparticle masses and the upper bounds on the chargino/neutralino pair production rates), the -physics observables such as the branching ratios for and , the discrepancy of the muon anomalous magnetic moment, the dark matter relic density and the LUX limits on the scattering rate of dark matter with nucleon. In getting the constraint from a certain observable which has an experimental central value, we use its latest measured result and require the NMSSM to explain the result at level.
Constraints from the direct searches for Higgs bosons at LEP, Tevatron and LHC. Especially we require that one CP-even Higgs boson acts as the SM-like Higgs boson discovered at the LHC. We implement these constraints with the packages HiggsSignal for Higgs data fit HiggsSignals () 333In our fit, we adopt a moderately wider range of the SM-like Higgs boson mass, , in comparison with the default uncertainty of for in the package HiggsSignal. This is because may induce a correction to at two-loop level Goodsell:2014pla (); Staub:2015aea (), which is not considered in the NMSSMTools. and HiggsBounds for non-standard Higgs boson search at colliders HiggsBounds ().
Constraints from the fine-tuning consideration: and .
Constraints from the preliminary analyses of the ATLAS and CMS groups in their direct searches for sparticles at the LHC Run-I. We implement these constraints by the packages FastLim Papucci:2014rja () and SModelS Kraml:2013mwa (). These two packages provide cut efficiencies or upper bounds on some sparticle production processes in simplified model framework, and thus enable us to impose the direct search bounds in an easy and fast way. In the appendix, we briefly introduce the two packages.
Constraints from the latest searches for electroweakinos and stops by the ATLAS collaboration at the LHC Run-I. We implement these constraints by detailed Monte Carlo simulation. Since we have to treat more than twenty thousand samples at this step, this process is rather time consuming in our calculation by clusters. In the appendix, we provide details of our simulation.
After analyzing the samples that survive the constraints, we find that they can be classified into four types: for Type I samples, corresponds to the SM-like Higgs boson and is Bino-dominated, while for Type II, III and IV samples, acts as the SM-like Higgs boson with being Bino-, Singlino- and Higgsino-dominated respectively. In Table 1, we list the favored parameter ranges for each type of samples before and after considering the constraints from the direct search experiments, i.e. the constraints (4) and (5). Note that in the last row of this table, the retaining ratio is defined by where is the number of the samples that satisfy the constraints (1), (2) and (3) in our scan, and is the number of the samples that further satisfy the constraints (4) and (5). This table indicates that although the direct search experiments are effective in excluding parameter points encountered in the scan, they scarcely change the ranges of the input parameters where and take rather low values, or equivalently speaking the NMSSM can naturally predict and even after considering the direct search constraints from LHC Run-I. The underlying reason for this phenomenology is that the exclusion capability of the direct search experiments depends not only on sparticle production rate, but also on the decay chain of the sparticle and the mass gap between the sparticle and its decay product. We checked that a large portion of the excluded samples are characterized by . In this case, there exist one moderately light neutralino and one moderately light chargino with both of them being Wino-dominated, and their associated production rate at the LHC is quite large so that the signal of the production after cuts may exceed its experimental upper bound (see appendix for more information). Among the four types of points, we also find that the lowest fine-tuning comes from type III and type IV samples, for which and may be as low as about 2. This character is shown in Fig.1, where we project type III and IV samples on plane. We emphasize that for the type IV samples with , is upper bounded by about . In this case, it is compressed spectrum among the Higgsino-dominated particles , and that helps the samples evade the direct search experiments.
Because the type IV samples were scarcely studied in previous literatures and also because they have similar phenomenology to that of the NS scenario in the MSSM, we in the following focus on this type of samples. In order to make the essential features of the samples clear, we only consider those that satisfy additionally the condition . Hereafter we call such samples collectively as the NS scenario with a Higgsino-dominated . As we will show below, in this scenario is upper bounded by about , so the condition is equivalent to . In this case, gauginos and sleptons affect little on the properties of the lightest three neutralino, and the lighter chargino.
3 Key features of the NS scenario with being Higgsino-dominated
In this section, we investigate the features of the NS scenario with being Higgsino-dominated. We are particulary interested in neutralino-chargino sector and stop sector since they play an important role in determining the fine tunings of the theory. In Table 2, we show the favored ranges of some quantities such as and stop masses. This table indicates that our scenario is featured by , , , , and , and among the ranges, only the lower bound of is shifted from to by the constraints from the direct search experiments. Moreover, we checked that in our scenario, the ratio is restricted in the range from about 1 to 1.5. In this case, Higgsinos and Singlino are approximately degenerated in mass, and consequently they mix strongly to form mass eigenstates.
In Fig.2, we show the mass spectrum of , , and in our scenario. This figure indicates that the mass splittings among the particles satisfy , and . We remind that these splittings are induced by the strong mixings between Higgsinos and Singlino, and significantly larger than those among , and in the NS scenario of the MSSM Baer:2012uy () . In Fig.3, we show the field components of the states , and respectively. As is expected, the , and components in are comparable in magnitude with the largest one coming from the component. We emphasize again that the large Singlino component, i.e. , can dilute the interactions of the Higgsino-dominated with other fields, and consequently can reach its right relic density. In this case, we checked that the main annihilation channels of in early universe include . As for , its largest component comes from either field (for most cases) or field (in rare cases), and in general the two components are comparable, which can be learned from the figure and also from Eq.(13). We checked that due to the spectrum and the mixings, the dominant decay of is , and that of is usually . By contrast, the possible dominant decay modes of are rather rich, which include . Since is Singlino-dominated, its production rate is rather low, and consequently its phenomenology is of less interest.
Next we turn to the properties of . From the interactions of presented in MSSM-1 (), one can infer that if is dominated and meanwhile , the relation should hold. On the other hand, if is dominated, prefers to decay into the Higgsino-dominated with . These features are exhibited in Fig.4, where we show the correlations between different decay rates of in our scenario. From Fig.4, one can also learn that the branching ratio of is less than . This is because is Singlino-dominated and its component is small. Moreover, we note that in our scenario is lower bounded by about , which is about less than that in the NS scenario of the MSSM Kobakhidze:2015scd (); Cao:2012rz (); Han:2013kga (). One reason for the difference is that in our scenario has richer decay modes.
4 Future detection of our scenario
4.1 Detection at 14 TeV LHC
From the analysis in last section, one can learn that the NS scenario with being Higgsino-dominated is characterized by predicting and sizable mass splittings among the Higgsino-dominated neutralinos and chargino, i.e. 30 GeV 70 GeV and 50 GeV 110 GeV. Although this kind of spectrum is allowed by the direct searches for the electroweakinos at LHC Run I, it is expected to be tightly constrained at the upgraded LHC.
|30 fb||300 fb|
|P1||(bin2) = 2.09||(bin1) = 1.75||(bin2) = 4.59||(bin1) = 2.54|
|P2||(bin6) = 2.88||(bin5) =1.44||(bin6) = 5.58||(bin2) = 2.96|
|P3||(bin6) = 1.71||(bin11) = 1.01||(bin6) = 3.31||(bin2) = 1. 82|
|P4||(bin14) = 0.32||(bin15) = 0.21||(bin16) = 0.42||(bin14) = 0.34|
We investigate this issue by considering the neutralino and chargino associated production processes at 14 TeV LHC. For simplicity we adopt 4 benchmark points listed in Table 3, which are discriminated by the values of (or equivalently ), and . Since for these points decays into plus an off-shell boson, and decays mainly into plus a boson (on-shell or off-shell), the signal region SR0 in the ATLAS direct searches for electroweakinos by trileptons and large signal Aad:2014nua (), which was proposed in the analysis Baer:1985at () and also briefly introduced in the appendix of this work, is most pertinent to explore those points. In our analysis, we simulate the processes to get their summed rate in each bin of the signal region, and present the result in the last four columns of Table 4. We also present in the table the backgrounds of the bins at 14 TeV LHC, which were obtained by detailed simulations done in Cao:2015efs (). With these results, we evaluate the significance for each bin, where and correspond to the number of signal and background events and is the assumed systematical uncertainty of the backgrounds. Assuming 30 fb and 300 fb integrated luminosity data at 14 TeV LHC, we present the best two signal bins and corresponding expected significances for P1, P2, P3 and P4 in Table 5. This table reveals following information:
With 30 fb integrated luminosity data, P1 and P2 can be excluded at 2 confidence level, and with 300 fb data P3 can also be excluded. In any case, the point P4 is hard to be excluded.
For each point, which signal bin is best for exclusion depends on the mass splittings among the neutralinos and chargino. For example, since for all the four points, the most effective bins usually require 80 or 110 GeV. For points P1, P2 and P3, the bins satisfying are preferred for exclusion since , and by contrast the bins with