The least constrained supersymmetry with R-parity violation

The strong constraints on the R-parity conserving supersymmetry (SUSY) from the LHC searches motivate us to consider the new models in which the low-scale SUSY is still allowed. We propose a kind of R-parity violating SUSY scenarios with a non-zero $U^c_2 D^c_2 D^c_3$ operator. Three relevant LHC searches are recasted to test the status of this scenario in terms of four simplified models, with either light stop-Bino, stop-Higgsino, sbottom-Bino or sbottom-Higgsino. Some difficult scenarios for the LHC SUSY searches in these simplified models are identified. By extrapolate the current LHC searches to the future 14 TeV LHC with integrated luminosity of 300 fb$^{-1}$, most of the difficult scenarios can be probed, except the sbottom-Bino simplified one. An improved search, which utilises the multiple b-jets as well as its kinematic features, is expected to probe the signature of this case.


I. INTRODUCTION
As one of the most promising candidates for new physics beyond the Standard Model (SM), supersymmetry (SUSY) [1,2] provides an elegant solution to the gauge hierarchy problem. In the Supersymmetric SMs (SSMs), the gauge coupling unification can be realized. In order to forbid the renormalizable superpotential terms which induce the fast proton decays, the Z 2 R-parity (R = (−1) (3B−L)+2S ) is introduced, where B, L, and S are baryon number, lepton number, and particle spin, respectively. Under the Rparity symmetry, all the SM particles are even while their superpartners are odd. Thus, the Lightest Supersymmetric Particle (LSP) will be stable. Especially, the neutralino LSP serves as the very promising weakly-interacting-massive-particle dark matter (DM) candidate, which can have the correct DM relic density as well [3].
However, the searches for R-parity conserving (RPC) SUSY signals at the LHC, which mainly rely on the large missing transverse energy (MET) in the final state, have given the strong constraints. The gluino/squark masses have been pushed to a couple of TeV scale [4,5], challenging the naturalness problem [6,7] and little hierachy problem [8,9] of the SUSY theories. On the other hand, the main goal for SUSY is to solve the gauge hierarchy problem, so R-parity is not mandatory. The R-parity violation (RPV) terms in the superpotential are [10]: where L i , E c i , Q i , U c i , D c i and H u denote the left-handed lepton, right-handed lepton, lefthanded doublet quarks, right-handed up-type quarks, right-handed down-type quarks and up-type higgs. The λ and λ" are antisymmetric in exchange of i → j and j → k respectively. In Eq. (1.1), the first three terms break the lepton number symmetry and the last term breaks the baryon number symmetry. Note that proton can still be stable as long as only the lepton number or baryon number symmetry is broken.
The RPV SUSY has been searched at the LHC [11,12] in several different channels, with special attention paid to the gluino and top squark productions. The signature of pure hadronic multi-jet [13] in the final state has been searched to constrain the gluino pair production if λ ijk, i =3 is non-zero. When λ 3jk = 0, there could be top quarks from the gluino decay, the leptonic decay of which gives leptons + multi-jet final state [14][15][16]. Searches for the same final state are also constraining the L i Q j D c k opeartor. These operators will also lead to stop either decaying into two jets or decaying into a lepton and a jet, which has been searched in resonant di-jet pair [17] and lepton-jet pair [18]. Finally, if the RPV couplings are small such that the R-hadrons are stable at the scale of detector size, there are searches for long-lived R-hadrons [19]. From those searches, we can observe that the bounds obtained for those operators giving leptons in the final state are quite stringent, e.g. gluino being excluded up to ∼ 2 TeV, stop being excluded up to ∼ 1 TeV. As a result, these scenarios will be also in tension with the naturalness problem, same as for the RPC case. But the bounds with U c i D c j D c k operator are much weaker due to the large hadronic activity expected at the LHC [20][21][22][23][24][25]. There are plenty of studies that try to improve the sensitivity for RPV scenario with non-zero λ ijk , by using jet substructure analysis on either neutralino jet [26] or top squark jet [27,28], and by multivariate analyses [29].
Among all possible λ ijk , the scenario with i = 3 will give top quark in the final state from the on-shell/off-shell neutralino decay. The leptonic mode of which will be stringently constrained. Moreover, terms with i, j, k = 1, 2 are constrained [30] by the low energy experiments such as single nucleon decay channels and neutron-antineutron oscillation. In this paper, we will consider the least constrained scenario, i.e. RPV with only none zero λ 223 . This coupling is only constrained by the renormalisation group and its size is allowed to be as large as 1. 25. In the experiment searches as well as some phenomenological studies, the RPV bounds on top/bottom squark are studied under the assumption that they are the LSP and 100% decay through the RPV operator. However, this might not be true in the SSMs inspired by a Grant Unified Theory (GUT) [31], where the lightest neutralino is the LSP. We will study the top/bottom squark bounds in the cases either there is Bino or Higgsino LSP. Then, the top squark can only decay into on-shell/off-shell top quark and a neutralino, which further decay through the RPV operator U c 2 D c 2 D c 3 , i.e., csb. While the bottom squark decay is more complicate, beside the RPC decay ofb → bχ 0 /b → t * χ± , there is also direct RPV decayb → cs.
This paper is organized as follows. In Sec. II, we introduce four simplified SUSY models with U c 2 D c 2 D c 3 R-parity violating operator. Their corresponding LHC signals will be discussed. In Sec. III, the current LHC sensitivities to those signals as well as their future prospects are studied. Our conclusions are given in Sec. IV. We also show the validation of our recasting of experimental searches in Appendices A-C.

II. THE SIMPLIFIED MODELS AND SIGNALS
We consider the simplified RPV SUSY models with following assumptions: • λ 223 is the only non-vanishing RPV coupling.
• The only light colored particle is a right-handed dominant bottom squark or top squark, while all the others are too heavy to be produced at the LHC.
• Inspired from SUSY GUT as well as SUSY naturalness, we assume there is either an Bino (B) or a Higgsino (H) that has mass below the sbottom/stop, acting as the LSP.
As a result, we have four versions of simplified models:t −B;t −H;b −B;b −H.
In the minimal SUSY framework, the tree-level mass matrix of the neutralino sector in the basis of (B,W 0 ,H 0 where M 1 and M 2 are soft mass parameters for Bino and Wino, µ is the bilinear Higgs mass in the superpotential, and tan β is the ratio between the vacuum expectation values of H u and H d , θ W is the weak mixing angle. The limit M 1 M 2 /µ gives the Bino LSP in our simplified model. While the Higgsino LSP is more involved. In the limit µ M 1 /M 2 , there will be two mass eigenstates for neutralinos that have masses close to µ. Both have similar amount of H u and H d component. Their mass difference is given by which is tiny in the decouple limit µ M 1 /M 2 . Moreover, the Higgsino has another component in the chargino sector. The chargino mass matrix can be written as The mass difference between the charged Higgsino and the lighter neutral Higgsino is thus given by It can be simplified further in the large tan β limit: To conclude, we will have two neutralinos and one chargino for the Higgsino LSP cases. All of those three particles have mass close to µ. For specification and simplicity, we will take mχ± = mχ0 2 + 1 = mχ0 1 + 1 GeV throughout this work. The dominant SUSY signal of these simplified models at the LHC are the sbottom/stop pair productions with their subsequent decays. Their productions are simply through the QCD couplings, thus with approximatively identical cross section for stop and sbottom. In Fig. 1, we plot the production cross section of sbottom at 8 TeV, 13 TeV and 14 TeV proton-proton collider.
The decays of stop/sbottom are more complicate. we will discuss each of the simplified models case by case.
•t −B: The only allowed channel for the stop decay ist → t * χ0 , withχ 0 =B and top quark being either on-shell or off-shell depending on the mass difference betweent andχ 0 , as shown in the left panel of Fig. 2.
•t −H: Since there is also charged Higgsino lighter than the stop, beside the channel t → t * χ0 1,2 , there is decay oft → bχ ± with subsequent decayχ ± → W * χ0 1 1 , shown in the right panel of Fig. 2. When the stop is right-handed dominating, the decay width of each channel is given by with the two-body phase space function λ(x, y, z) = x 2 + y 2 + z 2 − 2(xy + xz + yz) and Y t is the top quark Yukawa coupling. These two channels are comparable if they are kinematically allowed, while the later one is dominating when the mass difference between the stop and Higgsino is small (m˜t < m t + mH).
•b −B: Firstly, the sbottom can decay through the RPC channel with decay width where Y b is the bottom Yukawa coupling. In the mass limit m˜b m b /mB, the decay width can be estimated as Γ(b → bB 0 ) ∼ 0.013 × mb 8π . In contrast to thet −B simplified model, the sbottom can also decay directly through the RPV operator U c 2 D c 2 D c 3 . Its decay width can be written as Thus, the decay width in Eqs. (2.8) and (2.9) will be around the same size if the λ 223 ∼ O(0.1).
•b −H: This case is similar with thet −H simplified model. The sbottom can decay either throughb → bH 0 1,2 orb → tH ± . Comparing to Eqs. (2.6) and (2.7), the decay widths of both channels are proportional to bottom quark Yukawa coupling instead of top quark Yukawa coupling, for the pure right-handed sbottom 2 . In our setup, the neutralino will decay into three-body final states through an offshell squark (b/s/c). We should require that the decay length be within the detector. Otherwise, the neutralino will leave nothing inside the detector, behaving exactly the same as in RPC case. The RPV three body decay width of neutralino is [32] where we have assumed that all the quark masses are negligible, the phase space integral and C is coupling between theχ 0 1 − q −q. For mχ0 1 ∼ 100 GeV, C ∼ 0.1 and mq mχ0 1 , |λ 223 |/m 2 q > 8.0 × 10 −9 is required, in order for neutralino to decay within O(1) mm. For the heavier neutralinoχ 0 2 and the charginoχ ± in the Higgsino LSP case, both particles are assumed to be dominated by the RPC decay, i.e.,χ 0 2 → h * /Z * χ0 1 andχ ± → W * χ0 1 . Due the compressed spectrum of Higgsino sector, the final states from the offshell bosons (h * , Z * , W * ) are too soft to be detected and only theχ 0 1 is visible. As a result, all of the three Higgsinos leave the same signal at the detector, that are three jets and one of them is b-tagged. In fact, if the RPV decays ofχ 0 2 andχ ± are important, i.e., thẽ χ 0 2 → bcs andχ ± → tcs/ssb/ccb are opening. The detector signals remain the same, except for theχ ± → tcs channel, which produces an extra top quark in the final state.
in the final state. Considering this, we will give sbottom a little mixing of left-handed part and focus on the Γ(b → tH ± ) case. Besides, there is direct RPV channelb → sc, with its decay width given in Eq. (2.9) as well.

A. Analysis and results under 13 TeV data
As discussed in the previous section, our signals include dijet resonance (b → cs), multi-jet (t → bχ ± /b → bχ 0 ) and lepton + jets (t → tχ 0 /b → tχ ± ). Even though most of our specific signals have not been searched at the LHC yet, there are some existing searches for the similar final states which could already constrain our signal processes. We will recast three relevant RPV SUSY searches from ATLAS: a lepton plus high jet multiplicity search [14], pair-produced resonances in four-jet final states [17] and multijet final states [37]. The validations of our recasting are provided in appendices.
To derive the bounds from recasting, a variable R ai = N ai NP /N ai UL is defined in each signal region i of each analysis a, where N ai NP is the number of our signal events in the signal region i of analysis a obtained from our simulation and N ai UL is the observed 95% C.L. model independent upper limit provided in each experimental analysis. The maxima R max = max a,i {R ai } is defined as the most sensitive one from all of the searches. This means a signal point is excluded by the current search if R max > 1.  [14], [17], and [37], respectively.
In Fig. 3, we plot the R max values in color on the m˜t-mχ0 plane fort −B andt −H simplified model. The most sensitive search on each grid are indicated by the point shapes: stellar, circular and triangle correspond to the lepton plus high jet multiplicity analysis [14], pair-produced resonances in four-jet final state analysis [17] and multi-jet final state analysis [37], respectively.
In thet −B simplified model, because of the on-shell/off-shell top quark in the final state which could decay leptonically, the lepton plus jets search is the most sensitive one for most of the time. The constraint on this model is quite stringent, except for the regions with m˜t ∼ mχ0 or relatively light Bino. In the former region the lepton from top decay is too soft. While for too light Bino, the three jet from RPVχ 0 decay will be collimated, performing as a single jet in the detector. Thus, the jet multiplicity in the final state is reduced.
The bounds obtained in thet −H simplified model is much weaker, mainly because of the branching ratio suppression for each channel, i.e.,t → tχ 0 andt → bχ ± produce the final states with and without detectable lepton. From the figure, we can see that still the lepton plus jets search is most sensitive one in most region. This means the current searches are only sensitive to thet → tχ 0 while thet → bχ ± mode which produces an energetic b-jet is overlooked. For both theb −B andb − H simplified models, there is direct RPV sbottom decaỹ b → cs. At the LHC, there is a search [17] for stop pair which decay into sd or bs through non-zero λ 312 or λ 323 which coincide with our scenario when Br(b → cs) = 100%. The corresponding bounds are presented in the left panel of Fig. 4. Similar to the stop case, the sbottom with mass below ∼ 425 GeV has been excluded in this scenario.
In theb −B simplified model, beside the direct RPV decay, sbottom can decay into bχ 0 with subsequent RPV decayχ 0 → bcs. This channel is the most difficult channel respecting to current searches: 1) it does not produce any lepton in the final state; 2) for m˜b ∼ O(100) GeV, the final state jets are typically too soft to pass the jet selections in the multi-jet search. As we can see from the right panel of Fig. 4, the LHC searches are only able to exclude the corner with both light sbottom and neutralino. In this region, the sbottom cross section is large and the decay final states of neutralino are collimated, reconstructed as an single jet. So that the signal here is appeared to be similar as the dijet resonanceb → jj. The four-jet resonance search [17] provides the most sensitive probing in most region. Especially, for very light neutralino mχ ∼ 25 GeV, sbottom is excluded up to 425 GeV in this model, which is close to the limit obtained in the Br(b → cs) = 100% scenario.
We have also performed the test on the scenario with Br(b → cs) = Br(b → bχ 0 ) = 50%. Because both channels are dominantly constrained by the same search, i.e., dijet resonances search [17], we find the distributions of R max of this scenario is similar to the right panel of Fig. 4. Note that the rate of true dijet events is reduced to 25%.  handed sbottom mixing is taken to be 0.1 so thatb → tH ± dominates. The bounds are shown in the left panel of Fig. 5 which is similar to the left panel of Fig. 3. The lepton plus jets search is the most sensitive one, which exclude the region with m˜b − 500 GeV mH m˜b − m t . The scenario with comparable branching ratios of detecting RPV decaỹ b → cs and subsequently decayb → tH ± /bH 0 will be more difficult to probe, due to the branching ratio suppression. The corresponding bound with Br(b → cs) = 50% are shown in the right panel of Fig 5. It shows that the current search can only exclude the region with m˜b ∼ [400, 500] GeV and mH ∼ [100, 200] GeV.

B. Prospects with higher luminosity
From our above study, we have shown that the current searches are not yet able to excludes most of the parameter space, especially in thet −H simplified model, theb −B simplified model, and the Br(b → cs) = 50% scenario in theb −H simplified model. However, the R max values on the most of the grids in those scenarios are already around O(0.1). It will be interesting to see the prospects of the sensitivity at higher luminosity LHC. In the following, we will simply extrapolate the exclusion limits at current stage to that of the future 14 TeV LHC with integrated luminosity of 300 fb −1 .
The following assumptions as adopted in Ref. [38] are made • The definitions of signal regions are remained the same. Moreover, for both signal and background events, the selection efficiencies of each signal regions are almost kept the same from 13 TeV to 14 TeV.
• The statistical uncertainty of background is scaled by √ B, where B is the total number of background events in the most sensitive signal region, i.e., the one provides R max .
• The systematic uncertainty is proportional to the B. Even through the systematic uncertainty of background is around 10%-20% in the current analysis, we expect some improvements in the future. So, we take the systematic uncertainty to be 5% in the extrapolation. With these assumptions, the total number of signal and background events in a signal region is scaled by a factor of where σ sig/bkg (13/14 TeV) is the production cross sections of signal/background process at 13/14 TeV, and L /L 0 is the integrated luminosity of 13/14TeV. Note only the dominant background process in each analysis is considered to estimate the scaling of background cross section. That is tt + jet for the analysis in Ref. [14] and multi-jet for analyses in Refs. [17,37] respectively. The scaled total number of background events (N b ) together with the corresponding uncertainty (σ b ) in each signal region can be used to calculate the model independent 95% C.L. new physics upper limit (N limit ) in the signal region, which is defined as [39] 1 N where N = +∞ 0 L(N b |S, N b , σ b )P(S)dS is the normalization factor, P(S) is the prior probability of signal event number which is assumed to be uniform for all S. Taking into account the uncertainties of background (σ b ) and signal (σ s ), the likelihood is defined as 100. Finally, in each signal region, the ratio between the scaled number of signal events N s and the N limit is calculated. The maximal ratio R max 14 among all signal regions is used to test a given model.
The extrapolated exclusion bounds for these difficult scenarios are shown in Fig. 6. Due to the increased signal event number and smaller systematics that we have assumed at 14 TeV with integrated luminosity of 300 fb −1 , the branching ratio suppression in thẽ t −H simplified model becomes a less severe problem. There will be sufficient events with leptonic final states for the most of points with m˜t 1 TeV. So the search for lepton plus high jet multiplicity excludes most of the region, except those with relatively degenerate spectrum so that lepton is too soft to be detected and those with heavy stop and light neutralino so that jet multiplicity is low. The upper-right panel of Fig. 6 shows the R max 14 for the sbottom pair production with subsequent direct RPV decaỹ b → cs. The extrapolated dijet reconance search will push the bounds on sbottom mass to ∼ 750 GeV. In the higher mass region, where both the number and the energy of initial state radiated jets are increased, the multi-jet search becomes the most sensitive. As have been discussed above, theb −B is the most difficult model respecting to current searches. Moreover, including the direct RPV decay of sbottom does not change the current sensitivities in this model. In the lower-left panel of Fig. 6, we present the R max 14 distribution in the scenario with Br(b → cs) = 0%. (We verified that the scenario with Br(b → cs) = 50% gives the similar result.) The future prospects for this model is still weak. The lower sbottom mass region can only be constrained by the dijet resonant search, which is not optimized for the signature of this model. And the multi-jet search only constrains the high sbottom mass region. As a consequence, sbottom as light as ∼ 500 GeV can not be fully excluded. The search sensitivity to theb −H model is much better, when the leptons in the final state can be energetic. For our choice of sbottom mixing, the signature of Br(b → cs) = 0% scenario of theb −H simplified model is similar to that of thet −H simplified model. Thus, it has similar bound as being presented in the upper-left panel of Fig. 6. In the lower-right panel, we present the bound for the Br(b → cs) = 50% scenario of this model, the constraint of which is weaker than that of Br(b → cs) = 0% scenario, simply because of the branching ratio suppression. The lepton plus jet search can exclude the sbottom mass in this scenario up to ∼ 800 GeV.

IV. CONCLUSION
In this paper, we proposed a RPV SUSY scenario that is least constrained by current LHC searches and low energy experiments, in which only the U c 2 D c 2 D c 3 operator is nonzero. Motivated by naturalness argument, four simplified models with relatively light stop/sbottom are considered, i.e.,t −B,t −H,b −B, andb −H models.
Those simplified model can lead to collider signatures of multiple jets, dijet resonance as well as leptons plus jets if any on-shell/off-shell top quarks are produced. By recasting the relevant LHC searches on our simplified model, we found some difficult scenarios for the current searches, where the stop/sbottom masses are barely constrained. They aret −H simplified model, theb −B simplified model, and the Br(b → cs) = 50% scenario in theb −H simplified model. Next, we extrapolated those existing searches to higher energy and luminosity LHC, i.e., the 14 TeV 300 fb −1 LHC. Under our assumptions in the extrapolation, the future prospects of the LHC sensitivities to those difficult scenarios are promising. Especially, the stop/sbottom up to 1 TeV can be probed in thẽ t −H simplified model. However, since all current searches are not optimized on the signature of theb −B simplified model, the sbottom here is still very weakly constrained. The sbottom as light as ∼ 500 GeV can not be fully excluded. Note that the signature is featured by four b-jets and each two of them have the same origin. We expected that an improved search, which utilises these special features, can be sensitive to this model. As for the Br(b → cs) = 50% scenario in theb −H simplified model, the branching ratio suppression is substantial, and the sbottom with mass ∼ 600 GeV is still safe.
Ref. [14] searches for final state with multi-jets and a lepton at 13 TeV 36 fb −1 LHC. To validate our recasting, we take the second model in the paper, i.e., pp →gg →tt(→ bs)tt(→bs).
We generate events for pp →gg with MG5 aMC@NLO v2.6.0 interfaced to Pythia8 , which is used to decayg. In this step, the parton distribution function is provided by the NNPDF23LO . Then we do the detector simulation via Delphes-3.4.0 , with pile-up effect considered. The mass and detector parameters are M˜t = 1.0 TeV, Mg = 1.6 TeV and the b-tag efficiency 78%.
In this analysis, only the total number of backgrounds and their uncertainties, the observation number are given (we will take the data in Tab. II of Ref. [14] as an example). We used the Eq. (3.2) to calculate the new physics upper limit for each signal region. The uncertainty of the observed event number σ s is defined as σ s = 0.1N s . Our results are shown in Tab. I.
The final selection efficiencies of all signal region for our benchmark point have been given in the auxiliary Tab. II of Ref. [14], which refers to the row of "Exp" in our Tab. II.   The final selections of some signal regions on our benchmark point: the numbers in the row of "Exp" are given in the auxiliary Tab. II of Ref. [14] and the results from our simulation are given in the row of "Sim".
According to the Tab. I of Ref. [17], we present the corresponding numbers as well as efficiencies in Tab. IV, from which we could see that the recasting is successful.  The analysis cutflows in the experimental paper (Exp) and from our simulation (Sim). Benchmark point with m˜t = 500 GeV is chosen in this study.

Appendix C: Search for energetic muilti-jet final state
In Ref. [37], the massive supersymmetric particles in multi-jet final states are searched. To valid our recasting, we consider the gluino direct decay model as adopted in this experimental analysis.
We generate pp →gg with MG5 aMC@NLO v2.6.0 , theng fully decays to UDD quarks in Pythia8 , where the mass ofg is set to 1.1 TeV. Upper limit can be calculated using Eq. (3.2) as before, the results are given in Tab. V. The cuts flow of the analysis on the benchmark points have been provided in the Tab. 4 of Ref. [37]. For comparison, we present both the experimental result (Exp) and our simulated result (Sim) in Tab