Light stop decays into $W b \tilde{\chi}_1^0$ near the kinematic threshold

We investigate the decays of the light stop in scenarios with the lightest neutralino $\tilde{\chi}_1^0$ being the lightest supersymmetric particle, including flavour-violating (FV) effects. We analyse the region where the three-body decay $\tilde{t}_1\to W b \tilde{\chi}_1^0$ is kinematically allowed and provide a proper description of the transition region between the three-body decay and the four-body decay $\tilde{t}_1 \to\tilde{\chi}_1^0 b f \bar{f}'$. The improved treatment has been implemented in the Fortran package {\tt SUSY-HIT} and is used for the analysis of this region. A scan over the parameter range including all relevant experimental constraints reveals that the FV two-body decay into charm and $\tilde{\chi}_1^0$ can be as important as the three-, respectively, four-body decays if not dominant and therefore should be taken into account in order to complete the experimental searches for the light stop.


I. INTRODUCTION
The discovery of a new scalar particle by the LHC experiments ATLAS [1] and CMS [2] has marked a milestone for particle physics. The immediate investigation of its properties allowed to identify it as the Higgs boson, i.e. the quantum fluctuation associated with the Higgs mechanism. But still, the question remains open if it is the Higgs boson of the Standard Model (SM) or of some new physics (NP) extension beyond the SM (BSM). Among the numerous NP models that are investigated, supersymmetric theories [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] certainly belong to the best motivated and most intensely studied BSM scenarios. Based on a symmetry between fermionic and bosonic degrees of freedom each SM particle has a supersymmetric (SUSY) counterpart. The SUSY partners of the top quark, the stops, play a special role. The large top quark mass allows for a large splitting between the two stop mass eigenstatest 1 andt 2 with interesting phenomenological consequences. Thus, while the limits on the squarks of the first two generations are pushed to higher and higher values [18,19], light stops have not been excluded yet by the experiments. Stops play an important role in the corrections to the SM-like light Higgs boson mass of the Minimal Supersymmetric Extension of the SM (MSSM) and are crucial to shift its mass value from the tree-level upper bound given by the Z boson mass to the experimentally measured value of ∼ 125 GeV. Naturalness arguments favour the stops to be light as they significantly drive the amount of fine-tuning at the electroweak scale [20]. In the MSSM with five Higgs bosons, two neutral CP-even ones, h and H, one neutral CP-odd one, A, and two charged scalars H ± , the maximal mixing scenario optimally reduces the amount of fine-tuning [21] while ensuring the correct mass value * groeber@roma3.infn.it † milada.muehlleitner@kit.edu ‡ eva.popenda@psi.ch § alexander.wlotzka@kit.edu of h. Furthermore, light stops can help for the correct relic density through co-annihilation in scenarios with small mass differences of 15-30 GeV between the light stop and the lightest neutralinoχ 0 1 [22][23][24][25][26][27]. And last but not least, light stops are necessary for baryogenesis to generate the matter-antimatter asymmetry in the MSSM [28][29][30][31][32][33][34][35][36][37][38][39][40].
There exist numerous experimental analyses searching for stops in different mass windows. Light stops with masses below the kinematic threshold for the decay into a top quark and the lightest neutralino, assumed to be the lightest SUSY particle (LSP), can decay through the three-body decayt 1 → W bχ 0 1 into the LSP, a W boson and a bottom quark b. If thet 1 mass lies below the three-body decay threshold, the light stop, assumed to be the next-to-lightest SUSY particle (NLSP), can decay through a FV process into the LSP and a charm quark c or an up quark u,t 1 → (u/c)χ 0 1 [41,42]. Another competing decay channel in this mass regions is the four-body decayt 1 →χ 0 1 bff [43], where f and f stand for generic light fermions. Former bounds on the stop masses have been set by LEP [44,45] and Tevatron [46,47]. Searches based on charm tagging and monojets have been performed by ATLAS [48] and CMS [49]. More stringent bounds have been derived by ATLAS in decays into charm quarks or in compressed SUSY scenarios in [50] as well as in final states with one isolated lepton, jets and missing transverse momentum [51]. ATLAS searches in final states with two leptons have derived bounds on the stop mass under the assumption that it decays into a b-quark and an on-shell chargino, which decays via a real or virtual W boson, or that the stop decays into a top quark and the lightest neutralino [52]. The same decay modes have been taken in the analysis performed by CMS [53]. The latter analysis provides limits for various assumptions on the branching ratios, while the former analyses assume branching ratios of one in the respective final states.
In [54] we have reinterpreted the charm-tagged and monojet searches [49][50][51] by taking into account that the arXiv:1502.05935v1 [hep-ph] 20 Feb 2015 branching ratios for the FV two-body and for the fourbody decay can deviate significantly from one. This leads to considerably weakened exclusion bounds. In this work we investigate the transition region at the threshold of the three-body decayt 1 → W bχ 0 1 . In particular, we analyse in this threshold region the interplay between the FV two-body decay and the three-body decay above the threshold, respectively, the four-body decay just below the threshold. 1 It turns out that the two-body decay can still be significant here for certain parameter configurations and can hence be exploited to improve and/or complement analyses based on the three-body decay final states. We extend and refine former analyses [55][56][57] by including the recently computed SUSY-QCD corrections to the FV two-body decay [54] 2 and the FV treelevel couplings in the three-body decay as well as in the four-body decay where also a non-vanishing τ and bottom mass in the final state [54] are taken into account. Furthermore, the transition region between three-and four-body decays is consistently described by including a finite width in the W boson propagator, which becomes virtual below the three-body decay threshold. Finally, we check for the accordance with the LHC data on the Higgs boson search, the exclusion limits from SUSY searches as well as constraints from the relic density and B-physics and from electroweak precision measurements.
In Sec. II details on the calculation of the three-and four-body decay widths are given, followed in Sec. III by the description of the numerical analysis and the applied constraints. Our results are presented in Sec. IV. We conclude in Sec. V.

II. THREE-AND FOUR-BODY STOP DECAYS
We work in the framework of the MSSM with general flavour structure. Flavour-violating effects are constrained by experiment to be very small which can be naturally accounted for in the Minimal Flavour Violation (MFV) [59][60][61][62][63] approach e.g., where the only sources of FV are given by the CKM matrix elements. Flavour violation is induced through renormalisation group running. Due to the large mixing in the stop sector, the lightest uptype squarkũ 1 is then mostly stop-like. For convenience, we occasionally refer to it as the light stopt 1 in the following although it is understood that it has small flavour admixtures from the charm-and up-flavours. The threebody decay ofũ 1 into the lightest neutralino, a down-type fermion d i (i = 1, 2, 3), where i denotes the quark flavour, and a W boson,ũ 1 Note that we choose the parameters such that in the four-body decay only the diagrams with the intermediate W boson can become on-shell in the investigated region. 2 See also [58]. proceeds via down-type squark, chargino and up-type quark exchange. The Feynman diagrams are displayed in Fig. 1. The index s = 1, ..., 6 of the exchanged squark refers to one of the six squark mass eigenstates, which are no flavour eigenstates any more. In case of small FV as given in the MFV setup, the dominant final state is given by W bχ 0 1 . We have calculated the three-body decay with the general flavour structure by extending the results of [55,57] to all flavours. The full dependence on the bottom quark mass has been taken into account, whereas the first and second generation quark masses have been set to zero. The result for the decay width has been checked against a second, independent calculation by using FeynArts/FormCalc [64][65][66][67].
In the threshold region where the three-body decay mode of the light stop into W bχ 0 1 opens up, the off-shell effects of the W boson can be described by implementing the W boson width in the propagators of the W boson diagrams in the four-body decays Again in case of small FV the dominant final state is the one involving the b-quark, i.e. d i = b. The W boson width in the propagators introduces a gauge dependence. The width renders the W boson mass m W in the W boson propagators complex, whereas it is real in the corresponding Goldstone boson couplings, so that the cancellation of the gauge parameter dependence between the W boson and the associated Goldstone boson diagrams cannot take place any more. Possible solutions are given by the complex mass scheme [68], where a complex mass is introduced also in the Feynman rules, or by the overall-factor scheme [69,70], in which the whole tree-level amplitude is multiplied by where p W denotes the W boson four momentum and Γ W the W boson width. The product accounts for the maximal number of W propagators in the amplitude. We use the overall-factor scheme to ensure a gauge independent result. The drawback of this method is that close to the threshold the non-resonant contributions are neglected. We checked, however, explicitly, that in the scenarios found in our numerical analysis below, the effect of neglecting the non-resonant contribution is less than about 2% and hence acceptable. The three-body decay and the thus calculated four-body decay widths have been implemented in the SDECAY [71,72] routine of SUSY-HIT [73], where the SUSY Les Houches Accord (SLHA) [74] format has been extended to the SLHA2 format [75], as described in Ref. [54], to account for FV.
In order to ensure that the three-body and the fourbody decay widths match for mũ 1 − mχ0 1 mass differences above the kinematic threshold of an on-shell W boson, the W boson width must be computed in accordance with the loop order and the input values used for the computation of the four-body decay width. Thus, the tree-level W boson decay width is computed with massless first and second generation fermions, while the masses of the bottom quark and the τ lepton are kept finite.

III. NUMERICAL SETUP AND EXPERIMENTAL CONSTRAINTS
We have performed a random scan over the parameter space of the model with the same settings as in the U (2)inspired scan of Ref. [54]. We have applied the same constraints on the generated parameter points as in [54], but updated the branching ratio of the B 0 s → µ + µ − decay to the recently reported value Additionally we have checked for the dominant restrictions due to electroweak precision observables by throwing away all points which are outside the 2σ interval around the experimental value for the ρ-parameter ρ = 1.0004 ± 0.00024 [77] .
Among the parameter points fulfilling the constraints we have retained only those, for which the masses of the lightest up-type squarkũ 1 and the lightest neutralinoχ 0 1 comply with The mass window around the three-to four-body decay threshold has been chosen large enough to allow for studying all effects that emerge in the threshold region. Finally, for the parameter points above the threshold SModelS [78-80] based on the tools Phythia 6.4 [81], NLL-fast [82][83][84][85][86][87][88] and PySLHA [89], is used to ensure that all parameter points fulfill the exclusion bounds derived from direct searches by ATLAS and CMS [49][50][51][52][53]90]. Since the searches in the FV two-body decay channel are not covered by SModelS yet, for the parameter points below the threshold the procedure explained in [54] is used.
The scenarios surviving all constraints include chargino masses around 660 GeV, slepton masses of O(1 TeV) and charged Higgs masses in the range ∼ 400 to ∼ 1 TeV, so that the corresponding diagrams in the four-body decay with these particles in the intermediate propagators never go on-shell in the investigated threshold region.
IV. RESULTS Figure 2 shows the two-, three-, and four-body decay widths, respectively, for the parameter points of our scan which are in accordance with all applied constraints. The three-body decay, given by the green points, sets in at the the threshold mũ 1 − mχ0 1 = m W + m di . As expected, it approaches the four-body decay, illustrated by the blue points, for mũ 1 − mχ0 1 mass differences sufficiently above the threshold. 3 The relative size of the four-and the three-body decay widths is displayed in Fig. 3. It shows that the finite width effects are still sizeable 20 GeV above the threshold and therefore should be taken into account, as is done by including the total width of the W boson in the four-body decay. Note, that the scattering of the points at the upper end of the mass difference is subject to the numerical integration precision in the four-body decay. Furthermore, the remaining off-set between the four-and three-body decay at large mass differences is due to the finite value of the W boson width. As can be inferred from Fig. 2, the values of the twobody decay widths are equally distributed along the chosen range of the mũ 1 − mχ0 1 mass difference. While above the threshold the three-body decay dominates, close to the threshold the decay width for the two-body decay, which is shown in red, can be of similar size as the threeand four-body decay width, respectively. The branching ratio of the two-body decay is depicted in Fig. 4. With possible values as large as ∼ 40% at 20 GeV above the threshold, the two-body decay clearly is competitive with the other decay modes and thereby offers new discovery perspectives for light stops in this parameter region. In this region the charm is not soft any more and charm tagging could be used efficiently, as has been shown in [91], where a search for pair produced scalar partners of charm quarks was performed. Such large two-body decay widths are achieved in scenarios with relatively large FV as is the case in the U (2)-inspired scenarios investigated here. If such a set-up is realized by nature, Fig. 4  that it might not be possible to detect the light stop in the three-and four-body decay mode, respectively, if the masses of the light stop and the neutralino are such that they fall into the threshold regime. Hence, complementary searches in the two-and the three-, respectively, four-body decay mode are required in this case.
The two-body decay branching ratios for all scenarios of the random scan that passed the constraints are plotted in Fig. 5 in the mũ 1 -mχ0 1 mass plane. The upper and lower grey lines mark the borders of the interval defined in Eq. (6) and the color code indicates the value of the two-body decay branching ratio. While for the low mass region the parameter space is already very constrained such that no valid parameter points with stop masses lower than 260 GeV have been found, the fade out at high values of the stop and neutralino masses in Fig. 5 is due to the limited scan range of the input parameters of the model. The plot nicely illustrates the relative importance of the two-body decay in the four-to three-body transition region and underlines once more the necessity to take this decay channel into account in order to allow for a proper analysis of the stop decays in this mass range.

V. CONCLUSION
In this paper we have analysed decays of the light stop including the possibility of FV and with the lightest neutralino being the LSP. We investigated in particular the mass range where the three-body decay into W bχ 0 1 is kinematically allowed. We provide a proper description of this threshold region by resorting to the four-body decay intoχ 0 1 bff where the W boson total width has been taken into account in a gauge invariant way. The result-ing decay formula and the three-body decay with general flavour structure have been included in SUSY-HIT. We performed a scan over this threshold region where only the points in accordance with the constraints from the LHC Higgs and SUSY data, from the relic density and B physics measurements as well as from the electroweak precision data have been retained. The investigation of these scenarios revealed that the FV two-body decay into cχ 0 1 can be comparable to the three-, respectively, fourbody decay and even dominate for some parameter sets. In order to properly investigate this mass region, the experiments should therefore also investigate two-body decays with charm quarks in the final state, in order not to miss the light stop, which might be the first SUSY particle to be discovered at the LHC.

VI. ACKNOWLEGDEMENTS
We are grateful to Ben Allanach and Werner Porod for discussions on the flavour implementation in their codes. MMM would like to thank Filip Moortgat and Michael Spira for discussions on stop decays. AW acknowledges support by the "Karlsruhe School of Elementary Particle and Astroparticle Physics: Science and Technology (KSETA)".