Long-Lived Unstable Superparticles at the LHC

In various models of supersymmetry (SUSY), the lightest superparticle in the minimal SUSY standard model sector, which we call MSSM-LSP, becomes unstable. Then, we may observe the decay of the MSSM-LSP in the detector at the LHC experiment. We show that the discovery of such a decay process (and the determination of the lifetime of the MSSM-LSP) may be possible at the LHC even if the decay length of the MSSM-LSP is much longer than the size of the detector; sizable number of the MSSM-LSPs decay inside the detector if the lifetime is shorter than 0.01-1 millisec. We also discuss the implications of the study of the MSSM-LSP decay for several well-motivated SUSY models.

Supersymmetry (SUSY) is a well-motivated target of the LHC experiment. Indeed, not only the discovery but also detailed studies of the superparticles are possible at the LHC experiment if superparticles are within the kinematical reach [1,2]. In many of the studies, it is assumed that R-parity is conserved, and that the lightest superparticle (LSP) is the lightest neutralino. If so, all the produced superparticles cascade down to the lightest neutralino just after the production and SUSY events are characterized by large missing p T .
Even though the lightest superparticle in the minimal-SUSY-standard-model (MSSM) sector, which we call MSSM-LSP, is often assumed to be the lightest neutralino and is stable, it is not always the case. Various scenarios where the MSSM-LSP becomes unstable have been proposed. One important example is the gauge-mediated SUSY breaking scenario [3], where the SUSY breaking in the MSSM sector is mediated via the standard-model gauge interaction so that SUSY-induced flavor violation are strongly suppressed. In the gauge-mediated model, the gravitino becomes the LSP and the MSSM-LSP decays into the gravitino.
Another example is R-parity violation, with which the MSSM-LSP becomes unstable. Usually, R-parity conservation is assumed to realize the LSP dark matter scenario. However, LSP can be dark matter even if the R-parity is violated; if the R-parity violation is weak enough, the lifetime of the LSP becomes longer than the present age of the universe. This possibility becomes important when the gravitino is the LSP [4,5], because it has several advantages. In such a case, the thermal leptogenesis [6], which requires relatively high reheating temperature [7], may be possible without conflicting with the constraints from big-bang nucleosynthesis [8] and the overproduction of the gravitino [9]. In addition, if the gravitino is dark matter with R-parity violation, a fraction of gravitino dark matter decays until the present epoch. The decay becomes a source of the high energy cosmic rays. In particular, recently, it has been shown that the anomalous excesses of the γ-ray and positron fluxes observed by EGRET [10] and HEAT [11] experiments, respectively, can be simultaneously explained in the gravitino dark matter scenario with R-parity violation if the lifetime of the gravitino is about 10 26 sec [12,13]. In such a scenario, the MSSM-LSP decays mainly via the R-parity violating interaction with the lifetime of 10 −(5−6) sec. Discovery of the MSSM-LSP with such a lifetime may give us a hint to understand the origin of high energy cosmic rays.
The experimental search for the decay of the MSSM-LSP gives important test of the scenarios with unstable MSSM-LSP. Since the superparticles are expected to be copiously produced at the LHC, we may have a chance to find the signal of the decay of MSSM-LSP. With long-lived unstable MSSM-LSP, we may see the decay at the LHC experiment in the form of a displaced vertex from the interaction point, non-pointing particle, and/or a disappearance of high p T tracks. Discovery of the MSSM-LSP decay is very important to understand the property of the MSSM-LSP. However, if the decay length of the MSSM-LSP is much longer than the size of the detector, most of the MSSM-LSPs escape from the detector before they decay. In such a case, the typical signal of the SUSY events are the same as the case where the MSSM-LSP is stable, and the discovery of the decay becomes statistically non-trivial.
In this letter, we discuss the possibility of discovering the decay of the MSSM-LSP at the LHC experiment, paying particular attention to the case where the decay length is much longer than the size of the detector. We will show that the discovery may be possible if the lifetime is shorter than τ < ∼ 10 −(3−5) sec. Then, we consider the implication of the result for several types of models with unstable MSSM-LSP. Furthermore, if the decay is observed, the lifetime of the MSSM-LSP can be constrained [14,15]; we also discuss the possibility of determining the lifetime of long-lived MSSM-LSP.
Let us start with discussing the basic formulae. If the MSSM-LSP is unstable, a fraction of the MSSM-LSPs produced at the LHC experiment decay inside the detector. If the MSSM-LSP (with its lifetime τ ) has the velocity v, the decay probability before propagating the distance L is given by P dec (L) = 1 − e −L/vγτ , where γ = (1 − v 2 /c 2 ) −1/2 (with c ≃ 3.0 × 10 8 m/sec being the speed of light). Then, denoting the pseudo-rapidity of the MSSM-LSP as η ≡ − ln tan(θ/2) (with θ being the angle from the beam axis), the number of the MSSM-LSPs which decay inside the detector is given by where N tot is the total number of MSSM-LSP, l (max) (η) is the distance to the outer boundary of the detector from the interaction point, and f (η, v) ≡ N −1 tot dN/dηdv is the distribution function of the MSSM-LSP; dηdvf (η, v) = 1.
In our following discussion, it is convenient to define Then, when the size of the detector is much smaller than the decay length cτ , we obtain When cτ < ∼ N tot L (eff) , we expect several decay events inside the detector. Since, typically, v ∼ c and γ ∼ O(1 − 10), L (eff) becomes comparable to the size of the detector, as we will see in the following. Thus, roughly speaking, the number of the decay inside the detector is determined by the size of the detector and the total cross section for the SUSY events. In our analysis, for simplicity, we approximate the shape of the detector as a cylinder with the radius l where η edge = − ln tan(θ edge /2).
#1 The end-cap of the ATLAS detector covers only up to |η| < 2.7 [1]. We have checked that most of the MSSM-LSPs are within this region, and hence we do not impose a cut on η for simplicity. Now, we calculate how many MSSM-LSPs decay inside the detector. The details depend on the MSSM parameters, and on what the MSSM-LSP is. When the MSSM-LSP is unstable, charged (or even colored) MSSM-LSP is phenomenologically viable. In the following, we discuss two of the important cases; one is the case where the lightest neutralino χ 0 1 is the MSSM-LSP while in the other case, the lighter stauτ is the MSSM-LSP. For example, in the gauge-mediated model, they are two of the important candidates of the MSSM-LSP.
First, we consider the case where the lightest neutralino χ 0 1 is the MSSM-LSP, and is a long-lived unstable particle. Even though χ 0 1 is invisible, spectacular signal may arise if χ 0 1 decays inside the detector. For example, in the models mentioned above (i.e., the gaugemediated model or the model with R-parity violation), χ 0 1 decays into γ or Z boson and an invisible particle. If we can find a production of high energy γ or the decay products of Z from the point which is displaced from the interaction point, like non-pointing photon [15], it can be identified as the signal of the decay of χ 0 1 . (The study of these signals may require that the decay occurs at an inner region of the detector, which reduces the fiducial volume compared to Eqs. (4) and (5). For more details, see later discussion.) Signal of those events should be searched at the off-line analysis. In order to record those events, trigger may be an issue. One possibility is to use the missing p T trigger. Because two χ 0 1 s are produced in the SUSY events, even if one of χ 0 1 decays inside the detector to be the signal event, the other χ 0 1 is expected to escape from the detector. (Notice that we consider the case that cτ is much longer than the size of the detector.) Such a non-decaying χ 0 1 should be a source of large missing p T . With the MC analysis, we calculate the distribution of the p T of χ 0 1 . The distribution for mg = 1 TeV and 2 TeV are shown in Fig. 1. With the present choice of parameters, more than 80 − 90 % of χ 0 1 s have p T larger than 100 GeV for mg = 1 − 2 TeV. Thus, assuming that the escaping χ 0 1 is the dominant source of the missing p T , most of the signal events have large missing p T . Thus, if a relevant missing p T trigger is implemented, the event can be recorded.
In order to estimate how many χ 0 1 s decay inside the detector, we perform MC analysis. At the LHC experiment, MSSM-LSP is mostly from the productions of colored superparticles: pp →gg,gq, andqq ′ (withg andq being the gluino and squark, respectively). The cross sections for these processes as well as the decay chains of the superparticles depend on the MSSM parameters. Here, as a well-motivated example, we adopt the gauge-mediated model to calculate these quantities. (Notice that, even though we use the gauge-mediated model to fix the underlying parameters, it is just for simplicity.) In our study, the SUSY events are generated as follows: 1. Mass spectrum of the superparticles and their decay rates are calculated in the framework of the gauge-mediated model. Here, the simplest gauge-mediated model is adopted where the model is parametrized by tan β (i.e., the ratio of the vacuum expectation values (VEVs) of up-and down-type Higgs bosons), number of vector-like messenger multiplets (in 5 +5 representation of grand-unified SU(5) group) N 5 , the messenger scale M mess , and Λ which is the ratio of the F -component of the SUSY breaking field to its VEV.  In our analysis, we use ISAJET package [16] for the first step while, for second and third steps, HERWIG package [17] is used. For the study of the χ 0 1 -MSSM-LSP case, we take tan β = 20, N 5 = 1, M mess = 10 7 GeV, and Λ is fixed to determine the mass scale of MSSM particles (in particular, in the present analysis, the gluino mass). With the above choice of parameters, the MSSM-LSP is Bino-like lightest neutralino, and its mass is given by m χ 0 1 = 175 GeV, 360 GeV, and 500 GeV for the gluino mass mg = 1 TeV, 1.5 TeV, and 2 TeV, respectively. #2 With the MC analysis, we found that L (eff) is not sensitive to the mass spectrum of superparticles, and is ∼ 10 m; L (eff) = 8.5 m, 9.9 m, and 11.4 m for mg = 1 TeV, 1.5 TeV, #2 With the present choice of parameters, the Higgs mass becomes smaller than the present experimental bound of 114.4 GeV [18] when mg < ∼ 950 GeV. Because we choose the gauge-mediated model just as an example of the SUSY model to fix the mass spectrum of MSSM particles, and also because the Higgs mass is sensitive to the masses of stops, we do not take the Higgs-mass constraint so seriously and extend our study to the parameter region of mg < ∼ 950 GeV. If one is interested in the case of the gauge-mediated model, only the results for mg > ∼ 950 GeV are relevant.

Figure 2:
The lifetime of χ 0 and 2 TeV, respectively. We can see that L (eff) is slightly enhanced as mg increases, which is due to the decrease of the averaged velocity of the MSSM-LSP for larger value of mg. On the contrary, the total cross section for the SUSY events strongly depends on the masses of superparticles; the total cross section is given by 1240 fb, 80 fb, and 10 fb, for mg = 1 TeV, 1.5 GeV, and 2 TeV, respectively. In Fig. 2, we plot the value of τ which gives N dec = 10 (with L = 100 fb −1 ) as a function of the gluino mass. (Since N dec ∝ τ −1 , the lifetime which gives a different value of N dec can be calculated from the figure.) For mg = 1 TeV, 1.5 TeV, and 2 TeV, N dec ≥ 10 requires τ to be smaller than 7 × 10 −4 sec, 6 × 10 −5 sec, and 1 × 10 −5 sec, respectively. Thus, when the lifetime is shorter than 10 −(3−5) sec, the number of the decay of the MSSM-LSP inside the detector turns out to be larger than ∼ O(1).
Next, we consider the decay ofτ -MSSM-LSP. We expect several possibilities to find the signal of the decay ofτ . In the models we introduced,τ decays into tau lepton and an invisible particle. For example, in the gauge-mediated model, the invisible particle is the gravitino while, in the scenario with the R-parity violation, it is a neutrino. In those cases, we may find a displaced decay vertex by observing decay product(s) of the tau lepton. #3 In addition, ifτ propagates O(10 cm) or so, hits in inner trackers should exist. Then, if we do not observe corresponding hits in calorimeters or in muon detector, we may identify such a short track as a signal of theτ decay inside the detector. Furthermore, using the fact that two staus are produced, we can simply look for events with only oneτ track (which #3 Otherwise, if selectron (smuon) is the MSSM-LSP, it decays into electron (muon) and an invisible particle. Then, energetic charged tracks from displaced vertices are the target. may be useful to count the number of staus which decay very inside the detector for the determination of τ ). For the case ofτ -MSSM-LSP, the trigger may not be an issue because at least one high p T charged track (i.e.,τ ) exists in the SUSY event; then, we expect that we can use the muon trigger.
Here and hereafter, in the study of the case whereτ is the MSSM-LSP, we adopt the underlying parameters of tan β = 20, N 5 = 2, and M mess = 10 7 GeV, with whichτ becomes the MSSM-LSP. With such a choice of parameters, the mass ofτ becomes larger than 100 GeV when mg > ∼ 650 GeV, and is 170 GeV, 270 GeV, and 370 GeV, for mg = 1 TeV, 1.5 TeV, and 2 TeV, respectively. We follow the same procedure as the case of χ So far, we have seen that a sizable number of MSSM-LSPs decay inside the detector when τ < ∼ 10 −(3−5) sec. However, this does not necessarily mean that the decay of the MSSM-LSP can be easily observed. In particular, in our calculation of L (eff) , we have assumed that the typical size of the fiducial region is ∼ 10 m. (More accurately, see Eqs. (4) and (5).) This implies that, in the case of the ATLAS detector, the decay of the MSSM-LSP may have to be identified by using only the last layer of the muon chamber. Such an analysis requires very high efficiency of detecting signals of the decay in the muon detector. In some cases, more conservative procedure to confirm the decay may be necessary. For example, for the case of the χ 0 1 -MSSM-LSP, some possibilities are (i) to require the decay of χ 0 1 inside the inner tracker region so that we can detect non-pointing photon using the electromagnetic calorimeter [15], or (ii) to look for a decay in the calorimeters to see the decay vertex. For the case ofτ -MSSM-LSP, as we have mentioned, we can require (i) no hits in the muon detector for high p T charged particles observed by the inner detectors, or (ii) the discovery of the decay vertex ofτ in the calorimeters or inner trackers. Then, the size of the fiducial volume to study the decay is reduced. At the ATLAS detector, the inner radius of the muon chamber, the outer radius of the hadron calorimeter, and the outer radius of the inner tracker region are about 5 m, 4.25 m, and 103 cm, respectively [1]. Importantly, L (eff) given in Eq. (2) is proportional to the (typical) length of the fiducial region. Thus, if the MSSM-LSP is required to decay in the calorimeters or in the inner tracker, L (eff) is expected to be reduced by the factor of 2 − 10, and hence the maximum value of the lifetime with which sizable decay events are expected is. Even with such a smaller value of L (eff) , as Figs. 2 and 3 indicate, a significant number of the MSSM-LSP may decay in the fiducial region in the models we are interested in, as we see in the following. The understanding of the efficiency to find the signal of the decay should require extensive studies of the detector effects, which is beyond the scope of this letter. Here, we simply assume that the signals of the decay can be somehow identified in the following discussion.
Let us now consider implications of the search for the decay of the long-lived MSSM-LSP. First, we consider the gauge-mediated model. When the Bino-like neutralino is the MSSM-LSP, it decays into gravitino and a gauge boson (i.e., photon or Z-boson), and the lifetime of χ 0 1 is estimated as where m 3/2 is the gravitino mass. In the case whereτ is the MSSM-LSP, it decays into gravitino and the tau-lepton, and the lifetime ofτ is given by Thus, a sizable amount the MSSM-LSP decay is possible in both cases if m 3/2 < ∼ O(100 keV− 1 MeV); in particular, using the fact that too large mg is not preferred from the naturalness point of view, the number of the decay events can be as large as O(100) in such a case.
Another important case is with R-parity violation. In particular, with the introduction of the following bi-linear R-parity breaking interaction into the SUSY breaking terms: withL and H u being the slepton and the up-type Higgs boson, respectively, it was pointed out that the EGRET and HEAT anomalies can be simultaneously explained if the gravitino is the LSP [12,13]. (Here and hereafter, we neglect the generation index for sleptons for simplicity.) Such a scenario works for any kind of the MSSM-LSP as far as the lifetime of the gravitino is ∼ 10 26 sec. With the R-parity violating interaction given in Eq. (8), gravitino ψ µ dominantly decays as ψ µ → νZ or l ± W ∓ . When the gravitino is heavier than the weak bosons, the lifetime of the gravitino is estimated as [13] where κ = B RPV /m 2 ν (with mν being the sneutrino mass) is the VEV of the sneutrino in units of the VEV of the standard-model like Higgs boson. With the R-parity violation given in Eq. (8), the MSSM-LSP may also decay dominantly via the R-parity violating interaction. Then, when the Bino-like neutralino is the MSSM-LSP, it decays into a neutrino and a standard-model boson with the lifetime Whenτ is the MSSM-LSP, it decays into the tau lepton and a neutrino, and the lifetime is given by where mB is the mass of the Bino-like neutralino. Requiring τ 3/2 ∼ 10 26 sec to simultaneously explain the EGRET and HEAT anomalies, κ is required to be ∼ 10 −9 for m 3/2 ∼ 200 GeV, resulting in τ ∼ O(10 −6 sec) (O(10 −5 sec)) when the MSSM-LSP is Bino-like neutralino (stau). From the study of N dec given above, we expect sizable amount of the decay of the MSSM-LSP inside the detector with such a lifetime. In particular, when masses of superparticles are at most ∼ 1 TeV to solve the naturalness problem, the number of the MSSM-LSP decay inside the detector is expected to be ∼ 100 − 1000. Thus, the search for the decay of the long-lived MSSM-LSP should give an important test of the scenario to explain the origins of anomalous γ-ray and positron fluxes. So far, we have discussed the discovery of the decay of long-lived MSSM-LSP. Importantly, once the decay is found, we can also constrain the lifetime of the MSSM-LSP. This fact can be easily understood from Eq. (3); once the total number of the produced MSSM-LSP and the number of the decay in the detector are both determined, the lifetime τ is constrained using Eq. (3).
A relatively good determination of the lifetime is expected in particular when the MSSM-LSP isτ (or other charged superparticles) even if the decay length is much longer than the size of the detector [14]. #4 The tracking information about the long-lived charged particle #4 If the decay length cτ of the MSSM-LSP is comparable or smaller than the size of the detector, decrease of the decay point may be experimentally observed. Then, the lifetime is determined by using the distribution of the decay points when cτ > ∼ a few cm [19].
will be available if it travels transverse length longer than O(10 cm). Thus, once large amounts ofτ samples become available, we expect that the distribution function f can be directly determined from the experimental data. Since we consider the case where cτ is much longer than the size of the detector, most of the staus do not decay inside the detector, and are observed as energetic charged particles. Then, if enough amounts ofτ s are identified with the measurement of their momenta, we can determine f (η, v).
In discussing the identification of long-livedτ tracks, it should be noted that the stau may be confused with the muon in particular whenτ does not decay inside the detector. This fact makes the determination of N tot (or, more accurately, the number of events in which both ofτ s do not decay inside the detector) non-trivial. One possibility to distinguishτ from the muon is to use the time-of-flight information; for this purpose, the transition radiation tracker and the muon system are useful in the ATLAS detector. If the velocity ofτ is small enough, it takes sizable time to reach trackers, calorimeters, and muon chamber after the production. If the velocity information is combined with the momentum information,τ with small enough velocity can be distinguished from energetic muon. In our study, we require that the velocity of at least one of the two staus should be smaller than v max so that the event can be identified as a SUSY event; we assume that, with the velocity measurement,τ with v < v max can be distinguished from high p T muon whose velocity is almost the speed of light. We adopt several values of v max to see how the results depend on v max . On the contrary, if one of the staus decays inside the detector, it provides a striking signal, as we have discussed.
Based on the above argument, we assume that the SUSY events can be identified if there exists at least oneτ which escapes from the detector with v < v max , or if one ofτ s decays inside the detector. (Since we consider the case cτ ≫ L (eff) , we safely neglect the case where both ofτ s decay inside the detector.) Then, we expect that we can experimentally count the number of events where one of theτ s decays inside the detector and the other escapes from the detector with its velocity smaller than v max . We denote the number of such events as N can be calculated as a function of τ after the experimental determination of the velocity distribution ofτ with the above-mentioned type of SUSY events. Thus, with the measurement of N , a determination of τ is possible. Since we are interested in the case where N (v ′ <vmax) dec is much smaller than the total number ofτ s observed, the statistical error in the determination of the lifetime is dominantly from N If the correlation of the velocities of two staus is weak, N (v ′ <vmax) dec is given by where N (v<vmax ) tot is the number ofτ s which satisfy v < v max , which is also experimentally measurable. With our MC analysis, we have confirmed that Eq. (12) holds with a good accuracy. Thus, even though more accurate relation between N (v ′ <vmax) dec and τ can be obtained once the experimental data become available, we use Eq. (12) to estimate how well the lifetime can be constrained for simplicity.
In order to see the effect of the velocity cut, we show the distribution of vγ ofτ . The parameter Λ is taken to be 66 TeV and 141 TeV so that the gluino mass is given by mg = 1 TeV and 2 TeV, respectively. The result is shown in Fig. 4. From the figure, we can see thatτ acquires large velocity in average. Thus, if one imposes relatively severe cut on the velocity ofτ to reduce the muon background, significant amounts of the stau events will be also discarded. (See the following discussion.) Next, we calculate N (v<vmax) tot for v max = 0.8c and 0.9c; the lifetime which gives N (v<vmax) tot = 10 is also shown in Fig. 3. (Notice that N dec corresponds to N Assuming that all the decay events are identified, Fig. 3 shows contours on mg vs. τ plane on which the statistical uncertainty in the determination of τ is about 30 % (i.e., δτ = τ / √ 10). In order to determine τ at this level, the lifetime is required to be, for v max = c and 0.9c, for example, shorter than 1 × 10 −3 and 4 × 10 −4 sec (8 × 10 −5 and 4 × 10 −5 sec, 1 × 10 −5 and 8 × 10 −6 sec) for mg = 1 TeV (1.5 TeV, 2 TeV), respectively. Notice that, with a smaller input value of τ , better determination of the lifetime is expected.
As one can see, the sensitivity becomes worse as we require smaller value of v max . This is because most of the staus are produced with relatively large velocity. (See Fig. 4.) Importantly, the muon system of the ATLAS detector is expected to have a good time resolution of about 0.7 ns, and the velocity measurement is possible with the accuracy of δv/v 2 ≃ 0.03c −1 [1]. Thus, good separation ofτ from µ may be possible even with a relatively large value of v max . In addition, as we have mentioned, the SUSY event may be identified even with-out imposing the velocity cut if one of the staus decays inside the detector. Thus, if the total number of staus (with any velocity) can be somehow determined, the velocity cut is unnecessary. For example, if the correlation of the velocities of two staus is experimentally checked to be negligible, the total number of stau events can be calculated from the number of events with v 1 < v max < v 2 and that of v 1 , v 2 < v max , where v 1 and v 2 are velocities of two staus (with v 1 < v 2 ). In such a case, we can adopt the result with v max = c.
Such a measurement of the lifetime provide a quantitative information about underlying parameters. For example, in the gauge-mediated model, the measurement of the lifetime is possible when the gravitino is lighter than O(100 keV − 1 MeV), and consequently, the gravitino mass can be determined assuming thatτ decays into the gravitino. Then, we can obtain an important information about the SUSY breaking scale. #5 In addition, in the model with R-parity violation, the size of the R-parity violating coupling constant can be determined.
Finally, we comment on the case where the lightest neutralino is the MSSM-LSP. Even though the discovery of the decay of (long-lived) χ 0 1 , which is already very spectacular, may be possible, the precise determination of N tot , which is necessary for the determination of the lifetime of χ 0 1 , is very non-trivial. This is because χ 0 1 is neutral and we cannot directly count the number of χ 0 1 s which do not decay inside the detector. If the total number of SUSY events can be somehow estimated, however, the discovery of the decay of χ 0 1 gives significant information about the property of the MSSM-LSP. Once the SUSY events are found, we expect that the order-of-magnitude estimation of the number of SUSY events can be performed, which gives a bound on N tot . Then, the measurement of N dec gives some information about the lifetime.