An Updated Description of Heavy-Hadron Interactions in Geant-4

Exotic stable massive particles (SMP) are proposed in a number of scenarios of physics beyond the Standard Model. It is important that LHC experiments are able both to detect and extract the quantum numbers of any SMP with masses around the TeV scale. To do this, an understanding of the interactions of SMPs in matter is required. In this paper a Regge-based model of R-hadron scattering is extended and implemented in Geant-4. In addition, the implications of $R$-hadron scattering for collider searches are discussed.


Introduction
The observation of an exotic stable 1 massive particle (SMP) with electric, magnetic or colour charge (or combinations thereof) would be of fundamental significance. Searches are therefore routinely made for SMPs bound in matter, and for SMPs at cosmic ray and collider experiments [1,2,3,4,5]. Furthermore, SMPs with masses at the TeV scale are anticipated in a number of beyond-the-Standard-Model scenarios, such as supersymmetry and universal extra dimensions [4,5]. SMP searches will thus be a key component of the early data exploitation programme of the LHC experiments [6,7,8]. To facilitate such searches, models are needed of the interactions of SMP as they pass through matter. Although robust phenomenological approaches are available to predict the scattering of electromagnetically charged SMPs, it is not clear how the interactions of hadronic SMPs (hereafter termed R-hadrons 2 ) should be treated. This is due to uncertainties associated with the strong scattering processes and the hadronic mass hierarchy. In this work a model of R-hadron scattering [9] which is based on triple Regge phenomenology [10] is extended and implemented in Geant-4 [11]. Although the main purpose of this work is to aid searches at colliders, it is also envisaged that it will be useful for cosmic-ray searches [12]. 1 The term stable is taken to mean that the particle will not decay during its traversal of a detector. 2 The term R-hadron has its origin in the R-parity quantum number in supersymmetric theories and the work in this paper will be presented in the context of supersymmetric particles. However, the results given here are generally applicable to stable heavy exotic hadrons.
Hadronic interactions of R-hadrons at colliders can give rise to energy loss and a range of striking signatures. Scattering processes can thus potentially enhance the discovery capability of an experiment as well as challenging it's event-reconstruction algorithms [4]. For example, due to scattering effects, a SMP can possess different values of electric charge as it traverses a detector. Furthermore, hadronic interactions affect the rate at which SMPs lose energy and thus the potential of experiments to discover SMPs via the late decays of stopped SMPs [13,14].
To aid the development of search strategies, it is thus important that experiments have access to Monte Carlo models which span as fully as possible the range of conceivable signatures which could be associated with R-hadron production. At present, one model is available within Geant-4 [15]. This model, hereafter termed the generic model, is based on a black-disk approximation [16] and employs pragmatic assumptions regarding possible scattering processes and stable R-hadron states. The work presented in this paper concerns an approach, hereafter termed the triple Regge model, which uses the triple Regge formalism to predict the scattering of R-hadrons formed from stoplike (t) and sbottom-like (b) exotic colour-triplet squarks (q) and electrically neutral gluino-like (g) colour-octet particles [9]. Furthermore, the model uses well motivated assumptions regarding R-hadron mass hierarchies which differ from those employed by the generic model. This paper is organised as follows. First, we discuss mass hierarchies of R-hadrons. This is followed by a description of the triple Regge model and its implementation in Geant-4. Finally, model calculations of the energy loss and flavour composition for R-hadrons passing through iron are shown and discussed.

Exotic heavy hadrons and their masses
Fermionic mesons (qq ′ ,gqq ′ ) and bosonic baryons (qq ′ q ′′ ,gqq ′ q ′′ ) would be expected to be formed from stable squark and gluino states, where q, q ′ and q ′′ denote Standard Model quarks [17] 3 . In this work, with the exception of the lightest gluino baryonic state (described below), only R-hadrons comprising light valence quarks of type u and d are considered.
Calculations of R-hadron mass spectra have been made using a variety of approaches such as the bag model [18,19,20] and lattice QCD [21]. These predictions, together with with measurements of heavy hadron masses, make it possible to approximately determine those features of the mass hierarchies which are most relevant for the modelling of R-hadron scattering in matter. Of particular interest are the masses of the lowest-lying states, to which higher mass particles would be expected to decay before interacting hadronically, and the mass splitting between the lightest meson and baryon states.
Given the observed spin and flavour independence of interactions of heavy quarks with light quarks [22], the lightest squark-based meson and baryon states can be inferred [23] from the measured mass spectra of charm and bottom hadrons. Thus, using the mass spectra of the observed charm and bottom mesons as a guide, charged and neutral squark-based mesons would be approximately mass degenerate, and the lightest baryon state would be the scalar singletqud. This would be the exotic equivalent of the Λ c ,Λ b baryons. This picture is supported by calculations of hyperfine splittings of a few hundred MeV between thequd state and the vector statesquu,qdd and qud [16,23]. Furthermore, the mass difference between the stable baryon and mesonic states would be far less than 1 GeV and the decay of a baryon into a meson and proton would be kinematically forbidden [16,23].
The basic properties of gluino R-hadrons may not be similarly inferred from Standard Model hadronic mass spectra. A greater reliance on phenomenological approaches is therefore required to identify the stable states. Possible hadrons include gluino R-mesons (gqq 0 ,gud + ,gdū − ) 4 and gluino-gluon states (gg). Various calculations, such as in Ref. [18] (bag model), Ref. [21] (lattice QCD) and Ref. [16] (Potential model) estimate the masses of the gluino Rmesons to be the gluino mass plus 0.2 -0.7 GeV. The mass differences between the lowest lying states of each of these hadrons are less than the pion mass. Calculations of the lowest lying baryonic states using the bag model [19,20] predict that the lightest state is the flavour singletguds 0 , and that the "nucleon" R-baryons (gudd 0 , guud + ) are 0.2 -0.3 GeV heavier. The consequence of this is that even though these heavier states would be stable to strong decays, they would decay weakly into the singlet state over a time scale of ∼ 10 −10 s. The mass of theguds 0 state is typically in the range of the gluino mass plus 0.2 -0.7 GeV [18,20]. It is thus reasonable to assume that all gluino R-hadrons would be approximately degenerate in mass. Specifically none of the models allow for a R-meson / R-baryon mass splitting larger than the proton mass thus disallowing decays from baryonic to mesonic states.
For the work presented here, we use the squark and gluino R-hadron mass hierarchy assumptions given in Table 1. In addition, to avoid attempting calculations of poorly understood cascade decays, we use simplified mass spectra which comprise only those R-hadrons deemed to be stable. The underlying assumption is that any higher states would decay to the stable states before having an opportunity to interact.
Partly as a consequence of the assumed stable states, R-hadron event topologies can be anticipated which would be challenging for one of the most common collider search methods. The efficiencies of searches for slow muon-like objects [6,24,25,26,27,28] would be affected by the Rbaryon production processes which take place as R-hadron pass through matter. As discussed further in Section 3, such processes imply that a sbottom or gluino R-hadron, irrespective of its state as it entered a calorimeter, is likely to leave as an uncharged object and therefore escape detection in an outer muon detector.
Although well motivated mass-hierarchy assumptions were selected for this work, in the absence of experimental data, alternative scenarios should also be considered. The generic R-hadron scattering model makes more pragmatic assumptions regarding the mass hierarchies. This leads to the bulk of sbottom and gluino R-hadrons being charged after they leave a typical calorimeter [6]. Thus, taken together, the two models will aid in the development of a more complete set of collider searches for R-hadrons.

Scattering of R-hadrons in Matter
Although the fine details of R-hadron scattering are difficult to model, first-principle arguments can be used to build up a qualitative picture of the processes through which R-hadrons suffer energy loss and undergo charge and baryon number exchange.
The probability of an interaction between the heavy parton and a quark in the target nucleon is low since the cross section varies with the inverse square of the parton mass according to perturbative QCD [16]. One can thus use a picture of scattering of a stable non-interacting heavy parton accompanied by a coloured hadronic cloud of light constituents which take part in the scattering. Energy losses of a R-hadron in an interaction with a stationary nucleon will thus be determined by the kinetic energy of the light quark system in a R-hadron. At the LHC, for R-hadrons of masses above several hundred GeV the light quarks system's kinetic energy is typically of order GeV and small energy losses are thus anticipated. Another important feature of R-hadron scattering is that processes in which R-baryons are converted into R-mesons are suppressed due to kinematics and the absence of pions in material [16]. Thus, any mesons which convert to baryons will likely stay in this state during their passage through material. A number of models have been proposed [9,15,16,29,30] which are based on the above principles. In this section, brief summaries are given of the salient features of the triple Regge [9] and generic models [15,16]. This is followed by a description of the implementation of the triple Regge model in Geant-4.

Triple Regge model
Since the central picture is one of a low-energy light-quark system interacting with a stationary nucleon, R-hadron scattering can be treated with the phenomenology used to describe low-energy hadron-hadron scattering data [9,29,30], as is done in the triple Regge approach [9]. The triple Regge model was originally developed to describe the scattering of exotic hadrons containing heavy colourtriplet objects. Here, though, it has been extended to also treat gluino R-hadrons. This model assumes the stable states described in Section 2.
Using parameters fitted to low-energy hadron-hadron data, the triple Regge model makes predictions for Rhadron scattering cross sections, together with energy-loss calculations based on the triple Regge formalism. Figure 1 (left) shows the the model predictions of the scattering cross sections of a squark-based R-hadron off a stationary nucleon within a nucleus comprising equal numbers of neutrons and protons. The cross section formulae are given in Section 3.3 in which the implementation of the model in Geant-4 is described. The cross section is shown for different types of squark-based R-hadrons as a function of the Lorentz factor, γ. As can be seen, there is a large cross section for antibaryon (qūd) interactions which is due to a dominant annihilation process with a nucleon in the target.
It is also seen that, at lower values of γ (γ 10), the scattering cross section of squark-based R-hadrons containing a light valence antiquark (qū,qd) is larger than for antisquark-based R-mesons. This arises from the presence of Reggeon-exchange processes which are only permitted for R-hadrons containing a light antiquark. Owing to the presence of an additional light quark the scattering cross section for R-baryons (qud) is twice as large as for the mesons with light quarks.
Upon an interaction the probability that a R-meson becomes a baryon is around 10%. Once a R-hadron becomes a baryon it stays in this state. Another process which must be taken into account is the oscillation of a neutral mesonic squark-based R-hadron into its antiparticle [23,31]. Feynman diagrams of possible processes in which oscillations can occur are shown in Figure 2. Tree level gaugino (gluino and neutralino) exchange and oneloop charged current-chargino box diagrams are shown.
Since the conversion rate would be model dependent we allow two possibilities here: zero mixing, in which no oscillations take place and a maximal-mixing scenario in which there is a 50% probability that any neutral mesonic squark-based R-hadron which was produced would automatically be converted to its anti-particle. These mixing scenarios correspond to oscillation lengths which are infinite and zero, respectively. In Figure 1 (right) the predicted cross sections for gluino R-hadron species are shown. The baryon cross section is 50% greater than that for stop R-hadrons owing to the extra light quark in the gluino R-hadron. Similarly, since the gluino meson contains a light quark and antiquark, then both Regge and pomeron-exchange processes are available. The scattering cross section for gluino mesons at a fixed value of γ is the sum of the cross sections for a stop meson and anti-stop meson. Following Ref. [16], a gluino-gluon state is assumed to scatter in the same way as a gluino-based R-meson. Again, a baryon number transfer probability of 10% is used.

Generic model
The generic model is already available in Geant-4 [15] and is an update of earlier work [16]. In view of the inherent uncertainties associated with modelling R-hadron scattering a pragmatic approach is assumed in which the scattering rate is estimated with a constant geometric cross-section of 12 mb per light quark. All 2-to-2 and 2to-3 processes are allowed if they are kinematically feasible and charge conservation is respected. The proportion of 2-to-2 and 2-to-3 reactions is governed by phase space factors. The constant cross sections for stop and gluino R-meson and R-baryon interactions are superimposed on Figure 1 (left and right). This model assumes R-hadron mass hierarchies which differ from those used in the triple Regge model and predicts that the majority of baryons which are produced in scattering processes are electrically charged.

Implementation of triple Regge model in Geant-4
In the dynamical picture of R-hadron scattering used by Geant, the light quark system is decoupled from the heavy spectator parton before interacting with a nucleon according to Geant's parameterised model for light hadrons. In this way secondary particles and so-called black tracks are generated. Following the interaction, the light-quark system is recombined with the heavy parton.
To implement the triple Regge model in Geant-4, the software architecture used for the generic model was adapted. The scattering cross section was adapted from Ref. [9]. The pomeron and Regge parts of the cross section per nucleon are (in mb): Pomeron : σ P = 4.14 + 1.50 √ γ − 0.0545γ + 0.000822γ 3/2 (2) The scattering cross sections for squark and gluino Rhadrons 5 become: The available baryon states were chosen as shown in table 1 and the relative rates of the available processes reweighted to match the prescription in Section 3.1. This meant enforcing a 10% probability of exotic meson → baryon conversion as well as disabling the phase space weights of the generic model. The code is available for download [32]. Using single particle simulations, it was verified that the main features of the triple Regge model were well reproduced in the Geant-4 implementation. After the process reweighting, baryon number changing processes formed 10% of all meson interactions. The required charge exchange probabilities for meson-to-meson transformations were guaranteed by the degeneracy of the R-mesons and the availability of the same number and type of channels to every meson in a multiplet. It was also confirmed that the Geant-4 version reproduces the expected energy loss and charge exchange rates.

Results
To study the expected effects of scattering on R-hadrons, simulations were made of the passage through iron of Rhadrons of 300 GeV mass. The kinematic distributions of the R-hadrons were given by the Pythia generator [33] which simulated the direct pair production oftt,bb and gg in proton-proton collisions at 14 TeV centre-of-mass energy. The process through which the partonic final state was hadronised and R-hadrons were formed was modelled with customised Pythia-routines [34] based on the Lund string model [35]. A noteworthy free parameter in this program is the probability of the formation agg-state following the hadronisation of a gluino. This was set to the default value of 0.1 [34]. The influence of the value of this free parameter on the results shown in this Section is mentioned below.
For the results shown in this section, unless otherwise stated, the allowed R-mesons and baryons were assigned the masses prescribed in Table 1.  Figure 3 shows the predicted average energy loss per hadronic collision for squark-based R-mesons. There are only small differences between the generic and triple Regge models. The energy losses of squark and gluino-based Rbaryons in both approaches (not shown) are similar to the distributions shown in Figure 3.
The total energy loss for R-hadrons passing through 2m of iron is shown in Figure 4(a). The length of 2m was chosen since it corresponds to the typical depth of a calorimeter at a collider [36]. Gluino, stop and sbottombased R-hadrons which start in the statestd + ,bū − , and gud + , respectively are shown.
For a zero-mixing hypothesis, the stop-based R-hadron suffers an average energy loss (∼ 5.5 GeV), which is substantially greater than the energy losses for sbottom and gluino-based R-hadrons. This effect is due to the different R-mesons converting into baryonic states. Of the different stable baryon states, only the stop-based R-baryon is charged and therefore able to lose energy through continuous ionisation ; conversions into baryon states are studied in more detail below. As can be seen in Figure 4(a), the energy loss for the gluino-based R-hadron is slightly larger than that for the sbottom case owing to the typically greater number of hadronic interactions in the gluino case, as seen in Figure 4(b). It is also shown that mixing has only a small effect on the energy loss and the number of hadronic interactions of the squark-based R-hadrons.
While R-hadron energy loss is a useful possible signature in a collider search, the charge of the R-hadron after passing through matter is of arguably greater importance to the development of search strategies. As mentioned earlier, this is due to searches typically looking for muon-like candidates emerging from thick calorimeter material. In  The spectra are shown for gluino, stop, and sbottombased R-hadrons. Prior to traversal, the proportions of different hadronic states for a given heavy parton were calculated by the Lund string model as mentioned above. As can be seen, the fact that R-hadrons are most likely to be baryonic after traversing around 2m of iron, is explicitly shown here. Of the remaining mesonic R-hadrons, around one half would be electrically neutral. Thus, the majority of gluino and sbottom-based R-hadrons would be neutral after a passage of several metres in iron.
Varying the free parameter representing the probability of producing agg-state to the extreme values of 0 and 1 would clearly affect the relative ratios of different gluinobased R-hadron states formed in the primary interaction. However, the conclusion that a neutral baryon would be the dominant state after ∼ 2m of iron would be unaffected since agg-state is assumed to scatter in the same way as a gluino-based R-meson as mentioned in section 3.1. The effects of maximal mixing, while tending to slow down the meson-to-baryon conversion rate, similarly do not affect the conclusion that squark-based R-mesons will dominantly be converted to baryons after travelling ∼ 2m of iron. It is also implied in Figure 5 that R-hadrons can reverse the sign of their charge. For stop-based hadrons this occurs through mixing, whereby a positively charged meson can interact to become a neutral state which oscillates into its antiparticle before interacting again to become a negatively charged mesons. Furthermore, charge "flipping" can also occur for gluino-based R-hadrons since positively and negatively charged mesons can both be produced. Also shown is the fraction of the R-hadrons which give up all of their energy and are stopped in material. After 2m of iron, the fraction of stopped R-hadrons is typically 3-7% depending on the type of R-hadron and the mixing assumption.
Analogous flavour-composition distributions are shown in Figure 6 for antistop and antisbottom-based R-hadrons. As expected, the antibaryon states quickly die out owing to the large annihilation cross section and, in the absence of mixing, charged and neutral mesonic states are expected in roughly equal proportions. The effects of mixing allow the formation of stop and sbottom-based baryons and the same trends are thus observed as in Figure 5, albeit over longer traversal distances.

Implications of R-hadron Scattering on Searches at Colliders
Searches for stable R-hadrons at the Tevatron have exploited the expected penetrating muon-like behaviour of R-hadrons [24,25,26,27,28]. The most recent such search was made by the CDF experiment which reported a lower mass limit for stable stop squarks of around 250 GeV [28]. It is, however, not trivial to use this result to estimate mass limits on stable sbottom or gluino R-hadrons since, as discussed in this paper, such particles could dominantly enter a muon chamber as electrically neutral objects.
There has so far been only one SMP-search [25] by a Tevatron experiment at which limits were estimated for heavy stable fermionic particles with colour octet charge 6 . Under the assumption that the SMP would be electrically charged as it propagates through the detector this work excludes stable gluinos of masses less than around 200 GeV. Also using this assumption, a recent SMP-search at CDF [26] has been interpreted in Ref. [37] as providing a lower mass limit of ∼ 300 GeV.
If gluino R-hadrons do not behave as muon-like objects then other channels need to be investigated to estimate limits. CDF-measurements of events containing jets and missing transverse energy [38] were used to estimate a lower mass limit on stable gluinos of around 170 GeV [37]. However, this limit may only be relevant only for cases in which the R-hadrons are uncharged during their complete traversal of the CDF detector since the search applies a selection criteria which removes events containing isolated high momentum tracks. Similarly, the recent D0 measurement [39] of the rate of events with the monojet and missing transverse energy signature also applies selections which may suppress R-hadron events; in this case via the rejection of events containing calorimeter-muon candidates.
To make a rigorous estimate of gluino R-hadron mass limits requires dedicated detector simulations using several scattering models to estimate the effects of the various event selections. However, in the absence of this it can nevertheless be argued that the mass limits for stable gluinos may be well below 200 GeV. In a pessimistic scenario, lower mass limits on stable gluinos would be ∼ 30 GeV [40,41], as obtained at LEP-2. At LEP searches typically assumed the production of a substantial rate of electrically charged R-hadrons and information from inner tracking devices was primarily used. Similarly, the mass limits on stable sbottom-like objects would be around 90 GeV [40,42].
Hadronic interactions and the properties of the lightest R-hadrons can thus play a crucial role in determining detector signatures. Given the uncertainties on the properties of R-hadrons both at production and during their traversal of a detector it is therefore important that LHC experiments utilise fully the capabilities of their various subdetectors to look for a number of signatures.

Summary
An implementation and extension of model based on triple Regge phenomenology to squark and gluino-based R-hadron scattering has been implemented in Geant-4. The expected energy loss and the transformation of R-hadrons as they propagate through material was studied. The model is complementary to an existing Geant-4 approach for Rhadron scattering and will aid the development of search strategies based on a range of different signatures associated with R-hadron production.
The implications of R-hadron scattering on the results of previous searches were also discussed. In the context of the model discussed in this work, it was shown that mass limits on different types of R-hadrons may be lower than previously thought.
The location of the Geant-4 code together with instructions for running it can be found in Ref. [32].
David Milstead is a Royal Swedish Academy Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.