A hybrid method for determining particle masses at the Large Hadron Collider with fully identified cascade decays
Abstract:
A new technique for improving the precision of measurements of SUSY particle masses at the LHC is introduced. The technique involves kinematic fitting of events with two fully identified decay chains. We incorporate both event constraints and independent constraints provided by kinematic endpoints in experiment invariant mass distributions of SUSY decay products. Incorporation of the event specific information maximises the information used in the fit and is shown to reduce the mass measurement uncertainites by 30% compared to conventional fitting of experiment endpoint constraints for the SPS1a benchmark model.
IPMU070008
1 Introduction
Measurement of SUSY particle masses in Rparity conserving SUSY events at hadron colliders such as the LHC is complicated by a number of factors. First, imperfect detector hermeticity close to the beampipe and our ignorance of the initial state parton momentum prevent measurement of the final state momentum in the (beam)direction. This removes one kinematic constraint on the masses of particles generated in any given event. Second, the presence of an invisible Lightest SUSY Particle (LSP) at the end of each decay chain generates eight unknown fourmomentum components in every event. As a result the kinematics of SUSY events at hadron colliders are typically highly underconstrained.
Several approaches to this general problem have been documented. Given a sufficiently long decay chain, constraints on analytical combinations of SUSY particle (‘sparticle’) masses can be obtained from the positions of endpoints in distributions of invariant masses of combinations of visible SUSYdecay products (jets, leptons etc.) [1]. The system of equations may be solved with a numerical fit to obtain the individual masses, if enough constraints are provided [1, 2, 3].
An alternative class of techniques which does not rely on the endpoint information is also available. These techniques vary the assumed LSP momenta in a given event subject to event kinematic constraints in order to construct variables sensitive to the masses of sparticles present in the event. One such technique involves the construction of the variable [4, 2, 5], which is usually defined for events where the same decay chain appears in both ‘legs’ of the event. For given assumed LSP transverse momenta the maximum value of the transverse masses of the two legs is calculated. This quantity is then minimised by varying the LSP momenta subject to the event constraints to give the value of . The endpoint of the distribution depends on the test LSP mass and has recently been shown to display a kink structure when is equal to the true LSP mass [6, 7, 8, 9]. This shows that constraints are important for obtaining the absolute mass scale of SUSY particles.
Another, more involved technique of this kind is the mass relation method [10]. In this method the masses of sparticles in the decay chain are calculated for a range of LSP momenta. When the decay chain present in the event involves a large number of SUSY states, the calculated sparticle masses are constrained to lie on a hypersurface in the mass parameter space, and the intersection of several such hypersurfaces from different events determines the true sparticle masses. Alternatively the existence of solutions to the system of massshell conditions provided by underconstrained events can be used to constrain the individual masses [11].
The techniques described above all seek to determine the masses of particles appearing in kinematically underconstrained events by combining information from multiple events, either explicitly, as in the massrelation method, or implicitly by obtaining constraints from distributions of quantities derived from individual events. This paper describes a simple extension of these techniques (the hybrid method) in which information from multiple events is combined with event information in a kinematic fit to fully reconstruct events and hence constrain the masses of individual SUSY particles. In this paper, we especially emphasize a case where both legs contain the same cascade decay chain, so that events are more prominent over the background, and fewer unknown mass parameters are required. It is also possible to include constraints for this case, which provide independent information on the LSP mass. Full reconstruction of SUSY particle kinematics with this technique could also be useful for other purposes such as measuring the spinstatistics of SUSY states.
2 Description of technique
To simplify the description, we first examine the case where a specific decay chain appears in both legs of the event. Such ‘symmetric’ events will possess a considerably smaller branching ratio than ‘singleleg’ events containing only one such decay chain but the additional decay products can lead to striking signatures which strongly suppress the SM and SUSY backgrounds. Conversely, if such events are observable then it is also probable that significant numbers of singleleg events can also be observed. Both classes of event are used by the technique described here.
Our strategy is as follows. We start with constraints on sparticle masses obtained from kinematic endpoints in distributions of invariant mass combinations obtained from singleleg events, as described in Section 1. Because one of each of these distributions is obtained from the experimental dataset considered we call these constraints ‘experimentwise’. In addition to such constraints, we possess ‘eventwise’ constraints which are obtained from massshell conditions provided by the event visible decay products and LSP momenta, where the latter are further constrained by the measured event components. A kinematic fit to both the experimentwise and eventwise constraints can possess fewer degreesoffreedom than the simple experimentwise endpoint fit, potentially improving the sparticle mass precision.
In practice event visible momenta and values may be measured with limited precision causing each individual kinematic fit to generate mass values which potentially deviate more from the true masses than the equivalent quantities obtained from the experimentwise endpoint fit. Nevertheless because the dataset provides us with a number of uncorrelated candidate events, the means of all the fitted event mass values can indeed improve the mass measurement precision. The situation is analagous to measuring the mass of a resonance decaying to visible decay products  the mass value obtained from one event may be relatively inaccurate but the uncertainty is reduced by considering the mean mass obtained from all events.
It is instructive to consider at this point the roles played by the different constraints in the eventwise fits in this simplified case of symmetric events. Each leg of the event contributes four unknown LSP fourmomentum components to the degreesoffreedom of the event, while for a decay chain consisting of steps each leg contributes massshell conditions. Let us define the number of degreesoffreedom of the experimentwise endpoint fit to be , where is the number of sparticle masses and is the number of uncorrelated endpoint constraints. In this case the extra number of degreesoffreedom generated by including the eventwise information is , where is the number of unknown LSP fourmomentum components and is the number of eventwise constraints. The condition for inclusion of the eventwise information to potentially improve the mass precision is .
For symmetric events is given by for one leg and respectively or for two legs with or without constraints applied. Consequently if even considering one leg alone in conjunction with the endpoint constraints potentially improves the mass measurement precisions, while if improved precisions can potentially be obtained by using both legs with constraints.
It should be noted that given an unambiguous assignment of visible final states to decay chain steps, the technique described here could be extended to nonsymmetric events where some/all of the SUSY states appear in both legs of the event. In this case, for and steps in legs 1 and 2, is given by or when including or excluding the eventwise constraints. Consequently if improved mass measurement precisions can be obtained by using eventwise fits with constraints applied.
3 Example: step symmetric decay chain
3.1 General discussion
Let us now consider an example of mass measurement using symmetric events, for the specific case of . The decay chain present in both legs is
(1) 
where Greek letters denote SUSY states, Roman letters denote visible SM decay products and is the LSP. Denoting the two legs of the event with subscripts 1 and 2, the eight massshell conditions are:
(2)  
(3)  
(4)  
(5) 
To these constraints should be added the two constraints provided by the event components:
(6)  
(7) 
thus giving ten eventwise constraints in total (i.e. ). The number of unknown parameters appearing in these constraints is twelve – eight LSP fourmomentum components together with four unknown sparticle masses. With the definitions from Section 2 the number of extra unknown parameters present in the eventwise fit relative to the endpoint fit is therefore . Consequently indicating a potential gain over the endpoint fit. In practice the four massshell conditions for each leg may be solved analytically to give a locally invertible map from the four sparticle masses to the four components of the LSP four momentum. In this case and , however is clearly unchanged.
It is also interesting to consider the above example when constraints are not included in the eventwise fit. In this case and (or equivalently and if solving the massshell conditions) and hence . Each leg of the event is independent and the eventwise constraints just map sparticle masses to LSP four momenta. The eventwise fit is therefore equivalent to solving the kinematic endpoint mass constraints and no gain in mass precision is obtained. The situation changes when the constraints are applied because in this case the kinematics of the two legs are connected.
Note that an alternative, related, approach to this problem can also be taken. Namely, one merely utilises the equivalence of the fitted masses in the two legs of the event. In this case the constraints in Eqns. 5 and 7 reduce in number to six (since the masses appearing in Eqn. 5 are a priori unconstrained) and hence two degreesoffreedom remain (). Thus each event is underconstrained. The LSP fourmomenta which satisfy the constraints for a given event define a twodimensional surface in the fourdimensional , , , space. If a few events of this kind can be found, it is possible to solve for the masses, which are given by the coordinates of the point of intersection of all the event 2D surfaces in the 4D space. This approach is similar to the mass relation method described in the introduction and will be studied in full in a future paper.
3.2 Concrete example: decays at the SPS1a mSUGRA benchmark point
Let us now examine in detail a concrete realistion of the decay chain discussed above. At mSUGRA point SPS1a there is a significant branching ratio for the decay chain
(8) 
This chain provides 5 kinematic endpoint mass constraints from invariant mass combinations of jets and leptons [12]:

= 77.08 0.08(scale) 0.05(stat) GeV

= 431.1 4.3(scale) 2.4(stat) GeV

= 203.0 2.0(scale) 2.8(stat) GeV

= 380.3 3.8(scale) 1.8(stat) GeV

= 302.1 3.0(scale) 1.5(stat) GeV
For this study unbiased samples equivalent to 100 fb (one Monte Carlo ‘experiment’) of SPS1a signal events and background events were generated with HERWIG 6.4 [13, 14] and passed to a generic LHC detector simulation [15]. A lepton reconstruction efficiency of 90% was assumed.
Events were selected in which the above decay chain appears in both legs of the event with the following requirements:

2, with 100 GeV,

100 GeV,

max(100 GeV,),

4, where (isolated) and 6 GeV,

2 Opposite Sign Same Flavour (OSSF) lepton pairs. If the pairs are of different flavour both pairs must have . If both pairs are of the same flavour then one and only one of the two possible pairings must give two values which are both less than . These pairings allocate the leptons to each leg of the event.

One and only one possible pairing of the two leading jets with the two OSSF lepton pairs must give two values less than . These pairings allocate the jets to each leg of the event.

For each inferred leg of the event the maximum(minimum) of the two values must be less than . This ordering allocates the leptons to the and [2] positions in the decay chain.
The requirement of 4leptons in two OSSF pairs and two high jets consistent with kinematic endpoints, together with large , is effective at removing the majority of SM and SUSY backgrounds (see below).
Each selected event was fitted with MINUIT [16]. Free parameters were taken to be the four masses appearing in the decay chain: , , and . In the spirit of the discussion in Section 3.1 the massshell conditions and measured momenta of the visible decay products for each leg were solved to determine the LSP fourmomenta, giving two solutions for each leg. The minimisation function was defined by:
(9)  
where denotes an expected endpoint value derived from the masses in the eventwise fit with the formulae of Ref. [2], and denotes a ‘measured’ experimentwise endpoint value. The uncertainties in these ‘measured’ endpoints were those quoted above. The uncertainties on the measurements of the and components of , and , were given by where is the scalar sum of jet of the event. This function incorporating both eventwise constraints and experimentwise endpoint constraints was evaluated for each of the four pairs of momentum solutions obtained from solving the leg massshell conditions. Fitted masses were obtained when was minimsed for the event. Fitted masses were used in the subsequent analysis only if MINUIT judged the fit to have converged and .
Following application of the selection cuts described above and the requirements of fit convergence and low fit 38 SUSY ‘signal’ events with the above decay chain appearing in both legs were observed. 4 SUSY background events were observed, consisting of the above decay chain in both legs but with one or two leptonically decaying staus produced in the decays of the ’s. No background events were observed in 100 fb equivalent data. More SM background events may be expected in a real experiment, given that effects such as charge and lepton misidentification are not included in the fast detector simulation. The use of full GEANT detector simulation is required to model correctly these effects, which is beyond the scope of this paper, however given the hard kinematic selection cuts it is reasonable to assume that they are at least smaller than the negligible background considered above.
Each eventwise fit generated one set of values for the sparticle masses, namely those values which minimise the broad function in Eqn. 9. The distributions of these values for one Monte Carlo experiment are shown in Fig. 1. The widths of the distributions are greater than those which would be obtained were the constraints excluded from the fits – in that case each eventwise fit becomes equivalent to the experimentwise endpoint fit as explained in Section 3.1 and each distribution becomes a deltafunction located at the equivalent mass value. The key point is that despite the fact that the mass distributions in Fig. 1 are broader when constraints are used, the means of the distributions measure the sparticle masses more accurately. This may seem counterintuitive, however the reason is clear. The components for each event are measured with only limited precision, causing a potential shift in the minimum of the event function away from the true sparticle masses. There is no way to correct for this effect on an eventwise basis. By correctly parameterising the measurement precision in Eqn. 9 however we ensure that such shifts are on average unbiased when considering all events in the experiment and hence the means of the mass distributions compensate for this resolution effect, while also making use of the reduction in degreesoffreedom provided by the constraints. This compensation for resolution effects at the level of the experiment rather than individual events is also implicit in the endpoint method where gaussiansmeared endpoint functions are used when fitting the endpoints in order to compensate for experimental jet and lepton energy resolutions.
In order to demonstrate the performance of the technique and judge the uncertainties in the measurements the above procedure was repeated for 100 Monte Carlo experiments. For each experiment, kinematic endpoint positions were sampled from gaussians with means and sigmas given by the means and uncertainties listed above. The five sampled endpoint positions for each experiment were solved simultaneously with a MINUIT fit to give initial mass values for input to the MINUIT eventwise kinematic fits. For each experiment relative jet(lepton) energy scale values were sampled from gaussians of width 1%(0.1%) reflecting likely ultimate energy scale uncertainties at the LHC. Each experiment generated a set of sparticle mass histograms similar to those shown in Fig. 1. The means of these histograms for the 100 MC experiments were then used to construct likelihood histograms for the masses, shown in Fig. 2. The standard deviations of these histograms were taken to provide the uncertainties on the sparticle mass measurements.
Unbiased MC data equivalent to only one 100 fb experiment were available for this study. For this reason the same events were used for each MC experiment, with just the endpoint values and jet/lepton energy scales varying. The additional uncertainties in the final mass values expected from varying event samples were estimated from the mean statistical uncertainties in the mean experiment mass values as extracted from the eventwise distributions such as those shown in Fig. 1. We evaluated the experimentbyexperiment spread due to varying event samples as , where is the RMS of the eventwise distributions as shown in Fig. 1, and is the number of entries in each plot. These additional contributions were added in quadrature to the uncertainties obtained from the study. This approximation was checked with a second sample of SPS1a events equivalent to 100 different MC experiments, biased to force gluinos to decay to , or , to decay to and to decay to or . Excluding the SUSY background from this sample was estimated to bias the mean mass values by 1% and increase the uncertainties by 5%. The uncertainties increase when the SUSY background is excluded because the effect of the decrease in event statistics outweighs the reduction in bias caused by excluding the primarily background events, which have similar kinematics to the and signal events.
State  Input  EndPoint Fit  Hybrid Method,  Hybrid Method, no  

Mean  Error  Mean  Error  Mean  Error  
96.05  96.5  8.0  95.8(92.2)  5.3(5.5)  97.7(96.9)  7.6(8.0)  
142.97  143.3  7.9  142.2(138.7)  5.4(5.6)  144.5(143.8)  7.8(8.1)  
176.81  177.2  7.7  176.4(172.8)  5.3(5.4)  178.4(177.6)  7.6(7.9)  
537.2–543.0  540.4  12.6  540.7(534.8)  8.5(8.7)  542.9(541.4)  12.2(12.7) 
The results of this study are summarised in Table 1. For comparison purposes the analysis was initially carried out with the constraints removed from the function. The measurement precisions are consistent with those obtained from the conventional endpoint fitting method, as expected following the reasoning outlined in Section 3.1. The analysis was then repeated including the constraints, giving an overall improvement in sparticle mass precisions 30% for all four masses considered. A similar improvement was found when using the biased sample of nonrepeated events for different experiments. The measurement of mass differences is also improved, with for instance being measured with a precision of 4 GeV comparable with the natural widths of the light squarks ( 5 GeV at SPS1a) and their mass differences ( 6 GeV). Further improvement in mass measurement precision therefore probably requires that such effects be taken into account, for instance in the definition of the function in Eqn. 9.
4 Conclusions
A new technique for improving the precision of LHC mass measurements has been outlined in which experimentwise information, for instance invariant mass endpoint constraints, are combined with eventwise kinematic information such as constraints and measured fourmomenta of visible decay products. For the SPS1a model considered here the mass measurement precision was shown to improve by 30% for 100 fb of data. SUSY models with larger branching ratios for symmetric events containing the necessary decay chain, or with more indistinct or poorlymeasured kinematic endpoints, might be expected to benefit still further from application of the technique. The technique could be extended to nonsymmetric events where some/all of the SUSY states appear in both legs of the event – this strategy for measuring further SUSY masses will be examined in more detail in a future paper. Full reconstruction of SUSY particle kinematics with this technique could also potentially be useful for other purposes such as measuring the spinstatistics of SUSY states.
Acknowledgements
The authors wish to thank the organisers of the Les Houches 2007 Physics at TeVScale Colliders workshop, where the work described in this paper was first conceived. DRT wishes to acknowledge STFC for support.
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