Improving SUSY Spectrum Determinations at the LHC with the Wedgebox Technique
Abstract
The LHC has the potential not only to discover supersymmetry (SUSY), but also to permit fairly precise measurements of at least a portion of the sparticle spectrum. Proposed mass reconstruction methods rely upon either inverting invariant mass endpoint expressions or upon solving systems of massshell equations. These methodologies suffer from the weakness that one certain specific sparticle decay chain is assumed to account for all the events in the sample. Taking two examples of techniques utilizing massshell equations, it is found that also applying the wedgebox technique allows for the isolation of a purer event sample, thus avoiding errors, possibly catastrophic, due to mistaken assumptions about the decay chains involved and simultaneously improving accuracy. What is innovative is using endpoint measurements (via the wedgebox technique) to obtain a more homogeneous, wellunderstood sample set rather than just using said endpoints to constrain the values of the masses (here found by the massshell technique). The fusion of different established techniques in this manner represents a highly profitable option for LHC experimentalists who will soon have data to analyze.
Keywords:
keyword1, keyword20806.2492
1 Introduction
LHC experimentalists will soon determine whether or not Supersymmetry (SUSY) is a TeVscale phenomenon: if so, colored sparticles will probably be the first to be discerned, possibly soon followed by neutralinos, charginos, and sleptons if favorable decay channels are open, though measuring the masses of these latter colorless sparticles with percent precision will be challenging:1999fr (); Ball:2007zza (). The reason for this is that every Rparityconserving^{1}^{1}1If Rparity is not conserved, then it may be possible to fully reconstruct events. See Allanach:2001xz () for further details. SUSY event produces at least two invisible particles (the lightest SUSY particles, or LSPs) which carry away significant missing energy and make it impossible to reconstruct mass peaks. Therefore, many SUSY mass extraction techniques depend on precise measurement of invariant mass distribution endpoints. For a sparticle decaying into an LSP and a Standard Model (SM) fermion pair, either via a threebody decay or sequential twobody decays, it is straightforward to see how the endpoint of the difermion invariant mass distribution yields the mass difference between the decaying sparticle and the LSP (perhaps modified by the onmassshell intermediate for twobody decays) Paige:1996nx (); Hinchliffe:1996iu (); Gianotti:682494 (); Hinchliffe:1998ys (); Hinchliffe:1999zc (); Mura:1311179 (). Studies attempting to fully reconstruct the actual sparticle masses from invariant mass endpoint information rely on specific longer decay chains, typically Bachacou:1999zb (); Gjelsten:2004ki (); Gjelsten:2005aw (); Gjelsten:2005sv (); Gjelsten:2005vv (); Gjelsten:708246 (); Gjelsten:2006as (); Miller:2005zp (); Lytken:703773 (); Butterworth:2007ke (); Lester:2006cf (); Allanach:2000kt (); Tovey:2003ef (); Borjanovic:2005tv (); Tovey:2008ui () — each event would have two ’s (for instance) produced. It is then theoretically doable to construct enough invariant mass distributions to determine the sparticle masses; however, in practice endpoint measurement may be complicated by low event rates, fitting criteria, unaccountedfor (in the simulation) higherorder and radiativeHorsky:2008yi () effects, and (in particular) backgrounds.
Ideas on how to measure SUSY masses without relying on distribution endpoints have also been put forward. The work of Nojiri et al.Nojiri:2003tu (); Kawagoe:2004rz (), for example, uses massshell relations in a sufficiently long SUSY decay chain, e.g. ; if some of the masses are already known in this chain the others can in principle be found by solving massshell relations for a small sample of events^{2}^{2}2In practice, many events are still required., which may in fact lie far from the endpoint. Another method, due to Cheng et al.Cheng:2007xv (); Cheng:2008mg (); Cheng:2009fw (), starts from very similar looking massshell relations, but instead of assuming some masses and solving for the others, scans the whole massspace for points where these relations are most likely to be satisfied. Both methods, hereafter designated as ‘Mass Shell Techniques’ (MSTs)^{3}^{3}3Refs. Nojiri:2003tu (); Kawagoe:2004rz (); Nojiri:2007pq () use the name ‘mass relation method’ to refer to their technique., seem quite effective in obtaining percentlevel determination of the soughtafter SUSY masses, at least at the parameter points considered in those works.
The accuracy of both these MSTs hinges on one critical assumption: the decay topology of choice has been isolated. In the actual LHC data, the decay topology would have to be inferred, if this is at all possible, before proceeding; MSTs would thus appear to be excellent roads to SUSY mass reconstruction which, however, begin only at a point halfway to the destination.
The present work focuses on the first half of this road; i.e., isolating a desired decay topology^{4}^{4}4The recent paper Bai:2010hd () addresses similar issues from a somewhat different perspective., and on how this affects a subsequent MST analysis. As a first foray into this potentially quite thorny task, consider specific topologies studied in Nojiri:2003tu () and Cheng:2007xv (); Cheng:2008mg (); Cheng:2009fw () involving a pair of neutralinos () that subsequently decay to leptons (electrons and muons) via onshell sleptons. This situation is amenable to a wedgebox analysis Bisset:2005rn (); Bian:2006xx () which is based upon a scatter plot of the dielectron mass versus the dimuon mass . A key benefit of this technique is that it allows (at least partial) separation of individual events according to the specific pair whose production gave rise to them. Given sufficient statistics, events from each such decaytype fall in distinct, easilyrecognized zones of the wedgebox plot. The overall topology of the resulting wedgebox plot then tests for the significant presence of the various possible decay channels — which may for instance signal the meaningful presence of a decay channel erroneously assumed to have been insignificant as the basis for a MST analysis. Events can be selected from a specific zone of the wedgebox plot, preferably a zone populated by only one decay channel. This acts to maximize sample homogeneity and assure the basic MST assumption is satisfied.
Although the wedgebox technique relies on locating the endpoints of invariant mass distributions — just like the studies Bachacou:1999zb (); Gjelsten:2004ki (); Gjelsten:2005aw (); Gjelsten:2005sv (); Gjelsten:2005vv (); Gjelsten:708246 (); Gjelsten:2006as (); Miller:2005zp (); Lytken:703773 (); Butterworth:2007ke (); Lester:2006cf (); Allanach:2000kt (); Tovey:2003ef (); Borjanovic:2005tv (); Tovey:2008ui () mentioned earlier, the information sought is radically different: the wedgebox analysis is tailormade for event sample sets comprised of assorted produced sparticle pairs and multiple sparticle decay chains. The observed endpoints serve to delineate the zones and allow for selection of purer subset(s) from an overall sample set. (Using this endpoint information to determine a set of cuts is a far more rational course than that of arbitrarily choosing some numerical cutoff values to purify the data sample.) Virtually all previous studies presume such purification has already been accomplished either by an unspecified set of cuts or a fortuitous choice of SUSY input parameters. The wedgebox technique illustrates a concrete method of how to deal with the more general case, and the consequences which result and should not be ignored in a coupled analysis aiming to extract the sparticle masses.
This paper will show that MSTs, by construction unrelated to invariant mass endpoints, can nevertheless be improved by information contained in these endpoints — specifically via the wedgebox technique^{5}^{5}5Though our strategy is quite different from the ‘hybrid’ method of Nojiri:2007pq () which couples an MST with values for of endpoints.. The paper is organized as follows: Sections 2 and 3 concentrate on the ‘NMST’ method of Nojiri et al. and the ‘CMST’ method of Cheng et al., respectively; Section 4 then offers conclusions and some additional discussion on SUSY mass spectral analyses at the LHC.
2 The NMST Method of Nojiri et al.
For the NMST method, the focus will be on the decay of a heavy MSSM pseudoscalar Higgs boson as considered in Nojiri:2003tu (). The specific decay chain is
(1) 
where the Higgs boson decays to neutralinos () via onshell sleptons of the electron or muon variety (see Fig. 1).
Assuming the final leptons’ fourmomenta are known while the LSP’s escape detection, (1) implies six massshell constraints on the eight unknown components of LSP fourmomenta ( and ),
(2)  
(3)  
(4)  
(5)  
(6)  
(7) 
Nojiri et al. also posit two overall momentum conservation constraints
(8)  
(9) 
along directions ( , ) transverse to the beam (the direction), though this would appear to be contingent on the Higgs boson having no transverse momentum. If all the masses , , , , and are known in advance, one can solve the eight equations (2)(9) for the eight unknowns and reconstruct the Higgs boson mass via
(10) 
from just one Higgs boson event of the type (1). However, even in this idealized scenario which does not include detector resolution, particle widths, backgrounds, etc., there are two major caveats which prevent this procedure from being so straightforward:

There is a 4fold ambiguity in assigning labels to the leptons; this forms a combinatoric background.
So what one must do in practice is collect a number of events and deduce the correct value of from the maximum of the resulting distribution. In Nojiri:2003tu () a event sample (with no backgrounds) thus yielded a percentlevel determination of the Higgs boson mass.
2.1 Addition of the Wedgebox Technique
2.1.1 Box Topology
As shown in Nojiri:2003tu () the programme sketched in the previous section works fairly well at Snowmass Benchmark SPS1a Allanach:2002nj () where the dominant Higgs boson decays are via (1) with . Therein, 1000 events of the type
(11) 
were generated using the HERWIG6.4 Corcella:2000bw (); Corcella:2002jc (); Moretti:2002eu () event generator and passed through the detector simulator package ATLFAST RichterWas:683751 (). The only cuts required were that all four isolated leptons have , with two of the leptons also having while the other two have . Sameflavor events such as or were also included if one of the two possible pairings of OS leptons in such configurations gave a dilepton invariant mass beyond the kinematic endpoint (implying the other possible pairing is the correct one). Thirty percent of the original 1000 events passed the cuts and selection criteria, leaving 300 events for the NMST analysis, and these in turn yielded the correct Higgs boson mass with a resolution of only Nojiri:2003tu ().
The following set of cuts are herein adopted in a effort to reproduce these results using ISAJETBaer:1999sp (); Paige:2003mg () and a detector simulation which assumes a typical LHC experiment, as provided by private programs checked against results in the literature, in place of HERWIG and ALTFAST^{6}^{6}6The detector simulation of the calorimetry is based on a cell size of . Particle resolutions, adopted to approximate the CMS detector, are given by where for muons or hadrons with , for muons or hadrons with ; and for electrons and photons with . This simulation was also repeated using PYTHIA 6.4 Sjostrand:2006za () coupled with PGS 4 PGS4 (), yielding results very close to the ISAJET study. The CTEQ 6.1M Stump:2003yu () set of parton distribution functions is used with top and bottom quark masses set to and , respectively.:

Leptons must be hard ( for , respectively; ), and isolated (no tracks of other charged particles in a cone around the lepton, and less than of energy deposited into the electromagnetic calorimeter for around the lepton).

There must be missing energy in the range: .

No jets^{7}^{7}7Jets are defined by a cone algorithm with and must have . are present with a reconstructed energy greater than .
A sufficient sample of events is collected to represent an integrated LHC luminosity of , the same integrated luminosity as in Nojiri:2003tu (), though this study only finds about 200 events of the type (11) for this luminosity. About 11% of the generated events pass the cuts and selection criteria^{8}^{8}8This rises to about 16% if the jet and missing energy cuts are excluded, and to around 23.5% if sameflavor events meeting the criteria of Nojiri:2003tu () were also to be included. This is comparable to the 30% noted in Nojiri:2003tu ().
Fig. 2a shows the wedgebox plot at SPS1a for events. A ‘simple box’ topology consistent with the expected origin of lepton pairs is clear. Moreover, the number of flavorbalanced events () exceeds the number of flavorunbalanced events () at this parameter point Huang:2008qd (); this indicates that the events come primarily from a Higgs boson decay^{9}^{9}9Note that though in a simulation one can of course choose to only generate Higgs boson decay events, experimentalists lack this freedom. with decay topology (1). Though the final number of events passing cuts is somewhat small (only 40 compared to the which Nojiri:2003tu () estimates, for the reasons noted above) the number of events in the distribution of solutions is quite a bit larger: recall the bulleted caveats earlier which potentially can yield a = 16fold multiplicity factor. Though only a factor of 4 or so is observed, nonetheless a fairly clear peak^{10}^{10}10Apparently localized to within the binning size adopted in Fig. 2b. in the distribution emerges (see Fig. 2b) at the correct value^{11}^{11}11Also note here that the momentumconservation constraints (9) do not appear to be generally true; i.e., according to this analysis the parent Higgs boson is often generated with significant transverse momentum in the range . Surprisingly, this does not seem to affect the result. of .
Naturally, other processes might also generate events of the signal type. It is clearly incorrect to consider only events due to production without also including production. At SPS1a, and differ by less than 3 GeV, so the events may be expected to aid in the mass determination. However, only 6 events (as compared to 40 events) are found. In addition, 16 events from direct neutralino pair production and 15 events from production processes involving charginos (these two event categories are each produced via an electroweak vector gauge boson) and/or isolated leptons from heavyquark decays (in sparticlecontaining events) contribute to the background. Standard Model processes do not contribute any events. Fig. 3a shows the wedgebox plot at SPS1a including all these eventtypes. Fig. 3b then shows the specific neutralino pair that is generated in each event (irrespective of the production type). While the overwhelming majority of the events are due to as expected/hoped, a couple events, a event, and a event are also present. Somewhat fortuitously, all four of these non events fall outside of the box region delineated by the dashed lines in the lowerleft corner of Fig. 3(a,b), as do 6 of the 15 ‘Bad¡¯(mainly charginocontaining) events. Even considering possible additional experimental smearing, it should be apparent to experimentalists (who do not see colorcoded events) that such events are outside of the box. In this case, the inclusion or removal of these distinguishable wrongchain events does not markedly alter the distribution of solutions shown in Fig. 3c (which is little degraded from the only plot shown in Fig. 2b). However, it does indicate how reference to the exact (within experimental accuracy) numerical value for the edges of the box delivered by the wedgebox plot (or equivalently by an analysis of the two sameflavor dilepton invariant mass distributions) might aid in improving the NMST analysis. The worth of the wedgebox information becomes more pronounced when the topology is not so simple (as is often the case for allowable MSSM parameter sets). This more general situation is addressed next.
2.1.2 Wedge Topology
The SPS1a parameter point event set is dominated by events in which a pair is generated, consistent with the expected decay chain presumed by the NMST analysis in Nojiri:2003tu (). In principle, this NMST method should also work for a set of events for which . The complication comes when one considers that the experimentalists event set (consistent with the cuts already) specified will then in general consist of a mixture of different decay chains involving different generated pairs of neutralinos. As a case in point, consider the following MSSM parameter set:
 MSSM Test Point I

which has the mass spectrum shown in Table 1.
Particle  Mass (Point I) (GeV)  Mass(Point II) (Gev) 

Now the main production modes contributing to the signal are the Higgs boson channels , ‘direct’ neutralino pair production channels^{12}^{12}12As explained in Bian:2006xx (), direct channels are suppressed by isospin symmetry, while are phasespace suppressed. , and ‘mixed’ channels involving charginos, mainly production. A random sample of events^{13}^{13}13Note that for MSSM Test Point I the staus have been set more massive than the other sleptons to avoid taucontaining decays and generate more decays to the desired leptons. This is simply done by hand here for convenience. will therefore not be a clean collection of Higgs boson decays, which for the luminosity considered here () number about 150 against nearly 800 direct channel and 100 or so mixed channel events. This point then presents a more challenging case for applying the mass relation method.
However, the shape of the wedgebox plot at this point suggests selecting events via their decay topology: Fig. 4a shows a clear ’double wedge’ topology implying that events due to decays are confined to the innermost box bounded by kinematic edges at , while those due to events are enclosed in the legs of the short wedge terminating at . Events from decays span both of these regions and beyond up to the final kinematic edge at .
Strictly speaking, the structure of Fig. 4a does not uniquely lead to the particle assignments noted in the previous paragraph. In the MSSM, there are other processes that can generate edges in the wedgebox topology aside from those of the form , including and . The latter were given the appellation ‘stripes’ in Bisset:2005rn () wherein they are discussed in some detail. For instance, in principle, the shorter (longer) wedge could be due to () rather than due to () as described above. Separate studies of decay kinematics can potentially exclude such alternative possibilities. For parameter set choices examined herein, these other feasible decay modes have a totally negligible effect. More importantly, as will be discussed more in the next section, such ambiguities are largely irrelevant to the mass spectrum analyses described in this work.
As in Fig. 3(b,c), Fig. 4(b,c) is colorcoded^{14}^{14}14 Colorcoding is of course an unfair advantage available in simulations but unavailable to experimentalists. It is shown in the plots here to give the reader a clearer picture of what processes are actually occurring at significant rates. Information from this colorcoding does not enter into any of the analyses results (mass values) found in this work. to show the separate distributions of events from different production channels (A, H, or ’direct’ production DP) and by their assorted neutralinopair types; these distributions agree with remarks in the previous paragraph. Note, however, the presence of ‘Bad’ events which do not distribute themselves nicely within the kinematic bounds and which are typically due to chargino decays. Though nearly 10 percent of the total number of events are ‘Bad¡¯ events, about half of these are excluded by rejecting events outside the overall wedge structure, again nicely illustrating the strength of the wedgebox technique^{15}^{15}15Recently, Bai:2010hd () has also discussed ideas for excluding such bad events..
Without the assistance of the wedgebox plot, one might be led to assume that events with correspond mostly to the decay topology of (1) with . This, however, leads to a Higgs boson mass distribution as shown in Fig. 5a. There is neither a clear peak nor any kind of distinguishing feature near the correct value of . Evidently, what might seem the natural choice of using events from the densest region of the wedgebox plot is not optimal.
Instead, events should be selected from the most homogeneous zone of the wedgebox plot, which in this case consists of the outermost legs from to , corresponding mostly to (even without looking at Fig. 5b, Bian:2006xx () predicts that events in the outer wedge of a doublewedge plot come from Higgs boson decays). As seen in Fig. 5b, the NMST method now works splendidly, yielding an easilyidentified peak at the correct Higgs boson mass^{16}^{16}16Note from Fig. 4b that the contributions from and from are much more comparable than they were at SPS1a. At MSSM Test Point I the mass difference between and is around (as opposed to under at SPS1a), and their widths of are about 5 times larger than at SPS1a (as determined by ISAJET and incorporated into this simulation). These characteristics are reflected in the plot where the peak is spread over approximately two of the wide bins., even though now an additional sparticle mass plays into the equations. The goodness of fit is more surprising considering the number of ’bad’ events distributed throughout this zone; inspection reveals that these latter, however, often fail to give solutions to equations (2)(9), so they do not heavily interfere with the shape of the distribution.
Improvement of the NMST method via a wedgebox plot, at least for the Higgs boson decay topology considered here, is therefore quite straightforward. However, these improvements may not always be able to counter some of the shortcomings of the NMST model, namely, the assumptions of the known sparticle masses and zero net transverse momentum — Eqns. (9). The sparticle masses are assumed to be determined via some unspecified preceding analysis, and it would be more correct to attach uncertainties to these values rather than just input the exact values given by the simulator at this point in the parameter space. It is also quite possible that the mass of the heavier neutralino required for the analysis at MSSM Test Point I may be far less accurately determined (or left unknown) by said nameless preceding analyses than are the masses of the lighter sparticles (, , and ) which suffice at the SPS1a point. Thus a study in which both the Higgs boson mass and the mass of the heavier neutralino are simultaneously determined would certainly have merit. Better still would be to jettison reliance on such unnamed previous studies and to determine all the todate unknown beyondthestandardmodel particle masses in a selfcontained analysis. This is the aim of the CMST method to which this paper now turns (as opposed to piling more details into a fundamentallyweaker NMST analysis). The slightly more pedagogical goal of this NMST section has been to succinctly demonstrate the worth of a combined MST & wedgebox analysis without immediately introducing numerous subtle (and potentially distracting) issues inherent in an even more realistic and selfcontained study.
3 The CMST Method of Cheng et al.
Next consider the CMST method. The process treated by Cheng et al. Cheng:2007xv (); Cheng:2008mg (); Cheng:2009fw () is illustrated in Fig. 6 in which the masses , , , , and must satisfy a set of equations precisely analogous to (2)(9):
(12)  
(13)  
(14)  
(15)  
(16)  
(17)  
(18)  
(19) 
where the transverse momentum sums are assumed to be calculable from measurements of associated jet momenta (produced, though not shown, in the gray bubble in Fig. 6) and missing momentum (from the LSPs) necessary to balance the whole:
(20) 
The specific example considered in Cheng:2007xv (); Cheng:2008mg (); Cheng:2009fw () was production of two neutralinos via squarks in the MSSM, followed by decay via onshell smuons to muons and two LSPs:
(21) 
giving
(22) 
In the threedimensional space of masses , each event gives eight equations (12)(19) for the eight unknown LSP momenta and , assuming the outgoing muon momenta can be measured. The solution to this set of equations is again, as for the system of (2)(9), a quartic equation with 0, 2, or 4 real roots. In contrast to the discussion of the last section, however, rather than trying to find the point in space where the density of solutions () is maximized, instead the point where the gradient of is maximized (a heuristic argument for this is given in Cheng:2007xv (); Cheng:2008mg (); Cheng:2009fw ()) is sought.
In Cheng:2007xv (); Cheng:2008mg (); Cheng:2009fw () this method apparently works quite well at the MSSM parameter points studied, giving all relevant sparticle masses to a few percent after collection of data samples corresponding to of integrated LHC luminosity. However, to test the robustness of this technique and the extent to which augmentation with the wedgebox technique can improve results, consider a new MSSM parameter point (see Tab. 1 again for masses):
 MSSM Test Point II

3.1 Wedgebox selection
The wedgebox structure^{17}^{17}17 MSSM Test Point II is clearly
representative of the general case in the MSSM, where collisions
yield a ‘mixed bag’ of concomitant neutralino decays.
Events on this plot pass the same cuts as in Section 2.1,
minus the jet cut. of Fig. 7
is again generated using ISAJET
Baer:1999sp (); Paige:2003mg () and the event selection criteria mentioned earlier
(save no jet cut); the plot compartmentalized into four substantially
eventpopulated regions by the shown reddashed lines to which will
be applied the following nomenclature:
The Zone 1 box, with
,
is the most denselypopulated region of the wedgebox plot
and should include all
events.
Zone 2 is composed of two rectangles (the legs of wedges) running outwards
along both axes from the Zone 1 box — satisfying the condition that
and
.
Events due to and
not residing in
Zone 1 will fall^{18}^{18}18With the
()
events terminating at
(). The endlines at
are faintly discernible in Fig. 7; however, the forthcoming
analysis does not rely upon this. in Zone 2.
Zone 3, with
, lies outside
of Zones 1 & 2 and should only be populated by
,
and
events.
Consider first using fourlepton events across the whole wedgebox plot — this would include several different kinds of neutralino pair events. A scan was performed over the mass space, and the resulting values projected onto the plane. As Table 1 shows , Fig. 8 indicates that the gradient of maximizes along the line rather than at the desired true value: the CMST method fails badly in this attempt to find the sparticle masses. Since the wedgebox plot clearly indicates that , and production is substantial, the failure of the CMST is not surprising given the inhomogeneity of the data set.
Consider instead performing an analysis limited to events in Zone 3 — which should be more homogeneous. Before proceeding though one more feature of the wedgebox plot should be taken into account: there is a clear line at . This is due to neutralinos decaying to an LSP and a pair of leptons via an onshell rather than via a slepton^{19}^{19}19Although the missing energy cut should eliminate most SM , events, any such remnant background surviving would also populate this line.. The situation is greatly simplified if the events in Zone 3 are further curtailed to encompass only those for which . This truncated Zone 3 will be called Zone 3. This will exclude the onshell events along with those due to or . The remaining fairly homogeneous subset of events still yields 1000+ signal events (corresponding to of LHC integrated luminosity), 80% of which are in fact due to pair production (the remainder mostly involve colored sparticle decays into the heavier chargino, ).
At this juncture it is appropriate to revisit a statement made in the previous section — that ambiguities in identifying the particle identities from the wedgebox plot topology are largely irrelevant to the analysis to be performed. As noted in the case of MSSM Test Point I, the occurrence of ‘stripes¡¯ could mean that zones of the wedgebox plot should be reassigned. For instance, Zone 3 could be due to decays (and thus more correctly referred to as a region), while Zone 2 has many ‘stripe¡¯ events — decays. This is not true at this point in the parameter space. Since the goal is to select the region of the wedgebox plot with the purest event set, the outermost (up and to the right) clearly delineated region is chosen to perform the analysis. Said region will never be due to ¡®stripe¡¯ events — the isolated process will be of the form , where or . The mass value obtained from the analysis will be correct, the only question will be whether it is or . Further input may be required to resolve this uncertainty. Likewise, if the wedgebox plot topology was a single wedge^{20}^{20}20If the wedgebox plot is a single box, then this is virtually certainly , even if in principle one could attribute it to or with all other rates somehow suppressed., then the legs of the wedge could be due to or (in this latter case presumably something in the coupling heavily suppresses the rate for , and so such events make a negligible impact on the wedgebox plot). Again, an analysis of the type presented herein would yield a correct mass determination, only with some ambiguity as to the naming of the particle involved.
Additional improvements in the numerical results are possible if another piece of information is utilized: the actual value^{21}^{21}21It is acknowledged that this runs counter to the statement in the introductory remarks that the edge locations are used purely to delineate the data set and not to provide additional numerical information. However, it would be foolish to completely ignore this extra information that is readily accessible. The aim herein is to emphasize how the wedgebox can be used to purify a data set, not to exclude the use of additional useful information. for the edge of the box, . It is clear that this edge can be measured quite precisely via the wedgebox or the traditional onedimensional triangular mass distribution, yielding a relation between the three masses in the equation:
(23) 
This additional information enables one to scan for only and , with the mass calculated by solving the equation above.
Proceeding somewhat gradually, first consider applying an analysis incorporating to the neutralino events from Zone 3, but with no detector effects and with the simulation coopted to only include correct lepton placements in Fig. 6. The result, shown in Fig. 9, indicates that the desired physical mass is indeed given quite accurately by the point of steepest descent for .
Continuing on to the more realistic Zone 3 analysis including detector smearing and possible lepton placement errors leads to the result shown in Fig. 10. Clearly results are worse than in Fig. 9: the point of the steepest descent of is still in the neighborhood of the physical mass (there is in fact a local maximum quite close to the correct value, but it is not the global maximum indicated by the triangle), but it is obviously not a sharp maximum in the solutions space.
One reason for persisting inaccuracy is difficulty in applying the momentum conservation relation (20) — primarily due to detector smearing effects. With a simulation, one is able to compare the (’parton level’) net transverse momentum of the two neutralinos produced by the generator (prior to their decay), labeled as , to the sum of the momenta of the four leptons in the signal plus the missing momentum, designated as . If and actually match the quantities on the left and righthand sides of Eqns. (20), respectively, then they should be equal. However, as illustrated by Fig. 11, this is not the case. The difference arises from the detector smearing^{22}^{22}22Particle resolutions are as given in a previous footnote. of the lepton momenta (in addition the smearing of the momenta of the other observed particles alters the value calculated for the missing momentum^{23}^{23}23Another possible source of error is if the missing momentum is not solely due to the two LSP’s produced in the neutralino decays. There could also be SM neutrinos produced in some events. With cascade production, the initial gluinos or squarks will lead to quark jets with significant in addition to the desired neutralino pair. Decays of heavyflavored quarks within these quark jets may well yield such neutrinos, especially if heavyflavored sparticle production or decays of gluinos into heavyflavored quarks is enhanced, as is expected in some scenarios Kadala:2008uy (). How such neutrinos might affect a CMSTstyle analysis is currently under investigation WIP1 ().). The range of the imbalance between the experimentallymeasurable value of and the desired value of is . This is formidable in light of the fact that a small inaccuracy in (say ) can change the number of solutions at a given mass point by 10%.
If smearing effects could be eliminated, so that would essentially be equal to , fits to masses would be quite good (easily within 1%). Unfortunately, actual experiments cannot know how large the imbalance between and is in any given event: the best that can be done is to perform a scan over this uncertainty for each event, taking for instance . This is done in making Fig. 12, whose maximum does lie somewhat closer to the actual LSP mass; however, since the correct value of was assumed in this 1D projection, the result is somewhat better than what would be obtained in practice. The fitted LSP mass in Fig. 12 has a small error but with a slightly upbiased central value (primarily due to the fact that the neutralino pairs do not have a fixed CM energy).
4 Discussion & Conclusions
Use of a wedgebox plot to select the most homogeneous sample of events has been shown to increase the accuracy and efficacy of the NMST and CMST mass reconstruction methods. If the events analyzed do not mostly share the same decay topology, both methods fail. A wedgebox analysis can help ascertain whether or not this is the case. If the wedgebox is a simple box, then a mass reconstruction analysis can proceed with confidence^{24}^{24}24It is true that at the Snowmass Benchmark point SPS1a Allanach:2002nj (), a simple box does describe the wedgebox topology of a few processes studied at this point. But these SPS1a studies Gjelsten:2004ki (); Gjelsten:2005aw (); Gjelsten:2005sv (); Gjelsten:2005vv (); Gjelsten:708246 (); Gjelsten:2006as (); Tovey:2008ui (); Nojiri:2003tu () are certainly not representative of many other perfectly allowable parameter set choices and/or signature selections. There seems to have been some tendency to exaggerate, or at least extrapolate in an unsubstantiated way, the proven usefulness of various approaches. However, over much of the allowable MSSM parameter space the topology of the wedgebox plot is not merely a lone box — if a wedge or composite structure is observed, then selecting events from the legs of the wedge or the outer areas generally proves the most effective. The outermost (more to the right, more toward the top) ‘incorporated¡¯ portions of a wedgebox plot basically yield the purest data set. Here ‘incorporated portion¡¯ refers to a clearly delineated zone in the plot in agreement with the predictions of the underlying model. Thus, in Fig. 7 for instance, straggler events beyond , which are not due to neutralino decays, may be eliminated. Choosing the outermost portion must be weighed against the falling number of events populating such regions (recall though that here MST requirements are relatively modest, and the need for purity generally dominates). The triangular distribution of events within the dilepton mass distributions imply that an adequatelypopulated portion of the wedgebox plot will be welldelineated along its outer edges — which may be taken in this MSSM analysis as indicative that the events in this region are largely from neutralino processes rather than from other process.
Note that selecting events from the legs of a wedge runs counter to the choices made in all the NMST and CMST publications. In Nojiri:2007pq () for instance, some care is taken to describe the desirability of socalled ‘symmetric events’ — where both legs from the original parent particle contain the same intermediate particle states. The present work, on the other hand, makes the case that the benefits from using unsymmetric decay legs, e.g. efficient isolation of events with the same decay chain structure, may well trump the convenience of symmetric events in the subsequent MST analysis, and therefore unsymmetric events should not be ignored or viewed as an unnecessary complication.
The time scale required to collect a sufficient number of events to generate a wedgebox diagram is roughly comparable to that needed to perform an MST analysis. This is in spite of the fact that the wedgebox technique relies upon populating scatter plots while an MST analysis in principle only requires collection of enough events to simultaneously solve the requisite equations. In practice, ambiguity in assigning the leptons and multiple solutions to the resulting quartic equation (see bulleted items in Secn. 2 ) as well as experimental factors (also enumerated earlier) necessitate a far larger sample of events to perform either of the MST analyses discussed. Further, and even more compellingly, without augmentation by the wedgebox technique^{25}^{25}25And/or some other methodology yielding comparable information. The wedgebox technique does offer easily interpretable results, and the diparticle invariant mass distributions the technique relies upon have events which cluster near the endpoints, i.e., they have triangular distributions, which aid in obtaining clear results. Here it is perhaps worth noting that other invariant mass combinations or functions proposed which seem in theory to differentiate among event types are effectively of little use if the key region of the distribution is not adequately populated. The authors are unaware of other proposed equivalent techniques specifically designed for ascertaining which MSSM production and decay modes are represented in an experimental data set culled by excluded SM event types., applying an MST analysis to a quite limited data set is tantamount to wild speculation as to what SUSY channels are actually present and the results of such an analysis must be viewed most cautiously.
Scanning over the CM in a window can also enhance the data analysis. Assumptions that the partonic CM has no transverse momentum (as implied by equations (9) and (20) ) are basically incorrect; while in the NMST method this does not seem to matter, the CMST method is much more sensitive to this parameter. An ‘averaging’ over improves the result, but perhaps a more detailed analysis should eventually be performed as the latest set of structure functions and other knowledge of QCD becomes available.
The MST analyses presented here also assume that the decay chain involved is a series of twobody decays via intermediate onmassshell sleptons. This need not be the case, and the onmassshell assumption should be tested. This however is beyond the purview of the wedgebox technique. The dilepton distribution shapes for on and offshell decays are not identical Bachacou:1999zb (); Bisset:2005rn (), and this could be used to distinguish between the two possibilities; however, the effects of cuts, backgrounds and a finitesized data set must be considered. Ref. Chouridou:688033 () notes that distribution shapes for onshell (sequential twobody decays) and offshell (threebody decays) are effectively indistinguishable for some parameter choices. Also, Ref. Kitano:2006gv () finds that the shape of the dilepton distribution may be affected by the nature of the neutralinos (the extent to which they are gauginos or higgsinos), illustrating how dynamical issues arising from the nature of the coupling involved in a decay may not be separable from purely kinematic issues associated with the relevant masses.
So an alternative to a straightforward examination of the dilepton distribution shape is desirable Lester:2006cf (). The Decay Kinematic (DK) technique Kersting:2009ne (); Kang:2009sk (); Kang:2010hb () might offer such an alternative wherein crosscorrelating different invariant mass distributions resolves the onshell vs. offshell issue, though further studies are warranted. Another idea was put forward in Peskin:2008nw ()^{26}^{26}26See pages 50–51 therein., wherein a rudimentary sketch of a very Dalitzesque technique to look for the presence of twobody decay chains is presented — a realistic study applying this idea would be interesting. Refs. Lester:2006cf (); Lester:2005je () instead champion a ‘Markov chain’ approach to analyzing the event sample where “no assumption is made about the processes causing the observed endpoints.” Supposedly then the issue of whether the sleptons are onmassshell or offmassshell is rendered moot to the more modest goal of determining a region of parameter space consistent with the data in a nonMST analysis^{27}^{27}27Though suggested, this issue is not explored in any detail in either of these works..
The present work may be thought of as an early installment of a much grander programme to fuse all known kinematic mass reconstruction methods together. The makeup of this programme consists not only of combining fits from different methods for a static event sample set, but also of improving the composition of the event set under consideration. In the present work where the wedgebox technique is used to select (to the degree possible) events due to a specific neutralino pair^{28}^{28}28Situations in which may also be amenable to such analyses.. Once a fairly homogeneous event sample is obtained^{29}^{29}29An alternative track is attempted in Ref. Lester:2005je (), wherein the idea is to deal with all of the complexity of a mixed data set in the mass analysis program, rather than bifurcate the analysis into a purifying stage and then an analysis stage. The inherent weakness of this approach is that results from studying just the simplest subset of the events are impeded by the need to disentangle more confusing events. then it becomes straightforward to apply various mass reconstruction methods and crosscheck them. For example, in the case of Higgs boson decay considered here, one could try matching the NMST results presented here to those from a study of 4lepton invariant mass endpoints Huang:2008qd () — it would be especially instructive to compare results at the SPS1a parameter point, for example, where both techniques, in the total ignorance of sparticle masses, give poor results individually, but may give a stronger result in unisonWIP2 ().
An MST analysis is then an attractive option if enough massshell conditions can be found to match the number of mass components of the invisible finalstate particles, as is the case for the LSPgenerating SUSY decay chains considered herein. Further though, the present work shows how endpoint information funneled through the wedgebox technique can positively supplement such an MST analysis, as in the augmented CMST study presented in Secn. 3. No massreconstruction technique is immune from possessing potentially faulty assumptions, and so coupling several complementary analysis techniques will in general improve reliability as well as accuracy^{30}^{30}30Also minimizing overlapping information content between the analysis components will increase efficiency. Whether or not this is a significant issue would depend on how cpuintensive the techniques are and on the computing resources available..
Likewise, consideration of suitable inclusive variables, such as the variableLester:1999tx (); Barr:2003rg (); Lester:2007fq () and its variants Barr:2002ex (); Serna:2008zk (), to augment either an MST study Nojiri:2007pq () or an endpoint analysis Allanach:2000kt (); Tovey:2003ef (); Borjanovic:2005tv (); Tovey:2008ui (); Nojiri:2008hy (); Cho:2007dh (); Cho:2007qv (); Gripaios:2007is (); Barr:2007hy (); Ross:2007rm () has been shown to be beneficial, at least in some cases. Then there is the entire array of dynamical (and thus modeldependent) information associated with crosssections and the shapes(spread) of events plotted against one(or more) parameters. As noted earlier, kinematics can never really be totally divorced from the present dynamics. MSSM/mSUGRA studies combining information from crosssections Lester:2005je () or distribution shapes Ross:2007rm (); Gjelsten:2006tg (); Lester:2006yw () with that from an endpoint analysis have also been performed and are no doubt the vanguard of many more such studies, at least if initial LHC results prove favorable. And, when those first major blocks of data from the LHC become available, application of numerous analysis techniques — including the wedgebox techniques, would be a good idea.
5 Acknowledgements
This work was supported in part by the National Natural Science Foundation of China Grant No. 10875063 to MB and RL.
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