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if dark matter is produced at a hadron collider , its likely signature will be missing transverse momentum . to help determine the underlying origin of the observed dark - matter , it is important to measure the masses of the new particle states . mass determination is a key part to identifying the underlying theory which lies beyond the standard model . newly discovered particles could be kaluza - klein ( kk ) states from extra - dimensions , supersymmetric partners of known states , technicolor hadrons , or something else that we have not anticipated . models predict relationships between parameters : supersymmetry relates the couplings of current fermions to the couplings of new bosons and the supersymmetric particle masses reflect the origin of supersymmetry breaking ; masses of kk states tell us about the size of the extra dimensions . in general , mass determination of new particle states is central to discerning what lies beyond the standard model . if dark - matter particle states are discovered at the lhc , how will the masses of these new particles be measured ? determining the mass of the dark - matter particle ( or other states which decay to dark matter ) will be difficult because we do nt expect the dark - matter particles to leave tracks or have a displaced vertex , and because we do not know the rest frame of the initial parton collision .    if one has confidence about the specific model and process responsible for the observed events , then the magnitude of the process s cross - section constrains the mass . if one does not yet know the model , then model - independent techniques will be needed to determine properties of the newly discovered particles such as couplings , spin , phases , mixing angles , and masses . proposed methods for finding the gluino phase @xcite or spin @xcite , rely on our ability to make model - independent kinematic mass measurements . thus , model determination needs a suite of model - independent tools to provide initial constraints on the mass . there has been much recent work in developing model - independent mass determination tools . edges in invariant mass combinations provide information about mass differences or mass squared differences @xcite . if the hadron collider accesses many new states , then we may be able to determine the masses by combining the relationships provided by many different invariant - mass edges of a cascade decay @xcite . there is also a series of approaches called mass shell techniques ( mst ) where one uses an assumption about the topology and on - shell conditions to solve for the unknown masses . one mst variant assumes a long symmetric cascade decay chain and counts which multiplets of masses have solution to the most events @xcite , another assumes the masses in two events must be equal @xcite , and another hybrid combines a mst with the information from the many edges in cascade decays @xcite . variable is a simple example of such a hybrid technique . ] there has been extensive work using the so - called ` stransverse mass ' variable , @xmath6 , to determine the mass _ difference _ between parent particle s and a dark - matter candidate particle s mass given an assumed mass for the dark - matter candidate ( refs . @xcitehave more than 45 citations ) . if the final decay to the lightest state involves a three - body decay - body decay is a sufficient but not necessary condition . two - body decays can also display kinks @xcite provided the decaying particles have sufficiently large transverse boosts . ] as shown in fig . [ figeventtopology ] , then a ` kink ' in the @xmath2@xcite , will occur at a position which indicates the invisible particle mass , as described by cho , choi kim , park ( cckp ) @xcite and corroborated in refs . the ` kink ' in @xmath2 can be quantified by the constrained mass variable @xmath0 that we introduced in a previous letter @xcite . even in models where new invisible particles are nearly massless ( like the gravitino studied in @xcite ) , one would rather not just assume the mass of the lightest supersymmetric particle ( lsp ) , which is needed as an input to the traditional @xmath6 analysis , without measuring it in some model independent way . which approach turns out to be best is likely to depend on what scenario nature hands us , since the various techniques involve different assumptions . having different approaches also offers the advantage of providing a system of redundant checks .    in ref . @xcite , we introduced the @xmath0 kinematic variable which gives an event - by - event lower bound on the dark - matter particle s absolute mass given the mass difference between the dark matter candidate and its parent . in this paper , we introduce a complementary variable @xmath1 which gives an event - by - event _ upper _ bound on the same absolute mass . our study shows that this technique rivals other invisible - particle mass determination techniques in precision and accuracy . in this paper , we provides a demonstration of the use of the variable @xmath0 and @xmath1 in lhc conditions . the variables @xmath7 and @xmath8 give an event - by - event lower - bound and upper - bound respectively on the mass of @xmath9 assuming the topology in fig .  [ figeventtopology ] and the mass difference @xmath10 . to get the mass difference , we use events where @xmath9 decays into @xmath11 and two visible states via a three - body decay in which we can easily determine the mass difference from the end point of the visible - states invariant - mass distribution , @xmath12 . one might also conceive of a situation with @xmath0 supplementing an alternative technique that gives a tight constraint on the mass difference but may have multiple solutions or a weaker constraint on the mass scale @xcite@xcite . given this mass difference and enough statistics , @xmath0 s endpoint gives the mass of @xmath9 . however the main advantage of the @xmath0 method is that it does not rely simply on the position at the endpoint but it uses the additional information contained in events which lie far from the endpoint . as a result it gives a mass determination using significantly fewer events and is less sensitive to energy resolution and other errors . to illustrate the method , in this paper we study in detail the performance of the @xmath0 constrained mass variable in a specific supersymmetric model . we study events where each of the two branches have decay chains that end with a @xmath13 decaying to a @xmath14 and a pair of opposite - sign same - flavor(ossf ) leptons . thus the final states of interest contain four isolated leptons ( made up of two ossf pairs ) and missing transverse momentum . fig  [ figeventtopology ] defines the four momentum of the particle states with @xmath15 , @xmath16 , and the ossf pairs forming the visible particles @xmath17 . any decay products early in the decay chains of either branch are grouped into @xmath18 which we generically refer to as upstream transverse momentum ( utm ) . nonzero @xmath18 could be the result of initial state radiation ( isr ) or decays of heavier particles further up the decay chain . events with four leptons and missing transverse momentum have a very small standard - model background . to give a detailed illustration of the @xmath0 methods , we have chosen to analyze the benchmark point p1 from @xcite which corresponds to msugra with @xmath19 gev , @xmath20 gev , @xmath21 , @xmath22 , @xmath23 . our susy particle spectrum was calculated with isajet @xcite version  7.63 . we stress that the analysis technique employed applies generically to models involving decays to a massive particle state that leaves the detector unnoticed . a powerful feature of the @xmath0 distribution is that , with some mild assumptions , the shape away from the endpoint can be entirely determined from the unknown mass scale and quantities that are measured . the ideal shape fit against early data therefore provides an early mass estimate for the invisible particle . this study is meant to be a guide on how to overcome difficulties in establishing and fitting the shape : difficulties from combinatoric issues , from differing energy resolutions for the leptons , hadrons , and missing transverse momentum , from backgrounds , and from large upstream transverse momentum ( utm ) . as we shall discuss , utm actually provides surprising benefits . the paper is structured as follows : in section [ secub ] , we review @xmath0 and introduce the new observation that , in addition to an event - by - event lower bound on @xmath24 , large recoil against utm enables one also to obtain an event - by - event _ upper _ bound on @xmath24 . we call this quantity @xmath1 . section [ secmodeling ] describes the modeling and simulation employed . section [ secshapefactors ] discusses the implications of several effects on the shape of the distribution including the @xmath25 ( in our case @xmath26 ) distribution , the utm distribution , the backgrounds , combinatorics , energy resolution , and missing transverse momentum cuts . in section [ secperformance ] , we put these factors together and estimate the performance . we conclude in section [ secconclusions ] with a discussion about the performance in comparison to previous work . we will now review the definition of @xmath0 as providing an event - by - event lower bound on @xmath24 . in generalizing this framework , we find a new result that one can also obtain an upper bound on the mass @xmath24 when the two parent particles @xmath9 recoil against some large upstream transverse momentum @xmath27 . fig [ figeventtopology ] gives the relevant topology and the momentum assignments . the visible particles @xmath28 and @xmath29 and invisible particle @xmath11 are labeled with with momentum @xmath30 and @xmath31 ( which we group into @xmath32 ) and @xmath33 , respectively @xmath34 and @xmath35 in the other branch . we assume that the parent particle @xmath9 is the same in both branches so @xmath36 . any earlier decay products of either branch are grouped into the upstream transverse momentum ( utm ) 4-vector momentum , @xmath18 . we assume the two decay chains share a common end - state given in this diagram . all previous decay products are grouped into the upstream transverse momentum , @xmath18.,width=288 ]    in previous work with events of this topology , we @xcite showed how to find an event - by - event lower bound on the true mass of @xmath37 and @xmath24 . we assume that the mass difference @xmath38 can be accurately measured from the invariant mass edges @xmath39 or @xmath40 . for each event , the variable @xmath0 is the minimum value of the mass of @xmath9 ( the second lightest state ) after minimizing over the unknown division of the missing transverse momentum @xmath41 between the two dark - matter particles @xmath11 : @xmath42 there are eight unknowns corresponding to the four momentum components of @xmath33 and @xmath35 and five equations of constraint . because the true @xmath33 and @xmath35 must be within the domain over which we are minimizing , the function is guaranteed to be less than or equal to @xmath24 . one way of calculating @xmath0 for an event is to use @xmath43 @xcite , which provides a lower bound on the mass of @xmath9 for an assumed mass @xmath44 of @xmath11 . the true mass of @xmath9 lies along the line @xmath45 where we use @xmath46 to denote the possible masses of @xmath9 and to distinguish it from the true mass of @xmath9 denoted with @xmath24 . thus we can see that for @xmath44 to be compatible with an event , we must have @xmath47 .    for a given event , if one assumes a mass @xmath44 for @xmath11 , and if the inequality @xmath48 is satisfied , then there is no contradiction , and the event is compatible with this value of @xmath44 . if however , @xmath49 , then we have a contradiction , and the event excludes this value @xmath44 as a viable mass of @xmath11 . using this observation , @xmath0 can be found for each event by seeking an intersection between @xmath43 and @xmath50 @xcite . the lower bound on @xmath24 is given by @xmath51 where @xmath52 is a zero of @xmath53 in the case @xmath54 , the extreme events analyzed in cckp @xcite demonstrate that @xmath55 will only have one positive zero or no positive zeros , and the slope at a zero will always be negative . for no positive zeros , the lower bound is the trivial lower bound given by @xmath56 . note that a lower bound on the value of @xmath24 corresponds to a lower bound on the value of @xmath37 . the appendix in ref . @xcite shows that at the zeros of @xmath57 which satisfy eq([eqgprime ] ) , the momenta satisfy eqns([eqc1]-[eqc4 ] ) . if there is large upstream transverse momentum ( utm ) ( @xmath58 ) against which the system recoils , then we find a new result . using the @xmath6 method to calculate @xmath0 gives one the immediate ability to see that @xmath24 can also have an upper bound when requiring eqns([eqc1]-[eqc4 ] ) . this follows because for large utm the function @xmath57 may have two zeros has more than two zeros , but we have not encountered such cases in our simulations . ] which provides both an upper and a lower bound for @xmath24 from a single event . we have also found regions of parameter space where @xmath55 has a single zero but @xmath59 corresponding to an upper bound on the true mass of @xmath37 ( and @xmath24 ) and only the trivial lower bound of @xmath60 . we can can obtain some insight into the cases in which events with large utm provides upper bounds on the mass by studying a class of extreme event with two hard jets , @xmath61 and @xmath62 against which @xmath9 recoils ( @xmath63 ) . we will describe this extreme event and solve for the regions of parameter space for which one can analytically see the intersection points giving a lower bound and/or an upper bound . the event is extremal in that @xmath43 , which gives a lower bound on @xmath24 , actually gives the true value of @xmath24 when one selects @xmath44 equal to the true mass @xmath37 . the ideal event we consider is where a heavier state @xmath64 is pair - produced on shell at threshold . for simplicity we assume the lab - frame is the collision frame . assume that the @xmath64s , initially at rest , decay into visible massless jets @xmath61 , @xmath62 and the two @xmath9 states with the decay product momenta @xmath65 and @xmath66 . both jets have their momenta in the same transverse plane along the negative @xmath67-axis , and both @xmath9 s momentum are directed along the @xmath67-axis . finally , in the rest frame of the two @xmath9s , both decay such that the decay products visible states have their momentum @xmath68 and @xmath69 along the @xmath67-axis and both invisible massive states @xmath11 have their two momenta along the negative @xmath67-axis . in the lab frame , the four - vectors are given by @xmath70 for the event given by eqns([eqevent1a]-[eqevent1c ] ) , we can exactly calculate @xmath43 : @xmath71 this is found by calculating the transverse mass for each branch while assuming @xmath44 to be the mass of @xmath11 . the value of @xmath72 is chosen so that the transverse masses of the two branches are equal . substituting this value back into the transverse mass of either branch gives @xmath43 . fig [ figextrememt2 ] shows @xmath57 , given in eq([eqgchi ] ) , for several choices of @xmath73 for the process described by eqs([eqevent1a]-[eqevent1c ] ) with @xmath74 and @xmath75 . because @xmath64 is the parent of @xmath9 , we must have @xmath76 . if @xmath77 , then @xmath78 is larger than @xmath79 up until their point of intersection at @xmath80 . in this case their point of intersection provides a lower bound as illustrated by the dotted line in fig . [ figextrememt2 ] for the case with @xmath81 . for @xmath82 there are two solutions @xmath83 when @xmath84 , the function @xmath57 has only one zero with the lower bound equalling the upper bound at @xmath37 . the solid line in fig . [ figextrememt2 ] shows this case . between @xmath85 we again have two solutions but this time with @xmath86 the dashed line in fig . [ figextrememt2 ] shows this case with @xmath87 . for @xmath73 greater than this , we have @xmath88 and @xmath89 . shows @xmath57 for the extreme event in eq([eqevent1a]-[eqevent1c ] ) with @xmath74 and @xmath75 . the dotted line has @xmath81 and shows an event providing a lower bound on @xmath24 . the dashed line @xmath87 and shows an event with both a lower bound and an upper - bound on @xmath24 . the the solid line shows @xmath84 where the lower bound equals the upper bound.,width=384 ]    this example illustrates how @xmath0 can provide both a lower - bound and an upper - bound on the true mass for those events with large utm . the upper - bound distribution provides extra information that can also be used to improve early mass determination , and in what follows we will refer to the upper bound as @xmath1 . we now move on to discuss modeling and simulation of this new observation . as a specific example of the application of the @xmath0 method , we have chosen a supersymmetry model msugra , @xmath19 gev , @xmath20 gev , @xmath21 , @xmath22 , @xmath23 . from @xcite which we also used in @xcite . ] the spectrum used in the simulation has @xmath90 and @xmath91 . we have employed two simulation packages . one is a mathematica code that creates the ` ideal ' distributions based only on very simple assumptions and input data . the second is herwig  @xcite which simulates events based on a susy spectrum , mssm cross sections , decay chains , and appropriate parton distribution functions . if the simple mathematica simulator predicts ` ideal ' shapes that agree with herwig  generator , then one has reason to believe that the all the relevant factors relating to the shape are identified in the simple mathematica simulation . this is an important check in validating the benefits of fitting the @xmath0 and @xmath1 distribution shape as a method to measure the mass of new invisible particles produced at hadron colliders . our ` ideal ' distributions are produced from a home - grown monte carlo event generator written in mathematica . this generater serves to ensure that we understand the origin of the distribution shape . it also ensures that we have control over measuring the parameters needed to determine the mass without knowing the full model , coupling coefficients , or parton distribution functions . we also use this simulation to determine on what properties the ideal distributions depends . the simulator is used to create events satisfying the topology shown in fig  [ figeventtopology ] for a set of specified masses . we take as given the previously measured mass difference @xmath92 , which we use in all our simulations . we neglect finite widths of the particle states as most are in the sub gev range for the model we are considering . we neglect spin correlations between the two branches . we perform the simulations in the center - of - mass frame because @xmath0 and @xmath1 are transverse observables and are invariant under longitudinal boosts . the collision energy @xmath93 is distributed according to normalized distribution @xmath94 unless otherwise specified . the @xmath13 is produced with a uniform angular distribution , and all subsequent decays have uniform angular distribution in the rest frame of the parent . the utm is simulated by making @xmath27 equal to the utm with @xmath95 ( unless otherwise specified ) , and boosting the other four - vectors of the event such that the total transverse momentum is zero . as we will show , these simple assumptions capture the important elements of the process . being relatively model independent , they provide a means of determining the mass for various production mechanisms . if one were to assume detailed knowledge of the production process , it would possible to obtain a better mass determination by using a more complete simulation like herwigto provide the ` ideal ' distributions against which one compares with the data . here we concentrate on the more model independent simulation to demonstrate that it predicts the @xmath0 and @xmath1 distributions well - enough to perform the mass determination that we demonstrate in this case - study . the @xmath0 and @xmath1 distributions of herwig  events before smearing ( to simulate detector resolution ) is applied . the distributions end - points show @xmath96 gev . the top thick curve shows the net distribution , the next curve down shows the contribution of only the signal events , the bottom dashed curve shows the contribution of only background events.,title="fig:",width=297 ]  the @xmath0 and @xmath1 distributions of herwig  events before smearing ( to simulate detector resolution ) is applied . the distributions end - points show @xmath96 gev . the top thick curve shows the net distribution , the next curve down shows the contribution of only the signal events , the bottom dashed curve shows the contribution of only background events.,title="fig:",width=297 ]    in order to obtain a more realistic estimate of the problems associated with collision data , we generate samples of unweighted inclusive supersymmetric particle pair production , using the herwig  monte carlo program with lhc beam conditions . these samples produce a more realistic simulation of the event structure that would be obtained for the supersymmetric model studied here , including the ( leading order ) cross sections and parton distributions . it includes all supersymmetric processes and so contain the relevant background processes as well as the particular decay chain that we wish to study . charged leptons ( @xmath97 and @xmath98 ) produced in the decay of heavy objects ( susy particles and @xmath99 and @xmath100 bosons ) were selected for futher study provided they satisfied basic selection criteria on transverse momentum ( @xmath101  gev ) and pseudorapdity ( @xmath102 ) . leptons coming from hadron decays are usually contained within hadronic jets and so can be experimentally rejected with high efficiency using energy or track isolation criteria . this latter category of leptons was therefore not used in this study . the acceptance criterion used for the hadronic final state was @xmath103 . the detector energy resolution functions used are described in section  [ sec : detector ] . there are several factors that control or affect the shape of the @xmath0 and @xmath1 distributions . we divide the factors into those that affect the in - principle distribution and the factors that affect the observation of the distribution by the detector like energy resolution and selection cuts . the in - principle distribution of these events is influenced by the presence or absence of spin - correlations between the branches , the the @xmath26 distribution of the visible particles , any significant upstream transverse momentum ( utm ) against which the system is recoiling ( _ e.g. _ gluinos or squarks decaying further up the decay chain ) , and background coming from other new - physics processes or the standard model . as all these processes effectively occur at the interaction vertex , there are some combinatoric ambiguities . these are the factors that influence the in - principle distribution of events that impinges on the particle detector . the actual distribution recorded by the detector will depend on further factors . some factors we are able to regulate  for example cuts on the missing transverse momentum . other factors depend on how well we understand the detector s operation  such as the energy resolution and particle identification .    where the effect of such factors is significant , for example for the @xmath25 , @xmath27 , and background distributions , our approach has been to model their effect on the ideal distributions by using appropriate information from the ` data ' , much as one would do in a real lhc experiment . for the present our ` data ' are provided by herwig , rather than lhc events , but the principle is the same .      * * mass difference and mass scale *    the end - point of @xmath0 and @xmath1 distributions give the mass of @xmath13 . therefore the mass scale , @xmath104 , is a dominant factor in the shape of the ` ideal ' distribution . this is the reason we can use these distributions to determine the mass scale . fig [ figlotsofidealcurves ] shows the @xmath0 and @xmath1 distributions for five choices of @xmath104 assuming the herwig  generated @xmath26 and utm distributions . we show the @xmath0 and @xmath1 ideal distributions for five choices of @xmath104 assuming the herwig  generated @xmath26 and utm distributions.,width=384 ]    how does the shape change with mass scale ? the shape is typically sharply peaked at @xmath105 followed by a tail that ends at the mass of @xmath104 . the peak at @xmath56 is due to events that are compatible with @xmath106 . we say these events give the trivial constraint . because we bin the data , the height of the first bin depends on the bin size . as @xmath107 becomes larger , then the non - trivial events are distributed over a wider range and the endpoint becomes less clear . in general if all other things are equal , the larger the mass , the more events in the first bin and a longer flatter tail . the distribution also depends on the mass difference @xmath108 which we assume has been determined . we expect that experimentally one should be able to read off the mass difference from the @xmath26 kinematic end - point with very high precision . gjelsten , miller , and osland estimate this edge can be measured to better than @xmath109 @xcite using many different channels that lead to the same edge , and after modeling energy resolution and background . errors in the mass determination propagated from the error in the mass difference in the limit of @xmath110 are given approximately by @xmath111 where @xmath112 is the error in the determination of the mass difference @xmath108 . an error in @xmath108 will lead to an @xmath0 distribution with a shape and endpoint above or below the true mass in the direction indicated by eq([eqdeltammerroreffects ] ) .    to isolate this source of error from the uncertainty in the fit , we assume that the mass difference is known exactly in our stated results . in our case an uncertainty of @xmath113 would lead to an additional @xmath114 to be added in quadrature to the error from fitting .    * * spin correlations *    there are no spin correlation effects relevant to @xmath0 if the @xmath13 pairs are directly produced and the @xmath13 three - body decay is dominated by the @xmath100 boson ( _ e.g _ the sleptons are much heavier that @xmath13 ) @xcite . there are also no spin correlations if the @xmath13 parents are part of a longer decay chain which involves a scalar at some stage . in the model we are studying , there are enough vertices between the two @xmath13 decays that any correlation is very likely washed out , and we can treat their decays as uncorrelated . in the simple mathematica simulations , we have assumed no spin dependence in the production of the hypothetical ideal distribution .    * * input @xmath25 distributions *    the @xmath26 distribution affects the @xmath0 distribution . fig [ figmlldep ] shows two @xmath26 distributions and the corresponding @xmath0 distributions with @xmath110 ( no utm ) . the solid lines show the case where the three - body decay from @xmath13 to @xmath14 is completely dominated by a @xmath100 boson . the dashed line shows the case where the @xmath26 distribution is extracted from the ` realistic ' herwig  simulation . we can see that the @xmath0 distribution is affected most strongly in the first several non - zero bins . if we were to determine the mass only from the shape of these first several bins using only the @xmath100 contribution for the @xmath26 difference , we would estimate of the mass to be about @xmath115 gev below the true mass . this can be understood because the the shape change of the @xmath26 distribution effectively took events out of the first bin and spread them over the larger bins simulating the effect of a smaller mass . dependence of @xmath0 distribution on the @xmath26 distribution with @xmath54 . * left : * the @xmath26 distributions . * right : * the corresponding @xmath0 distributions . the solid curves show the case where the @xmath26 distribution when the three - body decay is dominated by the @xmath100 boson channel , and the dashed curves show the case where the @xmath26 distribution is taken directly from the herwig  simulation.,title="fig:",width=288 ]   dependence of @xmath0 distribution on the @xmath26 distribution with @xmath54 . * left : * the @xmath26 distributions . * right : * the corresponding @xmath0 distributions . the solid curves show the case where the @xmath26 distribution when the three - body decay is dominated by the @xmath100 boson channel , and the dashed curves show the case where the @xmath26 distribution is taken directly from the herwig  simulation.,title="fig:",width=288 ]    * * input upstream transverse momentum distribution *    as we discussed in section  [ secub ] , if there is a large upstream transverse momentum ( utm ) against which the two @xmath13 s recoil , then we have both an upper and lower bound on the mass scale . the left frame of fig .  [ figm2cisrideal ] shows the utm distribution observed in the ` realistic ' herwig  data . the right frame of fig . [ figm2cisrideal ] shows the @xmath0 and @xmath1 distributions for fixed utm ( @xmath27 ) of @xmath116 , @xmath117 , @xmath118 , @xmath119 , @xmath120 , and @xmath121 @xmath122 all with @xmath95 . as we discuss under the next bullet , we also find the distribution is not sensitive to the value of @xmath123 . for @xmath124 , these curves begin to approach a common shape . these are ideal @xmath0 upper and lower bound distributions where @xmath125 and @xmath126 . notice that there is no upper - bound curve for the case with zero @xmath27 utm . the utm makes the distribution have a sharper endpoint and thereby make the mass easier to determine . this is equivalent to having a sharper kink in @xmath2 in the presence of large utm @xcite . how do we determine @xmath27 from the data ? because we demand exactly four leptons ( two ossf pairs ) , we assume all other activity , basically the hadronic activity , in the detector is utm . the shape used in the ` ideal ' distribution is a superposition of the different fixed utm distributions , shown on right frame of fig . [ figm2cisrideal ] , weighted by the observed utm distribution , shown on the left frame of fig . [ figm2cisrideal ] . equivalently , we obtain the ideal distribution by selecting @xmath27 in the mathematica monte carlo according to the observed utm distribution . * left : * the utm distribution observed in the herwig  simulation . * right : * ideal @xmath0 upper bound and lower bound distribution for a range of upstream transverse momentum ( utm ) values ( @xmath127 ) where @xmath128 gev and @xmath129 gev . , title="fig:",width=297 ]   * left : * the utm distribution observed in the herwig  simulation . * right : * ideal @xmath0 upper bound and lower bound distribution for a range of upstream transverse momentum ( utm ) values ( @xmath127 ) where @xmath128 gev and @xmath129 gev . , title="fig:",width=297 ]    * * shape largely independent of parton distributions and collision energy *    in the limit where there is no utm , then @xmath0 is invariant under back - to - back boosts of the parent particles ; therefore , @xmath0 is also invariant to changes in the parton distribution functions . how much of this invariance survives in the presence of large utm ? the answer is that it remains largely independent of the parton collision energy and largely independent of the mass @xmath123 as shown in fig [ figdiff ] numerically . on the left frame , we show three distributions and in the right frame their difference with @xmath130 error bars calculated from @xmath131 events . the first distribution assumes @xmath132 , @xmath95 , @xmath93 distributed via @xmath133 . the second distribution assumes @xmath132 , @xmath134 , @xmath93 distributed via eq([eqsdep ] ) . the third distribution assumes @xmath132 , @xmath95 , and a fixed collision energy of @xmath135 . shows that even with large utm , the distribution is independent of @xmath123 and the parton collision energy . shown are three distributions and their difference calculated from @xmath131 events . ( 1 ) @xmath132 , @xmath95 , @xmath93 distributed via eq([eqsdep ] ) . ( 2 ) @xmath132 , @xmath134 , @xmath93 distributed via eq([eqsdep ] ) . ( 3 ) @xmath132 , @xmath95 , @xmath135.,title="fig:",width=297 ]   shows that even with large utm , the distribution is independent of @xmath123 and the parton collision energy . shown are three distributions and their difference calculated from @xmath131 events . ( 1 ) @xmath132 , @xmath95 , @xmath93 distributed via eq([eqsdep ] ) . ( 2 ) @xmath132 , @xmath134 , @xmath93 distributed via eq([eqsdep ] ) . ( 3 ) @xmath132 , @xmath95 , @xmath135.,title="fig:",width=297 ]    * * backgrounds *     the invariant mass of the ossf leptons from both branches of forming a dalitz - like wedgebox analysis . the events outside the @xmath136 signal rectangle provide control samples from which we estimate the background shape and magnitude . the dark events are signal , the lighter events are background.,width=480 ]    backgrounds affect the shape and , if not corrected for , could provide a systematic error in the estimated mass . in section [ secperformance ] we will see that the position of the minimum @xmath137 in a fit to @xmath138 is barely affected by the background . the main effect of the background is to shift the parabola up , giving a worse fit . to improve the fit , we may be able to estimate the @xmath0 and @xmath1 distribution and magnitude of the background from the data itself . we first discuss the sources of background , and then we describe a generic technique using a dalitz - like wedgebox analysis to estimate a background model which gives approximately the correct shape and magnitude of the background .    one reason we study the four - lepton with missing transverse momentum channel is because of the very - low standard - model background @xcite . a previous study @xcite estimates about @xmath139 standard - model four - lepton events ( two ossf pairs ) for @xmath140 with a @xmath141 gev cut . they suggest that we can further reduce the standard - model background by requiring several hadronic jets . because we expect very - little direct @xmath13 pair production , this would have very little effect on the number of signal events . also , because @xmath100s are a part of the intermediate states of these background processes , very few of these events will have @xmath26 significantly different from @xmath142 . what is the source of these standard - model backgrounds ? about @xmath143 is from @xmath100-pair production events with no invisible decay products , in which the missing transverse momentum can only arise from experimental particle identification and resolution errors . this implies that a slightly stronger @xmath41 cut could further eliminate this background . another @xmath144 are due to @xmath145 production . not explicitly discussed in their study but representing another possible souce of backgrounds are events containing heavy baryons which decay leptonically . if we assume @xmath146-quark hadrons decay to isolated leptons with a branching ratio of @xmath147 , then lhc @xmath148 production will lead to about @xmath149 events passing these cuts for @xmath140 where both ossf leptons pairs have @xmath150 . tau decays also provide a background for our specific process of interest . the process @xmath151 will be misidentified as @xmath152 or @xmath153 about @xmath154 of the time . because the @xmath155 decays introduce new sources of missing transverse momentum ( @xmath156 ) , these events will distort the @xmath0 calculation . this suggests that the dominant background to the @xmath157 hadrons will be from other susy processes . we now create a crude background model from which we estimate the magnitude and distribution of the background using the ` true ' herwig  data as a guide . we follow the suggestion of ref . @xcite and use a wedgebox analysis plotting the invariant mass @xmath158 against @xmath159 to supplement our knowledge of the background events mixed in with our signal events . this wedgebox analysis , seen in fig .  [ figmlldalitzlike ] for our herwig  simulation , shows patterns that tell about other susy states present . the presence of the strips along @xmath160 gev indicate that particle states are being created that decay to two leptons via an on - shell @xmath100 . the observation that the intensity changes above and below @xmath161 gev shows that many of the states produced have one branch that decays via a @xmath13 and the other branch decays via an on - shell @xmath100 . the lack of events immediately above and to the right of the @xmath162 coordinate but below and to the left of @xmath163 coordinate suggest that symmetric process are not responsible for this background . we also see the density of events in the block above @xmath164 and to the right of @xmath164 suggest a cascade decay with an endpoint near enough to @xmath165 that it is not distinguishable from @xmath142 . following this line of thinking , we model the background with a guess of an asymmetric set of events where one branch has new states @xmath64 , @xmath166 and @xmath11 with masses such that the @xmath26 endpoint is @xmath167 and the other branch is our @xmath13 decay . the masses one chooses to satisfy this edge did not prove important so long as the mass differences were reasonably sized ; we tried several different mass triplets ending with the lsp , and all gave similar answers . we now describe the background model used in our fits . one branch starts with a massive state with @xmath168 which decays to a lepton and a new state @xmath169 which in - turn decays to to a lepton and the lsp . the second branch has our signal decay with the @xmath13 decaying to @xmath14 and two leptons via a three - body decay . we added utm consistent with that observed in the events .    by matching the number of events seen outside the @xmath170 region , we estimate the number of the events within the signal cuts that are due to backgrounds . we estimate @xmath171 of the events with both ossf pairs satisfying @xmath170 are background events . the model also gives a reasonable distribution for these events . inspecting the actual herwig  results showed that actual fraction of background events was @xmath172 . if we let the fraction be free and minimize the @xmath137 with respect to the background fraction , we found a minimum at @xmath173 . our background model is simplistic and does not represent the actual processes , but it does a good job of accounting for the magnitude and the shape of the background mixed into our signal distribution . most of the herwig  background events came from @xmath99 and charginos which introduce extra sources of missing transverse momentum . never the less , the shape fit very accurately and the performance is discussed in section [ secperformance ] . it is encouraging that our estimate of the background shape and magnitude is relatively insensitive to details of the full spectrum . even ignoring the background , as we will see in section [ secperformance ] , still leads to a minimum @xmath137 at the correct mass .    * * combinatoric ambiguities *    if we assume that the full cascade effectively occurs at the primary vertex ( no displaced vertices ) , then the combinatoric question is a property of the ideal distribution produced in the collisions . there are no combinatoric issues if the two opposite - sign same - flavor lepton pairs are each different flavors . however if all four leptons are the same flavor , we have found that we can still identify unique branch assignments 90% of the time . the unique identification comes from the observation that both pairs must have an invariant mass @xmath26 less than the value of the @xmath174 edge . in @xmath175 of the events , there is only one combination that satisfies this requirement . this allows one to use @xmath176 of the four lepton events without ambiguity . the first @xmath177 are identified from the two ossf pairs being of different flavors and @xmath175 of the remaining can be identified by requiring both pairs satisfy @xmath178 on which branch . the events which remain ambiguous have two possible assignments , both of which are included with a weight of @xmath179 in the distribution . as just described , the ` ideal ' in - principle distribution is created from the observed @xmath26 distribution and the observed utm distribution . we include combinatoric effects from events with four leptons of like flavors . last , we can estimate the magnitude of background events and their @xmath0 and @xmath1 shape . we now modify the in - principle distribution to simulate the effects of the particle detector to form our final ` ideal ' distribution that includes all anticipated effects . the two main effects on the @xmath0 and @xmath1 distributions are the energy resolution and the @xmath41 cuts .    * * shape dependence on energy resolution *    energy resolution causes the @xmath0 and @xmath1 distributions to be smeared . here we assume the angular resolution is negligible . for both the mathematica monte carlo model and the herwig  events we simulate the detector s energy resolution by scaling the four vectors for electrons , muons , and hadrons by @xmath180 respectively @xcite@xcite . a more detailed detector simulation is of course possible , but since we do not know the true behavior of any lhc detector until the device begins taking data , a more sophisticated treatment would be of limited value here . in practice the dependence of the ideal distribution shapes on the missing transverse momentum resolution should reflect the actual estimated uncertainty of the missing transverse momentum of the observed events . smearing of the distributions decreases the area difference between two normalized distributions , thereby decreasing the precision with which one can determine the mass from a given number of signal events . this expanded uncertainty can be seen in section  [ secperformance ] . the @xmath0 calculations depend on the mass difference , the four - momenta of the four leptons , and the missing transverse momentum . as the lepton energy resolution is very tight , the missing transverse momentum s energy resolution is dominated by the hadronic energy resolution . we model the energy resolution of the utm as a hadronic jet . this significantly increases the uncertainty in the missing transverse momentum because hadrons have about five times the energy resolution error .    in our mathematica model , we represent the utm as a single four - vector @xmath18 , but in reality it will be the sum of many four - vectors . because we apply the energy resolution smearing to @xmath18 , if @xmath18 is small the simple mathematica model will have a smaller missing energy resolution . however , an events with almost @xmath116 utm could have a large missing momentum energy resolution if it has a lot of hadronic jets whose transverse momentum mostly cancels . fig [ figm2cisrideal ] shows that most of the time we have considerable hadronic utm , so this effect is a minor correction on our results .    * * shape dependence on missing transverse momentum cuts *    a key distinguishing feature of these events is missing transverse momentum . to eliminate the large number of standard model events with four - lepton and with no @xmath41 , we will need to cut on this parameter . [ figptcutdependence ] shows the herwig  simulation s missing transverse momentum versus the @xmath0 . a non - trivial @xmath0 requires substantial @xmath41 . small @xmath41 of less than about @xmath181 only affects the @xmath0 shape below about @xmath182 . the shape of the @xmath183 therefore will require a higher fidelity model from which to train the shapes . instead , we just choose to not fit bins with @xmath184 . all events near the end of @xmath1 distribution require significant @xmath41 , therefore @xmath41 cuts will not affect the part of this distribution which we fit . the number of events with no non - trivial upper - bounds will also be affected by @xmath41 cuts . we only fit the @xmath1 distribution up to about @xmath185 .     the missing transverse momentum vs @xmath0 values for herwig  data . this shows that a @xmath186 cut would not affect the distribution for @xmath187.,width=528 ] the result of @xmath137 fits to the data with differing assumptions for @xmath3 ( left pane ) and @xmath188 ( right pane ) . the thick line with filled squares shows the final result with all cuts , resolution error , combinatorics , and backgrounds included and estimated in the shape fitting . this gives us @xmath189 with @xmath190 events ( signal or background ) representing @xmath3 . after @xmath188 this improves to @xmath191 . the error - free best case gives @xmath192 . the correct value is @xmath90 . , title="fig:",width=297 ]   the result of @xmath137 fits to the data with differing assumptions for @xmath3 ( left pane ) and @xmath188 ( right pane ) . the thick line with filled squares shows the final result with all cuts , resolution error , combinatorics , and backgrounds included and estimated in the shape fitting . this gives us @xmath189 with @xmath190 events ( signal or background ) representing @xmath3 . after @xmath188 this improves to @xmath191 . the error - free best case gives @xmath192 . the correct value is @xmath90 . , title="fig:",width=297 ]    determining the mass based on the shape of the distribution enables one to use all the events and not just those near the end point . we fit both upper - bound and lower - bound shapes to the data as described in the appendix [ appendixchisqfitting ] . as one expects , fitting the lower - bound shape more tightly constrains the mass from below and fitting the upper - bound shape more tightly constrains the mass from above . combining the two gives approximately even uncertainty . we calculate ideal distributions assuming @xmath138 at five values @xmath193 , @xmath194 , @xmath195 , @xmath196 , and @xmath197 gev . we then fit a quadratic interpolation through the points . our uncertainties are based on the value where @xmath137 increases by @xmath28 from its minimum of this interpolation . this uncertainty estimate agrees with about @xmath198 of the results falling within that range after repeated runs . our uncertainty estimates do not include the error propagated from the uncertainty in the mass difference ( see eq([eqdeltammerroreffects ] ) ) . we present results for an early lhc run , about @xmath199 , and for the longest likely lhc run before an upgrade , about @xmath200 . after about @xmath201 , we have @xmath190 events ( about @xmath4 signal and @xmath202 background ) . after @xmath200 , we have about @xmath203 events ( about @xmath204 signal and @xmath205 background ) . only @xmath115 events out of @xmath204 are from direct pair production . most of our signal events follow at the end of different decay chains starting from gluinos or squarks . the upstream decay products produce significant utm against which the two @xmath13 parent particles recoil . first for the ideal case . after @xmath206 , using only signal events and no energy resolution , the @xmath137 fits to the predicted shapes give @xmath192 ( filled circles in fig . [ figperformanceplots ] ) . this mass determination can practically be read off from the endpoints seen in fig . [ figherwigresults ] ; the @xmath0 endpoint is near @xmath207 and subtracting the mass differences gives @xmath208 . we now explore how well we can do with fewer events and after incorporating the effects listed in section [ secshapefactors ] . how does background affect the fit ? if we ignore the existence of background in our sample , and we fit all the events to the signal - only shapes , then we find a poor fit shown as the empty circle curve in fig . [ figperformanceplots ] . by poor fit , we mean the @xmath137 is substantially larger than the @xmath209 bins being compared ( @xmath210 bins from each the upper - bound and lower - bound distributions ) . despite this worse fit , the shape fits still give a very accurate mass estimate : @xmath211 after @xmath199 and @xmath212 after @xmath200 . at this stage , we still assume perfect energy resolution and no missing transverse momentum cut . next , if we create a background model as described in section  [ secshapefactors ] , we are able to improve the @xmath137 fit to nearly @xmath28 per bin ; the mass estimate remains about the same , but the uncertainty increases by about @xmath213 . we find a small systematic shift ( smaller than the uncertainty ) in our mass prediction as we increase the fraction of the shape due to the background model vs the signal model . as we increased our fraction of background , we found the mass estimate was shifted down from @xmath214 at @xmath215 background to @xmath216 when we were at @xmath217 background . the best @xmath137 fit occurs with @xmath218 background ; which is very close to the @xmath219 we use from the estimate , but farther from the true background fraction of about @xmath220 . with @xmath188 of data , the systematic errors are all but eliminated with the endpoint dominating the mass estimate . these fits are shown as the triangles with dashed lines and give @xmath221 which after the full run becomes @xmath222 .    including energy resolution as described in section [ secshapefactors ] shows a large increase in the uncertainty . the dashed - line with empty square markers shows the @xmath137 fit when we include both a background model and the effect of including energy resolution . these fits are shown as the empty squares with dashed lines and give @xmath223 which after the full run becomes @xmath224 .    the final shape factor that we account for are the cuts associated with the missing transverse momentum . after we apply cuts requiring @xmath225 and fit only @xmath187 we have our final result shown by the thick lines with filled squares . this includes all cuts , resolution error , combinatorics and backgrounds . we find @xmath189 with @xmath190 events ( signal or background ) representing @xmath3 , and after @xmath188 this improves to @xmath191 . the true mass on which the herwig  simulation is based is @xmath226 , so all the estimates are within about @xmath227 of the true mass .     herwig  data for @xmath3 ( thick line ) and the smooth ideal expectation assuming @xmath228 generated by mathematica with all resolution , background , and combinatoric effects included ( thin line ) . the @xmath137 of this curve to the herwig  gives the solid - black square on the left frame of fig . [ figperformanceplots].,width=384 ]    fig [ figherwigwithexpectation ] shows the ideal curve expected if @xmath228 including all effects from energy resolution , background , combinatoric , and @xmath41 cuts . the @xmath137 corresponds to the solid square on the left pane of fig . [ figperformanceplots ] . the error in mass determination obtained with limited statistics can be estimated using poisson statistics . in our studies we find that , as one would expect , increasing the number of events by a factor of four , we bring down our error by about a factor of two . this means that one could expect @xmath229 after about @xmath230 which represents @xmath231 signal events and @xmath117 background events . despite adding some of the complicating effects one would encounter with real data , we have discovered other factors which demonstrate one could obtain an even better precision than we originally reported in ref . there , we used only our simple mathematica model that assumed @xmath54 and neglected most sources of realistic uncertainty . we assumed all the events could be modeled as being direct production without spin - correlations . with these simplifications , we argued the mass could be determined to @xmath232 using @xmath233 signal events .    in this paper , we performed a case study to show that that the relevant @xmath0 and @xmath1 shapes can be successfully determined from the mass difference @xmath56 , the @xmath26 distribution observation , the upstream transverse momentum ( utm ) distribution observation . we included and accounted for many realistic effects : we modeled the large energy - resolution error of hadronic jets . we also included the effects of backgrounds , @xmath41 cuts , and combinatorics . our signal and backgrounds were generated with herwig . we discussed how a dalitz - like plot can estimate the background fraction and shape . observed inputs were used in a simple model to determine ideal distribution shapes that makes no reference to parton distribution functions , cross sections , or other model - dependent factors that we are not likely to know early in the process .    despite these extra sources of uncertainty , we found a final mass determination of @xmath234 with about @xmath4 signal events which is still better than the appropriately scaled result from ref . the sources of the mass determination improvement are twofold : ( 1 ) the prediction and fitting of upper - bound distribution , and ( 2 ) the sharper end - point in the presence of large utm . under equivalent circumstances , the sharper endpoint is enough to give a factor of @xmath29 improvement in the uncertainty over the direct production case assumed in @xcite . fitting the upper bound tends to improve the determination by an additional factor of @xmath235 . this improvement is then used to fight the large hadronic - jet energy resolution and background uncertainty . mass determination using @xmath0 and @xmath1 applies to many other processes . we have focused on cases where the mass difference is given by the end - point of an @xmath26 distribution involving a three - body decay . if there is not a three - body decay , then the mass difference may be found by applying other mass determination techniques like the mass shell techniques ( mst ) @xcite or edges in cascade decays @xcite or @xmath6 at different stages in symmetric decay chains @xcite . how does our method s performance compare to previous mass determination methods ? firstly , this technique is more robust than the @xmath2 ` kink ' because in fitting to the shape of the distribution , it does not rely entirely on identification of the events near kinematic boundary . one can view @xmath0 and @xmath1 as variables that event - by - event quantify the ` kink ' . other than the ` kink ' technique , the previous techniques surveyed in the introduction apply to cases where there is no three - body decay from which to measure the mass difference directly . however , each of those techniques still constrains the mass difference with great accuracy . the technique of @xcite which uses edges from cascade decays determines the lsp mass to @xmath236 with about 500 thousand events from @xmath237 . the approach of @xcite assumes a pair of symmetric decay chains and assumes two events have the same structure . they reach @xmath238 using @xmath190 signal events after @xmath237 , but have a @xmath239 systematic bias that needs modeling to remove . by comparison , adjusting to @xmath190 signal events we achieve @xmath240 without a systematic bias after propagating an error of @xmath109 in the mass difference and with all discussed effects . uncertainty calculations differ amongst groups , some use repeated trial with new sets of monte carlo data , and others use @xmath137 . without a direct comparison under like circumstance , the optimal method is not clear ; but it is clear that fitting the @xmath0 and @xmath1 distributions can determine the mass of invisible particles at least as well , if not better than the other known methods in both accuracy and precision .    in summary , we have developed a mass determination technique , based on the constrained transverse mass , which is able to determine the mass of a dark - matter particle state produced at the lhc in events with large missing transverse momentum . the @xmath0 method , which bounds the mass from below , was supplemented by a new distribution @xmath1 which bounds the mass from above in events with large upstream transverse momentum . a particular advantage of the method is that it also obtains substantial information from events away from the end point allowing for a significant reduction in the error . the shape of the distribution away from the end - point can be determined without detailed knowledge of the underlying model , and as such , can provide an early estimate of the mass . once the underlying process and model generating the event has been identified the structure away from the end - point can be improved using , for example , herwig  to produce the process dependent shape . we performed a case - study simulation under lhc conditions to demonstrate that mass - determination by fitting the @xmath0 and @xmath1 distributions survives anticipated complications . with this fitting procedure it is possible to get an early measurement of the mass - with just 400 signal events in our case study we found we would determine @xmath241 . the ultimate accuracy obtainable by this method is @xmath242 . we conclude that this technique s precision is as good as , if not better than , the best existing techniques . we also want to thank chris lester for very helpful conversations and comments on the manuscript , and laura serna for reviewing the manuscript . ms acknowledges support from the united states air force institute of technology . this work was partly supported by the science and technology facilities council of the united kingdom . the views expressed in this paper are those of the authors and do not reflect the official policy or position of the united states air force , department of defense , or the us government . in order to determine @xmath138 we perform a @xmath243 fit between ideal distributions and the herwig  data . first for definitions : we define @xmath244 as the number of @xmath0 events in the region to be fit , and likewise @xmath245 is the number of @xmath1 events in the region to be fit . the @xmath0 of the events are grouped into bins ; @xmath246 counts the events in the @xmath247 bin . the variable @xmath248 is the normalized @xmath0 distribution of ideal events expected in bin @xmath249 as calculated with an assumed @xmath138 , the measured @xmath56 , the observed @xmath26 distribution , the observed utm distribution , and the appropriate detector simulator . we likewise define the upper bound distribution to be @xmath250 . we also define the background distribution for lower - bound and upper - bound distributions to be @xmath251 and @xmath252 and the fraction of the total events we estimate are from background @xmath253 .    assuming a poisson distribution , we assign an uncertainty , @xmath254 , to each bin @xmath249 given by @xmath255 and likewise for the upper - bound distribution . the second term has been added to ensure an appropriate weighting of bins with very few events that does not bias the fit towards or away from this end - point . in bins with few counts , normal poisson statistics does not apply has a large percent error when used as a predictor of the number of counts @xmath246 when @xmath256 is less than about 5 . ] . the @xmath243 is given by @xmath257 we calculate ideal distributions for @xmath258 . we fit quadratic interplant through the points . the minimum @xmath259 of the interplant is our estimate of @xmath138 . the amount @xmath138 changes for an increase in @xmath243 by one gives our @xmath260 uncertainty , @xmath261 , for @xmath138 @xcite . b.  c. allanach , c.  g. lester , m.  a. parker , and b.  r. webber , _ measuring sparticle masses in non - universal string inspired models at the lhc _ , _ jhep _ * 09 * ( 2000 ) 004 , [ http://xxx.lanl.gov/abs/hep-ph/0007009[hep-ph/0007009 ] ] . b.  c. allanach , j.  p. conlon , and c.  g. lester , _ measuring smuon - selectron mass splitting at the cern lhc and patterns of supersymmetry breaking _ , _ phys . * d77 * ( 2008 ) 076006 , [ http://xxx.lanl.gov/abs/0801.3666[0801.3666 ] ] .        c.  g. lester , _ constrained invariant mass distributions in cascade decays : the shape of the m(qll)-threshold and similar distributions _ , _ phys . _ * b655 * ( 2007 ) 3944 , [ http://xxx.lanl.gov/abs/hep-ph/0603171[hep-ph/0603171 ] ] . b.  k. gjelsten , d.  j. miller , p.  osland , and a.  r. raklev , _ mass determination in cascade decays using shape formulas _ , _ aip conf . * 903 * ( 2007 ) 257260 , [ http://xxx.lanl.gov/abs/hep-ph/0611259[hep-ph/0611259 ] ] . m.  m. nojiri , g.  polesello , and d.  r. tovey , _ a hybrid method for determining susy particle masses at the lhc with fully identified cascade decays _ , _ jhep _ * 05 * ( 2008 ) 014 , [ http://xxx.lanl.gov/abs/0712.2718[0712.2718 ] ] . c.  g. lester and d.  j. summers , _ measuring masses of semi - invisibly decaying particles pair produced at hadron colliders _ lett . _ * b463 * ( 1999 ) 99103 , [ http://xxx.lanl.gov/abs/hep-ph/9906349[hep-ph/9906349 ] ] . a.  j. barr , b.  gripaios , and c.  g. lester , _ weighing wimps with kinks at colliders : invisible particle mass measurements from endpoints _ , _ jhep _ * 02 * ( 2008 ) 014 , [ http://xxx.lanl.gov/abs/0711.4008 [ 0711.4008 ] ] . w.  s. cho , k.  choi , y.  g. kim , and c.  b. park , _ measuring superparticle masses at hadron collider using the transverse mass kink _ , _ jhep _ * 02 * ( 2008 ) 035 , [ http://xxx.lanl.gov/abs/0711.4526[0711.4526 ] ] . s.  moretti , k.  odagiri , p.  richardson , m.  h. seymour , and b.  r. webber , _ implementation of supersymmetric processes in the herwig event generator _ , _ jhep _ * 04 * ( 2002 ) 028 , [ http://xxx.lanl.gov/abs/hep-ph/0204123[hep-ph/0204123 ] ] . g.  marchesini _ et . _ , _ herwig : a monte carlo event generator for simulating hadron emission reactions with interfering gluons . version 5.1 - april 1991 _ , _ comput . phys . * 67 * ( 1992 ) 465508 . m.  bisset , n.  kersting , j.  li , f.  moortgat , and q.  xie , _ pair - produced heavy particle topologies : mssm neutralino properties at the lhc from gluino / squark cascade decays _ phys . j. _ * c45 * ( 2006 ) 477492 , [ http://xxx.lanl.gov/abs/hep-ph/0501157[hep-ph/0501157 ] ] . collaboration , s.  akhmadalev _ et . _ , _ hadron energy reconstruction for the atlas calorimetry in the framework of the non - parametrical method _ , _ nucl . instrum . _ * a480 * ( 2002 ) 508523 , [ http://xxx.lanl.gov/abs/hep-ex/0104002 [ hep - ex/0104002 ] ] .
we develop techniques to determine the mass scale of invisible particles pair - produced at hadron colliders . 
 we employ the constrained mass variable @xmath0 , which provides an event - by - event lower - bound to the mass scale given a mass difference . 
 we complement this variable with a new variable @xmath1 which provides an additional _ upper _ bound to the mass scale , and demonstrate its utility with a realistic case study of a supersymmetry model . 
 these variables together effectively quantify the ` kink ' in the function @xmath2 which has been proposed as a mass - determination technique for collider - produced dark matter . 
 an important advantage of the @xmath0 method is that it does not rely simply on the position at the endpoint , but it uses the additional information contained in events which lie far from the endpoint . 
 we found the mass by comparing the herwig  generated @xmath0 distribution to ideal distributions for different masses . 
 we find that for the case studied , with @xmath3 of integrated luminosity ( about @xmath4 signal events ) , the invisible particle s mass can be measured to a precision of @xmath5 . 
 we conclude that this technique s precision and accuracy is as good as , if not better than , the best known techniques for invisible - particle mass - determination at hadron colliders .