RECO level \sqrt{s}_{min} and subsystem \sqrt{s}_{min}: improved global inclusive variables for measuring the new physics mass scale in missing energy events at hadron colliders

The variable \sqrt{s}_{min} was originally proposed in arXiv:0812.1042 as a model-independent, global and fully inclusive measure of the new physics mass scale in missing energy events at hadron colliders. In the original incarnation of \sqrt{s}_{min}, however, the connection to the new physics mass scale was blurred by the effects of the underlying event, most notably initial state radiation and multiple parton interactions. In this paper we advertize two improved variants of the \sqrt{s}_{min} variable, which overcome this problem. First we show that by evaluating the \sqrt{s}_{min} variable at the RECO level, in terms of the reconstructed objects in the event, the effects from the underlying event are significantly diminished and the nice correlation between the peak in the \sqrt{s}_{min}^{(reco)} distribution and the new physics mass scale is restored. Secondly, the underlying event problem can be avoided altogether when the \sqrt{s}_{min} concept is applied to a subsystem of the event which does not involve any QCD jets. We supply an analytic formula for the resulting subsystem \sqrt{s}_{min}^{(sub)} variable and show that its peak exhibits the usual correlation with the mass scale of the particles produced in the subsystem. Finally, we contrast \sqrt{s}_{min} to other popular inclusive variables such as H_T, M_{Tgen} and M_{TTgen}. We illustrate our discussion with several examples from supersymmetry, and with dilepton events from top quark pair production.


The need for a universal, global and inclusive mass variable
It is generally believed that missing energy signatures offer the best bet for discovering new physics Beyond the Standard Model (BSM) at colliders. This belief is reinforced by the dark matter puzzle -the Standard Model (SM) does not contain a suitable dark matter candidate. If dark matter particles are produced at colliders, they will be invisible in the detector, and will in principle lead to missing energy and missing momentum. However, at hadron colliders the total energy and longitudinal momentum of the event are unknown. Therefore, the production of any invisible particles can only be inferred from an imbalance in the total transverse momentum. The measured missing transverse momentum P T then gives the sum of the transverse momenta of all invisible particles in the event.
Unfortunately, P T is the only measured quantity directly related to the invisible particles. Without any further model-dependent assumptions, it is in general very difficult if not impossible to make any definitive statements about the nature and properties of the missing particles. For example, leaving all theoretical prejudice aside, one would not be able to answer such basic and fundamental questions like [1][2][3][4][5]: How many invisible particles were produced in the event? Are all invisible particles SM neutrinos, or are there any new neutral, stable, weakly-interacting massive particles (WIMPs) among them? What are the masses of the new invisible particles? What are their spins? What are the masses of any (parent) particles which may have decayed to invisible particles?
The recent literature is abundant with numerous proposals 1 on how under particular circumstances one might be able to measure the masses of the invisible particles. Unfortunately, all of the proposed methods suffer from varying degrees of model-dependence 2 : • Limited applicability topology-wise. Most methods are model-dependent in the sense that each method crucially relies on the assumption of a very specific event topology.
One common flaw of all methods on the market is that they usually do not allow any SM neutrinos to enter the targeted event topology, and the missing energy is typically assumed to arise only as a result of the production of (two) new dark matter particles. Furthermore, each method has its own limitations. For example, the traditional invariant mass endpoint methods [10][11][12][13][14][15][16][17][18][19][20] require the identification of a sufficiently long cascade decay chain, with at least three successive two-body decays [21]. The polynomial methods of Refs. [22][23][24][25][26][27][28][29] also require such long decay chains and furthermore, the events must be symmetric, i.e. must have two identical decay chains per event, or else the decay chain must be even longer [21]. The recently popular M T 2 methods [30][31][32][33][34][35][36][37][38][39] do not require long decay chains [21], but typically assume that the parent particles are the same and decay to two identical invisible particles 3 . The limitations of the M CT methods [40][41][42] are rather similar. The kinematic cusp method [43] is limited to the so called "antler" event topology, which contains two symmetric one-step decay chains originating from a single s-channel resonance. In light of all these various assumptions, it is certainly desirable to have a universal method which can be applied to any event topology. To the best of our knowledge, the only such method in the literature is the one proposed in Ref. [1], where the √ s min variable was first introduced. The √ s min variable is defined in terms of the total energy E and 3-momentum P observed in the event, and thus does not make any reference to the actual event topology. In this sense √ s min is a universal variable which can be applied under any circumstances.
• Limited applicability signature-wise. As a rule, most of the proposed methods work well only if the corresponding signature contains some minimum number of high p T isolated leptons. Leptonic signatures have the twofold advantage of lower SM backgrounds and good lepton momentum measurement. The performance of the methods typically deteriorates as we lower the number of leptons in the signature. The most challenging 1 See Ref. [6] for a recent review. 2 Worse still, there are even fewer ideas for measuring the spins of the new particles in a truly modelindependent fashion [7][8][9]. 3 See [3,4] for a more general approach which avoids this assumption.
signature of multijets plus E T has rarely been studied in relation to mass and spin measurements (see, however [33,[44][45][46][47]). Unfortunately, at hadron colliders like the Tevatron and LHC, one typically expects strong production to dominate the new physics cross-sections, and this in turn guarantees the presence of some minimum number of jets in the signature. At the same time, a priori there are no theoretical arguments which would similarly guarantee the presence of any hard isolated leptons. Therefore, one would like to have a general, sufficiently inclusive method, which treats jets and leptons on an equal footing. The √ s min method of Ref. [1] satisfies this requirement as well, since it does not differentiate between the type of reconstructed objects. In fact, the original proposal of Ref. [1] did not require any object reconstruction at all, and used (muon-corrected) calorimeter energy measurements to define the observed E and P in the event.
• Combinatorics problem. Even if one correctly guesses the new physics event topology, and the signature happens to be abundant in hard isolated leptons, one still has to face the usual combinatorics problem of how to properly associate the reconstructed objects with the individual particles in the assumed event topology. Here we shall be careful to make the distinction between two different aspects of the combinatorics problem: -Partitioning ambiguity. As a prototypical example, consider a model of supersymmetry (SUSY) in which R-parity is conserved and the lightest supersymmetric particle (LSP) is neutral and stable. Each SUSY event contains two independent cascade decay chains, so first one must decide which reconstructed objects belong to the first decay chain and which belong to the second [32,48]. However, a priori there are no guiding principles on how to do this partitioning into subsets. The decision is further complicated by the inevitable presence of jets from initial state radiation, which have nothing to do with the SUSY cascades [49]; by final state radiation, which modifies the assumed event topology; and by the occasional overlapping of jets [50].
-Ordering ambiguity. Having separated the objects into two groups, one must still decide on the sequential ordering of the reconstructed objects along each decay chain. One well-known example of this problem is the ambiguity between the "near" and "far" lepton in the standard jet-lepton-lepton squark decay chain [20].
The severity of either one of these two combinatorics problems depends on the type of signature -simple signatures resulting from short decay chains suffer from less combinatorics but tend to have larger SM backgrounds. By the same token, more complex signatures, which result from longer decay chains, are easier to see over the SM backgrounds, but very quickly run into severe combinatorial problems. Thus ideally one would like to have a method which treats all objects in the event in a fully inclusive manner, so that neither of these two combinatorial issues can ever arise at all. The √ s min variable of Ref. [1] was proposed for exactly this reason, and is free of the partitioning and ordering combinatorial ambiguities.
• Limited use of the available experimental information. At hadron colliders, events with invisible particles in the final state present an additional challenge: the total energy and longitudinal momentum of the initial state in the event are unknown. On the other hand, the transverse momentum of the initial state is known, which has greatly motivated the use of transverse variables like the missing "transverse energy" E T , the scalar sum of transverse momenta H T , the transverse mass M T , the stransverse mass M T 2 [30], the contransverse mass M CT [40], etc. An unsettling feature of a purely transverse kinematical approach is that it completely ignores the measured longitudinal momentum components of the visible particles. In principle, the longitudinal momenta also carry a certain amount of information about the underlying physics, although it is difficult to see immediately how this information can be utilized. (For example, one cannot take advantage of longitudinal momentum conservation, because the longitudinal momentum of the initial state is unknown.) By defining the √ s min variable in a manifestly 1+3 Lorentz invariant way, Ref. [1] proposed one possible way to utilize the additional information encoded in the measured longitudinal momenta.
The above discussion makes it clear that the method of the √ s min variable has several unique advantages over all other known methods: it is completely general and universal, is fully inclusive, and to the fullest extent makes use of the available experimental information. In spite of these advantages, the √ s min variable has not yet found wide application. The one major perceived drawback of √ s min is its sensitivity to initial state radiation (ISR) and/or multiple parton interactions (MPI) [1,6,[51][52][53]. To see how this comes about, let us first review the formal definition of √ s min .

Definition of √ s min
Consider the most generic missing energy event topology shown in Fig. 1. As seen from the figure, in defining √ s min , one imagines a completely general setup -each event contains some number n vis of Standard Model (SM) particles X i , i = 1, 2, . . . , n vis , which are visible in the detector, i.e. their energies and momenta are in principle measured. Examples of such visible SM particles are the basic reconstructed objects, e.g. jets, photons, electrons and muons. The visible particles X i are denoted in Fig. 1 with solid black lines and may originate either from ISR, or from the hard scattering and subsequent cascade decays (indicated with the greenshaded ellipse). In turn, the missing transverse momentum P T arises from a certain number n inv of stable neutral particles χ i , i = 1, 2, . . . , n inv , which are invisible in the detector. In general, the set of invisible particles consists of some number n χ of BSM particles (indicated with the red dashed lines), as well as some number n ν = n inv − n χ of SM neutrinos (denoted with the black dashed lines). As already mentioned earlier, the P T measurement alone does not reveal the number n inv of missing particles, nor how many of them are neutrinos and how many are BSM (dark matter) particles. This general setup also allows the identities and the masses m i of the BSM invisible particles χ i , (i = 1, 2, . . . , n χ ) in principle to be different, as in models with several different species of dark matter particles [54][55][56][57]. Of course, the Figure 1: The generic event topology used to define the √ s min variable in Ref. [1]. Black (red) lines correspond to SM (BSM) particles. The solid lines denote SM particles X i , i = 1, 2, . . . , n vis , which are visible in the detector, e.g. jets, electrons, muons and photons. The SM particles may originate either from initial state radiation (ISR), or from the hard scattering and subsequent cascade decays (indicated with the green-shaded ellipse). The dashed lines denote neutral stable particles χ i , i = 1, 2, . . . , n inv , which are invisible in the detector. In general, the set of invisible particles consists of some number n χ of BSM particles (indicated with the red dashed lines), as well as some number n ν = n inv − n χ of SM neutrinos (denoted with the black dashed lines). The identities and the masses m i of the BSM invisible particles χ i , (i = 1, 2, . . . , n χ ) do not necessarily have to be all the same, i.e. we allow for the simultaneous production of several different species of dark matter particles. The global event variables describing the visible particles are: the total energy E, the transverse components P x and P y and the longitudinal component P z of the total visible momentum P . The only experimentally available information regarding the invisible particles is the missing transverse momentum P T . neutrino masses can be safely taken to be zero m i = 0, for i = n χ + 1, n χ + 2, . . . , n inv . (1.1) Given this very general setup, Ref. [1] asked the following question: What is the minimum value √ s min of the parton-level Mandelstam invariant mass variable √ s which is consistent with the observed visible 4-momentum vector P µ ≡ (E, P )? As it turned out, the answer to this question is given by the universal formula [1] √ where the mass parameter M is nothing but the total mass of all invisible particles in the event: and the second equality follows from the assumption of vanishing neutrino masses (1.1). The result (1.2) can be equivalently rewritten in a more symmetric form in terms of the total visible invariant mass M defined as Notice that in spite of the complete arbitrariness of the invisible particle sector at this point, the definition of √ s min depends on a single unknown parameter M -the sum of all the masses of the invisible particles in the event. For future reference, one should keep in mind that transverse momentum conservation at this point implies that The main result from Ref. [1] was that in the absence of ISR and MPI, the peak in the √ s min distribution nicely correlates with the mass threshold of the newly produced particles. This observation provides one generic relation between the total mass of the produced particles and the total mass M of the invisible particles. Based on several SUSY examples involving fully hadronic signatures in symmetric as well as asymmetric topologies, Ref. [1] showed that the accuracy of this measurement rivals the one achieved with the more traditional M T 2 methods.

1.3
√ s min and the underlying event problem At the same time, it was also recognized that effects from the underlying event (UE), most notably ISR and MPI, severely jeopardize this measurement. The problem is that in the presence of the UE, the √ s min variable would be measuring the total energy of the full system shown in Fig. 1, while for studying any new physics we are mostly interested in the energy of the hard scattering, as represented by the green-shaded ellipse in Fig. 1. The inclusion of the UE causes a drastic shift of the peak of the √ s min distribution to higher values, often by as much as a few TeV [1,51,52]. As a result, it appeared that unless effects from the underlying event could somehow be compensated for, the proposed measurement of the √ s min peak would be of no practical value. The main purpose of this paper is to propose two fresh new approaches to dealing with the underlying event problem which has plagued the √ s min variable and prevented its more widespread use in hadron collider physics applications. But before we discuss the two new ideas put forth in this paper, we first briefly mention the two existing proposals in the literature on how to deal with the underlying event problem.
First, it was recognized in Ref. [1] that the contributions from the underlying event tend to be in the forward region, i.e. at large values of |η|. Correspondingly, by choosing a suitable cut |η| < η max , designed to eliminate contributions from the very forward regions, one could in principle restore the proper behavior of the √ s min distribution [1]. Unfortunately, there are no a priori guidelines on how to choose the appropriate value of η max , therefore this approach introduces an uncontrollable systematic error and has not been pursued further in the literature.
An alternative approach was proposed in Refs. [51,52], which pointed out that the ISR effects on √ s min are in principle calculable in QCD from first principles. The calculations presented in Refs. [51,52] could then be used to "unfold" the ISR effects and correct for the shift in the peak of the √ s min distribution. Unfortunately, in this analytical approach, the MPI effects would still be unaccounted for, and would have to be modeled and validated separately by some other means. While such an approach may eventually bear fruit at some point in the future, we shall not pursue it here.
We see that, for one reason or another, both of these strategies appear unsatisfactory. Therefore, here we shall pursue two different approaches. We shall propose two new variants of the √ s min variable, which we label √ s min are unharmed by the effects from the underlying event, thus resurrecting the original idea of Ref. [1] to use the peak in the √ s min distribution as a first, quick, model-independent estimate of the new physics mass scale. In Section 7 we compare the performance of √ s min against some other inclusive variables which are commonly used in hadron collider physics for the purpose of estimating the new physics mass scale. Section 8 is reserved for our main summary and conclusions.

Definition of the RECO level variable
In the first approach, we shall not modify the original definition of √ s min and will continue to use the usual equation (1.2) (or its equivalent (1.4)), preserving the desired universal, global and inclusive character of the √ s min variable. Then we shall concentrate on the question, how should one calculate the observable quantities E, P and P T entering the defining equations (1.2) and (1.4).
The previous √ s min studies [1,51,52] used calorimeter-based measurements of the total visible energy E and momentum P as follows. The total visible energy in the calorimeter E (cal) is simply a scalar sum over all calorimeter deposits where the index α labels the calorimeter towers, and E α is the energy deposit in the α tower. As usual, since muons do not deposit significantly in the calorimeters, the measured E α should first be corrected for the energy of any muons which might be present in the event and happen to pass through the corresponding tower α. The three components of the total visible momentum P were also measured from the calorimeters as where θ α and ϕ α are correspondingly the polar and azimuthal angular coordinates of the α calorimeter tower. The missing transverse momentum can similarly be measured from the calorimeter as (see eq. (1.6)) P T (cal) ≡ − P T (cal) . (2.5) Using these calorimeter-based measurements (2.1-2.5), one can make the identification in the definition (1.2) and construct the corresponding "calorimeter-based" √ s min variable as √ s This was precisely the quantity which was studied in [1,51,52] and shown to exhibit extreme sensitivity to the physics of the underlying event.
Here we propose to evaluate the visible quantities E and P at the RECO level, i.e. in terms of the reconstructed objects, namely jets, muons, electrons and photons 4 . To be precise, let there be N obj reconstructed objects in the event, with energies E i and 3-momenta P i , i = 1, 2, . . . , N obj , correspondingly. Then in place of (2.6-2.8), let us instead identify and correspondingly define a "RECO-level" √ s min variable as which can also be rewritten in analogy to (1.4) as √ s where P T (reco) and P T (reco) are related as in eq. (2.12) and the RECO-level total visible mass What are the benefits from the new RECO-level √ s min definitions (2.13,2.14) in comparison to the old calorimeter-based √ s min definition in (2.9)? In order to understand the basic idea, it is worth comparing the calorimeter-based missing transverse momentum P T (which in the literature is commonly referred to as "missing transverse energy" E T ) and the analogous RECO-level variable H T , the "missing H T ". The H T vector is defined as the negative of the vector sum of the transverse momenta of all reconstructed objects in the event: Then it is clear that in terms of our notation here, H T is nothing but P T (reco) . It is known that H T performs better than E T [58]. First, H T is less affected by a number of adverse instrumental factors such as: electronic noise, faulty calorimeter cells, pile-up, etc. These effects tend to populate the calorimeter uniformly with unclustered energy, which will later fail the basic quality cuts during object reconstruction. In contrast, the true missing momentum is dominated by clustered energy, which will be successfully captured during reconstruction. Another advantage of H T is that one can easily apply the known jet energy corrections to account for the nonlinear detector response. For both of these reasons, CMS is now using H T at both the trigger level and offline [58]. min does not solve the underlying event problem completely and as a matter of principle. Every now and then the underlying event will still produce a well-defined jet, which will have to be included in the calculation of √ s (reco) min . Because of this effect, we cannot any more guarantee that √ s min provides a lower bound on the true value √ s true of the center-of-mass energy of the hard scattering -the additional jets formed out of ISR, pile-up, and so on, will sometimes cause √ s (reco) min to exceed √ s true . Nevertheless we find that this effect modifies only the shape of the √ s (reco) min distribution, but leaves the location of its peak largely intact. To the extent that one is mostly interested in the peak location, √ s (reco) min should already be good enough for all practical purposes.
. . , n sub , originating from within the subsystem, while the remaining n vis − n sub visible particles X n sub +1 , . . . , X nvis are created upstream, outside the subsystem. The subsystem results from the production and decays of a certain number of parent particles P j , j = 1, 2, . . . , n p , (some of) which may decay semi-invisibly. All invisible particles χ 1 , . . . , χ ninv are then assumed to originate from within the subsystem.

Definition of the subsystem variable
In this section we propose an alternative modification of the original √ s min variable, which solves the underlying event problem completely and as a matter of principle. The downside of this approach is that it is not as general and universal as the one discussed in the previous section, and can be applied only in cases where one can unambiguously identify a subsystem of the original event topology which is untouched by the underlying event. The basic idea is schematically illustrated in Fig. 2, which is nothing but a slight rearrangement of Fig. 1 exhibiting a well defined subsystem (delineated by the black rectangle). The original n vis visible particle X i from Fig. 1 have now been divided into two groups as follows: 1. There are n sub visible particles X 1 , . . . , X n sub originating from within the subsystem.
Their total energy and total momentum are denoted by E (sub) and P (sub) , correspond-ingly. The subsystem particles are chosen so that to guarantee that they could not have come from the underlying event.
2. The remaining n vis − n sub visible particles X n sub +1 , . . . , X n vis are created upstream (outside the subsystem) and have total energy E (up) and total momentum P (up) . The upstream particles may originate from the underlying event or from decays of heavier particles upstream -this distinction is inconsequential at this point.
We also assume that all invisible particles χ 1 , . . . , χ n inv originate from within the subsystem, i.e. that no invisible particles are created upstream. In effect, all we have done in Fig. 2 is to partition the original measured values of the total visible energy E and 3-momentum P from Fig. 1 into two separate components as Notice that now the missing transverse momentum is defined as while the total visible invariant mass M (sub) of the subsystem is given by At this point the reader may be wondering what are the guiding principles for categorizing a given visible particle X i as a subsystem or an upstream particle. Since our goal is to identify a subsystem which is shielded from the effects of the underlying event, the safest way to do the partition of the visible particles is to require that all QCD jets belong to the upstream particles, while the subsystem particles consist of objects which are unlikely to come from the underlying event, such as isolated electrons, photons and muons (and possibly identified τ -jets and, to a lesser extent, tagged b-jets).
With those preliminaries, we are now ready to ask the usual √ s min question: Given the measured values of E (up) , E (sub) , P (up) and P (sub) , what is the minimum value √ s (sub) min of the subsystem Mandelstam invariant mass variable √ s (sub) , which is consistent with those measurements? Proceeding as in [1], once again we find a very simple universal answer, which, with the help of (3.3) and (3.4), can be equivalently written in several different ways as follows: where in the last line we have introduced the Lorentz 1+2 vectors As usual, the length of a 1+2 vector is computed as Before we proceed to the examples of the next few sections, as a sanity check of the obtained result it is useful to consider some limiting cases. First, by taking the upstream visible particles to be an empty set, i.e. P T (up) → 0, we recover the usual expression for √ s min given in eqs. (1.2,1.4). Next, consider a case with no invisible particles, i.e. M = 0 and correspondingly, P T = 0. In that case we obtain that √ s (sub) min = M (sub) , which is of course the correct result. Finally, suppose that there are no visible subsystem particles, i.e. E (sub) = P (sub) = M (sub) = 0. In that case we obtain √ s (sub) min = M , which is also the correct answer.
As we shall see, the subsystem concept of Fig. 2 will be most useful when the subsystem results from the production and decays of a certain number n p of parent particles P j with masses M P j , j = 1, 2, . . . , n p , correspondingly. Then the total combined mass of all parent particles is given by By the conjecture of ref. [1], the location of the peak of the √ s (sub) min ( M ) distribution will provide an approximate measurement of M p as a function of the unknown parameter M . By construction, the obtained relationship M p ( M ) will then be completely insensitive to the effects from the underlying event.
At this point it may seem that by excluding all QCD jets from the subsystem, we have significantly narrowed down the number of potential applications of the √ s min variable can prove to be useful in its own right, and in a wide variety of contexts. To see this, note that the typical hadron collider signatures of the most popular new physics models (supersymmetry, extra dimensions, Little Higgs, etc.) are precisely of the form exhibited in Fig. 2. One typically considers production of colored particles (squarks, gluinos, KK-quarks, etc.) whose cross-sections dominate. In turn, these colored particles shed their color charge by emitting jets and decaying to lighter, uncolored particles in an electroweak sector. The decays of the latter often involve electromagnetic objects, which could be targeted for selection in the subsystem. The √ s (sub) min variable would then be the perfect tool for studying the mass scales in the electroweak sector (in the context of supersymmetry, for example, the electroweak sector is composed of the charginos, neutralinos and sleptons).
Before we move on to some specific examples illustrating these ideas, one last comment is in order. One may wonder whether the √ s (sub) min variable should be computed at the RECO-level or from the calorimeter. Since the subsystem will usually be defined in terms of reconstructed objects, the more logical option is to calculate √ s . However, to streamline our notation, in what follows we shall always omit the "reco" part of the superscript and will always implicitly assume that √ s (sub) min is computed at RECO-level.

SM example: dilepton events from tt production
In this and the next two sections we illustrate the properties of the new variables √ s (reco) min and √ s (sub) min with some specific examples. In this section we discuss an example taken from the Standard Model, which is guaranteed to be available for early studies at the LHC. We consider dilepton events from tt pair production, where both W 's decay leptonically. In this event topology, there are two missing particles (two neutrinos). Therefore, these events very closely resemble the typical SUSY-like events, in which there are two missing dark matter particles. In the next two sections, we shall also consider some SUSY examples. In all cases, we perform detailed event simulation, including the effects from the underlying event and detector resolution.

Event simulation details
Events are generated with PYTHIA [59] (using its default model of the underlying event) at an LHC of 14 TeV, and then reconstructed with the PGS detector simulation package [60]. We have made certain modifications in the publicly available version of PGS to better match it to the CMS detector. For example, we take the hadronic calorimeter resolution to be [61] while the electromagnetic calorimeter resolution is [61] σ E where the energy E is measured in GeV, S = 3.63% is the stochastic term, N = 0.124 is the noise and C = 0.26% is the constant term. Muons are reconstructed within |η| < 2.4, and we use the muon global reconstruction efficiency quoted in [61]. We use default p T cuts on the reconstructed objects as follows: 3 GeV for muons, 10 GeV for electrons and photons, and 15 GeV for jets. For the tt example presented in this section, we use the approximate next-to-next-toleading order tt cross-section of σ tt = 894 ± 4 +73+12 −46−12 pb at a top mass of m t = 175 GeV [62]. For the SUSY examples in the next two sections we use leading order cross-sections.
Since our examples are meant for illustration purposes only, we do not include any backgrounds to the processes being considered, nor do we require any specific triggers. A detailed study of the dilepton tt signature including all those effects will appear elsewhere [63].
min variable We first consider SUSY-like missing energy events arising from tt production, where each W -boson is forced to decay leptonically (to an electron or a muon). We do not impose any trigger or offline requirements, and simply plot directly the output from PGS 5 . We show various √ s quantities of interest in Fig. 3, setting M = 0, since in this case the missing particles are neutrinos and are massless. The dotted (yellow-shaded) histogram represents the true √ s distribution of the tt pair. It quickly rises at the tt mass threshold and then eventually falls off at large √ s due to the parton density function suppression. Because the top quarks are typically produced with some boost, the √ s true distribution in Fig. 3 peaks a little bit above threshold: It is clear that if one could directly measure the √ s true distribution, or at least its onset, the tt mass scale will be easily revealed. Unfortunately, the escaping neutrinos make such a measurement impossible, unless one is willing to make additional model-dependent assumptions 6 . 5 Therefore, our plots in this subsection are normalized to a total number of events equal to σ tt × BR(W → e, µ) 2 .
6 For example, one can use the known values of the neutrino, W and top masses to solve for the neutrino kinematics (up to discrete ambiguities). However, this method assumes that the full mass spectrum is already known, and furthermore, uses the knowledge of the top decay topology to perfectly solve the combinatorics problem discussed in the Introduction. As an example, consider a case where the lepton is produced first and the b-quark second, i.e. when the top first decays to a lepton and a leptoquark, which in turn decays to a neutrino and a b-quark. The kinematic method would then be using the wrong on-shell conditions. The advantage of the √ s min approach is that it is fully inclusive and does not make any reference to the actual decay topology.  min variable: the blue (red) histogram is obtained by switching off (on) the underlying event (ISR and MPI). These curves reveal two very interesting phenomena. First, without the UE, the peak of the √ s (cal) min distribution (blue histogram) is very close to the parent mass threshold [1]: The main observation of Ref. [1] was that this correlation offers an alternative, fully inclusive and model-independent, method of estimating the mass scale M p of the parent particles, even when some of their decay products are invisible and not seen in the detector. Unfortunately, the "no UE" limit of eq. (4.5) is unphysical, and the corresponding √ s (cal) min distribution (blue histogram in in Fig. 3) is unobservable. What is worse, when one tries to measure the √ s (cal) min distribution in the presence of the UE (red histogram in Fig. 3), the resulting peak is very far from the physical threshold: In the tt example of Fig. 3, the shift is on the order of 1 TeV! It appears therefore that in practice the √ s (cal) min peak would be uncorrelated with any physical mass scale, and instead would be completely determined by the (uninteresting) physics of the underlying event. Once the nice model-independent correlation of eq. (4.5) is destroyed by the UE, it becomes of only academic value [1,6,[51][52][53].
However, Fig. 3 also suggests the solution to this difficult problem. If we look at the distribution of the √ s (reco) min variable (black solid histogram), we see that its peak has returned to the desired value: √ s thus resurrecting the original proposal of Ref. [1]. In order to measure physical mass thresholds, one simply needs to investigate the distribution of the inclusive √ s (reco) min variable, which is calculated at RECO-level. Each peak in that distribution signals the opening of a new channel, and from (4.7) the location of the peak provides an immediate estimate of the total mass of all particles involved in the production. Of course, the √ s (reco) min distribution is now not as sharply peaked as the unphysical "no UE" case of √ s (cal) min , but as long as its peak is found in the right location, the method of Ref. [1] becomes viable once again.
Our first main result is therefore nicely summarized in Fig. 3, which shows a total of 4 distributions, 3 of which are either unphysical (the blue histogram of √ s min version. As already anticipated in Sec. 2, the basic idea is that energy deposits which are due to hard particles originating from the hard scattering, tend to be clustered, while the energy deposits due to the UE tend to be more uniformly spread throughout the detector. In order to see this pictorially, in Figs. 4 and 5 we show a series of calorimeter maps of the combined ECAL+HCAL energy deposits as a function of the pseudorapidity η and azimuthal angle φ. Since the calorimeter in PGS is segmented in cells of (∆η, ∆φ) = (0.1, 0.1), each calorimeter tower is represented by a square pixel, which is color-coded according to the amount of energy present in the tower. We have chosen the color scheme so that larger deposits correspond to darker colors.
Each calorimeter map figure below has four panels. In the upper two panels the calorimeter is filled at the parton level directly from PYTHIA. This corresponds to a perfect detector, where we ignore any smearing effects due to the finite energy resolution. The lower two plots in each figure show the corresponding results after PGS simulation. Thus by comparing the plots in the upper row to those in the bottom row, one can see the effect of the detector resolution. While the finite detector resolution does play some role, we find that it is of no particular importance for understanding the reason behind the big swings in the √ s min peaks observed in Fig. 3.
Let us instead concentrate on comparing the plots in the left column versus those in the right column. The left plots show the absolute energy deposit E α in the α calorimeter  Figure 4: PGS calorimeter map of the energy deposits, as a function of pseudorapidity η and azimuthal angle φ, for a dilepton tt event with only two reconstructed jets. At the parton level, this particular event has two b-quarks and two electrons. The location of a b-quark (electron, muon) is marked with the letter "q" ("e", "µ"). A grey circle delineates (the cone of) a reconstructed jet, while a green dotted circle denotes a reconstructed lepton. In the upper two plots the calorimeter is filled at the parton level directly from PYTHIA, while the lower two plots contain results after PGS simulation. The left plots show absolute energy deposits E α , while in the right plots the energy in each tower is shown projected on the transverse plane as E α cos θ α . tower, while in the right plots this energy is shown projected on the transverse plane as E α cos θ α . The difference between the left and the right plots is quite striking. The plots on the left exhibit lots of energy, which is deposited mostly in the forward calorimeter cells (at large |η|) [1]. The plots on the right, on the other hand, show only a few clusters of energy, concentrated mostly in the central part of the detector. Those energy clusters give rise to the objects (jets, electrons and photons) which are reconstructed from the calorimeter. Furthermore, each energy cluster can be easily identified with a parton-level particle in the top decay chain. In order to exhibit this correlation, in Figs. 4 and 5 we use the following notation for the parton-level particles: a b-quark (electron, muon) is marked with the letter "q" ("e", "µ"). A grey circle delineates (the cone of) a reconstructed jet, while a green dotted circle marks a reconstructed lepton (electron or muon). The lepton isolation requirement implies that green circles should be void of large energy deposits off-center, and indeed we observe this to be the case.
In particular, Fig. 4 shows a bare-bone dilepton tt event with just two reconstructed jets and two reconstructed leptons (which happen to be both electrons). As seen in the figure, the two jets can be easily traced back to the two b-quarks at the parton level, and there are no additional reconstructed jets due to the UE activity. Because the event is so clean and simple, one might expect to obtain a reasonable value for √ s min , i.e. close to the tt threshold. However, this is not the case, if we use the calorimeter-based measurement √ s (cal) min . As seen in Table 1, the measured value of √ s (cal) min is very far off -on the order of 1 TeV, even in the case of a perfect detector. The reason for this discrepancy is now easy to understand from Fig. 4. Recall that √ s (cal) min is defined in terms of the total energy E (cal) in the calorimeter, which in turn is dominated by the large deposits in the forward region, which came from the underlying event. More importantly, those contributions are more or less equally spread over the forward and backward region of the detector, leading to cancellations in the calculation of the corresponding longitudinal P z(cal) momentum component. As a result, the first term in (2.9) becomes completely dominated by the UE contributions [51].
Let us now see how the calculation of √ s (reco) min is affected by the UE. Since object reconstruction is done with the help of minimum transverse cuts (for clustering and object id), the relevant calorimeter plots are the maps on the right side in Fig. 4. We see that the large forward energy deposits which were causing the large shift in √ s (cal) min are not incorporated into any reconstructed objects, and thus do not contribute to the √ s (reco) min calculation at all. In effect, the RECO-level prescription for calculating √ s min is leaving out precisely the unwanted contributions from the UE, while keeping the relevant contributions from the hard scattering. As seen from Table 1, the calculated value of √ s (reco) min for that event is 363 GeV, which is indeed very close to the tt threshold. It is also smaller than the true √ s value of 427 GeV in that event, which is to be expected, since by design √ s min ≤ √ s, and this event does not have any extra ISR jets to spoil this relation.
It is instructive to consider another, more complex tt dilepton event, such as the one shown in Fig. 5 Fig. 3 indicates that this happens quite often -the tail of the √ s (reco) min distribution is more populated than the (yellow-shaded) √ s true distribution. This should be no cause for concern. First of all, we are only interested in the peak of the √ s (reco) min distribution, and we do not need to make any comparisons between √ s (reco) min and √ s true . Second, any such comparison would be meaningless, since the value of √ s true is a priori unknown, and unobservable.
min variable Before concluding this section, we shall use the tt example to also illustrate the idea of the subsystem √ s (sub) min variable developed in Sec. 3. Dilepton tt events are a perfect testing ground for this idea, since the W W subsystem decays leptonically, without any jet activity. We therefore define the subsystem as the two hard isolated leptons resulting from the decays of the W -bosons. Correspondingly, we require two reconstructed leptons (electrons or muons) at the PGS level 7 , and plot the distribution of the leptonic subsystem √ s (sub) min variable in Fig. 6. As before, the dotted (yellow-shaded) histogram represents the true √ s distribution of the W + W − pair. As expected, it quickly rises at the W W threshold (denoted by the vertical arrow), then falls off at large √ s. Since the √ s (W W ) true distribution is unobservable, the best we can do is to study the corresponding √ s (sub) min distribution shown with the solid black histogram. In this subsystem example, all UE activity is lumped together with the upstream b-jets from the top quarks decays, and thus has no bearing on the properties of the leptonic √ s (sub) min . In particular, we find that the value of √ s (sub) min is always smaller than the true √ s true . More importantly, Fig. 6 demonstrates that the peak in the √ s (sub) min distribution is found precisely at the mass threshold of the particles (in this case the two W bosons) which initiated the subsystem. Therefore, in analogy to (4.7) we can also write √ s is the combined mass of all the parents initiating the subsystem. Fig. 6 shows that in the tt example just considered, this relation holds to a very high degree of accuracy.
This example should not leave the reader with the impression that hadronic jets are never allowed to be part of the subsystem. On the contrary -the subsystem may very well include reconstructed jets as well. The tt case considered here in fact provides a perfect example to illustrate the idea. Let us reconsider the tt dilepton sample, and redefine the subsystem so that we now target the two top quarks as the parents initiating the subsystem. Correspondingly, in addition to the two leptons, let us allow the subsystem to include two jets, presumably coming from the two top quark decays. Unfortunately, in doing so, we must face a variant of the partitioning 8 combinatorial problem discussed in the introduction: as seen in Fig. 7, the typical jet multiplicity in the events is relatively high, and we must therefore specify the exact procedure how to select the two jets which would enter the subsystem. We shall consider three different approaches.
• B-tagging. We can use the fact that the jets from top quark decay are b-jets, while the jets from ISR are typically light flavor jets. Therefore, by requiring exactly two b-tags, and including only the two b-tagged jets as part of the subsystem, we can significantly increase the probability of selecting the correct jets. Of course, ISR will sometimes also contribute b-tagged jets from gluon splitting, but that happens rather rarely and the corresponding contribution can be suppressed by a further invariant mass cut on the two b-jets. The resulting √ s (sub) min distribution for the subsystem of 2 leptons and 2 b-tagged jets is shown in Fig. 8 with the black histogram. We see that, as expected, the distribution peaks at the tt threshold and this time provides a measurement of the  The disadvantage of this method is the loss in statistics: compare the normalization of the black histogram in Fig. 8 after applying the two b-tags, to the dotted (yellowshaded) distribution of the true tt distribution in the selected inclusive dilepton sample (without b-tags).
• Selection by jet p T . Here one can use the fact that the jets from top decays are on average harder than the jets from ISR. Correspondingly, by choosing the two highest p T jets (regardless of b-tagging), one also increases the probability to select the correct jet pair. The corresponding distribution is shown in Fig. 8 with the blue histogram, and is also seen to peak at the tt threshold. An important advantage of this method is that one does not have to pay the price of reduced statistics due to the two additional b-tags.
• No selection. The most conservative approach would be to apply no selection criteria on the jets, and include all reconstructed jets in the subsystem. Then the subsystem √ s (sub) min variable essentially reverts back to the RECO-level inclusive variable √ s (reco) min already discussed in the previous subsection. Not surprisingly, we find the peak of its distribution (red histogram in Fig. 8) near the tt threshold as well.
All three of these examples show that jets can also be usefully incorporated into the subsystem. The only question is whether one can find a reliable way of preferentially selecting jets which are more likely to originate from within the intended subsystem, as opposed to Figure 8: The same as Fig. 6, but in addition to the two leptons, the subsystem now also includes: exactly two b-tagged jets (black histogram); the two highest p T jets (blue histogram); or all jets (red histogram). The dotted (yellow-shaded) histogram gives the true √ s distribution of the tt pair.
from the outside. As we see in Fig. 8, in the tt case this is quite possible, although in general it may be difficult in other settings, like the SUSY examples discussed in the next section.
5. An exclusive SUSY example: multijet events from gluino production Since √ s min is a fully inclusive variable, arguably its biggest advantage is that it can be applied to purely jetty events with large jet multiplicities, where no other method on the market would seem to work. In order to simulate such a challenging case, we consider gluino pair production in supersymmetry, with each gluino forced to undergo a cascade decay chain involving only QCD jets and nothing else. For concreteness, we revisit the setup of Ref. [1], where two different possibilities for the gluino decays were considered: • In one scenario, the gluinog is forced to undergo a two-stage cascade decay to the LSP.
Therefore, gluino pair production will nominally result in 8 jet events. Of course, as shown in Fig. 9, the actual number of reconstructed jets in such events is even higher, due to the effects of ISR, FSR and/or string fragmentation. As seen from the figure, each such event has on average ∼ 10 jets, presenting a formidable combinatorics problem. We suspect that all 9 mass reconstruction methods on the market are doomed if they were to face such a scenario. It is therefore of particular interest to see how well the √ s min method (which is advertized as universally applicable) would fare under such dire circumstances.
• In the second scenario, the gluino decays directly to the LSP via a three-body decaỹ so that gluino pair-production events would nominally have 4 jets and missing energy.
For concreteness, in each scenario we fix the mass spectrum as was done in [1]: we use the approximate gaugino unification relations to relate the gaugino and neutralino masses as We can then vary one of these masses, and choose the other two in accord with these relations. Since we assume three-body decays in (5.2) and (5.1), we do not need to specify the SUSY scalar mass parameters, which can be taken to be very large. In addition, as implied by (5.3), we imagine that the lightest two neutralinos are gaugino-like, so that we do not have to specify the higgsino mass parameter either, and it can be taken to be very large as well. After these preliminaries, our results for these two scenarios are shown in Figs. 10 and 11, correspondingly. In Fig. 10 (Fig. 11) we consider the 8-jet signature arising from (5.1) 9 With the possible exception of the MT gen method of Ref. [32], see Section 7 below.  (the 4-jet signature arising from (5.2)). In both figures, panels (a) correspond to a light mass spectrum mg = 600 GeV, mχ0 2 = 200 GeV and mχ0 1 = 100 GeV; while panels (b) correspond to a heavy mass spectrum mg = 2400 GeV, mχ0 2 = 800 GeV and mχ0 1 = 400 GeV. Each plot shows the same four distributions as in Fig. 3. The  Figure 12: The same as Fig. 4, but for a SUSY event of gluino pair production, with each gluino forced to decay to 4 jets and the LSP as in (5.1). The SUSY mass spectrum is as in Figs. 10(a) and 11(a): mg = 600 GeV, mχ0 2 = 200 GeV and mχ0 1 = 100 GeV. As in Figs. 4 and 5, the circles denote jets reconstructed in PGS, and here "q" marks the location of a quark from a gluino decay chain. Therefore, a circle without a "q" inside corresponds to a jet resulting from ISR or FSR, while a letter "q" without an accompanying circle represents a quark in the gluino decay chain which was not subsequently reconstructed as a jet.
2mg. If the effects of the UE are ignored, the position of this threshold is given rather well by the peak of the √ s (cal) min distribution (blue histogram). Unfortunately, the UE shifts the peak in √ s min (black histogram) is stable against UE contamination, and its peak is still in the right place (near M p ).
Having already seen a similar behavior in the tt example of the previous section, these results may not seem very impressive, until one realizes just how complicated those events are. For illustration, Fig. 12 shows the previously discussed calorimeter maps for one particular "8 jet" event. This event happens to have 11 reconstructed jets, which is consistent with the typical jet multiplicity seen in Fig. 9. The values of the √ s quantities of interest for this event are listed in Table 1. We see that the RECO prescription for calculating √ s min is able to compensate for a shift in √ s of more than 1.5 TeV! A casual look at Fig. 12 should be enough to convince the reader just how daunting the task of mass reconstruction in such events is. In this sense, the ease with which the √ s min method reveals the gluino mass scale in Figs. 10 and 11 is quite impressive.

An inclusive SUSY example: GMSB study point GM1b
In the Introduction we already mentioned that √ s min is a fully inclusive variable. Here we would like to point out that there are two different aspects of the inclusivity property of √ s min : • min makes no reference to any reconstructed objects at all). This makes √ s min a very convenient variable to use in those cases where the newly produced particles have many possible decay modes, and restricting oneself to a single exclusive signature would cause loss in statistics. For illustration, consider the gluino pair production example from the previous section. Even though we are always producing the same type of parent particles (two gluinos), in general they can have several different decay modes, leading to a very diverse sample of events with varying number of jets and leptons. Nevertheless, the √ s (reco) min distribution, plotted over this whole signal sample, will still be able to pinpoint the gluino mass scale, as explained in Sec. 5.
• Event-wise inclusivity: √ s min is inclusive also with regards to the type of events, i.e. the type of new particle production. For simplicity, in our previous examples we have been considering only one production mechanism at a time, but this is not really necessary -√ s min can also be applied in the case of several simultaneous production mechanisms.
In order to illustrate the last point, in this section we shall consider the simultaneous production of the full spectrum of SUSY particles at a particular benchmark point. We chose the GM1b CMS study point [64], which is nothing but a minimal gauge-mediated SUSYbreaking (GMSB) scenario on the SPS8 Snowmass slope [65]. The input parameters are Λ=80 TeV, M mes =160 TeV, N mes =1, tan β = 15 and µ > 0. The physical mass spectrum is given in Table 2. Point GM1b is characterized by a neutralino NLSP, which promptly decays   (predominantly) to a photon and a gravitino. Therefore, a typical event has two hard photons and missing energy, which provide good handles for suppressing the SM backgrounds.
We now consider inclusive production of all SUSY subprocesses and plot the √ s min distributions of interest in Fig. 13. As usual, the dotted yellow-shaded histogram is the true √ s distribution of the parent pair of SUSY particles produced at the top of each decay chain. Since we do not fix the production subprocess, the identity of the parent particles varies from event to event. Naturally, the most common parent particles are the ones with the highest production cross-sections. For point GM1b, at a 14 TeV LHC, strong SUSY production dominates, and is 87% of the total cross-section. A few of the dominant subprocesses and their cross-sections are listed in Table 3.  Table 3: Cross-sections (in pb) and parent mass thresholds (in GeV) for the dominant production processes at the GM1b study point. The listed squark cross-sections are summed over the light squark flavors and conjugate states. The total SUSY cross-section at point GM1b is 9.4 pb.
The true √ s distribution in Fig. 13 exhibits an interesting double-peak structure, which is easy to understand as follows. As we have seen in the exclusive examples from Secs. 4 and 5, at hadron colliders the particles tend to be produced with √ s close to their mass threshold. As seen in Table 2, the particle spectrum of the GM1b point can be broadly divided (according to mass) into two groups of superpartners: electroweak sector (the lightest charginoχ ± 1 , second-to-lightest neutralinoχ 0 2 and sleptons) with a mass scale on the order of 200 GeV and a strong sector (squarks and gluino) with masses of order 700−900 GeV. The first peak in the true √ s distribution (near √ s ∼ 500 GeV) arises from the pair production of two particles from the electroweak sector, while the second, broader peak in the range of √ s ∼ 1500 − 2300 GeV is due to the pair production of two colored superpartners 10 . Each one of those peaks is made up of several contributions from different individual subprocesses, but because their mass thresholds 11 are so close, in the figure they cannot be individually resolved, and appear as a single bump.
If one could somehow directly observe the true √ s SUSY distribution (the dotted yellowshaded histogram in Fig. 13), this would lead to some very interesting conclusions. First, from the presence of two separate peaks one would know immediately that there are two widely separated scales in the problem. Second, the normalization of each peak would indicate the relative size of the total inclusive cross-sections (in this example, of the particles in the electroweak sector versus those in the strong sector). Finally, the broadness of each peak is indicative of the total number of contributing subprocesses, as well as the typical mass splittings of the particles within each sector. It may appear surprising that one is able to draw so many conclusions from a single distribution of an inclusive variable, but this just comes to show the importance of √ s as one of the fundamental collider physics variables. Unfortunately, because of the missing energy due to the escaping invisible particles, the true √ s distribution cannot be observed, and the best one can do to approximate it is to look at the distributions of our inclusive √ s min variables discussed in Sec. 2: the calorimeter-based √ s min (dotted red histogram). We can immediately see the detrimental effects of the UE: first, the electroweak production peak has been almost completely smeared out, while the strong production peak has been 10 The attentive reader may also notice two barely visible bumps (near 950 GeV and 1150 GeV) reflecting the associated production of one colored and one uncolored particle:gχ ± 1 ,gχ 0 2 andqχ ± 1 ,qχ 0 2 , correspondingly. 11 A few individual mass thresholds are indicated by vertical arrows in Fig. 13. Figure 14: The same as Fig. 6, but for the GMSB SUSY example considered in Fig. 13. Here the subsystem is defined in terms of the two hard photons resulting from the twoχ 0 1 →G + γ decays. The vertical arrow marks the onset for inclusiveχ 0 1χ 0 1 production.
shifted upwards by more than a TeV! This behavior is not too surprising, since the same effect was already encountered in our previous examples in Secs. 4 and 5. Fortunately, we now also know the solution to this problem: one needs to consider the RECO-level variable √ s (reco) min instead, which tracks the true √ s distribution much better. We can see evidence of this in Fig. 13  min does show two separate peaks (indicating that SUSY production takes place at two different mass scales), the peaks are in their proper locations (relative to the missing mass scale M ), and have the correct relative width, hinting at the size of the mass splittings in each sector. We thus conclude that all of the interesting physics conclusions that one would be able to reach from looking at the true √ s distributions, can still be made based on the inclusive distribution of our RECO-level √ s (reco) min variable. Before concluding this section, we shall take the opportunity to use the GM1b example to also illustrate the √ s (sub) min variable proposed in Sec. 3. As already mentioned, the GM1b study point corresponds to a GMSB scenario with a promptly decaying Bino-likeχ 0 1 NLSP. Most events therefore contain two hard photons from the twoχ 0 1 decays to gravitinos. Then it is quite natural to define the exclusive subsystem in Fig. 2 in terms of these two photons. We notice that the peak of the √ s (sub) min distribution nicely reveals the location of the neutralino mass threshold, and from there the neutralino mass itself. We see that the method of √ s (sub) min provides a very simple way of measuring the NLSP mass in such GMSB scenarios (for an alternative approach based on M T 2 , see [66]).

Comparison to other inclusive collider variables
Having discussed the newly proposed variables √ s (reco) min and √ s (sub) min in various settings in Secs. 4-6, we shall now compare them to some other global inclusive variables which have been discussed in the literature in relation to determining a mass scale of the new physics. For simplicity here we shall concentrate only on the most model-independent variables, which do not suffer from the topological and combinatorial ambiguities mentioned in the Introduction.
At the moment, there are only a handful of such variables. Depending on the treatment of the unknown masses of the invisible particles, they can be classified into one of the following two categories: • Variables which do not depend on an unknown invisible mass parameter. The most popular members of this class are the "missing H T " variable Here we follow the notation from Sec. 2, where P T i is the measured transverse momentum of the i-th reconstructed object in the event (i = 1, 2, . . . , N obj ). The main advantage of H T and H T is their simplicity: both are very general, and are defined purely in terms of observed quantities, without any unknown mass parameters. The downside of H T and H T is that they cannot be directly correlated with any physical mass scale in a model-independent way 12 .
• Variables which exhibit dependence on one or more invisible mass parameters. As two representatives from this class we shall consider M T gen from Ref. [32] and √ s (reco) min from Sec. 2 here. We shall not repeat the technical definition of M T gen , and instead refer the uninitiated reader to the original paper [32]. Suffice it to say that the method of M T gen starts out by assuming exactly two decay chains in each event. The arising combinatorial problem is then solved by brute force -by considering all possible partitions of the event into two sides, computing M T 2 for each such partition, and taking the minimum value of M T 2 found in the process. Both M T gen and √ s (reco) min introduce a priori unknown parameters related to the mass scale of the missing particles produced in the event. In the case of √ s (reco) min , this is simply the single parameter M , measuring the total invisible mass (in the sense of a scalar sum as defined in eq. (1.3)). The M T gen variable, on the other hand, must in principle introduce two separate missing mass parameters M 1 and M 2 (one for each side of the event). However, the existing applications of M T gen in the literature have typically made the assumption that M 1 = M 2 , although this is not really necessary and one could just as easily work in terms of two separate inputs M 1 and M 2 [3,4]. The inconvenience of having to deal with unknown mass parameters in the case of M T gen and √ s (reco) min is greatly compensated by the luxury of being able to relate certain features of their distributions to a fundamental physical mass scale in a robust, model-independent way. In particular, the upper endpoint M (max) T gen of the M T gen distribution gives the larger of the two parent masses max{M P 1 , M P 2 } [71]. Therefore, if the two parent masses are the same, i.e. M P 1 = M P 2 , then the parent mass threshold M p = M P 1 + M P 2 is simply given by On the other hand, as we have already seen in Secs. 4-6, the peak of the √ s (reco) min is similarly correlated with the parent mass threshold, see eq. (4.7).
In principle, all four 13 of these variables are inclusive both object-wise and event-wise. It is therefore of interest to compare them with respect to: In Fig. 15 we first revisit the case of the dilepton tt sample discussed in Sec. 4. In addition to the true √ s (yellow shaded) and √ s (reco) min (black) distribution already appearing in Fig. 3, we now also plot the distributions of 2M T gen (red dots), H T (green dots) and H T (blue dots), all calculated at the RECO-level. For completeness, in Fig. 15 we also show a variant of M T gen , called M T T gen (magenta dots), where all visible particle momenta are first projected on the transverse plane, before computing M T gen in the usual way [32] 14 . All results include the full simulation of the underlying event. For plotting convenience, the H T distribution is shown scaled down by a factor of 2. 13 We caution the reader that HT is often defined in a more narrow sense than eq. (7.2). For example, sometimes the HT term is omitted, sometimes the sum in eq. (7.2) is limited to the reconstructed jets only; or to the four highest pT jets only; or to all jets, but starting from the second-highest pT one. 14 We caution the reader that the definition of MT T gen cannot be found in the published version of Ref. [32] -the MT T gen discussion was added in a recent replacement on the archive, which appeared more than two years after the original publication. Based on the results from Fig. 15, we can now address the question, which inclusive distribution shows the best correlation with the parent mass scale (in this case the parent mass scale is the tt mass threshold M p = 2m t = 350 GeV marked by the vertical dotted line in Fig. 15). Let us begin with the two variables, H T and H T , which do not depend on any unknown mass parameters. Fig. 15 reveals that the H T distribution peaks very far from threshold, and therefore does not reveal much information about the new physics mass scale. Consequently, any attempt at extracting new physics parameters out of the missing energy distribution alone, must make some additional model-dependent assumptions [72]. On the other hand, the H T distribution appears to correlate better with M p , since its peak is relatively close to the tt threshold. However, this relationship is purely empirical, and it is difficult to know what is the associated systematic error.
Moving on to the variables which carry a dependence on a missing mass parameter, √ s (reco) min , 2M T gen and 2M T T gen , we see that all three are affected to some extent by the presence of the UE. In particular, the distributions of 2M T gen and 2M T T gen are now smeared and extend significantly beyond their expected endpoint (7.3). Not surprisingly, the UE has a larger impact on 2M T gen than on 2M T T gen . In either case, there is no obvious endpoint. Nevertheless, one could in principle try to extract an endpoint through a straight-line fit, for example, but it is clear that the obtained value will be wrong by a certain amount (depending Figure 16: The same as Fig. 15, but for the gluino pair production example from Sec. 5, with each gluino decaying to 4 jets as in (5.1). We use the light SUSY mass spectrum from Fig. 10(a). The vertical dotted line now shows thegg mass threshold M p = 2mg = 1200 GeV. on the chosen region for fitting and on the associated backgrounds). All these difficulties with 2M T gen and 2M T T gen are simply a reflection of the challenge of measuring a mass scale from an endpoint as in (7.3), instead of from a peak as in (4.7). By comparison, the determination of the new physics mass scale from the √ s (reco) min distribution is much more robust. As shown in Fig. 15, the √ s (reco) min peak is barely affected by the UE, and is still found precisely in the right location.
All of the above discussion can be directly applied to the SUSY examples considered in Sec. 5 as well. As an illustration, Figs. 16 and 17 revisit two of the gluino examples from Section 5. In both figures, we consider gluino pair-production with a light SUSY spectrum (mχ0 1 = 100 GeV, mχ0 2 = 200 GeV and mg = 600 GeV). Then in Fig. 16 each gluino decays to 4 jets as in eq. (5.1), while in Fig. 17 each gluino decays to 2 jets as in eq. (5.2). (Thus Fig. 16 is the analogue of Fig. 10(a), while Fig. 17 is the analogue of Fig. 11(a).) The conclusions from Figs. 16 and 17 are very similar. Both figures confirm that H T is not very helpful in determining the gluino mass scale M p = 2mg = 1200 GeV (indicated by the vertical dotted line). The H T distribution, on the other hand, has a nice well-defined peak, but the location of the H T peak always underestimates the gluino mass scale (by about 250 GeV in each case). Figs. 16 and 17 also confirm the effect already seen in Fig. 15: that the underlying event causes the 2M T gen and 2M T T gen distributions to extend well beyond their upper kinematic endpoint, thus violating (7.3) and making the corresponding extraction of M p rather problematic. In fact, just by looking at Figs. 16 and 17, one might be tempted to