Alan J Barr
email: a.barr@physics.ox.ac.uk

Christopher G Lester
email: lester@hep.phy.cam.ac.uk

A Review of the Mass Measurement Techniques proposed for the Large Hadron Collider

We review the methods which have been proposed for measuring masses of new particles at the Large Hadron Collider paying particular attention to the kinematical techniques suitable for extracting mass information when invisible particles are expected.

INTRODUCTION AND SCOPE

This review principally concerns itself with kinematic methods of mass reconstruction, and in particular those that have been considered for use with hadron colliders, notably the Large Hadron Collider (LHC). Specifically, kinematic methods demand that at least some particles are sufficiently close to their mass shells that their energy-momentum Lorentz invariant p μ p μ ≈ m 2 can be used to constrain their masses. Such methods aim to determine, to bound, or to otherwise constrain p in order to learn about m. Assuming momentum and energy conservation, one also can learn about the four-momenta of -and hence constrain the masses of -particles which are not directly observed experimentally. Two important examples are [START_REF] Alitti | A precise determination of the W and Z masses at the CERN pp collider[END_REF] unstable particles which decay and (2) weakly interacting particles which, though stable, do not interact with the detector.

It is of course true that when further information -beyond the purely kinematic -is either known or assumed that one could use that information too. With sufficient theoretical and experimental understanding, and provided the calculation is tractable, one could obtain maximal information about an event by comparing its statistical likelihood under different mass (or spin or other) hypotheses. The ability to numerically marginalise over uncertain information (such as momentum components of invisible particles) has made such calculations computationally feasible. This approach -sometimes called the Matrix Element method -has been employed at the CERN SppS (e.g. [START_REF] Alitti | A precise determination of the W and Z masses at the CERN pp collider[END_REF]), LEP (e.g. [START_REF] Acciarri | Measurement of mass and width of the W boson at LEP[END_REF]) and the Tevatron (e.g. [START_REF] Abazov | Measurement of the top quark mass in the lepton + jets final state with the matrix element method[END_REF]), and proposals exist for the LHC (e.g. [START_REF] Alwall | Measuring sparticles with the matrix element[END_REF]).

Such methods are ideal for making precise statements about parameters when some confidence about the underlying model has been gained. They are however unwieldy in the early stages of an investigation, when there are usually very many interesting hypotheses to test, each with a wide range of allowed parameters. Another non-kinematic method of obtaining information is the investigation of masses with virtual particles, far from their mass shells. A well-known example is the constraint on the Higgs Boson mass (assuming e.g. the Standard Model as the underlying theory) from loop contributions to electroweak observables [START_REF] Aleph | Precision Electroweak Measurements and Constraints on the Standard Model[END_REF].

The main advantage of the kinematic approaches reviewed in this article is that they make very few assumptions about the details of the underlying physical model (gauge groups, spins etc). This means that they can provide rather robust information, and act as a first step towards understanding the underlying theory. (or may not) have proceeded via one or more on-mass-shell intermediates. References to sections should be considered indicative rather than exhaustive.

Outline of the Review

Many, though not all, mass measurement techniques can be broken down into three phases:

(1) the postulation of a hypothesis or hypotheses about the 'decay topology' -by which we mean the sequence of decays which involve the particles whose masses are to be determined, (2) the identification of the most appropriate final-state observables, and (3) the construction of constraints or measurements of the target particle masses, using those observables.

In what remains of the introduction, we say a few more words about these three phases. In Section 1.2 we will outline in more detail what we mean by decay topologies and kinematic hypothesis. Indeed, readers new to the field may find that section a useful starting point from which to determine which mass measurement techniques are relevant to their needs. We will go on to talk in general terms about the basic observables which we believe are in our remit, and will then comment on the nature of the secondary observables which are derived from them and which are subsequently used to build the mass measurement constraints. Subsequently we will note some of the issues that present recurrent challenges faced by mass determination methods of all kinds -in particular ambiguities of the final or internal state and of the possibility that multiple decay chains may be present in a single event.

Thereafter, the main part of the review is divided up into sections which, to first order, each cover the mass measurement techniques which are appropriate to a specific decay topology, i.e. to a specific set of assumptions about the identities of the particles participating in the observed decays, and the kinematical structure of those decays.

Broadly speaking, the review starts with the decay topologies or hypotheses which make the fewest assumptions -for example some assume nothing more than "that momentum is conserved" -and from there the review progresses to the hypotheses which make progressively greater numbers of assumptions. For example one of the last techniques discussed assumes that the experimenter is able to identify samples of quintuples of events, each of which share the same five-particle topology and particle content (though differing in kinematics), and which together amount to an over-constrained system from which the unknown masses can be determined.

The most significant step-changes in the number of assumptions made by the increasingly complex hypothesis in the main part of the view concern the following: (1) increasing numbers of decay products, (2) increasing numbers of invisible decay products, (3) increasing lengths of "decay chain", (4) increasing numbers of decay chains present in each event, [START_REF] Aleph | Precision Electroweak Measurements and Constraints on the Standard Model[END_REF] assumptions making use of pairs of decay chains related (in single events) by a mutual interaction (recoil), and [START_REF] Aad | The ATLAS Experiment at the CERN Large Hadron Collider[END_REF] assumptions requiring increasingly pure samples of events, and [START_REF] Adolphi | The CMS experiment at the CERN LHC[END_REF] assumptions requiring greater control of detector acceptances and efficiencies and background shapes over wide ranges.

In the appendix we gather together some definitions of kinematic variables and useful results.

Decay Topologies or Hypotheses

In the literature the shorthand "topology" is used to indicate a sequence of decays of heavy objects to lighter ones. The constituents of what is often referred to as the "final state" may actually have further non-trivial dynamics (e.g. showering and hadronisation of quarks or gluons to jets). They might be indistinguishable (e.g. identical leptons) or indeed may be unobserved (e.g. neutrinos).

A list of some topologies and links to corresponding sections of this review can be found in Table I on Page 5. Note that hypotheses of a particular event topology may differ in their details yet still provide a correct description of that event. For example hypothesis (1) might be that particle A decayed to a final state comprising particles X, Y and Z, without specifying any details about the intermediate mechanism. A refinement, hypothesis [START_REF] Acciarri | Measurement of mass and width of the W boson at LEP[END_REF], might assume an explicit form of the decay chain, e.g. A → BX followed by the decay B → Y Z. Provided that both hypotheses do indeed correspond to the event observed it will be possible to extract more information from the more detailed hypothesis (2): for example the mass of the intermediate particle B.

The topologies of Table I are therefore not all mutually exclusive. There is in fact a tension between the desire to assume more details about topology -in order to obtain more information about the event -and the need, in the presence of many competing processes, to propose something general enough that it stands a respectable chance of matching the observed event.

As can be seen from Table I, parts (though by no means all) of this review are concerned with topologies in which two particles are believed to have been produced in the initial state. This is partially motivated by the expectation that in many models, new particles are odd under a Z 2 parity (such as R-parity for supersymmetry) under which Standard Model particles are all even.

The lightest such parity-odd particle, if stable for cosmological timescales, is expected to be weakly interacting, therefore to be unobserved by the apparatus. Invisible particles (which are not unique to such models) are represented by dashed lines in Table I.

Observables and other quantities defined per-event

The main information which is obtained from a hadron collider event is the momentum and energy of the observable particles which impinge on the active volume. Particles which are detected will have their reconstructed momenta and energy smeared by the experimental resolutions, and this smearing will need to be understood in calibration channels and modelled by those interpreting the experiments. Detailed understanding of the detector response is clearly necessary to perform precision measurements, and is also a key component approaches which combine information from different sources (e.g. different events) in order to over-constrain the kinematics.

Not all particles are necessarily observed -for example neutrinos and any other weaklyinteracting particles are expected to pass through the apparatus undetected. What is more, it is not usually possible to reconstruct particles with small angles to the beam-pipe -for example the hermetic region of the LHC general purpose detectors [START_REF] Aad | The ATLAS Experiment at the CERN Large Hadron Collider[END_REF][START_REF] Adolphi | The CMS experiment at the CERN LHC[END_REF] is restricted to a fiducial pseudorapidity of |η| < ∼ 5 where η =log tan(θ/2) and θ is the angle relative to one of the beam directions.

The incoming parton momenta are generally not known in hadron-hadron collisions, so the centre-of-mass energy and the longitudinal boost of the centre-of-mass frame are not fixed by the initial conditions. The sum of the momenta of any invisible particles can be inferred from conservation of momentum.

Except in the very special case of central exclusive production 1 the only information which can be obtained about invisible particles' momenta is the sum of the components perpendicular to the beams:

p invisible T ≈ / p T ≡ -p vis T
where the second sum is over the visible transverse momenta of all final-state particles. The first equality is only approximate since particles at large |η| will be undetected (though visible in principle), and because of experimental smearing of the p vis T . Most heavy particles decay sufficiently rapidly that they do not travel macroscopically measurable distances. Familiar exceptions include τ leptons and B hadrons which can travel macroscopic distances from their production location. If decay products can be reconstructed to secondary vertices away from the primary interaction point, the additional information can help in particle identification, and in kinematic reconstruction. Some examples of using the kinematic information from secondary vertices can be found in Section 3.10.

In addition to measuring directions, energies and momenta, the majority of hadron collider experiments can make statements about the identities of the particles they detect by using hardware designed for that purpose. Muon detectors placed after hadron calorimeters are a prime example of a means by which a species of particle can be identified with a high degree of confidence.

Nonetheless, many types of paricles (especially those with similar properties) cannot be identified without dedicated hardware such as Čerenkov detectors [START_REF] Roberts | A new type of Cerenkov detector for the accurate measurement of particle velocity and direction[END_REF].

The typical set of observables therefore consists of the four-momenta of a set of objects, some of which are individual particles, and some of which may be groups of particles (e.g. hadronic jets).

Each of these objects have an identification hypothesis or hypotheses with associated probabilities.

Together with a hypothesis about the topology, these observables can be used to make inferences about the properties (and particularly for this review, masses) of the particles.

The uses to which our observables are put tend to be restricted by the invariance of space-time under rotations and Lorentz boosts, or by the approximate axisymmetry of the LHC detectors, or by the lack of knowledge of the centre-of-momentum frame of the primary interaction due to the composite nature of the colliding protons. As a result, our primary observables tend to be combined into secondary derived quantities (which are themselves invariant under general boosts or axial boosts or general rotations or rotations about the beam axis, or combinations thereof) and these are in turn used as building blocks or inputs to more complicated tertiary methods and variables. The secondary derived quantities of which we speak include the invariant masses and transverse masses which, though discussed later, will already be familiar to most readers. However, we give advanced warning, that there are sometimes situations in which it is useful to form derived quantities which lack some of the usual symmetries of space-time or the detector. For example, the first such quantity we will come across is m C (the "contralinear" invariant mass) which is decidedly deviant under Lorentz boosts of any kind, but is nonetheless useful.

Constraints and quantities defined per-dataset and hybrids

As well as having observables and derived quantities which come from individual events, it is natural to expect also observables that are formed from samples containing large numbers of events.

These could be called per-dataset variables. One classic example of a class of such variables, which we will discuss in more detail later, are kinematic endpoints. Typically the position of a kinematic endpoint places a constraint on some relationship of the masses of the particles involved in the decays that generated the endpoint. A second example would be a mass constraint coming from a fit to the shape of a differential distribution constructed from a large number of events. Furthermore, one can even conceive of hybrid variables, by which we mean variables which mix together pieces of information from per-dataset and per-event into something more powerful. The resulting hybrids appear to be defined "per-event", but in fact make use of global properties of the dataset as a whole. The most common reason for doing this is the desire to apply, to individual events, one or more constraints of the type which cannot be deduced from any single event, but which can be deduced from the set of all events.

In this review we will try to draw a distinction between per-event, per-dataset and hybrid variables. We will leave further discussion of the merits and drawbacks of hybrid variables until later (Section 4.9).

Ambiguities

It is often the case that final state particles cannot be uniquely attributed to particular positions in the hypothesised decay chain. This may simply be due to there being repeated identical particles in the final state. A second source of ambiguity can arise from initial state radiation (ISR). Any high-scale process at a hadron collider will inevitably be accompanied by jets due to ISR, and so mass measurement techniques, particularly those using jets, need to be robust with respect to its presence. An extended discussion of complications caused by ISR can be found in Section 4.8 in the context of one particular event topology. A third class of ambiguity can result from there being alternative internal particle assignments that leave the identities of particles in the final state permuted. For example, consider the supersymmetric decay chain q → χ0 2 q → ˜ ∓ ± q → χ0 1 ∓ ± q.

In this chain the charge of the intermediate slepton is not known and so one cannot tell whether the positively-charged lepton originated from the decay of the neutralino or from the subsequent decay of the slepton. A fourth source of ambiguity can arise from lack of certainty as to whether the decay topology hypothesised for a given event actually reflects reality. For example, the decay chain above (in which the slepton is an on-shell resonance) has the same final particle content as a similar chain in which the slepton is much heavier than the χ0 2 , thus forcing the χ0 2 to decay by an effective three body decay rather than via two successive two body decays.

There is substantial variability in the extent to which mass measurement techniques which have been proposed in the literature choose to address the challenges presented by such ambiguities. Some are developed with these ambiguities in mind from the start, while others do not address them at all and merely hope that ways will be found to address them in the future. The reader is encouraged to think critically about the assumptions made in each of the techniques reviewed herein, and to consider the ways in which they may or may not be sensitive to unresolved ambiguities. In particular, it should not be assumed that the presence of a technique in this review guarantees that it can be used in practice, or that any technique can produce a definitive answer that is not strongly dependent on one or more untestable assumptions made at its core.

Spins

The spins of the participating particles and the chiralities of their couplings can play important roles in the dynamics of the decays. In most cases, the effects of the particles spins on experimentally-accessible distributions are small, but various analyses have been proposed which indicate sensitivity to spins in a variety of cascade decays. Angular correlations in variables other than cascade decays have also been studied [START_REF] Barr | Measuring slepton spin at the LHC[END_REF][START_REF] Buckley | Discriminating spin through quantum interference[END_REF][START_REF] Buckley | Quantum interference effects among helicities at LEP-II and Tevatron[END_REF][START_REF] Alves | Deciphering the spin of new resonances in Higgsless models[END_REF][START_REF] Buckley | Determining spin through quantum azimuthal-angle correlations[END_REF][START_REF] Boudjema | A model independent spin analysis of fundamental particles using azimuthal asymmetries[END_REF]. A separate review article on the subject of spin determination methods has recently been published [START_REF] Wang | A review of spin determination at the LHC[END_REF] and we refer the reader to that article for more details.

VARIABLES FOR PARTICLE PRODUCTION AT OR NEAR THRESHOLD

If one wishes to make very few assumptions about the type of interaction, the decay topology, and the types or particles involved, then the best one can generally do is to construct an observable which (because it is constructed out of quantities proportional to energy) scales approximately as the energy of the centre-of-mass of the collision.

The distribution functions of momenta of partons within protons ("PDFs") are largely rapidly falling functions of the momentum fraction x, so above threshold, cross-sections tend to decrease with the centre-of-mass energy of the parton-parton system, ŝ1/2 . This means that heavy particles can often be expected to be produced at or near threshold, and the energy of the collision can be expected to give a good indication of the mass scale of the particles produced.

A variety of variables sensitive to the overall mass-energy scale have been proposed. Since the momentum of the parton-parton centre-of-mass generally cannot be known when invisible particles have been produced, the majority are constructed from only those momentum components perpendicular to the beam pipe.

In the context of supersymmetry, the simplest mass-scale measurements are those deriving from ad-hoc variables that have some kind of correlation, even if only approximate, with the masses or mass differences of the primary particles produced in the interaction. Unfortunately there is little standardisation of nomenclature in this area: variables with the same experimental definition can have more than one name depending on which collaboration uses them (for example, both ATLAS and CMS have at different times defined an identical variable, though the former called their variable M eff while the latter named theirs H T ) and even worse, neither collaboration has stuck to a single definition of either of these variables for any great length of time. For example, the earliest [START_REF] Hinchliffe | Precision SUSY measurements at CERN LHC[END_REF] definitions of M eff within ATLAS, which remained in use for more than a decade [START_REF] Aad | Expected Performance of the ATLAS Experiment -Detector[END_REF], defined the "effective mass" variable in terms of the scalar sum of the four highest p T jets and the missing transverse momentum as follows:

M eff = i=1,4 |p T,i | + |/ p T |. ( 1
)
More recent ATLAS work [START_REF]Early supersymmetry searches in channels with jets and missing transverse momentum with the ATLAS detector[END_REF] has re-defined M eff as the sum of the hightest "n" p T jets, where n depends on the analysis channel. The peak of the supersymmetric component of the original M eff distribution of ( 1) and [START_REF] Hinchliffe | Precision SUSY measurements at CERN LHC[END_REF][START_REF] Aad | Expected Performance of the ATLAS Experiment -Detector[END_REF] was found to correlate at the O(10%) level with a characteristic SUSY mass-scale m SUSY ≡ min(m g, m ũR ) for models drawn from the 5-parameter constrained Minimal Supersymmetric Standard Model (cMSSM). A more general MSSM study [START_REF] Tovey | Measuring the SUSY mass scale at the LHC[END_REF], found that the scalar sum over all jets given by

M est = i |p T,i | + |/ p T | (2)
had a peak position which correlates well with a cross-section-weighted SUSY mass scale -after the latter was corrected by the (a priori unknown) mass of the lightest supersymmetric particle.

Within CMS, the analogous variable is called H T whose name, we believe, comes from earlier use at the Tevatron [START_REF]Search for gluinos and scalar quarks in pp collisions at √ s = 1.8tev using the missing energy plus multijets signature[END_REF]. In 2006 a single document, the CMS technical design report [47], defines H T in two different ways! The "jets and missing energy searches" section defines H T in terms of the scalar sum of the second, third and fourth (but not the first) jet energies and the missing transverse momentum according to

H T = E T (2) + E T (3) + E T (4) + |/ p T | ( 3
)
where E T (i) is the transverse energy of the ith jet, and

E T = E sin θ. ( 4
)
In the trigger section of the same document [47] H T is defined differently as the scalar sum of the E T values of all jets, excluding the missing transverse momentum:

H T = E T (1) + E T (2) + E T (3) + • • • . ( 5
)
In later CMS work [START_REF] Collaboration | Performance of methods for data-driven background estimation in susy searches[END_REF] the definition of H T has changed for a third time, and is now the scalar sum of the transverse momenta of all jets:

H T = p T (1) + p T (2) + p T (3) + • • • . (6)
Regardless of the specific definition used, the implicit assumption behind variables such as M eff and H T is that in a hadron collider particles tend to be produced near threshold. Any particles produced exactly at at rest, and which decay in a semi-invisible two-body decay, produce visible daughters with transverse momenta p T = |p T | less than the two-body decay momentum,

p T ≤ p * = λ 1 2 (m A , m B , m C ) 2m A (7)
where m A is the mass of the parent, m B and m C are the masses of the two daughters and

λ(a, b, c) = a 2 -(b + c) 2 a 2 -(b -c) 2 .
The inequality in [START_REF] Adolphi | The CMS experiment at the CERN LHC[END_REF] becomes an equality in the case where there is no momentum along the direction of the beam pipe. Thus the transverse momentum distribution contains information about the mass scale of any heavy particles produced, though the above inequality will be smeared by detector resolution, by production of heavy states well above threshold and by recoil of the parent against initial state radiation.

As discussed above, the centre-of-mass energy of the (parton-parton) collision ŝ1/2 is sensitive to the mass scale of heavy particles even if few details about their decay topology are known.

When invisible particles are produced, there is insufficient information to reconstruct ŝ1/2 for any particular event, but it will be bounded from below by the observable

ŝ1/2 min = (E 2 -P 2 Z ) 1 2 + (/ p 2 T + M 2 invis ) 1 2
where M invis is the sum of the mass of all invisible particles thought to have been produced [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF].

It has been noted [START_REF] Papaefstathiou | Effects of QCD radiation on inclusive variables for determining the scale of new physics at hadron colliders[END_REF] that though ŝ1/2 min and other similar variables are very heavily modified by initial state radiation, the amount of modification is nonetheless calculable. This is all we shall say about ŝ1/2 for the moment, however we will return to ŝ1/2 min in more detail in Section 4.8 where we discuss how it might concretely be used to measure masses of pair-produced events.

We note in passing that [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF] does not confine itself to suggesting that ŝ1/2 min be used only to place strict bounds on ŝ1/2 , or to place constraints on particle masses in the manner described in Section 4.8. On the contrary, [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF] advances a number of quite different potential uses for ŝ1/2 min , some not even related to mass measurement, which are not discussed further in this review.

VARIABLES FOR SINGLE CASCADE DECAY CHAINS

Decays to two visible particles ("two-body visible")

The simplest examples of kinematic mass reconstruction, e.g in the case of Z → e + e -, are familiar. The decay topology can be written A → BC (Figure 1a) where capital letters are used to label particles, and corresponding lower case letters represent their four-momenta. The parent (Z boson) mass can be obtained from the straightforward construction of the "invariant mass" from the square of the sum of the visible four-momenta:

m 2 bc = (b + c) 2 . ( 8
)
One therefore obtains an event-by-event estimator of the Z boson mass, and can form a distribution which may be calibrated by comparison to calculations and/or Monte Carlo simulations (Figure 2a).

Decays to a visible and an invisible particle ("two-body semi-invisible")

A more interesting case, because the final state contains missing information, can be found by considering leptonic W boson decay (Figure 1b). For W → ν, the topology is again A → BC, but the neutrino is essentially invisible. Henceforth we will denote invisible particles with a slash;

writing this now as A → B / C. Although the three-momentum of the neutrino is not observed, its transverse momentum / c T may typically be inferred from energy momentum conservation in the transverse plane if there are no other invisible particles in the event. For each event there is some range of values of m W which are consistent with the observables b, / c T , and the known mass of the lepton m B and the (negligible) mass of the neutrino m / C . The boundary of the allowed domain is conveniently found by the explicit construction of the transverse mass, M T [53][START_REF] Arnison | Experimental observation of isolated large transverse energy electrons with associated missing energy at s 1/2 = 540 GeV[END_REF][START_REF] Banner | Observation of single isolated electrons of high transverse momentum in events with missing transverse energy at the CERN pp collider[END_REF][START_REF] Smith | Transverse mass and width of the W boson[END_REF][START_REF] Barger | Improved transverse-mass variable for detecting higgs-boson decays into Z pairs[END_REF]: The (lower case) "transverse energy" quantities e for each particle are defined by

M 2 T ≡ m 2 B + m 2 / C + 2 e b e / c -b T • / c T . ( 9
)
e 2 = m 2 + p 2 T . ( 10
)
These e are equal to the E T quantities (also denoted "transverse energy") defined in (4) in the massless limit. That the function in Equation 9gives the largest value of m W consistent with the observations is noted in [START_REF] Cheng | Minimal kinematic constraints and M T 2[END_REF][START_REF] Barr | Transverse masses and kinematic constraints: from the boundary to the crease[END_REF]. While the results of hypothesising incorrect values for the mass of one of the daughter particles are of great interest -and are explored further in Section 4.2one can also obtain a simple but equally important result when the correct values of the daughter particles masses are assumed. For the true values of m B and m / C and in the approximation where the widths are narrow and experimental resolutions small, the inequality

M T ≤ m A ( 11
)
is satisfied by construction, with equality when the relative rapidity of the daughter particles vanishes. Therefore a histogram of values of M T , for many events with the same topology, should populate some regions (corresponding to allowed values of m W ) but not other regions, corresponding to disallowed values of m W . The mass could then be determined from the boundary of the populated region -the kinematic endpoint or edge. In practice, background events, finite-width effects and experimental resolutions smear the edge, so precise determinations of the W ± mass The "dilepton" decay topology. The particle labelled Z is assumed to be unobserved by the detector.

using this method need to model such effects (see [START_REF] Arnison | Experimental observation of isolated large transverse energy electrons with associated missing energy at s 1/2 = 540 GeV[END_REF][START_REF] Banner | Observation of single isolated electrons of high transverse momentum in events with missing transverse energy at the CERN pp collider[END_REF] and others subsequently including the example in fig. 2b).

Fully visible three-body decays

Techniques for analysing three-body decays of the type shown in Figure 3, i.e. where all three daughters are visible, can be most conveniently analysed using the tried-and-tested method of the Dalitz plot [START_REF] Dalitz | On the analysis of τ -meson data and the nature of the τ -meson[END_REF][START_REF] Amsler | Review of particle physics[END_REF]. This plot projects the momenta onto a surface (usually {m 2 BC , m 2 BD }) which is uniformly populated for a three-body decay with a constant matrix element. Intermediate resonances can be observed as bands in these plots for particular values of invariant mass. Angular momentum multipoles can be determined from the rank of the spherical tensor needed to reproduce the observed angular distributions.

Attempts to reproduce the desirable features of the Dalitz plot when invisible particles are unobserved are revisited in Section 5.4.

The dilepton edge: two successive two-body decays

An example of a hypothesis used for the partial reconstruction of one part of an event is the topology shown in Figure 4. This is sometimes called the "dilepton" topology, since it was first studied in the context of the LHC [START_REF] Paige | Determining SUSY particle masses at LHC[END_REF] for the case of the supersymmetric decay χ0

2 → ql ± ˜ ∓ → ql ± l ∓ χ0
1 . The kinematics are most easily studied in the rest-frame of particle B (the slepton in the example above) in which if the masses are fixed, the sizes of the momenta of the final state FIG. 5: An example "dilepton" distribution (taken from [START_REF] Paige | Determining SUSY particle masses at LHC[END_REF]) for the topology shown in Figure 4. In this example, the kinematic endpoint is at approximately 100 GeV.

particles X, Y and Z are fixed. The invariant mass of the visible system, m XY , then depends only on the angle θ between X and Y . In the limit of small masses of X and Y (which is approximately true for the dilepton case), the density of states is proportional to m XY up to a maximum at

(m max XY ) 2 = (m 2 A -m 2 B )(m 2 B -m 2 Z ) m 2 B ( 12
)
when θ = π. Plotting a distribution of m XY one therefore obtains a triangular distribution, such as the one shown in Figure 5. The maximum endpoint of this distribution can be measured, giving one constraint on the three variables, m A , m B , and m Z . It is worth noting in this context that while the endpoint of the sequential two-body decay (12) constrains differences in squared masses, the equivalent single step three-body decay A → XY Z would have an endpoint at m max XY = m Am Z , so would constrain the difference in unsquared masses.

Examples of applications include sensitivity for multiple kinematic endpoints from competing decay chains [START_REF]ATLAS Detector and Physics Peformance TDR[END_REF], calculations of the m distribution shapes [START_REF] Smillie | Distinguishing spins in supersymmetric and Universal Extra Dimension models at the Large Hadron Collider[END_REF][START_REF] Athanasiou | Distinguishing spins in decay chains at the Large Hadron Collider[END_REF][START_REF] Nojiri | Neutralino decays at the LHC[END_REF][START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Birkedal | Slepton mass measurements at the LHC[END_REF], tests of lepton universality [START_REF] Goto | Study of the slepton non-universality at the CERN Large Hadron Collider[END_REF][START_REF] Hinchliffe | Lepton flavor violation at the LHC[END_REF][START_REF] Allanach | Measuring smuon-selectron mass splitting at the CERN LHC and patterns of supersymmetry breaking[END_REF], and an examination of pairs of such dilepton chains [START_REF] Bian | Wedgebox analysis of four-lepton events from neutralino pair production at the LHC[END_REF][START_REF] Bisset | Improving SUSY Spectrum Determinations at the LHC with Wedgebox and Hidden Threshold Techniques[END_REF].

If individual lepton flavour numbers are assumed to be conserved then in the dilepton case the signal can be expected to be found in opposite-sign same-flavour (OSSF) pairs (e + e -and μ + μ -).

Backgrounds from e.g. t t will not have lepton flavour correlations, and so an estimate of the OSSF background distribution (resulting from such flavour-uncorrelated sources) can be obtained from the opposite-sign, different flavour (OSDF) e ± μ ∓ distribution [START_REF]ATLAS Detector and Physics Peformance TDR[END_REF].

The di-tau invariant mass was investigated in [START_REF] Hinchliffe | Measurements in SUGRA models with large tan(beta) at LHC[END_REF]. This last case is not strictly an example of FIG. 6: The "ditau" decay topology. the topology of Figure 4 since each tau decay also generates invisible particles (neutrinos), so the appropriate topology is that of Figure 6. More about chains with multiple invisible particles can be found in Section 4.6. Helicity effects in tau distributions are discussed in [START_REF] Choi | tau polarization in SUSY cascade decays[END_REF][START_REF] Mawatari | Tau polarization in SUSY cascade decays at LHC[END_REF][START_REF] Godbole | Using Tau Polarization to probe the Stau Co-annihilation Region of mSUGRA Model at LHC[END_REF].

Constraints from the qll-like chain

If the "dilepton" topology of Section 3.4 is extended by one two-body decay, we reach a chain having three successive two-body decays, resulting in a final state consisting of three visible (frequently but not always light) particles, and one (frequently but not always massive) invisible particle.

The most frequently considered context in which this topology is used is the decay of q →

q χ0 2 → ql ± ˜ ∓ → ql ± l ∓ χ0 1 q χ0 1 χ0 2 q ˜ ∓ R l ± near l ∓ far
which has led to this chain being known as the "qll-chain". In fact the qll case is really a special one in the sense that it assumes particular identities of particles, and hence admits only particular possibilities for ambiguities. This chain was first suggested as a means of measuring sparticle masses in [START_REF] Hinchliffe | Precision SUSY measurements at CERN LHC[END_REF][START_REF] Bachacou | Measurements of masses in SUGRA models at CERN LHC[END_REF]. These early works proposed that, following on from the di-lepton edge technique described above, other one-dimensional invariant mass distributions be plotted involving the quark (or rather jet) momenta in addition to the momenta of the leptons. As before, relativistic kinematics impose an upper limit on any particular invariant mass distribution, and the position of any particular upper limit (or more generally kinematic end-point, or in some cases just "end-point") may be established as a function of the masses of the particles involved in the chain. As always, these kinematic end-point positions are valid only if the events are from the topology considered, and will be smeared by detector resolution effects. Events from "backgrounds" may have almost any invariant mass.

Conventionally, the lepton produced first in the decay of the heavier neutralino is called the "near lepton" (near to the quark) and is notated l near , while the lepton produced second in the decay of the slepton is called the "far lepton" and is notated l far [START_REF] Lester | Model independent sparticle mass measurements at ATLAS[END_REF][START_REF] Allanach | Measuring sparticle masses in non-universal string inspired models at the LHC[END_REF]. For the concrete case of the qll it is not possible in a single event (in isolation from any other information) to determine which observed lepton is l near and which is l far and thus it is not possible to construct an invariant mass distribution consisting of exclusively of the combination m lnearq (or that of m l far q ). The early qllchain studies [START_REF] Hinchliffe | Precision SUSY measurements at CERN LHC[END_REF][START_REF] Bachacou | Measurements of masses in SUGRA models at CERN LHC[END_REF][START_REF]ATLAS Detector and Physics Peformance TDR[END_REF] elected to put to one side the issue of this ambiguity (c.f. general discussion in section section 1.5). Subsequent attempts at addressing the issue of this ambiguity established the need to build mass constraints out of kinematic end-points of distributions which were truly "observable". For example, the first such attempts [START_REF] Lester | Model independent sparticle mass measurements at ATLAS[END_REF][START_REF] Allanach | Measuring sparticle masses in non-universal string inspired models at the LHC[END_REF] proposed that the distributions of and m max lq(low) . From a kinematic perspective, though not from a spin perspective (see Section 1.6), there is no point in using m ql + and m ql -in place of m lq(high) and m lq(low) : even though either pairing is "observable", the Majorana nature of the neutralino makes the two distributions identical.

m ll , m llq , m lq(high) ≡ max{m ql + , m ql -}
Note that many of the invariant mass combinations that can be constructed are not independent of the others. For example, in the limit of massless visible particles,

m 2 ll + m 2 lq(low) + m 2 lq(high) = m 2 ll + m 2 ql + + m 2 ql -= m 2 llq .
In some but not all cases, this can lead to the kinematic end-points themselves being related. For example (m max ll ) 2 + (m max lq(high) ) 2 = (m max llq ) 2 over some but not all parts of mass-space [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF].

There is definitely a clear benefit to be derived from critically (re-)examining the choices of one-dimensional distributions used to constrain the qll-chain for there is no reason to believe that the "traditional" choices of endpoint [START_REF] Lester | Model independent sparticle mass measurements at ATLAS[END_REF][START_REF] Allanach | Measuring sparticle masses in non-universal string inspired models at the LHC[END_REF] are optimal in any sense -indeed it is very unlikely that the "traditional" choices are optimal by any definition as a measure of optimality was never part of their design. For example, [START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF] point out that it may be preferable to look for two endpoints (i.e. the l ± near q edge and the l ± far q edge) in the "union" distribution of m ql -∪ m ql + rather than to split this distribution into the l ± q high and l ± q low components, as the resulting inversion space has only a twofold rather than a threefold ambiguity. Similarly, it can be advantageous to look for maxima in linear combinations of invariant masses. For example [START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF] investigates the properties of kinematic endpoints of distributions of (m 2α ql -+ m 2α ql + )

1 α for different values of α and finds merit in the particular case m ql -+ m ql + (in this regard note the kinematic end-point of the "l ± q sum-edge"

in Table II on Page 58).

One consequence of moving from technically unobservable distributions (like that of m lnearq ) to observable distributions (like that of m lq(high) ) can be that the locations of the kinematic end-points may become piecewise-smooth functions of the unknown masses [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Lester | Model independent sparticle mass measurements at ATLAS[END_REF][START_REF] Allanach | Measuring sparticle masses in non-universal string inspired models at the LHC[END_REF][START_REF] Gjelsten | Measurement of the gluino mass via cascade decays for SPS 1a[END_REF]. Furthermore, such invariant mass distributions can evolve non-trivial shapes, and can acquire undesirable features (so called "feet") near the end points which might in some cases make end-point measurement prone to large systematic errors [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Gjelsten | Measurement of the gluino mass via cascade decays for SPS 1a[END_REF][START_REF] Miller | Invariant mass distributions in cascade decays[END_REF]. Local non-differentiability of end-point position need not, in itself, be a problem for mass determination (note that piecewise-smooth functions like |x| + 2x can have well defined inverses) however it can be a visible symptom of a separate issue which is of concern in certain cases: ambiguity in end-point inversion, discussed below. Accordingly there has been some recent interest in alternative observable distributions for which end-point positions are smooth functions of the masses [START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF].

Ambiguity in end-point "inversion". Very often one finds oneself in the unfortunate position of having too few observables to constrain all the parameters of a model. On other occasions one may find oneself with a much larger number of independent measurements, sufficient to overconstrain a model. In this fortunate position, one potentially has the power to rule out a model, or else to give strong constrains on the parameters of a model which is consistent.

Occasionally one may find oneself in the very special situation in which the number of independent observables or measurement happens to match exactly the number of free parameters (e.g. masses) in the model. In such situations it can be very hard to resist the temptation to search for analytic or closed-form "inversions" : i.e. solutions for the parameters (e.g. masses) in terms of the observables or measurements (e.g. the position of the end-points). Many such "inversions" have been published published for different sets of observables for the qll-chain [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF][START_REF] Miller | Invariant mass distributions in cascade decays[END_REF][START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF][START_REF] Costanzo | Supersymmetric particle mass measurement with invariant mass correlations[END_REF].

For some sets of end-point measurements the inversion process may yield a single set of consistent masses -hopefully the correct ones -while for some other sets of end-point methods there may be more than one set of consistent masses (of which one is hopefully correct while the others are spurious).

In fact the previous statement applies to the idealised situation in which detector resolution is perfect. While the performance of some "inversions" degrade rapidly in response to even small amounts of experimental smearing/resolution, others are much more tolerant. It seems that there is much scope for future work to determine which inversions are best suited to experimental appli-cation and which are not.

For more detailed discussion see [START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF] and [START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF]. In particular these papers pose the further question: "Can one find sets of distributions whose end-points always yield the smallest number of spurious solutions?" and in answering this yield entirely new sets of invariant mass distributions for the qll-chain.

There are benefits, clearly, in widening our understanding of what features in data drive our mass constraints. Looking at endpoint inversion formulae (and minimal sets of invertible endpoints) is one way that can be accomplished. Nevertheless, it should be remembered that the issue of analytic invertibility alone must not drive the choice of variables used. Frequently there will be other more important issues to address which might include: (1) which end-points are easiest to observe (dependent on slope and shape near the end-point; relative numbers of signal and background events near the edge; the degree to which the background shape and size may be independently predicted); (2) which are least smeared by detector resolutions; and (3) which are least sensitive to cuts and acceptance or things which can affect systematic uncertainties. Furthermore, it seems very likely that the best measurements will be made by putting together the largest possible number of pieces of (sometimes overlapping) evidence in a joint numerical fit, rather than by inverting a set of equations for a particular set of constraints at the expense of other observables.

The reader who is not convinced that there is much work yet to be done in identifying better (or at least additional) means of constraining masses in the qll-chain would be well advised to review the cautionary tale of the hitherto undiscussed lower kinematic endpoint known as the l + l -q threshold. Most of the "traditional" sets of endpoints [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Lester | Model independent sparticle mass measurements at ATLAS[END_REF][START_REF] Allanach | Measuring sparticle masses in non-universal string inspired models at the LHC[END_REF][START_REF] Miller | Invariant mass distributions in cascade decays[END_REF] as well as some of the new proposals [START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF] rely to a lesser or indeed greater extent on the l + l -q threshold proposed first in [START_REF]ATLAS Detector and Physics Peformance TDR[END_REF]. This is the lower end-point of the m llq distribution under the additional constraint that m ll < m max ll / (2). This lower end-point is notorious2 for having experimental systematic errors associated with its measurement (in part due to the shape [START_REF] Lester | Constrained invariant mass distributions in cascade decays: The shape of the 'm qll threshold' and similar distributions[END_REF] at turn-on being concave) which are in some cases much larger than those required to make use of the constraint it provides. Such an end-point may turn out to be just the sort of measurement that looks good on paper but turns out to be under poor experimental control.

Moving away from one-dimensional constraints. It is often suggested, particularly by the experimental community, that one-dimensional distributions of variables like m ll , m qqll and FIG. 7: The "gluino" decay chain.

the other Lorentz invariants discussed above offer the simplest, and probably the most easily measurable distributions from which to extract information about the masses of the parents. But is this suggestion correct? It is certainly being challenged. These one-dimensional invariant mass distributions can all be thought of as "projections" of the higher-dimensional space in which the measurements live, onto a single dimension. The full three-dimensional shape of the qll-chain has been noted in [START_REF] Costanzo | Supersymmetric particle mass measurement with invariant mass correlations[END_REF], and there are many promising proposals to use fits to structures in observables of two (and higher) dimensions in order to gain information from correlations that are not otherwise available in one-dimensional distributions [START_REF] Bian | Wedgebox analysis of four-lepton events from neutralino pair production at the LHC[END_REF][START_REF] Bisset | Improving SUSY Spectrum Determinations at the LHC with Wedgebox and Hidden Threshold Techniques[END_REF][START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF][START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF][START_REF] Costanzo | Supersymmetric particle mass measurement with invariant mass correlations[END_REF]. In principle there is a lot more information available in these higher-dimensional distributions -but whether that information is easier or harder to extract than that from the one-dimensional distributions will depend to a large extent on the relative degree to which the systematic uncertainties can be understood/controlled by the experimental collaborations in the two cases.

Constraints from the qqll-like chain

Adding a further two-body decay to the "qll" chain produces the topology shown in Figure 7, which sometimes called the "qqll" or "gluino" chain since the most studied example has been

g → q q → qq χ0 2 → qql ± ˜ ∓ → qql ± l ∓ χ0 1 .
Many of the kinematic endpoints for this longer chain can be found in the results of section section 3.4 and section section 3.5 (or relabellings thereof). The new endpoints, including the maximum of the four-body qqll distribution have been calculated using massless approximation for the visible particles [START_REF] Gjelsten | Measurement of the gluino mass via cascade decays for SPS 1a[END_REF] assuming all particles on the backbone are on mass-shell. 3 These can depend on any of the other five masses in the problem (g, q, χ0 2 , ˜ and χ0

1 ). The same chain has been used to put constraints on the spin of the gluino [START_REF] Alves | It's a gluino[END_REF]. When dealing with chains of this length containing many jets in the final state, most studies have found it necessary to have addional information about the jets (for example bottom-quark tags) to reduce ambiguities due to combinatorics and ISR etc. FIG. 8: The "dark matter sandwich" decay chain.

Other chains containing successive two-body decays

One interesting chain which has been studied (to the best of our knowledge) only in [START_REF] Agashe | Distinguishing Dark Matter Stabilization Symmetries Using Multiple Kinematic Edges and Cusps[END_REF] is the "dark matter sandwich" topology of Figure 8. What makes this chain different from qll-like chains is that the missing invisible particle (the dark matter particle) emerges at the mid-point of the decay chain, rather like the filling in a sandwich. In the context in which it was investigated, the particles on the back bone were allowed arbitrary masses, while the two visible ejecta were treated as massless. As such the invariant mass distribution of the two visible particles depends on five unknown masses (the four backbone masses and the sandwiched invisible particle's mass).

Formulae for the position of the kinematic endpoint and the differential shape of the invariant mass distribution of the invariant mass of the pair of visible ejecta may be found in [START_REF] Agashe | Distinguishing Dark Matter Stabilization Symmetries Using Multiple Kinematic Edges and Cusps[END_REF].

These distributions are notable for having kinematic cusps -places in the differential distribution where two curves with different slopes come together. Further discussion on such cusps can be found in Sections 3.11 and 4.11.

When backbone sparticles are off mass-shell in multi-step decay chains

It is worth asking whether we would be able to tell if an observed ensemble of similar multiparticle final states is likely to have come from events containing a long series of 2-body decays, or whether it might (for example) have originated from a somewhat shorter decay chain with more particles emitted at a smaller number of vertices. This question can be rephrased as asking whether the narrow-width approximation is valid for all the intermediate states.

In the context of the discussion above, it may be noted that not all sparticles on the "backbone" need be on their mass-shell. It is possible to imagine scenarios in which (for example) the sleptons are heavier than the second-lightest neutralino. The second-lightest neutralino would then not be able to decay via an on-shell slepton, though a three body decay via a highly virtual off-shell slepton might still be possible, illustrated in Figure 9. The observed final state particle content (a jet and two opposite sign sane flavour leptons) offers no clue as to whether the decay topology has a virtual or an on-shell slepton in the backbone. This is a problem, because the positions of the kinematic endpoints of the usual invariant mass distributions are entirely different functions of the masses of the sparticles involved. Consequently it might be possible, if events coming from the on-shell scenario were analysed using the off-shell hypothesis (or vice versa) to obtain entirely spurious mass measurements. Fortunately, the way that the event-space is populated (i.e. the shapes of the distributions [15, 20, 30, 65, 80-82, 84, 87-89]) and the relationships between the positions of the kinematic endpoints [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Gjelsten | Measurement of the gluino mass via cascade decays for SPS 1a[END_REF][START_REF] Miller | Invariant mass distributions in cascade decays[END_REF][START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF][START_REF] Lester | Three body kinematic endpoints in SUSY models with non-universal Higgs masses[END_REF], betray clues as to the nature of the topology and can permit the type of the decays (two-body versus three-body or similar) to be determined correctly under favourable circumstances.

The qqll-chain is as susceptible as the qll-chain to ambiguities introduced from not knowing which (if any) of the particles on the backbone are on or off mass shell. It is possibly the case that the only work which has considered the qqll-chain with off-mass-shell particles on the backbone is an incomplete undergraduate project [START_REF] Freedman | Student project[END_REF].

We round off this section by noting an observation of [START_REF] Agashe | Distinguishing Dark Matter Stabilization Symmetries Using Multiple Kinematic Edges and Cusps[END_REF] regarding a particular class of models considered therein in which a single massive particle could decay via multi-body decays into two or more visible particles and either one or two invisibles particles of identical and unknown mass.

These two topologies are shown in Figure 10. It was noted therein that the distribution of the invariant mass of the visible system would show a double endpoint structure -one endpoint being the difference between the mass of the parent and the invisible daughter: (m max vis ) 1 = m Am C , and the other being (m max vis ) 2 = m A -2m C . Were such a double endpoint structure to be observed, and were one prepared to hypothesise the underlying structure of Figure 10, the authors of [START_REF] Agashe | Distinguishing Dark Matter Stabilization Symmetries Using Multiple Kinematic Edges and Cusps[END_REF] propose that one use the linear combination of endpoints 2(m max vis ) 1 -(m max vis ) 2 to measure the parent mass m A , while the combination (m max vis ) 1 -(m max vis ) 2 would measure the invisible particles' mass m C . FIG. 10: Occasionally there are models which can produce both of the above topologies in the decays of a single type of particle. For such models, variables have been proposed which constrain the unknown masses.

See the end of Section 3.8.

Directly reconstructible

Most of the discussion until this point has involved final states for which at least one of the daughter particles is expected to go undetected. When the particle(s) of interest decay to a set of daughters all of which are visible, then determining the mass of the parent(s) should generally more straightforward (at least in principle). A simple example of the fully-visible case, A → BC (Figure 1a) was discussed in Section 3.1, and the three-body case A → BCD in Section 3.3.

However, even when all the particles are visible the kinematical reconstruction is not necessarily trivial. For example it is often difficult to assign the visible particles to the appropriate decay, particularly if there is a large number of final-state objects or much initial state radiation. Other ambiguities can arise when attempting to associate final state hadronic jets to particular types of decay, since jets are themselves composite objects. Some examples of papers considering these more difficult cases are surveyed in what follows.

Combinatorial complications.

Even when all of the final state particles can be identified, the task of reconstructing the masses of the parent particles can be far from trivial. In events with many objects (jets, leptons, . . . ) in the final state the attempt to associate such objects to particular parents involves considering a factorially large number of different possible combinations. Though one can attempt to resolve some of these ambiguities in particular cases by appealing to e.g. lepton number conservation [START_REF]ATLAS Detector and Physics Peformance TDR[END_REF][START_REF] Barr | The precision determination of invisible-particle masses at the LHC[END_REF][START_REF] Serna | A short comparison between m T 2 and m CT[END_REF] or by looking for correlated kinematic features [START_REF] Bian | Wedgebox analysis of four-lepton events from neutralino pair production at the LHC[END_REF][START_REF] Bisset | Improving SUSY Spectrum Determinations at the LHC with Wedgebox and Hidden Threshold Techniques[END_REF], often there is no alternative but to assume that all kinematic combinations are possible.

A particularly difficult case -because very many jets are expected in the final state -is baryonnumber violating, R-parity violating Supersymmetry [START_REF] Allanach | Measuring supersymmetric particle masses at the LHC in scenarios with baryon-number R-parity violating couplings[END_REF][START_REF] Butterworth | Discovering baryon-number violating neutralino decays at the LHC[END_REF]. If the baryon-number-violating couplings are small, then the Supersymmetric decay chain proceeds as in the R-parity conserving FIG. 11: The topology explored in [START_REF] Allanach | Measuring supersymmetric particle masses at the LHC in scenarios with baryon-number R-parity violating couplings[END_REF].

case, but then each of the two lightest supersymmetric particles decays into three (different-flavour) quarks: i.e. χ0

1 → qqq. One can attempt to reconstruct the χ0 1 mass from three-jet invariant mass combinations. However in a hadron collider one expects (as well as the six jets from the two χ0

1 decays) further jets from cascade decays, so the combinatorial background from wrong jet combinations can be very significant. The first attempts to reconstruct such complex topologies made use of leptons from the cascade decays (Figure 11) to simultaneously form invariant mass combinations for several heavy particles [START_REF] Allanach | Measuring supersymmetric particle masses at the LHC in scenarios with baryon-number R-parity violating couplings[END_REF].

It has been shown that for one can reconstruct the heavy particle masses in such cases without relying on the existence of leptons in the cascade decays. That analysis made use of more sophisticated jet algorithms to determine the scale at which a single merged jet from the 3-quark system (from each χ0

1 decay) can be resolved into sub-jets [START_REF] Butterworth | Discovering baryon-number violating neutralino decays at the LHC[END_REF]. Similar sub-jet analyses have been proposed in reconstruction of other boosted heavy objects, with recent examples including searches for boosted Higgs bosons decaying to b b when produced in association with a vector boson [START_REF] Butterworth | Jet substructure as a new Higgs search channel at the LHC[END_REF] or in association with t t [START_REF] Plehn | Fat Jets for a Light Higgs[END_REF]. Anther example of using jet substructure to improve the mass resolution (and hence signal to background discrimination) of heavy objects can be found in the context of highly boosted top quarks [START_REF]Reconstruction of high mass tt resonances in the lepton+jets channel[END_REF].

Mass from velocity of metastable particles

When charged massive stable particles traverse the detector their mass can by determined from simultaneous measurements of momentum and velocity (β). The momentum measurement is usually obtained in the same way as for a muon -i.e. be determining the bending radius of the particle in an externally applied magnetic field. The particle's velocity can be found from precision timing information, or from measurements of energy loss ( dE dx ) or from a combination of both methods. When the mass of the metastable particle has been determined, the full 4vectors of all instances of that particle can be determined event-by-event. This allows the mass of its parents/ancestors to be reconstructed by forming invariant masses along appropriate cascade decay chains.

LHC-related studies have considered the case of heavy leptons [START_REF] Aad | Expected Performance of the ATLAS Experiment -Detector[END_REF][START_REF] Nisati | On the possible detection of massive stable exotic particles at the LHC[END_REF][START_REF] Polesello | Reconstruction of quasi-stable charged sleptons in the ATLAS muon spectrometer[END_REF][START_REF] Kazana | Study of the NLSP from the GMSB models in the CMS detector at the LHC[END_REF][START_REF] Ambrosanio | Measuring the SUSY breaking scale at the LHC in the slepton NLSP scenario of GMSB models[END_REF][START_REF] Ambrosanio | SUSY long-lived massive particles: Detection and physics at the LHC[END_REF][START_REF] Allanach | Detecting exotic heavy leptons at the Large Hadron Collider[END_REF][START_REF] Ellis | Gravitino dark matter scenarios with massive metastable charged sparticles at the LHC[END_REF][START_REF] Raklev | Massive Metastable Charged (S)Particles at the LHC[END_REF] and bound states of heavy coloured objects (so-called R-hadrons) [START_REF] Aad | Expected Performance of the ATLAS Experiment -Detector[END_REF][START_REF] Kilian | Split supersymmetry at colliders[END_REF][START_REF] Hewett | Signatures of long-lived gluinos in split supersymmetry[END_REF][START_REF] Kraan | Discovery potential of R-hadrons with the ATLAS detector[END_REF][START_REF] Hellman | A strategy to detect and identify gluino R-hadrons with the ATLAS experiment at the LHC[END_REF]. Slow-moving particles present particular experimental difficulties because the delay in reaching the outer parts of the detectors (the muon chambers) means they risk being identified with the wrong bunch crossing. The experimental issues associated with triggering and reconstructing such particles have been addressed and are understood [START_REF] Tarem | Can ATLAS avoid missing the long lived stau?[END_REF][START_REF] Tarem | Trigger and reconstruction for heavy long-lived charged particles with the ATLAS detector[END_REF]. For more details on searches and measurements of massive stable particles we refer the reader to a recent review paper dedicated to that topic [START_REF] Fairbairn | Stable massive particles at colliders[END_REF].

Using spatial as well as momentum information

If invisible long-lived particles decay within the detector then the location of the decay vertex in space can be used to provide constraints on the kinematics. Examples of models predicting such displaced vertices include bilinear [START_REF] De Campos | Probing bilinear R-parity violating supergravity at the LHC[END_REF] or baryon-violating [START_REF] Allanach | Extracting the flavour structure of a baryon-number R-parity violating coupling at the LHC[END_REF] R-parity violating supersymmetry, and anomaly-mediated supersymmetry [START_REF] Barr | Discovering anomaly-mediated supersymmetry at the LHC[END_REF].

A demonstration of how the position of the secondary vertex can be combined with direct kinematic information has been given in the context of a gauge-mediated supersymmetry breaking model [START_REF] Kawagoe | Study of the gauge mediation signal with non-pointing photons at the CERN LHC[END_REF]. Cascades terminating in the decay χ0

1 → γ G were considered, which (provided they occur within the tracking volume) produce photons which detectable in the calorimeter but which do not point back to the primary interaction point. The position, arrival time and momentum direction of the photons are used to determine the photon momentum, allowing the (invisible) gravitino momentum to be completely determined. Knowing both the photon and the gravitino momentum, the kinematics of the rest of the decay chain can also be determined.

Multiply branched trees

One way to extract mass information from multi-branched graphs (such as that shown in fig. 12a) is by treating them with the same methods as single decays of unknown internal structure to nbody final states (i.e. ignoring the existence of on-shell intermediate particles). For example one get a good measurement of the Higgs boson mass by constructing a transverse mass variable M T for the decay H → W + W -→ + ν -ν treating it just like a single four-body decay [START_REF] Barr | Measuring the Higgs boson mass in dileptonic W-boson decays at hadron colliders[END_REF]. This method has some merits, but does not make use of the full kinematic information available in such topologies. Multi-branched trees can have multiple on-shell constraints and so can contain a rich spectrum of possible Lorentz invariants. Plotting correlations between appropriate combinations can dilute backgrounds, thereby improving the measurability of kinematic endpoints [START_REF] Huang | Model-independent SUSY masses from 4-lepton kinematic invariants at the LHC[END_REF][START_REF] Huang | Hidden Thresholds: A Technique for Reconstructing New Physics Masses at Hadron Colliders[END_REF].

These decay topologies have also bring to light other interesting features. In particular one can observe in projected variables kinematical cusps [START_REF] Han | Kinematic Cusps: Determining the Missing Particle Mass at the LHC[END_REF]. These features are places in the differential distribution where two curves with different slopes come together, as shown in Figure 12b. Such cusps are a general feature of kinematic distributions not just multi-branched trees. For example one can find a [START_REF] Agashe | Distinguishing Dark Matter Stabilization Symmetries Using Multiple Kinematic Edges and Cusps[END_REF] for a cusp in topology described in Section 3.7. The source of these cusps, as well as the other singularity structures -endpoints and thresholds -are discussed further in Section 4.11.

Kinematic cusps could well be as useful for extracting mass information as endpoints. In fact because the differential distributions are typically populated by large numbers of events near these cusps, uncertainties in cusp positions might well be smaller than endpoint positions.

In these multi-branched decays -as elsewhere -it is possible to use the extra kinematic constraints to select events in which all final state-momenta are well-constrained despite the presence of invisible particles [START_REF] Choi | Reconstructing the Higgs boson in dileptonic W decays at hadron collider[END_REF] (see also Section 5.4).

The contransverse mass

The invariant mass [START_REF] Albrow | Prospects for diffractive and forward physics at the LHC[END_REF] has the property that it is not modified under any operation which transforms all of the particles with the same boost p i → Λ β p i . It is also possible to construct variables which are invariant when different boosts are applied to their constituent particles. In particular, one can construct a variable which is constructed from the sum of two arbitrary Lorentz vectors a and b and which is invariant under equal and opposite boosts of those vectors

b → Λ β b (13) c → Λ -β c. ( 14
)
A variable which satisfies this back-to-back boost invariance condition was defined in [START_REF] Tovey | On measuring the masses of pair-produced semi-invisibly decaying particles at hadron colliders[END_REF],

M 2 C = m 2 B + m 2 C + 2(E b E c + b • c)
where the bold quantities again represent the Euclidean three-vector momenta. Note the plus sign before the dot product which distinguishes M C from the invariant mass (8). In the limit when m B and m C are negligible, and the visible particles have originated from the decay A → BC, one can

see that M C = √ 4E b E c = 2p
* where p * was defined in [START_REF] Adolphi | The CMS experiment at the CERN LHC[END_REF].

Because of our ignorance (in a hadron collider) of the z-momentum of the initial state it is useful to define the related quantity constructed from purely transverse quantities

M 2 CT = m 2 B + m 2 C + 2 (e b e c + b T • c T ) ( 15
)
where e is defined in [START_REF] Roberts | A new type of Cerenkov detector for the accurate measurement of particle velocity and direction[END_REF]. This quantity is known as the contransverse mass.

Neither M C nor M CT have found much application for single two-body decays, but they have interesting invariance properties when pairs of identical semi-invisible decays (see Section 4.5) are produced back-to-back in the transverse plane. The resulting contra-linear invariance properties are of interest because back-to-back configurations with extremal M CT can be generated from the threshold-production ('at rest') configuration by applying the transformations (13) to the respective parent particles.

VARIABLES FOR PAIRS OF CASCADE DECAY CHAINS

We have seen that the transverse mass M T (Equation 9) is useful in situations involving

A → B / C
where / C is the only invisible particle in the event. This begs the question: "What comparable tools can be employed in situations where there are two identical invisible particles in each event -such as might arise in models with stable or meta-stable weakly interacting particles whose creation is protected by a multiplicative quantum number?" (R-parity conserving supersymmetry [START_REF] Dimopoulos | Softly Broken Supersymmetry and SU(5)[END_REF] and universal extra-dimensional models [START_REF] Cheng | Bosonic supersymmetry? Getting fooled at the CERN LHC[END_REF] being just two examples of such models.)

Techniques for extracting mass information from pairs of cascade decays are described in this section. First we introduce the sorts of event-topology which are relevant to this question and the notation convention we will use when describing them.

The topologies of interest in this section (shown in Figure 13) share the common feature that each event is composed of two "sides" -where each "side" consists of a decay chain which terminates in an invisible particle and one or more visible particles. For obvious reasons we call such events "double sided". The sides need not be identical (Figure 13(e) provides an asymmetric example, and we will discuss asymmetric examples in more detail in section section 4.7) though topologies with identical chains on each side have historically been the most studied.

To distinguish the sides of the particles when discussing events of these types, we used unprimed indices for the particles on one side, and primed indices for particles on the other. Where possible we use letters nearer the beginning of the alphabet for the most senior parent particles and letters nearer the end of the alphabet for the most junior daughter particles. For example, the simplest double sided topology (Figure 13(a)) might be denoted

(A → B / C) + (A → B / C ).
If a pair of particles is produced in the collision, and then each of these goes on to decay, there are both additional constraints and additional complications compared to the single decay case.

New combinatorial ambiguities arise, since it is no longer generally possible to associate a particular visible particle with one or other of these decay chains. In addition there are constraints which link information between the two cascades -for example the missing transverse momentum is usually assumed to be equal to the sum of the momenta of any invisible particles from both decay chains.

In this section we describe the simplest non-trivial example of a pair of decay chains -that being an identical pair of single-step decays, with each decay producing one visible and one invisible G labels 'upstream' particles, as defined in the text.

daughter (Figure 14). We examine the kinematic constraints for that case, and then go on to examine more complicated topologies including multi-step cascade decays and non-identical chains.

Identical semi-invisible pair decays: M T 2

We already saw in Section 3.2 that the transverse mass could be applied in circumstances where there is a single mother particle (frequently the particle whose mass we hope to bound) decaying in one or more steps ultimately into a single invisible particle (whose mass we may not know) and one or more visible particles.

The M T 2 variable [START_REF] Lester | Measuring masses of semiinvisibly decaying particles pair produced at hadron colliders[END_REF] (also known as the stransverse mass) 4is the analogue of the transverse mass which can be applied in the situations where there are not one but two parent particles, each undergoing decays to a single invisible particle (whose mass we may not know) and one or more visible particles. The most general topology of this type may be seen in Figure 14 while specific examples may be seen in Figure 13.

The usual definition of M T 2 in this case written for the general case shown in Figure 14 The sixth and final input is a pair of hypothesised masses for the invisible particles ( m / C and m / C ). To distinguish the real from hypothesised masses, the latter have been given a tilde. In principle these two hypothesised masses could be taken to be different from each other (see Section 4.7) however in practice most studies take them to be identical. When both hypothesised masses are taken to be identical that common value is often denoted by χ. In these terms, the usual definition of M T 2 is as follows:5

M T 2 (m B , m B , b T , b T , / p T ; χ) ≡ min / c T +/ c T =/ p T max M T , M T . ( 16
)
where M T is the transverse mass constructed from m B , m / C (= χ), b T and / c T , while M T is the transverse mass constructed from m B , m / C (= χ), b T and / c T , and where the minimisation is over all hypothesised transverse momenta / c T and / c T for the invisible particles which sum to the observed missing transverse momentum. In Equation 16the dependence on χ (or equivalently on m / C and m / C in the case that they differ) has been separated from the dependence on the other inputs by a semi-colon to emphasise that the quantities to the left of the semi-colon are observables, while χ to the right is instead a parameter. M T 2 might thus be better described not as an observable in the usual sense, but rather as an "observable function" -in this case a function of χ.

There are many parallels between the stransverse and the transverse mass. Most importantly (as was first mentioned in section 3.2) the transverse mass can be viewed in two different but equivalent ways: either as an event-by-event lower bound on the mass of the parent particle (in terms of a mass hypotheses for the invisible particle), or as a curve delineating the boundary between the regions of the two-dimensional space of the unknown parent and daughter masses which are -or are not -consistent with a particular event. The same two interpretations are valid for the stransverse mass:

In the first interpretation, most frequently used in the case that particles A and A (though not event-by-event lower bound for m A in terms of a hypothesis (i.e. χ) for the mass of the invisible particles. It may be shown [START_REF] Lester | Measuring masses of semiinvisibly decaying particles pair produced at hadron colliders[END_REF][START_REF] Barr | The Truth behind the glamour[END_REF] that it is possible to saturate this bound with certain kinematic configurations. A typical usage pattern therefore would be to plot a histogram of M T 2 (χ), over all events, with the intention of identifying a clear kinematic end-point in that distribution located at m A -at least for the case where χ is chosen to be equal to the true value m / C . This technique has been used by the CDF collaboration to measure the top quark mass in the dilepton channel [START_REF]Simultaneous template-based top quark mass measurement in the lepton+jets and dilepton channels using 3.2fb -1 of CDF data (the dilepton channel uses a new variable m T 2 .)[END_REF] and has been suggested for the same use at the LHC [START_REF] Cho | Measuring the top quark mass with m T 2 at the LHC[END_REF]. The freedom to re-evaluate M T 2 at different values of χ corresponds to the need to obtain different bounds on m A under differing assumptions about the mass of the invisible particles that A and A decayed into.

The second interpretation of M T 2 is that it (or more specifically the functional form of the curve M T 2 (χ)) describes, for each event, the boundary between the region of (parent, daughter)mass space that is consistent with that event and the region that is inconsistent with that event in the manner indicated in Figure 15. The first explicit proof of this property was recorded in [START_REF] Cheng | Minimal kinematic constraints and M T 2[END_REF] and similar ideas have been expressed elsewhere [START_REF] Baumgart | Discovery and measurement of sleptons, Binos, and Winos with a Z[END_REF]. Viewing M T 2 as a "boundary of a consistent region of mass space" is a powerful idea, not only because it provides a different way of understanding M T 2 , but also because it allows us to see that the transverse mass and even the ordinary invariant mass could similarly have been defined as such boundaries. Indeed, once it has been seen that the transverse mass M T could have been defined as the boundary of an allowed region, the proof that the stransverse mass is such a boundary follows immediately from its usual min max definition [START_REF] Battaglia | Contrasting supersymmetry and universal extra dimensions at the CLIC multi-TeV e+ e-collider[END_REF]. The interpretation of M T 2 as a boundary also shows us that generalisations of M T 2 -for example to situations with dissimilar parent masses -would ideally be constructed so that they give the boundary of the consistent region of an "extended" mass-space with a higher dimensionality.

Dependence of M T 2 on the WIMP mass(es)

Different approaches can be made to the problem of the dependence of M T 2 on the a priori unknown parameter χ, the hypothesis for the mass of the invisible particles. If one is using M T 2 as a bound on the mass of the parent particle, one possibility would be to take the most conservative value -i.e. to set χ = 0. Since

M T 2 (χ = 0) < M T 2 (χ > 0) ≤ m A
using a trial value χ = 0 will return a value which is certainly less than m A , the mass of the parent. This conservative approach has been shown to be useful when using M T 2 as a tool to distinguish events which are not consistent with particular Standard Model decay topologies [START_REF] Barr | Measuring slepton spin at the LHC[END_REF][START_REF] Lytken | Prospects for slepton searches with ATLAS[END_REF][START_REF] Barr | The race for supersymmetry: using M T 2 for discovery[END_REF], because the invisible particles of the Standard Model -the neutrinos -do indeed have very small masses and so satisfy χ ≈ 0. The problem with assuming χ = 0 is that for m / C = 0 the bound is not saturated; while M T 2 (χ = 0) < m A for all events, there are no events for which M T 2 (χ = 0) approaches m A , so one cannot use the end-point of the M T 2 (χ = 0) as a measurement of m A . The first example of M T 2 being used in LHC data to separate expected standard model backgrounds from potential signals from supersymmetry is shown in Figure 16.

To measure masses we want to use the property that if the correct hypothesis is made for the mass of the invisible particle, then M T 2 returns a value ≤ m A , with equality for some state configurations. The dependence of the M T 2 distribution on the unknown mass of the invisible daughter particle χ is therefore important. For a distribution of interest to depend upon an unknown parameter might be seen as a disadvantage. But it is possible to turn this argument on its head; the fact that the distribution of M T 2 (χ) depends on χ might allow us to simultaneously extract both the mass of the parent and the mass of the invisible daughter.

To see how the dependence of M T 2 (χ) on χ can be made to help us, consider the envelope of the maximum of the curves M T 2 (χ) over all events. The individual bounding curves from different events will generally have different shapes, but all must share the property (by construction) that

M T 2 (χ = m / C ) ≤ m A , ( 17
)
with the equality satisfied for some set of kinematic configurations. In general different events will be maximal for χ < m / C and for χ > m / C . However the property [START_REF] Battaglia | Contrasting supersymmetry and universal extra dimensions at colliders[END_REF] means that the bounding

d dχ max events M T 2 (χ) χ=m / C - = d dχ max events M T 2 (χ) χ=m / C + .
This feature was first spotted in simulations of pairs of three-body gluino decays g → q q χ0 1 [START_REF] Cho | Transverse mass for pairs of gluinos[END_REF] (see also Figure 17) but has also been explored for simpler and more complex topologies [START_REF] Gripaios | Transverse observables and mass determination at hadron colliders[END_REF][START_REF] Barr | Weighing WIMPs with kinks at colliders: Invisible particle mass measurements from endpoints[END_REF][START_REF] Cho | Measuring superparticle masses at hadron collider using the transverse mass kink[END_REF].

The existence of this 'kink' in the M T 2 endpoint makes it tempting to infer that it will be straightforward to extract both the parent and the invisible-daughter masses. However for a substantial change in the gradient d dχ max events M T 2 (χ) at (m / C , m A ) there must be contributions from events with substantially different properties. Pairs of two-body decays in which the sum of the parents' transverse momenta is zero, and which have fixed m A will not produce kinks because the kinematics are so constrained that the gradients at χ = m / C ± have to be equal [START_REF] Barr | Weighing WIMPs with kinks at colliders: Invisible particle mass measurements from endpoints[END_REF].. The event-by-event changes which lead to measurable 'kinks' in the end-point function can come from the two different sources below [START_REF] Cho | Transverse mass for pairs of gluinos[END_REF][START_REF] Gripaios | Transverse observables and mass determination at hadron colliders[END_REF][START_REF] Barr | Weighing WIMPs with kinks at colliders: Invisible particle mass measurements from endpoints[END_REF]. FIG. 17: The points show how a measured upper bound of the M T 2 distribution for pairs of three-body decays (g → q q χ0 1 ) depends on the a prior unknown mass of the invisible particle. The straighter red (more curved blue) line shows the configuration which is maximal for χ greater than (less than) m / C . The area above and to the left of both curves gives the domain of allowed values of (m χ0

1 , m g ). Notice the change in gradient in the envelope curve near [START_REF]Reconstruction of high mass tt resonances in the lepton+jets channel[END_REF]780); the coordinate of the position of this kink corresponds to the masses of the neutralino and the gluino used in the simulation. Adapted from [START_REF] Cho | Transverse mass for pairs of gluinos[END_REF].

• Different values of m B , the invariant mass of the visible-particle subsystem, will lead to different boundary curves in the space (m / C , m A ). This mechanism is the dominant source of the kink seen in fig. 17, where the q q invariant mass changes significantly between events.

Other topologies in which the visible system is a composite constructed from the sum of two or more visible particles will share this behaviour.

• When one allows the two-parent center-of-mass to be boosted, the extremal boundary curves correspond to configurations with arbitrarily large parent momenta. The bounding curves for χ < m / C (χ > m / C ) come from events in which the invisible particle is emitted parallel to (anti-parallel to) the boost direction. Systems with finite boosts have correspondingly less-pronounced kinks.

The two-parent center-of-mass frame can be expected to have a small transverse boost unless the parents were themselves created from previous decays, or there was large initial state radiation. The negative sum of the transverse momenta of the parents -i.e. the momentum against which those parents are recoiling -is often known as the 'upstream' momentum, P UT and is represented in Figure 14 by the label G.

Since only a small number of signal events are expected to contribute near the kinematic endpoint, it might be considered difficult to extract information from the χ dependence of max events M T 2 (χ) without having a very good knowledge of the backgrounds. Nevertheless attempts to measure the invisible particle mass from the kink (or variables characterising the location of that kink) have shown some promise. The position of the kink of the first sort (coming from a variable-mass visible particle system) was successfully captured in simulations [START_REF] Cho | Transverse mass for pairs of gluinos[END_REF].

A method for exploiting the second (boost-generated) kink in the case where the invariant mass of the visible particle system is fixed has been proposed in [START_REF] Konar | Superpartner mass measurements with 1D decomposed[END_REF] and further explored in [START_REF] Cohen | Extracting the Dark Matter Mass from Single Stage Cascade Decays at the LHC[END_REF]. The central observation of that method is that one can construct one-dimensional analogues of M T 2 using only the components of the visible momenta parallel to (perpendicular to) the upstream momentum direction. Because the perpendicular analogue M T 2⊥ has a distribution with an endpoint (and indeed a shape) which is independent of P UT it can be used as a 'control' sample against which the M T 2 distribution (constructed from components parallel to P UT ) can be compared.

For a pair of two-body decays the experimental problems in understanding the behaviour of the parallel and the perpendicular endpoints are likely to be considerable. For that case, the fraction of near-extremal events will be small, the backgrounds important, and the systematic uncertainties in fitting the shapes are likely to be significant. It is not yet clear whether the method could be practical in such cases.

We note that while these 'kinks' have been most frequently studied for topologies containing a pair of decay chains, the same effects also generate kinks in single decays or decay chains [START_REF] Cho | Transverse mass for pairs of gluinos[END_REF][START_REF] Gripaios | Transverse observables and mass determination at hadron colliders[END_REF][START_REF] Barr | Weighing WIMPs with kinks at colliders: Invisible particle mass measurements from endpoints[END_REF],

or in asymmetric decay chains (for which see Section 4.7).

The mathematical structure of these 'kinks' is discussed further in Section 4.11.

Decomposing M T 2 with respect to upstream momenta: (M T 2⊥ and M T 2 )

It can be useful to decompose M T 2 into "components"6 which are perpendicular or parallel to the upstream transverse momentum. These components are called, respectively, M T 2⊥ and M T 2 [START_REF] Konar | Superpartner mass measurements with 1D decomposed[END_REF]. One advantage of this decomposition is that the component perpendicular to the upstream transverse momentum, M T 2⊥ , has no dependence on the magnitude of the recoil supplied by the upstream transverse momentum for any value of the trial mass χ. 7 A second advantage of this decomposition is that under a reasonable set of circumstances, the shape of the differential distribution of M T 2⊥ becomes fully determined (i.e. it does not depend on unknown parameters such as the unknown centre of mass energy) [START_REF] Konar | Superpartner mass measurements with 1D decomposed[END_REF]. This existence of this universal shape might therefore make it possible to fit the distribution more accurately and make it possible to extract masses by a secondary step [START_REF] Konar | Superpartner mass measurements with 1D decomposed[END_REF]. The kinematic endpoints of the M T 2⊥ and M T 2 distributions may be found in the appendix in equations (A.20) and (A. [START_REF] Meade | Top partners at the LHC: Spin and mass measurement[END_REF]). Compare the related properties of M CT ⊥ and M CT discussed in Section 4.5.

Identical chains of decays

If one is willing to assume that the visible particles originate from two identical two-step decays of identical sparticles, i.e. then several M T 2 variables can be calculated for each event [START_REF] Serna | A short comparison between m T 2 and m CT[END_REF][START_REF] Burns | Using subsystem m T 2 for complete mass determinations in decay chains with missing energy at hadron colliders[END_REF] (first suggested in the context of the variable M CT in [START_REF] Tovey | On measuring the masses of pair-produced semi-invisibly decaying particles at hadron colliders[END_REF]). Using the endpoints of three versions of M T 2 , changing what is interpreted as visible transverse momentum, missing transverse momentum and upstream momentum one can, in combination with the dilepton endpoint, identify the correct masses (assuming perfect resolution and no combinatoric ambiguity). In [START_REF] Burns | Using subsystem m T 2 for complete mass determinations in decay chains with missing energy at hadron colliders[END_REF] a similar approach to forming M T 2 subsystems is proposed, and the origin of kinks in the maxima of the various distributions is explored. Several methods of extracting particle masses using multiple M T 2 distributions are introduced, including a hybrid method that uses the dilepton endpoint. The kink analysis is discussed in more detail in this review in section section 4.2 (Page 34), and the hybrid method in section section 4.9 (Page 45).

An alternative to finding limits is to make hypotheses about the particle masses, and then for each mass hypothesis to count the number of events for which there are real positive solutions for the energies of the unseen particles ('consistent events') [START_REF] Cohen | Extracting the Dark Matter Mass from Single Stage Cascade Decays at the LHC[END_REF][START_REF] Cheng | Mass determination in SUSY-like events with missing energy[END_REF]. A region is formed in threedimensional mass space which is consistent with all events, and this tends to a minimum volume as the number of events approaches infinity. The correct masses correspond to a point at the region's tip. By looking for the point at which the number of consistent events is a maximum, one can obtain an estimate for all three masses. A computer library called WIMPMASS [START_REF] Cheng | WIMPMASS library[END_REF] that facilitates implementation of the techniques of [START_REF] Cheng | Mass determination in SUSY-like events with missing energy[END_REF] may be downloaded.

Events containing slightly longer pairs of identical chains are amenable to other treatments. For example, under the hypothesis that the invisible particles are massless (relevant in many GMSB models with gravitinos in the final state) events containing a pair of identical decay chains of the form are fully reconstructible even though there are two unseen particles in the final state. Such a reconstruction is demonstrated in the appendix of [START_REF] Hinchliffe | Measurements in gauge mediated SUSY breaking models at LHC[END_REF] using a GMSB-like scenario as an example with two copies of the chain χ0

2 → ˜ ∓ ± → χ0 1 ∓ ± → Gγ ∓ ± i.
If one drops the assumption that the unobserved final state particles are massless, then there are too few constraints from a single event to reconstruct the event. However we will return to this double-chain later in Section 5 where we will see that there are a number of techniques that would permit the masses to be recovered if one is prepared to consider more than one event simultaneously.

Pair decays with small upstream momentum: M CT

Identical pairs of semi-invisible decays in which the parents had zero upstream momentum (i.e. Figure 14 with recoil momentum g=0) have interesting properties if the visible daughters are used as inputs to the M CT variable [START_REF] Tovey | On measuring the masses of pair-produced semi-invisibly decaying particles at hadron colliders[END_REF]. The definition of M CT used for pair decays of the above type uses only the momenta of the two visible decay products (or systems of products) and is as follows:

M 2 CT = m 2 B + m 2 B + 2 e b e b + b T • b T ( 18
)
where e b is once again defined as in Equation [START_REF] Roberts | A new type of Cerenkov detector for the accurate measurement of particle velocity and direction[END_REF]. Take note of the subtle difference between the definition of M CT in equation ( 18) for pair decays, and the definition of M CT for in equation [START_REF] Smillie | Distinguishing spins in supersymmetric and Universal Extra Dimension models at the Large Hadron Collider[END_REF].

The definition in (18)8 uses only half of the final state momenta (namely those of the two systems B and B in Figure 14 which are visible!) while the definition in equation ( 15)9 uses all the final state momenta of a single two-body decay.

M CT enjoys the property that it is invariant under equal and opposite boosts in the transverse plane of the primed and un-primed systems. This insensitivity is a welcome feature as equal-mass parent particles produced in hadron collisions will have (in the absence of initial-state radiation) back-to-back transverse boosts, and the magnitude of those boosts will be unknown and unmeasurable if there are invisible daughters.

It has been shown [START_REF] Serna | A short comparison between m T 2 and m CT[END_REF] that M CT is equal to M T 2 in the special case where χ = 0, the visible particles are massless, and the upstream transverse momentum is zero. Since it has also been shown [START_REF] Cheng | Minimal kinematic constraints and M T 2[END_REF] that M T 2 delineates the boundary between allowed and disallowed regions in mass space, we can see that M CT has the same bounding property in mass space under these conditions.

Although M CT does not quite describe the boundary of the allowed region of mass space when the event contains non-zero upstream momentum it is nevertheless bounded above by a value M max CT which is calculable for any boost, so a "boost-corrected" M CT can be used to recover a good determination of the masses in sequential decays [START_REF] Polesello | Supersymmetric particle mass measurement with the boost-corrected contransverse mass[END_REF]. The combination of masses determined by this maximum value of this contransverse mass is (in the limit where m B = m B = 0, and P UT =0)

M max CT = m 2 A -m 2 / C m A = 2p * .
To within a factor of two, this endpoint is therefore telling us the momentum of the daughters in the rest frame of the parent. This simple dependence of M max CT on the unknown parameter m / C may make distributions of M CT very convenient for later interpretation, since the endpoint can be measured for one hypothesised value of m / C (for example 0), and later reinterpreted for other trial invisible-particle masses. This can be compared to the behaviour of M T 2 which (though it exactly describes the bound in mass space for any trial mass m / C and under arbitrary boosts of the parents) has a non-trivial (and boost-modified) dependence on the invisible particle trial mass, χ.

Because of the ease of reinterpreting M CT for different m / C , one could suggest that even in the case of arbitrarily boosted parents (for which the variable does not quite describe the boundary of a domain in mass space) the M CT distribution could still be the most suitable choice for presenting endpoints kinematics relevant to mass measurements [START_REF] Polesello | Supersymmetric particle mass measurement with the boost-corrected contransverse mass[END_REF].

Two one-dimensional decompositions of M CT were proposed in [START_REF] Matchev | A general method for determining the masses of semi-invisibly decaying particles at hadron colliders[END_REF] and named M CT ⊥ and M CT . These were constructed from the components of the visible momenta in the directions respectively perpendicular to and parallel to the upstream transverse momentum two-vector. Since the projection of a vector onto a plane is not changed by any boost which is perpendicular to that plane, the distributions of M CT ⊥ is unmodified by the magnitude of the boost which the visible systems acquire from recoil against the upstream transverse momentum. Indeed the M CT ⊥ distribution (or at least that part of it which has M CT ⊥ > 0) has a universal shape in the absence of spin correlations [START_REF] Matchev | A general method for determining the masses of semi-invisibly decaying particles at hadron colliders[END_REF]. This distribution does not depend on, for example, ŝ or the longitudinal boost of the parents, which it is claimed should make it much easier to fit in order to extract the kinematic limit M max CT ⊥ . The maximum of the M CT distribution does depend on the boost, but does so in a simple calculable way [START_REF] Matchev | A general method for determining the masses of semi-invisibly decaying particles at hadron colliders[END_REF]. A summary of some of these results may be found in the Appendix in equations (A.22) and (A.23). Compare also the related properties of M T 2⊥ and M T 2 discussed in Section 4.3.

In [START_REF] Cho | Shining on buried new particles[END_REF] a variant has proposed which considers separately the M CT for each side of a pair-decay event, and then uses a M T 2 -like construction but now using M CT rather than M T on each branch

M CT 2 = min / c T +/ c T =/ p T max M CT , M CT .
With a judicious choice of mC , the resultant variable has a Jacobian which increases the density of events near the kinematic endpoint M max CT 2 . The shape of the distributions for typical Standard Model backgrounds were not investigated in [START_REF] Cho | Shining on buried new particles[END_REF], but provided that the backgrounds near the peaked endpoint structures can be reduced [START_REF] Barr | A comment on 'Amplification of endpoint structure for new particle mass measurement at the LHC[END_REF] then this variable could increase the observability of kinematic end-points and the precision with which they might be determined.

Multiple invisible daughters per chain

The generalisation of M T 2 to cases with more than one invisible particle in each decay chain have been considered in [START_REF] Barr | Discovering anomaly-mediated supersymmetry at the LHC[END_REF][START_REF] Barr | Measuring the Higgs boson mass in dileptonic W-boson decays at hadron colliders[END_REF][START_REF] Chang | Neutrino alternatives for missing energy events at colliders[END_REF] In [START_REF] Chang | Neutrino alternatives for missing energy events at colliders[END_REF] it is recognised that while chains containing multiple light invisible particles will generally have different kinematic properties to those containing smaller numbers of heavier invisible particles, nevertheless there are cases (such as the decay of a neutralino to multiple neutrinos χ → ν . . . ν) for which the presence of multiple invisibles would be very difficult to infer.

Non-identical decay chains

If the two decay chains do not contain identical mass particles along their length then the results above need some modification. In some cases it may possible to find parts of the chains (particular decays) which are identical and to apply the identical-chain methods of the previous sections to those subsystems [START_REF] Nojiri | Handling jets + missing E T channel using inclusive m T 2[END_REF].

The generalisation of the above methods to two-chain processes with non-identical masses has been considered in [START_REF] Barr | Transverse masses and kinematic constraints: from the boundary to the crease[END_REF][START_REF] Konar | Dark matter particle spectroscopy at the LHC: Generalizing m T 2 to asymmetric event topologies[END_REF]. For example in [START_REF] Barr | Transverse masses and kinematic constraints: from the boundary to the crease[END_REF] it was shown that in a pair decay with differentmass parents one can hypothesise a value for the ratio of the parents masses m A /m A , and produce a distribution which is sensitive to their product m A m A . In principle one can also determine the correct value of the input ratio m A /m A from a kink structure in the endpoint of this distribution in a manner reminiscent of, but different to that discussed in Section 4.2 (when changes due to a different input parameter -the hypothesised mass χ of the invisible daughters -were considered).

A similar method can be employed to measure the mass of non-identical mass invisible particles: by using either the inverse of the transverse mass (M -1 T ) [START_REF] Barr | Transverse masses and kinematic constraints: from the boundary to the crease[END_REF] or by constructing a variant of M T 2 with two different invisible particle masses [START_REF] Barr | Transverse masses and kinematic constraints: from the boundary to the crease[END_REF][START_REF] Konar | Dark matter particle spectroscopy at the LHC: Generalizing m T 2 to asymmetric event topologies[END_REF]. With either variant, ridges or creases in the domain of consistent masses are found. These crease structures -which are generalisations of the 'kinks' discussed in Section 4.2 -intersect at the special point in the parameter space for which the assumed masses (or relationships between masses) were correctly hypothesised. In [START_REF] Konar | Dark matter particle spectroscopy at the LHC: Generalizing m T 2 to asymmetric event topologies[END_REF] the latter method is studied in detail, the P UT dependence of the modified M T 2 distribution determined, and the configurations of the bounding events described.

Inclusive pair-decay variables

For double sided event topologies in which one (or both) of the two equal mass parent particles generates a large number of visible particles in its decay (e.g. as suggested in Figure 14) it is rea-sonable to ask the question "Could one, in a real detector, tell which of the observed/reconstructed final state particles or calorimetric energy deposits belong to B (i.e. to one side) and which belong to B (i.e. to the other side)?" In some particular cases (principally those in which A and A are guaranteed to be produced with large and opposite boosts) the answer to the above question might be "yes", as the constituents of B and B might be found in opposite hemispheres. This leads to so-called "hemispheric M T 2 " techniques [START_REF] Nojiri | Inclusive transverse mass analysis for squark and gluino mass determination[END_REF], previously investigated in [47,[START_REF] Barnett | Discovering supersymmetry with like sign dileptons[END_REF][START_REF] Baer | Signals for minimal supergravity at the CERN large hadron collider: Multi-jet plus missing energy channel[END_REF]. The remainder of the time the answer to the above question is likely to be "no", in which case there are a few alternative approaches. One such approach, which is specifically targeted at analysing these pair decays, is to define the inclusive variable M T Gen [START_REF] Lester | Mass scale measurements in pair-production at colliders[END_REF]. After we have discussed M T Gen , we will discuss an alternative approach involving ŝ1/2 min . It may be shown that M T Gen , canonically defined [START_REF] Lester | Mass scale measurements in pair-production at colliders[END_REF] as the smallest value of M T 2 obtained after trying all possible partitions of the visible momenta (excluding those visible momenta deriving from G in Figure 14 in cases where it is possible to determine which they are) between the two sides, again has an interpretation as the boundary. The equivalence between these two different definitions of M T Gen was shown in [START_REF] Barr | Transverse masses and kinematic constraints: from the boundary to the crease[END_REF] using insights from [START_REF] Cheng | Minimal kinematic constraints and M T 2[END_REF]. In this case M T Gen is the boundary of the region of parent/invisible mass-space which is consistent with the hypothesis that the visible momenta were, in some unknown order, derived from A or A in association with two invisible particles. The M T Gen endpoint thus provides a constraint which tells you the parent particle mass as a function of a hypothesised daughter particle masses, in just the same way that M T 2 does. In short, M T Gen is the natural generalisation of M T 2 to events in which you believe there is a pair of parent particles but whose momenta you cannot be sure of uniquely assigning to their respective parents.

During the process of scanning all possible partitions of the visible momenta into two groups (one for each side of the event) M T Gen has to combine observed particle four-momenta together into resultant transverse-momenta for input to M T 2 . There are a variety of ways in which this combination can be done, each of which has merits and demerits. M T Gen is not unique in needing to combine four-momenta to produce transverse-momenta, but we will only discuss this issue in the context of M T Gen . Options for determining the 1+2 vector for the visible system (all of which have the same space-like components p T but differ in their time-like component) include [155]: (a) summing the Lorentz four vectors (E, p) then ignoring the final p z ; (b) summing the Lorentz 1+2 vectors (e, p T ); (c) projecting the total energy onto the transverse plane (using E T = |E| sin θ) after summing the constituent four-vectors; (d) as for (c) but projecting each constituent before summing. Which of these combinations is used for combining momenta has important additional consequences for the dependence of these variables on initial state radiation and/or multiple parton interactions as will be discussed shortly.

A second and alternative approach to analysing pair-decays inclusively involves returning to ŝ1/2 min [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF], which the reader will recall has already featured in this review in Section 2 where it was discussed as a global event variable. All that is needed to re-use ŝ1/2 min in the context of the pair-decay hypothesis it is to revisit how it should be interpreted. For example, [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF] noted that when sparticlepair decays are generated without initial state radiation and without multiple parton interactions, then there is a strong correlation between (a) the position of the peak of the ŝ1/2 min distribution and (b) the sum of the masses of the two parent particles. One might hope to use this correlation to measure those parent particle masses. 10 It is noted in [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF] that this correlation has a status closer to a "conjecture" or a "rule of thumb" than a "necessity", since it results from a fortuitous (and somewhat process dependent) cancellation between two effects with quite different origin [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF]. The first effect is that the true value of ŝ1/2 exceeds the value it would take for threshold production by an unknown and almost certainly different amount in each event (i.e. ŝ1/2 > 2m parent ) with the mean offset carrying some production-process dependence. The second effect is that the bound on ŝ1/2 provided by ŝ1/2 min seems to be lower than ŝ1/2 by approximately the same amount as the excess in the first effect. In consequence we have a somewhat process dependent coincidence, with two quite different effects approximately cancelling out, leading to the correlation between the peak of the ŝ1/2 min distribution and 2m parent . 11 The above correlation was demonstrated in simulations without initial state radiation. It is noted in [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF][START_REF] Papaefstathiou | Effects of QCD radiation on inclusive variables for determining the scale of new physics at hadron colliders[END_REF] that the effect of inclusion of initial state radiation (ISR) and multiple parton interactions (MPI) can be to substantially increase the measured values of ŝ1/2 and to substantially modify the shape and position of the peak of the ŝ1/2 min distribution by large factors (order 2). This 10 ŝ1/2 min shares with MT 2, MT Gen and similar variables the property that it measures parent masses for a given hypothesised mass of the invisible daughters. So when we say "this correlation can be used to the masses of the two parent particles" we mean for suitable hypothesised invisible particle masses. 11 Note the difference here between ŝ1/2 min and MT Gen . The construction of the MT Gen variable makes direct use of the pairwise hypothesis, whereas the construction of ŝ1/2 min does not. However both can have the pairwise hypothesis applied to their interpretation, and consequently both can potentially say useful things about pairwise events. Whereas an MT Gen measurement places a direct and calculable bound on the parent masses in each event (subject to the pairwise hypothesis), a ŝ1/2 min measurement of those parent masses is reliant on a correlation based on a fortuitous cancellation between quite different and process dependent effects). This does not mean that either variable is better or worse than they other. The two variables are simply different. Which you should use depends on which assumptions you do or do not wish to make and at what level you wish to make them. For example, if you are prepared to make a strong pairwise hypothesis about the pairwise nature of all the events in your sample, then you should perhaps use MT Gen since it can make more use of that hypothesis. On the other hand, it might be advantageous to avoid making the pairwise assumption at construction, so that you can interpret the same set of events in many ways. In which case ŝ1/2 min may be more useful.

means that, in the presence of ISR, the correlation spotted in [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF] requires either accurate modelling of such effects, or alternatively requires care to reduce the effect of ISR (by for example restricting attention to momenta in the small |η| region).

Sensitivity to initial state radiation and multiple parton interactions is by no means unique to ŝ1/2 min . For example, M T Gen distributions also can gain undesirable high tails when ISR or other momenta not deriving from the parents are "mistakenly" identified as the products of the parents and used as inputs. Indeed inclusive variables tend, by construction, to be sensitive to all objects in the final state, and so will, to a greater extent than more selective variables, tend to contain contributions from initial state QCD radiation as well as decay products from heavy object decays. Inclusive variables which combine four momenta using options (a) and (c) above are often particularly sensitive to ISR, etc, since these momentum combination schemes are more sensitive to the appearance of a small number of momenta at high rapidity. For all such variables, the contributions from ISR and MPI will need to be well modelled before the variables can be used in precision constraints.

It has been shown that QCD radiation and multiple parton interactions can play an important role in modifying inclusive distributions [START_REF] Konar | √ ŝmin : A global inclusive variable for determining the mass scale of new physics in events with missing energy at hadron colliders[END_REF][START_REF] Papaefstathiou | Effects of QCD radiation on inclusive variables for determining the scale of new physics at hadron colliders[END_REF]156]. In [START_REF] Papaefstathiou | Effects of QCD radiation on inclusive variables for determining the scale of new physics at hadron colliders[END_REF] it is even argued such dependence should be regarded as beneficial, since it allows us to test not only our understanding of QCD, but also our understanding of the link between the scale of the QCD radiation and the mass scale of the particles that were involved in its generation. In this sense, accurate modelling of the sensitivity of inclusive variables to this radiation might in itself lead to indirect mass determinations or constraints.

One final approach to controlling the effect of ISR is to attempt to find ways of "removing" it from the analysis. Such techniques presuppose that the decay chain of interest can be well reconstructed from only a subset of the jets in the final state. In such a case, this information can be used to discard some jets (under the assumption that they had a high probability of coming from initial-state radiation) and improve the mass reconstruction [START_REF] Alwall | Novel reconstruction technique for New Physics processes with initial state radiation[END_REF].

Hybrid variables

The concept of hybrid (as opposed to per-event and per-dataset variables) was introduced in Section 1.4. What use might hybrid variables be for pair decays? No single event containing a qll-chain of Section 3.5 can generate the constraint on the slepton and neutralino masses seen in Equation 12-however such a constraint can emerge from the "dilepton edge" by consideration of the sample of events as a whole. In a scenario where it is foreseen that such a constraint could be obtained from a large sample of events, one might profit from constructing a variable which "reinterprets" each event in the light of that constraint. Ideally this process injects valuable additional information into each event, and the resulting hybrid variables are more powerful than their nonhybrid relatives. This supposes, however, that the information injected is "good" or "relevant" to the event into which it was injected, which need not be true. Successful application of hybrid variables is thus limited by the degree of homogeneity in the samples of events to which they might be applied, the extent to which the events in those samples satisfy the "injection hypothesis", and the degree to which both these requirements could be verified.

We will now look at two hybrid variables that apply the dilepton edge constraint to pair decays, M 2C and M 3C .

The hybrid variables M 2C [START_REF] Barr | The precision determination of invisible-particle masses at the LHC[END_REF][START_REF] Ross | Mass determination of new states at hadron colliders[END_REF] and M 3C [START_REF] Barr | Precision Determination of Invisible-Particle Masses at the CERN LHC: II[END_REF] use as their hybrid "ingredient" the mass relationships obtained from the "dilepton edge" of the qll-chain. Using this ingredient, together with M T 2 , these variables reconstruct event-by-event lower and/or upper bounds on the LSP mass.

The only important difference between M 2C and M 3C is the topology to which each variable is applied. Both have the same final state, but the internals are different. The topologies for each variable are shown side-by-side in Figure 18. M 2C assumes a pair of identical particles, decaying identically to one LSP and two SM particles. M 3C requires two sequential two-body decays per branch, with an intermediate real particle. In both cases, the "daughter" particles are commonly taken to be the lightest neutralino, and the "parent" particles are taken to be the second lightest neutralino. The slepton, if it is on shell, is treated as neither a parent nor a daughter but as an internal particle.

It is easy to see conceptually how M 2C generates an upper and/or a lower bound on the LSP mass for each event using the constraint coming from the dilepton edge. We have already seen in Figure 15 that, in the absence of any additional "hybrid" information, the nature of the "ordinary"

M T 2 constraint is to identify the region of (parent,daughter)-mass space which is compatible with the event. This allowed region was shaded and labelled with a in Figure 15. With only the information from M T 2 , all we know is that the true parent and daughter mass combination is somewhere in this region. If we now introduce the hybrid constraint from the dilepton edge on the same set of axes we get one of the cases shown in Figure 19. The constraint from the dilepton edge is always the straight blue (dot-dash or solid) line passing through the point at which the parent and daughter particles take the correct, though unknown, masses, with

Δ = m A -m B = m max .
If we apply the M T 2 constraint and the dilepton edge constraint simultaneously, we can see that the point where the true parent and daughter masses lie must be found somewhere on the part of the straight line which overlaps the shaded M T 2 region. This smaller allowed region is indicated by the solid (rather than dot-dashed) sections of the straight blue lines in Figure 19. It will be noted that, depending on the shape of the M T 2 boundary, the allowed region of joint constraint may be either finite or unbounded in extent. If finite, there is always an upper bound on the neutralino masses (both parent and daughter), and sometimes a lower bound too. If unbounded in extent, there may be only a lower bound on the neutralino masses, or if unlucky, no bound at all. Which situation one finds depends very much on the kinematics of the individual events, and depends in particular on the net transverse momentum in the lab-frame of the pair of parent neutralinos (P UT ). Where a lower bound and/or upper bound for the mass of the lightest neutralino exists, it is called M 2C,LB and/or M 2C,U B as appropriate.

Analogous definitions and results apply to the case of M 3C , except that in this case the sequence of two-body decays means that the dilepton edge provides a constraint (12) on the differences of squared masses of participating particles rather than the mass difference Δ.

Alternatively, instead of generating hybrid variables per se, one can instead talk of hybrid techniques, where again one combines constraints (such as those from a dilepton edge) with what would otherwise be under-constrained event topologies to reconstruct masses. A fine example of this includes [START_REF] Nojiri | A hybrid method for determining SUSY particle masses at the LHC with fully identified cascade decays[END_REF] which combines kinematic edges with events of the type shown in Figure 13(d).

Going beyond pairs of decay chains

It is straightforward to generalise the kinematical boundary method to chains involving more than two decay chains. For example one can define an analogue of the M T 2 variable (16) suitable for the case where three parents are produced, and each decays to a system of visible particles and invisible particles. One can construct the best bound possible, without knowing the splitting of the invisible momentum between chains, but subject to the constraint that the momenta of the invisibles should sum to the total missing momentum [START_REF] Barr | The race for supersymmetry: using M T 2 for discovery[END_REF]155]

M T 3 ≡ min / c T +/ c T +/ c T =/ p T max M T , M T , M T .
There are obvious generalisations to more than three cascades, but such constructions have not received much attention in the literature.

Singularity variables

It has been noted [START_REF] Kim | Algebraic singularity method for mass measurement with missing energy[END_REF] that all of the kinematic endpoints identified in Section 3 and 4 (e.g. m ll , M T , M T 2 , M CT , . . . ) as well as the cusps described in Section 3.7 and 3.11 are points where the projection from momentum space into the variable of choice becomes singular provided that the correct hypothesis has been made for the relevant masses. It is shown in [START_REF] Kim | Algebraic singularity method for mass measurement with missing energy[END_REF] that one can systematically identify all the singularities. It is also possible to construct normalised variables (singularity coordinates) locally perpendicular to all such singularity structures in a procedural manner which, though somewhat involved, is well defined. These variables can be constructed for different trial masses and the most singular behaviour sought.

VARIABLES BASED ON SUPPOSITIONS RELATING TO MULTIPLE EVENTS

We have, by now, seen many examples of events or topologies in which it is not possible to determine the full kinematics of a single event in isolation from any others. The problem has been that a typical event can be expected to contain far more unknowns (e.g. the components of the momenta of the invisible particles, and the masses of the unseen internal resonances) than can be constrained by the available observations. However, although each event in isolation may contain too little information to allow kinematics and unknown masses to be fully determined, it is sometimes the case that one or more events taken together with some joint assumptions can overcome this hurdle. In practice, this requires three conditions to be satisfied: (1) that you have a sample of events in which a sufficiently large fraction can be expected to share a common topology and particle content, (2) that the "unknowns" may be divided into those which are "shared" among all events (e.g. model parameters such as unknown masses) and those which are "independent" (e.g. kinematical variables such as the four momenta of the LSPs in each event), and (3) that the number of independent (kinematical) unknowns per event is smaller than the number of constraints that may be applied to each event by observation and/or hypothesis. If these three conditions are satisfied, then when a sufficiently large sample of events is considered as a whole, the joint system should become over-constrained and it should be possible to determine not only the unknown masses of the participating particles, but also information about the momentum components of the unseen particles in each event.

Some methods taking this approach look quite literally at pairs (or triples or quintuples) of events -indeed however many are necessary to get an over or exact constraint -and attempt thereby to use each such pair (or . . . ) to gain multiple independent measurements of model parameters [START_REF] Nojiri | Proposal for a new reconstruction technique for SUSY processes at the LHC[END_REF][START_REF] Kawagoe | A new SUSY mass reconstruction method at the CERN LHC[END_REF][START_REF] Cheng | Accurate mass determinations in decay chains with missing energy[END_REF][START_REF] Cheng | Accurate Mass Determinations in Decay Chains with Missing Energy: II[END_REF]. We will discuss some of these in Section 5.1. Other studies shy away from such an approach, preferring to derive a measurement from the sample of events "as a whole" rather than from pairings. In this latter category come not only all matrix element methods and model dependent fits (of which there are too many to produce a definite list 12 ) but also methods that form approximate goodness of fit minimisations. We will mention one such method [START_REF] Webber | Mass determination in sequential particle decay chains[END_REF] in Section 5.2.

There is still much debate and little consensus as to whether it is better to work with event "pairs" (etc) or to work with the sample of events in its entirety. There are arguments and proponents on both sides -and it is hoped that both methods will be tested on the LHC data.

Methods looking at small groups of events

One of the first studies to attempt to extract masses by looking at small groups of events that individually would be unconstrained was the "mass relation method" [START_REF] Nojiri | Proposal for a new reconstruction technique for SUSY processes at the LHC[END_REF][START_REF] Kawagoe | A new SUSY mass reconstruction method at the CERN LHC[END_REF]. This considered long decay chains of the form (specifically they considered a final state containing two opposite sign same family leptons, two jets and a neutralino coming from a gluino in the initial state). The key idea is that the particle masses can be fully determined -to within a set of discrete choice-ambiguities -if sufficiently many events are considered in combination. In principle, events would need to be considered five-at-a-time.

However to simplify the presentation of the method, the authors made the assumption that the three lightest sparticle masses were already known, and this allows them to use events in pairs. The authors noted that it would be possible to extend this method to shorter chains if more than one were present in each event, as the missing momentum constraint would then couple the momenta of the invisible particles and thus couple the constraints on the chains.

Indeed, though couched in a somewhat different language, the so-called "polynomial" method of [START_REF] Cheng | Accurate mass determinations in decay chains with missing energy[END_REF][START_REF] Cheng | Accurate Mass Determinations in Decay Chains with Missing Energy: II[END_REF] can be thought of as extending the mass relation method to identical pair decays of the form make use of "shape" information one needs a much better understanding of detector acceptances and efficiencies over the full range of the differential distribution in question than one does if one is merely fitting a local property such as a resonance. Similarly, fitting the shape of a "signal" distribution over a wide range (a range in which the background distributions might have very non-trivial differential distributions of their own) places much more stringent requirements on the experimenter's understanding of the size and shape of the underlying backgrounds. 14Matrix element methods, also known as likelihood methods or shape methods, have a long history of making use of all the events in a sample to constrain a set of model parameters -not only in particle physics, but in all areas of the sciences. In fact, if the underlying model (for both new physics and relevant backgrounds) is well understood, and the only remaining question is the determination of some parameters within that model, then no method can beat the matrix element or likelihood methods for their ability to extract information from data [START_REF] Neyman | On the problem of the most efficient tests of statistical hypotheses[END_REF]. In these methods, the basic idea is that if (within the confines of a fully parametrised model) it is possible to determine the probability with which any given set of observables is likely to arise, or (equivalently), if one can predict the shape of the differential distributions of certain observables as a function of the parameters of the model in question, then it is possible to do whichever of the following is most desired: (1) to determine which set of model parameters make the data most likely, (2) to determine confidence intervals for some of the model parameters, (3) to sample from the posterior distribution of model parameters induced by the likelihood of the data given an appropriate prior. Indeed, there are yet more ways that results of a shape based analysis could be interpreted or presented -but the common feature of each is their dependence on a well understood likelihood (the probability of the data given a model). If the parameters of the model in question are masses (or if the parameters may be used to derive masses) then these techniques perform model dependent mass measurements.

For example, [START_REF] Alwall | Measuring sparticles with the matrix element[END_REF] used a matrix element method to place a constraint on the parent and daughter masses in the topology of the type shown in Figure 13(a). It is interesting to note that the shape of the constraint obtained in [START_REF] Alwall | Measuring sparticles with the matrix element[END_REF] bears many similarities to that obtained from the corresponding M T 2 analyses. Possible reasons for this similarity are suggested in [START_REF] Barr | Transverse masses and kinematic constraints: from the boundary to the crease[END_REF].

One ingredient required by analyses that make use of shapes is, therefore, an ability to predict the shape of differential distributions of useful observables. In some cases, there is no alternative to using event generators to calculate the shapes of such distributions by Monte Carlo methods.

There are instances, however, where the shapes of distributions can be calculated analytically to provide useful insights into either their nature or into the sets of circumstances in which the shapes are most useful [START_REF] Smillie | Distinguishing spins in supersymmetric and Universal Extra Dimension models at the Large Hadron Collider[END_REF][START_REF] Athanasiou | Distinguishing spins in decay chains at the Large Hadron Collider[END_REF][START_REF] Burns | A general method for model-independent measurements of particle spins, couplings and mixing angles in cascade decays with missing energy at hadron colliders[END_REF][START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Gjelsten | Measurement of the gluino mass via cascade decays for SPS 1a[END_REF][START_REF] Miller | Invariant mass distributions in cascade decays[END_REF][START_REF] Lester | Constrained invariant mass distributions in cascade decays: The shape of the 'm qll threshold' and similar distributions[END_REF][START_REF] Matchev | A general method for determining the masses of semi-invisibly decaying particles at hadron colliders[END_REF][START_REF] Konar | Superpartner mass measurements with 1D decomposed[END_REF][START_REF] Barr | The precision determination of invisible-particle masses at the LHC[END_REF][START_REF] Ross | Mass determination of new states at hadron colliders[END_REF][START_REF] Barr | Precision Determination of Invisible-Particle Masses at the CERN LHC: II[END_REF][START_REF] Kraml | Same-sign top quarks as signature of light stops at the LHC[END_REF].

It should be noted that the shapes of some of those differential distributions are sensitive to the spins and couplings of the particles involved, and that in fact most spin-sensitive analyses rely on this as their only means of extracting information about those spins [START_REF] Barr | Using lepton charge asymmetry to investigate the spin of supersymmetric particles at the LHC[END_REF][START_REF] Smillie | Distinguishing spins in supersymmetric and Universal Extra Dimension models at the Large Hadron Collider[END_REF][START_REF] Athanasiou | Distinguishing spins in decay chains at the Large Hadron Collider[END_REF][START_REF] Burns | A general method for model-independent measurements of particle spins, couplings and mixing angles in cascade decays with missing energy at hadron colliders[END_REF][START_REF] Barr | Measuring slepton spin at the LHC[END_REF].

In practice, matrix element and likelihood methods tend to be applied only in the mature stages of an analysis. For example, currently they are being used for top quark mass measurements [170,[START_REF] Aaltonen | Precise measurement of the top-quark mass in the lepton + jets topology at cdf ii[END_REF] and Higgs boson searches [START_REF] Collaboration | Search for standard model higgs boson production in association with a w boson using matrix element techniques with 4.3 fb-1 of cdf data[END_REF] at the Tevatron. Such techniques are less likely to perform well in the early stages when there remains some debate as to the nature of the model being fitted -and in particular when the distribution of the background is poorly understood. As a result, there is also a large number of methods that are not strictly based on the statistical likelihood, but which nevertheless try to determine model parameters from some form of fit over model parameters to some aspect of the data.

Are all methods that look at large homogeneous samples of events based on shapes or differential distributions? Fortunately not! For example, though the mass relation method was initially propoposed [START_REF] Nojiri | Proposal for a new reconstruction technique for SUSY processes at the LHC[END_REF] as a method that should consider pairs of events (thus leading to its discussion earlier in Section 5.1) it has been noted, for example in Section X of [START_REF] Allanach | Les Houches 'Physics at TeV Colliders 2003' Beyond the Standard Model Working Group: Summary report[END_REF], that the mass relation method can trivially be extended to work on all events, and indeed becomes less arbitrary in the process. With similarities to the mass relation method and to the methods of [START_REF] Cheng | Mass determination in SUSY-like events with missing energy[END_REF][START_REF] Cheng | Accurate mass determinations in decay chains with missing energy[END_REF][START_REF] Cheng | Accurate Mass Determinations in Decay Chains with Missing Energy: II[END_REF],

another homogeneous event sample technique not based on event shapes has been proposed in [START_REF] Webber | Mass determination in sequential particle decay chains[END_REF] which aims to use linear algebra to efficiently determine unknown masses from events which are individually under-(but collectively over-) constrained, by asking us to solve the kinematics of each event in terms of a minimal number of assumptions, and then minimise a goodness of fit through variation of these assumptions. In the example used in [START_REF] Webber | Mass determination in sequential particle decay chains[END_REF] there are sufficient constraints to permit us to determine the four-momenta of both of the invisible particles provided we are prepared to to hypothesise all eight participating masses. A goodness of fit function which compares the squares of those LSP momenta to the hypothesised LSP masses can then be constructed and subsequently minimised over all possible choices of the set of eight masses. This results in an overall "best fit for the participating masses" which uses all events democratically. Though this is not strictly a likelihood method, one can think of methods in this class as attempting to form a "heuristic" likelihood -by which we mean a function which loosely shares some of the properties of a real likelihood, in particular it is large near "good" values of the model parameters, and small For example, the general point is made in [START_REF] Kersting | A Simple Mass Reconstruction Technique for SUSY particles at the LHC[END_REF][START_REF] Kang | Neutralino Reconstruction at the LHC from Decay-frame Kinematics[END_REF] that events in particular corners of kinematic phase space can be "more reconstructible" than general events in the bulk. For example, [START_REF] Kersting | A Simple Mass Reconstruction Technique for SUSY particles at the LHC[END_REF] investigates di-chargino production, with each chargino decaying to an opposite sign di-lepton pair and a lighter neutralino. This is a topology of the form in Figure 13(b). In general these events are not reconstructible, however the subset of events in which two di-lepton invariant masses (one from each "side") are close to their upper kinematic limit are forced to adopt particular kinematic configurations: in these "extremal" events, the decay products of the heavier charginos may be collinear in the chargino rest frame, for example. If one selects only events near kinematic endpoints, one can therefore make use of this additional kinematic information to render the events sufficiently reconstructible that masses may be determined. The same trick was applied in Section 20.2.4.1, in the Supersymmetry chapter of [START_REF]ATLAS Detector and Physics Peformance TDR[END_REF].

Another example of performing "approximate" reconstruction near kinematic endpoints is the so-called "MAOS Method"15 of [START_REF] Cho | M T 2 -assisted on-shell reconstruction of missing momenta and its application to spin measurement at the LHC[END_REF][START_REF] Choi | Reconstructing the Higgs boson in dileptonic W decays at hadron collider[END_REF]. Here it is noted that for events that are near their M T 2 endpoint (Section 4.1) the missing particles' momenta are constrained to be similar to the values selected by the assignment (or "splitting" of / p T ) that determines the value of M T 2 . It has been demonstrated that by selecting such near-endpoint events, and by using the "approximate momentum reconstruction" implied by the splitting selected by the M T 2 minimisation, then it is possible to reconstruct not only the masses of the particles involved, but also place strong constraints on their spins. This is demonstrated in the the context of a supersymmetric model [START_REF] Cho | M T 2 -assisted on-shell reconstruction of missing momenta and its application to spin measurement at the LHC[END_REF] and for the determination of the Higgs boson mass in the channel h → W + W -→ + -νν [START_REF] Choi | Reconstructing the Higgs boson in dileptonic W decays at hadron collider[END_REF].

Not all "approximate reconstructions" are motivated by edges of phase space -an alternative approach is to construct variables from the kinematic configuration which has the greatest likelihood, or from a weighted average of possible configurations, perhaps weighted by a prior motivated by a Monte Carlo simulation. This approach has been employed in template-based measurements of the top quark mass in the di-leptonic channel at the Tevatron [START_REF][END_REF][START_REF] Fatakia | A measurement of the mass of the top quark in the di-lepton channels using the D0 detector at fermilab[END_REF][START_REF] Adelman | Measurement of the top quark mass using the template method in the lepton plus jets channel with in situ W → jj calibration at CDF-II[END_REF][START_REF] Abulencia | Measurement of the top quark mass using template methods on dilepton events in proton antiproton collisions at √ s = 1.96 TeV[END_REF][START_REF] Brandt | Measurement of the mass of the top quark in dilepton final states with the D0 detector[END_REF][START_REF] Meyer | Measurement of the Top Quark Mass using Dilepton Events and a Neutrino Weighting Algorithm with the D0 Experiment at the Tevatron (Run II)[END_REF][START_REF][END_REF].

CONCLUSIONS

The story may be apocryphal, but it has been said that prior to the establishment of the quark model, new particles were being found at such an alarming rate that it was seriously proposed that a Nobel Prize ought to be awarded to the first physicist who couldn't discover a new particle.

In the 1950s and early 1960s particle physics may have been expanding into a theoretical vacuum driven by an excess of experimental results. In the case of mass measurement techniques for the LHC, however, the process seems to have been turned upside down. The earliest LHC specific techniques were proposed in 1996, or thereabouts [START_REF] Hinchliffe | Precision SUSY measurements at CERN LHC[END_REF][START_REF] Paige | Determining SUSY particle masses at LHC[END_REF], and in the course of the intervening 15 years they have been developed beyond all recognition. All of this has happened in an almost complete absence of data against which to test these techniques. The Tevatron collaborations are owed a debt of thanks both for inventing their own methods, but also for acting as a testbed for some LHC proposals, for example for mass determination in the dileptonic t t system.

March of 2010 saw the first collisions at the LHC with centre of mass energies of 7 TeV, and so the long wait for that data is now over. Those who have invested considerable effort in developing mass measurement techniques are looking on in expectation, waiting to see what the data will bring.

In very little time, experimental collaborations will dash the hopes of phenomenologists the world over by refusing to release any plots derived from any of the mass measurement variables which are more complicated than mere invariant masses, as they will be too busy tearing themselves apart in debates over how best to measure the photon reconstruction efficiency in the pseudorapidity range 1.4 < |η| < 1.5. The development of mass measurement techniques, which has already seen a period of incredible productivity over the last 15 years, is thus assured a second wind.

While writing this review, nothing was more disheartening than finding the words "We propose a new variable . . . " in one of the abstracts circulated in the daily arXiv digest for hep-ph. We are pleased to be able to confirm that we, ourselves, have managed to create no new variables during the course of this review.

where χ, the hypothesis for m / C is also used in e χ = χ 2 + c 2 T . Contransverse mass (Section 3.12, 4.

A.2. Kinematic endpoints of cascade decay chains

The kinematic endpoints for the decay chain q → χ0 2 → ˜ ∓ ± near q → χ0 1 ∓ far ± near q (and other chains with the same topology) can be found in Table II.

A.3. Some properties of two-body decays and variables related to them

This section summarises some frequently-used results relating to two-body semi-invisible decays of a single particle. In this section our notation assumes that the decay is labelled

A → B / C
and that upstream transverse momentum is defined to be

P UT = -a T .

Transverse momentum:

For any P UT recoil, with the correct invisible particle mass hypothesis χ = m / C the momentum of each daughter in the parent rest frame is

p * = λ 1 2 (m A , m B , m / C ) 2m A (A.

A.4. Endpoints for pairs of semi-invisible decays

This section summarises endpoint formulae for kinematic variables used for pairs of semi-invisible decays. In this section our notation assumes that the decays are labelled

A → B / C A → B / C
and that the upstream transverse momentum is P UT = -(a T + a T ). 16 After [START_REF] Burns | Using subsystem m T 2 for complete mass determinations in decay chains with missing energy at hadron colliders[END_REF].

FIG. 1 :

 1 FIG.1: Two very simple decay topologies.

  FIG. 2: (a) Dilepton invariant mass distribution for the process pp → Z/γ → + -. From [51]. (b) Transverse mass distribution for pp → W → eν. The W boson mass is determined from a fit to the range indicated with the double-headed horizontal arrow. From [52].

FIG. 3 :

 3 FIG.3: A single particle "A" decaying to three visible particles "B", "C" and "D".

  and m lq(low) ≡ min{m ql + , m ql -} be used. along with other variables (discussed later) to measure the corresponding kinematic endpoints m max ll , m max llq , m max lq(high)

FIG. 9 :

 9 FIG.9:The qll decay chain with a terminal three-body decay.

  FIG. 12: (a) A branched decay chain of the sort considered in Section 3.11. (b) Examples of kinematic cusps. Adapted from [117].

FIG. 13 :

 13 FIG.13: Examples of the "dual-sided" event topologies discussed in Section 4.

FIG. 14 :

 14 FIG.14: Generic M T 2 configuration. The sets labelled B and B may correspond to individual particles, or groups of visibles. In the latter case the visible could result from internal cascade decays within the 'blobs'.

  casts the variable as a function of six things. The first four are straight-forward, being the invariant masses (m B and m B ) and the transverse momenta (b T and b T ) of the visible final state particles, or collections thereof, on each side of the event. The fifth input is the observed missing transverse momentum in the event, often denoted / p T . If G in Figure 14 is taken to represent the totality of all other visible momenta in the event regardless of source, then / p T is equivalent to -(g T + b T + b T ). Whether or not / p T is "useful" is dependent on how closely it resembles / c T + / c T , which depends on how many other invisible particles there are in the event and on the detector reconstruction resolution for g T , b T and b T .

FIG. 15 :

 15 FIG.15:The nature of the M T 2 constraint from a single event. The region above the dashed line (marked ) is consistent with the constraints, while that below and to the right of the line (marked) is inconsistent. Similar regions can be drawn for a single decay chain where the regions are bounded by M T .

FIG. 16 :

 16 FIG.16: This figure, reproduced from[START_REF]Early supersymmetry searches in channels with jets and missing transverse momentum with the ATLAS detector[END_REF], shows the preliminary leading dijet M T 2 distribution for the first 70 inverse nanobarns of ATLAS data. The dotted line shows the shape of a potential SUSY signal in a model with a large amount of strongly interacting sparticle production. Note that the QCD dijet background constitutes the majority of the events passing cuts, but that as it lacks a mass scale the majority of those events fall at very low M T 2 values. This contrasts with the behavour of top-pair and potential SUSY events which have high mass scales and occur at high M T 2 values.

  . The transverse mass (and hence the stransverse mass M T 2 ) remains bounded below by a minimum value m < and above by m > (= m A for χ = m / C ). In the case of a n-body decay to a set of visible particles and a set of invisible particles, A → B + C + . . . + / X + / Y + . . . the minimum value of is simply the sum of the masses of the daughters m B + m C + . . . + m / X + m / Y + . . .. With larger numbers of invisible particles produced, the fraction of states near M T = m A (or M T 2 = m A for the two-chain case) is reduced. In such cases the end-point might only be inferred from a measurement of the shape of the distribution. Sequential decays producing invisibles at each step further restrict the range of M T (and hence M T 2 ).

FIG. 18 :

 18 FIG. 18: Event topologies required (a) by M 2C and (b) by M 3C .

FIG. 19 :

 19 FIG. 19: The four potential combinations of constraint coming from M T 2 and the dilepton edge which together contribute to M 2C are shown. The mass difference Δ = m Am B is assumed to have been measured independently, e.g. from the endpoint of the dilepton invariant mass distribution in three-body decays A → B + -. Knowledge of Δ constrains the (m A , m B ) solution to lie on the diagonal line. The constraint from the pair decay kinematics gives the separate M T 2 constraint, for which the solution must lie above and to the left of the dashed line. The intersection of the shaded region with the diagonal line then gives an event-by-event allowed solution space.

5 ): M 2 CT = m 2 b + m 2 b. 7 )

 52227 + 2(e b e b + b T • b T ). (AStransverse mass (Section 4.1 et sequens):M T 2 = min / c T +/ c T =/ p T max M T , M T .(A.8)

2 = a 4 + b 4 + c 4 -(A. 15 )p 2 -P 2 UT: 2 -P 2 UT:

 2444152222 , b, c) = a 2 -(b + c) 2 a 2 -(bc) 2a 2 b 2 -2a 2 c 2 -2b 2 c 2 . (A.10)For m B = 0 (A.9) simplifies top * = m 2 Am 2 / C 2m A : m B = 0 (A.11) For single particle production and two-body decay with P UT = 0 b max T = p * : P UT = 0. (A.12) For fixed P UT and m B = 0, b max T = p * e y : m B = 0 (A.13) where sinh y = |P UT |/m A and p * for m B = 0 is given in Equation A.11. For fixed P UT and m B = 0, b max T = m B sinh(y + η) (A.14) where sinh η = p * /m B . Transverse mass: for any P UT recoil, with the correct invisible particle mass hypothesis M max T = m A : χ = m / C For fixed P UT , and an arbitrary mass hypothesis χ = m / C and with m B =0, 16 * e -y + (|P UT | + p * e -y ) 2 + χ 2 χ < m / C , m B = 0 p * e +y + (|P UT |p * e +y ) 2 + χ 2 χ > m / C , m B = 0 (A.16) where sinh y = |P UT |/m A and p * for m B = 0 is given in Equation A.11.

TABLE I :

 I Cartoons indicating various decay topologies, and relevant sections of this review. Dashed lines

indicate 'invisible' particles which traverse the apparatus undectected. Blobs indicate decays which may

For pp → ppX the centre-of-mass four-vector can be fully reconstructed from the outgoing proton momenta if dedicated detectors are installed at very high |η|[START_REF] Albrow | Prospects for diffractive and forward physics at the LHC[END_REF][START_REF] Albrow | The FP420 R&D Project: Higgs and New Physics with forward protons at the LHC[END_REF].

A discussion of the experimental drawbacks of the l + l -q threshold distribution may be found in Section 1.2 of[START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF].

Contrast how little has been written[START_REF] Freedman | Student project[END_REF] about the case where some particles on the backbone of the qqll-chain are off mass-shell.

The nick-name "stransverse mass" arose as a shortened form of "supersymmetric transverse mass" as MT 2 was originally applied most frequently to supersymmetric events in cases where the transverse mass was no longer usable.

Computer libraries that can evaluate MT 2 may be found in[START_REF] Barr | Oxbridge stransverse mass library[END_REF] and in[START_REF] Cheng | UCD stransverse mass library[END_REF] The library of[START_REF] Cheng | UCD stransverse mass library[END_REF] can only compute MT 2 using the bisection algorithm of[START_REF] Cheng | Minimal kinematic constraints and M T 2[END_REF], but it is very simple to use and is not dependent on external packages. It is also distributed as part of the WIMPMASS library[START_REF] Cheng | WIMPMASS library[END_REF]. The library of[START_REF] Barr | Oxbridge stransverse mass library[END_REF] contains algorithms for a larger number of variables (including MT Gen , M2C , etc, as well as a copy of the algorithm in[START_REF] Cheng | Minimal kinematic constraints and M T 2[END_REF][START_REF] Cheng | UCD stransverse mass library[END_REF]) but depends on the external Minuit2 library[129,[START_REF] James | Minuit: A System for Function Minimization and Analysis of the Parameter Errors and Correlations[END_REF].

We use the term "components" figuratively since MT 2 is not a vector. Strictly it is the transverse momenta which are inputs to MT 2 which are resolved into components.

Recall that this was not true for MT 2 , and in fact the dependence of MT 2 at unphysical values of χ was a necessary ingredient for forming one of the two types of MT 2 "kink" (see Section 4.2).

Historically this was how MCT was first defined in[START_REF] Tovey | On measuring the masses of pair-produced semi-invisibly decaying particles at hadron colliders[END_REF].

So far as we are aware, active use of MCT for single particle systems does not seem to have been encouraged prior to[START_REF] Cho | Shining on buried new particles[END_REF].

Nevertheless, it is worth mentioning[START_REF] Alwall | Measuring sparticles with the matrix element[END_REF] as an example of a matrix element method providing an alternative to the purely kinematic methods of Section 4.1 for measuring particle masses in pair production.

i.e. the distinction between observables based on global properties of distributions (such as their shapes) and observables based on local properties (endpoints) or small numbers of events.

In contrast a narrow structure, such as a resonance, can often be fitted using a sideband technique with comparatively little understanding of the backgrounds.

"MAOS" stands for "MT 2 Assisted On Shell"

Acknowledgments

We are grateful for very helpful discussions with (amongst others) Johan Alwall, Kiwoon Choi, Christopher Cowden, Ben Gripaios, Rupert Gwenlan, Nick Kersting, Teng Jian Khoo, Ian Woo Kim, Yeong Gyun Kim, Kyoungchul Kong, Jae Sik Lee, Bob McElrath, Konstantin Matchev, David Miller, Mihoko Nojiri, Per Osland, Giacomo Polesello, Mario Serna, Kazuki Sakurai, Dan Tovey and Bryan Webber, and colleagues from the Cambridge Supersymmetry Working Group and the Dalitz Institute (Oxford). We are grateful to Alex Pinder for preparing some of the material for this review. AJB acknowledges the support of the Science and Technology Facilities Council, and thanks Dr Damaris Koch and the Master and Fellows of Peterhouse, Cambridge for their hospitality.

. This method takes pairs of events and finds the mass hypotheses that are compatible with them.

Perhaps surprisingly (given the large number of particles in the final state) the evidence provided in [START_REF] Cheng | Accurate mass determinations in decay chains with missing energy[END_REF][START_REF] Cheng | Accurate Mass Determinations in Decay Chains with Missing Energy: II[END_REF] suggests that combinatorial ambiguities may not present insurmountable problems for such techniques. Computer libraries and Mathematica notebooks are freely available [START_REF] Cheng | WIMPMASS library[END_REF] which provide routines that may be used to determine the masses which are consistent with pairs of events of this type.

Matrix element methods, distribution shapes, and combining events

The majority of the mass measurement methods described previously have been based on conclusions derived from small numbers of events, or from the local properties of distributions. For example: invariant masses were used to measure the mass of single particles decaying to visible products; local properties of distributions (for example the location of discrete features, such as a kinematic endpoints) were used to measure certain relationships between masses; and the solutions of sets of simultaneous equations derived from a small number of events were used to find many masses at once under the assumption that the events were homogeneous.

It should be noted, however, that there are other techniques which typically become powerful only when looking at much larger numbers of events. In the main, these are methods sensitive to the global (or at least non-local) shapes of differential distributions of observables, though there are other examples (e.g. [START_REF] Webber | Mass determination in sequential particle decay chains[END_REF]) which we will comment on which do not conform to this pattern. It should be admitted that the distinction being drawn here 13 is perhaps not as clear cut as we are suggesting -one might argue, for example, that many events are needed to see a kinematic endpoint, or that only by knowing or using the shape of the distribution near an endpoint can the endpoint be reliably fitted, and that therefore the use of kinematic endpoints requires a good understanding of the non-local properties of distributions and/or large numbers of events. Nonetheless, we think that the distinction is a useful one to draw in the sense that, from an experimental perspective, to elsewhere.

Model fitting and adding other observables

To improve our determination of the masses we may well be willing to make assumptions about the physics beyond hypothesising topologies and decays. For example some of the first variables explored in this review (in Section 2) were those which already made the additional (often implicit) assumption that heavy particles are produced near their kinematic threshold, so with E ≈ m.

Assumptions about the nature of the incoming partons allow us to turn initial-state radiation to our advantage. QCD radiation adds particles to the final state by an amount which can be calculated, and hence could potentially be exploited to provide information on heavy object masses [START_REF] Papaefstathiou | Effects of QCD radiation on inclusive variables for determining the scale of new physics at hadron colliders[END_REF] -though more detailed work is needed to check the practicality of this proposal.

If a particular model is assumed one can combine multiple measurements of different topologies to fit for the masses. The constraints from mass or kinematic edge measurements have been interpreted in the context of supersymmetric models by several groups [START_REF] Allanach | Measuring sparticle masses in non-universal string inspired models at the LHC[END_REF][START_REF] Lafaye | SFITTER: SUSY parameter analysis at LHC and LC[END_REF][START_REF] Bechtle | Fittino, a program for determining MSSM parameters from collider observables using an iterative method[END_REF][START_REF] Bechtle | Constraining SUSY models with Fittino using measurements before, with and beyond the LHC[END_REF][START_REF] Roszkowski | Efficient reconstruction of CMSSM parameters from LHC data -A case study[END_REF]. If constraints from a future electron-positron collider can be added [START_REF] Weiglein | Physics interplay of the LHC and the ILC[END_REF] then the near-degenerate directions in the LHC-only measurements can be resolved, leading to improved mass constraints even on those particles not directly observable at any future e + e -machine.

Alternatively, if one is willing to make suppositions about the couplings of any new particles (as might be reasonable for example for a supersymmetric model) one could further constrain the masses using only LHC data. The number of events observed is usually a strong function of the mass (both because of the parton distribution functions and the explicit dependence of the matrix element on kinematic variables such as ŝ). If one is willing to entertain hypotheses about the couplings involved, it is possible to interpret the measured cross-sections as constraints in mass space [START_REF] Lester | Determining SUSY model parameters and masses at the LHC using cross-sections, kinematic edges and other observables[END_REF][START_REF] Dreiner | SUSY parameter determination at the LHC using cross sections and kinematic edges[END_REF], considerably improving the overall mass scale determination even for a modest integrated luminosity of ∼ 1fb -1 , and with modest uncertainties in the predicted cross sections of order 20%.

Alternative approaches to under-constrained events

We have already seen that when invisible particles are produced it is not always possible to fully reconstruct the event kinematics. Nonetheless, there are sometimes ways in which it is possible to perform approximate kinematic reconstructions.

¯A. SOME FREQUENTLY USED DEFINITIONS AND FORMULAE

For convenience we include definitions of some of the more frequently-used kinematic variables, and reproduce some of the most important kinematic endpoint formulae.

A.1. Summary of simple kinematic variables

Bold font symbols indicate Euclidean momentum vectors in three (or two transverse) dimensions. A subscript T indicates a quantity built from transverse momentum components.

Invariant mass (Section 3.1):

(A.1)

Transverse energy (Section 3.2)

Effective mass (Section 2) -a typical definition is a sum over the leading n jets:

where historically n has always been taken to be "4", but more recently n has become dependent on the analysis channel. Note that H T , the analogue in CMS and at the Tevatron, has a number of different definitions (see equations ( 3), ( 5) and ( 6)), of which the one most frequently used at present is

TABLE II: (Containing results from [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Allanach | Measuring sparticle masses in non-universal string inspired models at the LHC[END_REF][START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF][START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF].) This table lists the absolute kinematic endpoints of invariant mass distributions formed from decay chains of the type q → χ0 2 → ˜ ∓ ± near q → χ0 1 ∓ far ± near q for known particle masses. The shorthand notation used is:

and q = m 2 q and X is m 2 h or m 2 Z as appropriate. The visible particles are assumed to have negligible mass. Inversion formulae (i.e. masses in terms of endpoints) for certain subsets of the above endpoints are published in [START_REF] Gjelsten | Measurement of SUSY masses via cascade decays for SPS 1a[END_REF][START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF][START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF].

Note that this table is by no means exhaustive: many other interesting endpoints have been proposed for the qll-type chain (including those of [START_REF] Matchev | Precise reconstruction of sparticle masses without ambiguities[END_REF][START_REF] Burns | Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy[END_REF]) and for other types of chain [START_REF] Agashe | Distinguishing Dark Matter Stabilization Symmetries Using Multiple Kinematic Edges and Cusps[END_REF].

Stransverse mass: for any P UT , for the correct invisible particle mass hypothesis,

For fixed P UT and m B = 0 the maximum value of M T 2 for any invisible particle mass hypothesis χ is given by Equation A.16 but now with P UT → P UT /2 [START_REF] Burns | Using subsystem m T 2 for complete mass determinations in decay chains with missing energy at hadron colliders[END_REF]:

where sinh y 2 = |P UT |/2m A . Formulae for M max T 2 with m B = 0 are given in [START_REF] Cho | Transverse mass for pairs of gluinos[END_REF] for P UT = 0, and in [START_REF] Burns | Using subsystem m T 2 for complete mass determinations in decay chains with missing energy at hadron colliders[END_REF] for arbitrary P UT .

The 1-D decomposed versions of M T 2 (Section 4.3) constructed from components of momenta parallel to (perpendicular to) P UT [START_REF] Konar | Superpartner mass measurements with 1D decomposed[END_REF] have endpoints: and likewise for the projections in the same limit [START_REF] Matchev | A general method for determining the masses of semi-invisibly decaying particles at hadron colliders[END_REF]: