Constrained $\sqrt{\hat{S}_{min}}$ and reconstructing with semi-invisible production at hadron colliders

Mass variable \sqrt{\hat{S}_{min}} and its variants were constructed by minimising the parton level center of mass energy that is consistent with all inclusive measurements. They were proposed to have the ability to measure mass scale of new physics in a fully model independent way. In this work we relax the criteria by assuming the availability of partial informations of new physics events and thus constraining this mass variable even further. Starting with two different classes of production topology, i.e. antler and non-antler, we demonstrate the usefulness of these variables to constrain the unknown masses. This discussion is illustrated with different examples, from the standard model Higgs production and beyond standard model resonance productions leading to semi-invisible production. We also utilise these constrains to reconstruct the semi-invisible events with the momenta of invisible particles and thus improving the measurements to reveal the properties of new physics.


Introduction
The Standard Model (SM) is now essentially complete after CMS [3] and ATLAS [4] found its last missing bit, lone neutral scalar of the model, the Higgs boson. The SM is so far extremely successful in explaining the fundamental particles and the interactions between them. However some of the unresolved theoretical questions together with very convincing experimental observations, such as dark matter, neutrino oscillation and several others compel us to believe that the SM can not be the complete description. Numerous models beyond Standard Model (BSM) was constructed to accommodate some of these phenomena with a general belief that the scale of new physics is just around the corner at few to multi-TeV level. Unfortunately, large hadron collider (LHC) has not observed any indication of new physics so far. Now, if any of these TeV scale BSM theories exists in nature then it can manifest its signature at the next LHC run. A scenario with positive signal essentially necessitates the determination of the new particle mass, spin and coupling etc associated with that new physics.
Recently popular theoretically appealing BSM theories are the ones which accommodate the thermal relic dark matter as stable and weakly interacting massive particle (WIMP) estimating the tightly constrained observed amount of dark matter density [5]. Hence, this stability of the dark matter in most of the BSM theory is ensured by some discrete symmetry, such as Z 2 symmetry in supersymmetry or many other scenario. Once this symmetry is respected, all the heavy BSM particles in such model has to be produced in pairs; subsequently decaying into some lighter BSM resonance together with SM particles (which may or may not be detected and measured at the detector) in multiple steps of successive decay. Typically at the end of each decay chain lightest BSM particle is produced which is the dark matter particle of that model and escape the detection. Hence, at least two massive and lightest BSM particles remain hidden in these events. The only way to know their presence is the observation of sizable P T in the detector calculated from the imbalance of transverse visible momenta produced in such events. The reconstruction of a dark matter signal at hadronic collider is challenging because of the partial knowledge of the incoming parton momenta further burdened with multiple massive final state particles of unknown mass goes undetected keeping no individual momentum informations at the detector.
There has been several studies under gone into mass and spin determination in the context of semi-invisible production at the hadronic collider 1 and we classify them based on the topology information as follows: Exclusive variables are defined based on the topology of the production mechanism and decay processes under consideration. Identical signatures consists of visibles and invisibles in the final state can be originated from very different topologies which is deeply related to the stabilising symmetry of the dark matter (DM). Shape of the visible invariant mass can effectively carry informations on topology along with the mass spectrum [9] of the decay chain. Underlying DM stabilising symmetry can also be probed [10][11][12] using kinematic edge and cusp in the invariant mass distributions and from the shapes of transverse mass variable M T 2 . Even the assumption of one particular underlying symmetry allows some fixed number of different topologies from which the correct one can be identified comparing suitable kinematic variables [13]. One expects that the ignorance of the correct topology can add difficulties in solving combinatorial ambiguity [14][15][16][17] which is one source of complexity in mass determination methods, more prominently available when associated with long decay chain. This ambiguity can be originated from two different sources. Firstly, allocation of the final state particles to the correct decay chain, i.e. from which side of the decay chain some particular states is produced. Secondly, the ordering of the assigned particle in a single decay chain. The hemisphere method [18] and PT vs M methods [15] are introduced to reduce the this ambiguity in assigning the correct final state particles to the corresponding decay chain. However, the ordering of the particles left unresolved. The M T 2 variable together with invariant mass are also shown to reduce the combinatorics significantly [16]. In the literature several classes of exclusive variables are defined assuming that the correct knowledge of topology is available and anticipating that the combinatorial ambiguity can be controlled. The exclusive mass determination methods can be categorised as follows • Edge measurement method : Based on the idea of constructing all possible invariant masses out of visible decay products in each decay chain [19][20][21][22][23][24][25]. Each invariant mass has an endpoint which is experimentally observable and these endpoints are related to the unknown masses in the decay chain. To evaluate all the unknown masses by inverting the the equations in terms of measured endpoints, one needs sufficient number of independent endpoint measurements. So essentially a long decay chain in necessary to have unique measurement of all the unknown masses. However this criteria inevitably invites combinatorial ambiguity thereby reducing the effectiveness of the method. This method also does not use all the available informations like missing transverse momentum P T in the event.
• Polynomial method : One tries to utilise all the available information in the event of particular topology and solve for the unknown masses and momentums [26][27][28][29][30] considering on-shell cascade decay. In the literature, typically the production of two heavy invisible particle is considered at the final state, assuming Z 2 type of DM stabilising symmetry in the theory. All the unknown invisible momenta components are solved utilising mass-shell constraints and missing P T constraints in the event. It can be shown that one needs to consider long decay chains to solve for all unknowns in the event.
Combinatorial ambiguity naturally arises here from the requirement of the long decay chain. Moreover, resulting invisible momenta remain ambiguous due to existence of multiple solutions originating from non-linear mass-shell constraints [30,31].
• Transverse mass variable: Rather than considering full event information, transverse projection of momenta is considered during calculation. Contrary to previous cases, even small decay chain can constrain the masses realistically. There are many variants of transverse mass variable exist in the literature such as M T 2 [32][33][34][35][36][37][38][39][40], M sub T 2 [41], M CT 2 [42,43], 1D orthogonal decomposition of M T 2 (M T 2⊥ and M T 2 ) [44], asymmetric M T 2 [45,46] and M approx T 2 [47], M CT [48][49][50], and variants M CT ⊥ and M CT [51] etc. Among these broad class of transverse mass-bound variables, we briefly discuss some properties of M T 2 which is studied widely in the literature. This variable is defined as the constrained minimisation of maximum of two transverse masses M T from both sides of the decay chain. The minimisation is done over all possible partitions of missing transverse momenta where as, satisfying the P T constraint. M T 2 (m inv ) expressed as a function of the unknown invisible particle mass, can have an experimentally observed upper bound over many events. This provides a useful correlation between the trial invisible massm inv and measured upper bound M max T 2 , which represents the corresponding mass of the ancestor particle (commonly called as mother or parent) responsible for producing all the visible and the invisible particles within the (sub)system. This correlation also satisfy the true yet unknown mass parameters fulfilling the crucial equality M max T 2 (m true inv ) = m true mother . Interestingly, one can measure the true mass of both mother and daughter simultaneously by identifying a discontinuity (kink) arises due to additional conditions like two step decay chain [36,37], extra upfront P T from ISR [38,39] or in subsystem context [41]. Extracting these kinematic endpoint is occasionally troublesome with thinly populated events at the endpoint, and in presence of backgrounds. Available on-shell constraints of intermediate particles can be exploited in the (1+3) dimensional variable M 2 [52,53] to improve number the events appearing at the tail of these distributions.
Global and inclusive variable are independent of topology information and hence, do not require any information about the production mechanism of the particles in the event. Variables are constructed using only visible particles momenta and missing transverse momenta in the event. Several of them were well known and utilised for long as event selection variables e.g. H T [54], total visible invariant mass M [55], effective mass M ef f , total transverse component of invisible momentum E T , total visible energy E and total transverse energy E T in the event. Lately introducedŝ min [1] and its variantsŝ sub min andŝ reco min [2] are also constructed as global and inclusive variable for measuring mass scale of new physics. Being a topology independent variable, they are also applicable to any possible decay chain without worrying about the symmetric and asymmetric topology and simple analytical form is also available. Thus any generic topology can be assimilated without affected by the combinatorial ambiguity. We would discuss further about these variables in Sec. 2.
In this present work our objective is to demonstrate the usefulness of partial event informations including the topology in the variables likeŝ min which were constructed as global one. After an introduction of analytic form for these variables in Sec. 2, we discuss effects of additional constrains for two topology classes based on the production and decay of heavy resonance. Antler topology being one important topology for single resonant production, have considerable discovery potential in different SM and BSM modes. This class can be constrained significantly and some interesting features can be noticed. In Sec. 3 we motivate and discuss the constraints. We would notice the constrained variableŝ cons min can notably improve the distribution. Second interesting takeaway from these constraints is that not only the minimum quantityŝ min , but one can also construct a maximum quantityŝ max which is also bounded. Hence, one additional variableŝ cons max can be defined and finite since these constraints play a critical role. After construction of these constrained variables we display how they were restricted in several events. In the next section, Sec. 4 we consider similar variables in case of non-antler topology. In Sec. 5 we turn to the capability of these variable in reconstructing the events. M T 2 assisted method (known as MAOS) [56,57] is proposed earlier by utilising the transverse components of the invisible momenta obtaining from the minimisation of this transverse mass. Whereas, longitudinal components are solved using the on-shell constraints and thus having two fold ambiguity from each decay chain. Recently proposed some of the (1+3) dimensional M 2 [53] variables can lead to unique momentum reconstruction for symmetric topology where additional constraints over equality of mother and on-shell relative take a pivotal role. Here in the reconstruction from the minimisation ofŝ does not rely on any particular topology and can be used for any of the symmetric or asymmetric cases for a unique solution. We would further demonstrate that the inclusion of additional constraints also improve these reconstructed momenta. We summarize and conclude at the end.

2.
√ŝ mass-bound variables without additional constraints Let us start by discussing briefly about the variable √ŝ min which was first introduced [1] to determine the mass scale associated with any generic process (or event topology) involving missing particles. It is inspired from the fact that the precise knowledge of partonic system center of mass (CM) energy √ŝ carries the kinematic informations like masses of heavy resonance, or threshold of pair productions at the hadron collider. Hence, one may like to know the distribution of this variable even approximately, after recognising the fact that there is no way we can completely reconstruct the event, or extract all the momentum informations in case of general semi-invisible productions at the hadron collider. Utilising all the experimentally observed quantities, best one gets the minimum partonic CM energy which is compatible (or consistent) with the observed visible momenta and missing transverse momentum. Although general event topology can have a wide diversity in production mechanism of visibles and invisibles and also number of them, it emerged that the final minimisation leads to a rather simple and versatile functional form for √ŝ min . This variable was further extended [2] to apply in general subsystems, and also utilised reconstructed events to safe guard the generic variables from underlying events. Subsequently these √ŝ variables were shown and classified [7,58] as M 1 type of mass-bound variables represented in a compact nomenclature of M ... class of variables. Wide variety within this class are constructed systematically considering different projection methods, additional second projection [44,51], and considering different orders of the operations. Interestingly, most of the existing mass variables devised based on different utility can be accommodated in this unified picture, leaving many more new variable elements in this class hitherto unexplored.
One can simplify the discussion under the following assumptions which are rather common in wide class of BSM models: (i)The DM stabilisation is respected by discrete Z 2 symmetry. As a result, all BSM particles in the theory would produce in pair leading to two stable DM particles in the final state. They stay invisible in the detector resulting missing transverse energy as their combined footprint. (ii)There is only one DM candidate in the theory, or if multiple DM particles are there then they are degenerate in mass. One can note that even after making these two assumption the variableŝ min remains global and inclusive.
Under these assumptions analytic expression and properties of this mass-bound variable √ŝ min can be discussed using the non-antler topology displayed 2 in Fig. 1. In the (sub)system under consideration, two mothers denoted by the B 1 and B 2 either produced in hard scattering at the hadron collider, or starting point in the subsystem from a event with longer decay chain. Eventually each of these mothers decays to produce two visible and one invisible particle. The topology can also contain intermediate particles which may be on-shell or off-shell, symbolising into the blue bulb to show the final products only. Momenta p j of these visible SM particles V j (j = 1, . . . , 4) represented by blue lines can be measured at the detector. On the contrary, the invisible particles X i (i = 1, 2) in black dashed lines are of BSM nature with individual mass m i , and 3-momenta q i . The partonic Mandelstam variable for this topology is given by,ŝ Here, n inv = 2 is the number of invisible particles, E v = j e v j and P v z = j p z j are total 2 In general, there can be any number of visibles including asymmetric production topology or asymmetric invisibles (e.g. as in [46]) in the final state, but here we restrict our discussion for simplicity and as a reference for proceeding discussion in following sections. We refer ref. [1,2] for a most generic representation for which these √ŝ variables are constructed and applicable. Figure 1: Representative for a simple non-antler topology where after production of two heavy mother particles B α , each of them leading to single invisible massive particle X i together with number of visibles V j in the final state. The blue bulb represents the intermediate particle which may be offshell or on-shell. The visible particles are SM particles measurable at the detector and represented by blue lines denoted by V 1 , V 2 , V 3 and V 4 respectively. The invisible particles are represented by black dashed lines denoted by X 1 and X 2 respectively. The momenta of visibles and invisibles are denoted by p i , i = 1,2,3,4 and q j , j = 1,2.
energy and total longitudinal component of the visible momenta. In the above equation missing transverse momentum constraints P T = i q iT are also taken into account. Clearly even in this simplified case, there are 3n inv = 6 unknown momenta components, as well as unknown invisible mass with only two constraints from missing transverse momentum. So one can not hope to calculate true values ofŝ involved event by event. But it is important to realise that there is an absolute minimum exist forŝ in each event which also satisfy all these observable. By minimizingŝ with respect to unknown momenta q i subject to the missing P T constraints one gets m is a dimensionless mass fraction varies between 0 and 1 and is given by f replacing the above expression of q iT and q iz inŝ one gets the final form ofŝ min as One can follow from the computation that theŝ min does not assume any particular event topology or the DM stabilising symmetry of the model. Based on common BSM scenario, we restrict our description (also in Fig. 1) assuming Z 2 symmetry, so that, pair production of BSM particles are considered producing two invisible massive particles in the final states. From minimisation conditions in Eq. 2.2 and 2.3 one can infer that each DM particle carries a fraction of missing momenta, proportional to the corresponding mass fraction f (i) m . However finalŝ min is simply a function of total M inv irrespective to this fraction. Once we assume a pair of invisibles in the final state with same mass (or both massless), then this fraction f (i) m comes out as 1/2 for any choice of trial mass including the true invisible mass. The invisible momenta at the minimisation are, These invisible momenta calculated from minimisation may not represent that of the true event. However the uniqueness of these momenta can be useful to study the semi-invisible decays involved both in SM as well as BSM scenario. Momentum reconstruction can be exploited to analyse the properties of top quark decaying invisibly in the SM, whereas DM motivated BSM models are commonplace where uniqueness of invisible momenta can help to study decays with different topology. One can notice that the invisible momenta constructed throughŝ min are always parallel to each other with a magnitude proportional to mass fraction. Here we investigate how the partial knowledge of event information can improve the variablê s min and also reconstructed momentum obtained from it. In our further discussion, we divide the production topology as two kinds, such as, antler topology and non-antler topology. We discussŝ min with and without putting on-shell constraints in both kinds of these topology.

Antler topology and constrained variable
Antler topology is very common and well motivated in SM Higgs production. Resonant Higgs production and its semi-invisible decays into W-boson, h → W W * → lνlν or through τ decay h → τ τ → wν τ wν τ are some of the interesting channels. Several popular BSM scenario also have these production, such as, supersymmetric (SUSY) heavy Higgs decays through neutralinos H →χ 0 2χ 0 2 → Zχ 0 1 Zχ 0 1 [59], or SUSY Z production with leptonic decay Z →˜ +˜ − → −χ0 1 +χ0 1 [60,61]. In the model of universal extra dimension (UED) one can produce resonant second excitation states decaying into couple of lighter states, like [62,63]. Other class of examples being resonant exotics production with their semi-invisible SM decay. Doubly charged scalar in hadron collider can decay with one of the dominant decay channel into w-pair, φ ++ → w + w + → + ν + ν [64]. Similarly heavy Higgs or heavy Z can have SM semi-invisible decay H/Z → tt → bbw + w − → bb + ν −ν . Some of these antler topology was studied [65][66][67] showing that the invariant mass, transverse momenta and angular correlations constructed out of visible decay products are effective to measure the invisible and intermediate particle masses whereas heavy resonant mass is assumed to be known. Cusp and kink structures appeared in the distributions of these variables and their positions are also related to the unknown masses. Missing transverse Figure 2: Representative for a simple antler topology where A is a Z 2 parity even heavy resonance produces and decays to two daughter particles B 1 and B 2 and each of which finally decays to two SM visibles V j and one invisible particle X i . momentum constraints are not used in these study. Production mechanism in a linear collider (e + e − ) with a fixed center of mass energy is also very similar to this antler decay topology and semi-invisible decays can be studied in similar fashion [68].
Representative diagram for antler topology is shown in Fig. 2. Parity even heavy resonant state A, produced through on-shell production at the hadron collider, promptly decays to pair of parity odd particles B 1 and B 2 . In this simplified picture, each B subsequently decays same way as we have described earlier in Fig. 1, and thus producing couple of visible with an indivisible daughter. We also keep the same notation for momentum assignment associated to all final particles. Before defining theŝ min in presence of the additional constraints, we first list all the constraints available for this present topology. Apart from the antler resonance mass-shell constraint at some fixed value of theŝ depending upon resonant mass M A , additional mass equations and missing transverse momentum relations for this topology can be put together as, {constraints}: Among all constraints defined in the Eqs. 3.2-3.4, the variableŝ min is already satisfies last four constraints comprising two missing P T components and two mass-shell constraints from invisible daughters. In other words, new variables are further constrained with mass-shell relations of intermediate parents.
The true value of partonic Mandelstam variable for antler topology is the mass of the heavy resonance, that is √ŝ T rue = M A , once the heavy resonance produced on-shell and having narrow decay width. Hence, any mass bound variable constructed by minimisation, such as,ŝ min for antler topology needs to be bounded from above satisfying the relation s min ≤ŝ T rue . This end point can be measured from the endpoint at the distribution over many events. Constrained variableŝ cons min also satisfy similar relationŝ cons min ≤ŝ T rue , having endpoint at theŝ T rue . However additional intermediate particle mass-shell constraints ensure a larger value ofŝ cons min overŝ min for each event. This inequality would also reflect in the mass variable distributions contributing larger number of events at the endpoint of the distribution.
As we discussed earlier, a rather striking consequence of this additional mass-shell constraints are that they also permit us to construct a finite upper mass bound variable, which is meaningless otherwise. We define this constrained variableŝ cons max as the maximum partonic Mandelstam variable,ŝ cons max = max satisfying all the available constraints in the event listed in Eqs. 3.2-3.4. One can point out that this variable has a lower bound at theŝ T rue , so that the distribution of theŝ cons max will start at this value. More significantly with a large number of events at this threshold. Interesting point about these constrainedŝ variables are that the reconstructed momenta from their minimisation(maximisation) not only unique but they also improve over the same calculated throughŝ min . In case ofŝ cons min , better momentum reconstruction ensured by the points closer to its endpoint. Similarly,ŝ cons max gives better reconstruction from the points associated to its threshold. These points would be discussed further in the Sec. 5, where these correlations would be more evident. Finally, the definitions of differentŝ variables, after imposing different constrains, ensures the hierarchy among these mass variables: To illustrate the properties of these constraint variables, first we consider a simple example of SM Higgs production through gluon fusion at the hadron collider. Higgs boson decays further semi-invisibly through tau pair production, h → τ τ → wν τ wν τ . To compare with the representative diagram for the antler production in Fig. 2, τ being the intermediate particle of constrainedŝ. Neutrino ν τ is the invisible particle X i . We considered hadronic (leptonic) decay mode for the W boson thereafter to consider two invisibles (four invisibles tested in next example) in the final state. The distribution ofŝ min andŝ cons min are shown in the Fig. 3(left). Red binned histogram shows the distribution forŝ min , which can be calculated numerically or using analytical expression. Blue histogram shows the distribution of constrainedŝ cons min . As expected the the endpoint of both theŝ min andŝ cons min distributions are at the √ŝ T rue = M h = 125 GeV for a choice of invisible mass as zero. Evidently larger number of events at the endpoint for theŝ cons min distribution with a sharper drop can be measured more precisely. This is even more important once corresponding background also considered together. The Fig. 3(right) demonstrates the distribution of other constraint variableŝ cons max which has a threshold atŝ T rue with considerable number of events at the threshold.
It is expected that the endpoint of kinematic distribution would be less populated if one has more number of invisibles in the event. This is because increasing number of invisibles would increase the number of unknown momentums restricted with the same constraints. Following our previous example, we now consider the four invisible particle by decaying both w leptonically and demonstrated correspondingŝ min andŝ cons min distributions in Fig. 4(left). The red histogram shows the distribution forŝ cons min , whereas the green binned histogram showŝ s min to compare the effect due to the extra constraints. These distribution confirms that the number of events at the endpoint are considerably low as one increases the number of invisible particle in the final state. Although constrained variable can improve the situation only slightly, overall both these distributions forms a narrow tail rather than the sharp endpoint.
We further study one more interesting example from the resonant production of exotic doubly charged scalar [64] production at the hadron collider following its decay into dominant decay channel producing pair of w, which in turn decays leptonically. Hence, the resonant sub-system under consideration is φ ++ → w + w + → + ν + ν . In hadron collider this exotic state φ ++ can be produced associated with charged w − which mainly decays hadronically and it is possible to disentangle from the antler sub-system producing lepton pair from the exotic decay. We choose to use corresponding subsystem variableŝ sub min for our analysis. Here analytical expressions for the invisible particle momenta remain same except the modified form for P T which includes the visible contribution from non-sub-system [2]. The distributions for theŝ sub min and the constrained variableŝ sub,cons min are demonstrated in the Fig. 4(right). Dark binned histogram represents the distribution forŝ sub min which can be calculated both analytically or using numerical minimisation. The cyan colored histogram is the distribution forŝ sub,cons min utilising extra w mass-shell constrains, and minimised numerically. One can note that theŝ sub,cons min is performing better in getting the endpoint at φ ++ mass. Observed small tail is because of finite width from φ ++ and these extra constraints ensures that theŝ sub,cons min distribution starts from a threshold at the scale of 2m w .

Non-antler topology and constrained variables
Non-antler topology is extremely common in most of the BSM theories and also abundant in SM. This topology is already described in the Fig. 1, where B i are the parent particles produced in pair. After a cascade of decay each side of the decay chain produces number of visibles along with a massive invisible particle X i . Detailed discussion on the behaviour of s min as a mass bound variable is done extensively for this kind of topology. Here we would illustrate the constrained variables in the light of additional on-shell constraints. Unlike using GeV respectively. Non-antler heavy parent particle pair production must have a true parton level CM energy distribution starting from a threshold value of total parents mass as shown by yellow colored histogram. As a consequence of additional constraintsŝ cons min distribution also poses this same threshold, however with a considerable number of events at the threshold. these exact on-shell constraints for the parents mass, which is primarily one would like to know through these mass bound variables, ref. [53] uses constraints from the equality of two parents mass. Following our analysis in previous section, we continue using these mass-shell constraints with an expectation of improved momentum reconstruction.
As a consequence of on-shell constraints, one can expectŝ cons min distribution would start from a threshold at the sum of parents mass. This is contrary to the unconstrainedŝ min distribution which exhibits peak at that position giving an excellent correlation for the new physics mass scale. This is demonstrated in Fig. 5 where distributions forŝ min (black) andŝ cons min (blue) are plotted using a toy model of non-antler pair production at the the hadron collider, with parent and invisible mass as 300 GeV and 200 GeV respectively. Unlike antler decay topology where heavy particle resonant production form a near delta function at the parton level center of mass energy, here heavy parent particle pair production has a distribution starting from a threshold value of total parents mass as shown by yellow colored histogram. As we note that theŝ cons min distribution also poses this same threshold, however with a considerable number of events at the threshold. We will follow further in the next section to show the improvements in invisible momentum construction as a presence of these constraints and the events contributing at the threshold. Analogous to the variables constructed for antler topology, one can follow the similar hierarchy among all the constrained s mass variables after imposing different constrains: s min ≤ŝ cons min ≤ŝ T rue ≤ŝ cons max .  Figure 6: Some example events demonstrating the invisible momentum reconstruction in case of antler topology through minimisation during construction of differentŝ variables. Each color shaded region is representing the allowed phase space by additional constrains in the unknown invisible momentum parameter space. In both of the plot yellow elliptical region is constrained area describing q 1z ( q 1T ) and green elliptical region is constrained area for q 2z ( q 1T ). Intersection region between these two constrained ellipse, shaded in white, is eligible for containing all the constraintŝ( q 1T ) parameters as well as the trueŝ. Two other ends in this overlapping region would typically represent theŝ cons min andŝ cons max , with trueŝ in between them. Sinceŝ min does not have this additional constrain, it would be outside overlapping region and far from trueŝ. Inside the overlapping regionŝ contours are also presented where true CM energy is matches with the value of Higgs mass. The left figure shows one example event whereŝ T rue is closer toŝ cons min . This contributes at the endpoint of thê s cons min distribution and also giving better momentum reconstruction. The right figure shows another event whereŝ T rue is close to theŝ cons max contributing at the threshold of this distribution with better momentum reconstruction.

Event reconstruction capability
In this section we describe the invisible momentum reconstruction capability using mass variablesŝ min and improvement in it accounting for additional constraints in the context of antler and non-antler decay topology. Analytic expressions for invisible momenta components from theŝ min was already discussed in Sec. 2. It was also argued that these reconstructed invisible momenta usingŝ min are unique irrespective of any topology considered. Note that these reconstructed momenta from the minimisation ofŝ min are not the true momenta, but approximated momenta consistent with the observables in such event. These calculated momenta can be correlated with the true values of them to find the reconstruction efficiency similar to the other reconstruction methods like MAOS [56,57].
To describe the consequence of the constraints given in Eqs terms of their transverse components q iT . We get, where M vi is the invariant mass of visibles in the i-th decay chain, i = 1, 2. Missing transverse momentum constraints further permit us to rewrite them in terms of a single invisible particle transverse momentum components, which we choose as q 1T for our examples. By simplifying the right hand side of the Eq. 5.1, one gets the equation of ellipse in terms of the transverse momenta and the parameters outside the ellipse are not physical with the given event. Two elliptical allowed regions for each event correspond to two side of decay chain and these two regions can not be completely disjoint from each other. All the available constraints in an event are satisfied only at the intersection region between them. Different situations can emerge for this overlapping region. Two ellipses may intersect each other over a finite region or a point (touching each other). In other case one ellipse may contain the other ellipse. In Fig. 6 we consider such constrained regions demonstrated for two different events in antler topology. Each color shaded region is representing the allowed phase space by additional constrains in the unknown invisible momentum parameter space. Overlapping region between these two constrained ellipse is shaded in white whereŝ contours are also presented. One can identify the minimum value from this intersection region asŝ cons min and the | using both unconstrained (red) and constrained (blue)ŝ min methods. Left (right) plot displays the momentum reconstruction capability in antler topology for transverse (longitudinal) components of momentum. In each figure shown histograms one directly fromŝ min for which agrees with corresponding analytical form, also shown histograms using constrained minimisationŝ con min to compare the improvements over unconstrained ones. maximum value asŝ cons max which reside at opposite ends within this region. Sinceŝ T rue also satisfies all the constraints in the event it must also remain in the intersection region and in between these those two constrained points 3 . Since theŝ min variable does not satisfy all additional constraints in the event, it would lie outside the intersection region and relatively far from true value. The left figure display one typical example event whereŝ T rue is closer toŝ cons min . This contributes at the endpoint of theŝ cons min distribution and also giving better momentum reconstruction. The right figure shows another event whereŝ T rue is close to thê s cons max contributing at the threshold of this distribution with better momentum reconstruction. In both figure, we depicted different colored dots for the the position (invisible momentums during minimusation or maximisation) of allŝ variables together with actualŝ correspond to that particular event. One can even read the corresponding values of these mass variables from their contours plotted within intersecting region. Similarly in Fig. 7 we have shown the momentum reconstruction capability ofŝ cons min andŝ min in an example of Non-Antler topology. The yellow and green shaded regions represents constrained q 1z ( q 1T ) and q 2z ( q 1T ) respectively and their intersection region is suitable for constrainedŝ. The red, orange and black point shows the true momenta and reconstructed momenta given byŝ min ,ŝ cons min respectively. We are now in a position to quantify the capability of momentum reconstruction.  In each figure one histogram (in red bins) is shown forŝ min which agrees with corresponding analytical form. Also histograms with blue bins plotted in the same figure to display the momentum reconstruction capability using constrained minimisationŝ cons min pointing out improvements over unconstrained ones.
We discussed the additional constraints inŝ min to choose the minimisation that gives reconstructed invisible momenta closer to their true values. To understand this consequence better, we look into the movements of these calculated momenta once we impose the constraints. In Fig. 9 we demonstrate through a correlation plot of constructed invisible momentum versus the corresponding true momentum taking few random representative event points. In both plots, each red dot point represents the calculated momentum derived from theŝ min against the corresponding true momentum for each event. Similarly, green dots are for corresponding momentum derived from theŝ cons min . The purple arrows connecting from one red dot to other green dot represent the shift in the derived momentum once extra constraints are imposed. Since the true momentum is always same for a particular event, shifts due to minimisation in different mass variables are only horizontal. These arrows represent the degree of change due to constraints, shifting calculated momentums towards the diagonal true momentum points. Diagonal blue points the simply correlate true momenta with true momenta in each event to give the perspective how derived momenta composed against the true values. Left (right) plot corresponds to the transverse (longitudinal) momenta derived fromŝ min andŝ cons min .

Summary and conclusions
Large Hadron Collider started its extremely successful journey finding the long sought scalar. With no substantial evidence for BSM, expectation is high for next run of LHC. In the light of dark matter models, missing energy signals would be looked very carefully. Thê s min variants of mass variables were designed for prompt finding of mass scale in a model independent way for any complex topology of BSM events associated with semi-invisible final production. In the present analysis, we proposed to exploit additional partial informations available in the event as constraints to improve the finding. We classified our discussions based on two different class of simple production topology widely available both in SM and BSM production, which are, antler and non-antler. Different SM as well as new physics predicts antler production processes, including important Higgs production in the hadron collider. These topology can be constrained significantly using additional intermediate mass-shell conditions. We have demonstrated with different examples to show that the constrained variableŝ cons min can significantly improve the distribution and the measurements. More interestingly, these additional constraints ensures a finite upper value of theŝ variable, defined as,ŝ cons max which is not well-defined and finite in the unconstrained picture. Hence, this new variable also can be exploited up to some extend. Apart from considering different BSM example to demonstrate these variable in the context of sub-system topology and in the difficult signatures with more invisible final states in antler topology, we also demonstrated effect of these additional constraints in a simple non-antler topology.
To clarify the effects of these constraints in the invisible momenta parameter space, we choose phenomenological examples explicitly demonstrating how these mass variables are restricted and pushed towards the true values ofŝ, together with their choice the invisible momentums closer to that of true one. Hence, one can consider to quantify the capability of reconstructing the invisible momenta in present scenario. We constructed and shown the efficiency of momentum reconstruction using these constrainedŝ variables which predicts a unique momenta associated with each of these mass-bound variables in each event. In conclusion, we explored the utility of additional informations we may already have, during exploration of theŝ type mass variables which can be exploited with missing energy data at the LHC.