Mass reconstruction with M_{2} under constraint in semi-invisible production at a hadron collider

Mass reconstruction with under constraint in semi-invisible production at a hadron collider

Partha Konar Physical Research Laboratory, Ahmedabad-380009, India    Abhaya Kumar Swain Physical Research Laboratory, Ahmedabad-380009, India

The mass-constraining variable , a -dimensional natural successor of extremely popular , possesses an array of rich features having the ability to use on-shell mass constraints in semi-invisible production at a hadron collider. In this work, we investigate the consequence of applying a heavy resonance mass-shell constraint in the context of a semi-invisible antler decay topology produced at the LHC. Our proposed variable, under additional constraint, develops a new kink solution at the true masses. This enables one to determine the invisible particle mass simultaneously with the parent particle mass from these events. We analyze in a way to measure this kink optimally, exploring the origin and the properties of such interesting characteristics. We also study the event reconstruction capability inferred from this new variable and find that the resulting momenta are unique and well correlated with true invisible particle momenta.

Beyond Standard Model, Hadronic Colliders, Particle and resonance production
12.60.-i, 14.80.Ly, 11.80.Cr

I Introduction

The Large Hadron Collider (LHC), after the successful completion of its first run and the remarkable achievement of the Higgs boson discovery Chatrchyan:2012ufa (); :2012gk (), has already entered into its second phase. Upgraded with higher energy and luminosity, the main physics goal is to explore the multi-TeV scale associated with the physics beyond the Standard Model (BSM). Although the LHC has not reported any clinching evidence for new physics so far, expectations are running high for possible new physics signals in soon unless such signatures are already hidden inside the LHC data. Any scenario with a positive outcome essentially demands the measurements of the mass, coupling and spin of new BSM particles. However, this is going to be complex since many of the very likely scenarios with a wide class of BSM models have embraced the concept of thermal relic dark matter (DM) as some stable exotic member within them. Since these massive DM particles are colorless, electrically neutral and weakly interacting, once produced in the collider, they do not leave any trace at the detector. Hence, one needs to rely on the experimentally challenging signature of missing transverse momenta from the imbalance of total transverse momentum, accounting for all visible decay products in each event from an already adverse jetty environment of the hadron collider. Moreover, the DM in many models is expected to produce in pairs because of its stabilizing symmetry,111There can be a DM stabilizing symmetry other than which allows more than one DM particle per vertex and kinematic variables Agashe:2010gt (); Agashe:2010tu (); Cho:2012er (); Giudice:2011ib (); Agashe:2012fs () were used to distinguish between the DM stabilizing symmetries, but we stick to the popular parity for simplicity and extendability. commonly associated with the parity, which makes the event reconstruction even more challenging.

Keeping an eye on the above encumbrances, many ideas have already been developed for the determination of mass and spin. For some recent reviews, see Refs. Barr:2010zj (); Barr:2011xt () for different mass measurement techniques. To give a brief and sketchy classification for the mass measurement techniques, parts of which are also crucial for our study, one looks into the following:(a) In the end-point method Hinchliffe:1996iu (); Gjelsten:2004ki (); Allanach:2000kt (); Nojiri:2000wq (); Gjelsten:2005aw (); Burns:2009zi (); Matchev:2009iw (), the end point of the independently possible invariant mass constructed out of all combinations of visible particles is related to the unknown masses involved in a decay chain. Thus, a sufficient number of end-point measurements are needed to pin down all the unknown masses, which is only possible for a longer decay chain. (b) The polynomial method Nojiri:2003tu (); Kawagoe:2004rz (); Cheng:2007xv (); Nojiri:2008ir (); Cheng:2008mg () tries to solve for all unknowns (unknown masses and invisible particle four momenta) in the event, using all available constraints. Again all unknown masses can be determined only if one has at least a three-step decay chain. (c) The transverse mass methods are based on transverse mass variables defined by the minimum parent mass consistent with minimal kinematic constraints in the event, for a given trial mass of the invisible particle. Here constraints consist of the equality of parent mass, mass-shell relations and the missing transverse momenta from undetected invisible particles. Several interesting features of these variables received lots of attention recently. The transverse mass variables were further extended with many variants, such as,  Lester:1999tx (); Barr:2003rg (); Meade:2006dw (); Lester:2007fq (); Cho:2007qv (); Cho:2007dh (); Barr:2007hy (); Gripaios:2007is (); Nojiri:2008hy (), subsystem  Burns:2008va (), and its sister  Konar:2009wn (),  Cho:2009ve (); Cho:2010vz (), asymmetric  Barr:2009jv (); Konar:2009qr (), contransverse mass  Tovey:2008ui (); Polesello:2009rn (); Serna:2008zk (), and its sister  Matchev:2009ad (). In Ref. Mahbubani:2012kx (); Cho:2014naa () it is shown that the -dimensional generalization of can be useful, where one can apply the equality of parent mass and/or equality of relative222Relative is defined by any resonance other than parent and daughter realised in some particular topology. mass constraints to improve the number of events at the end point. (d) Inclusive variables are the ones whose construction do not take into account the production mechanism or topology of the new particles and only depend on the final state visible particles momenta and missing transverse momenta in the event. There are many inclusive variables, such as  Tovey:2000wk (), total visible invariant mass  Hubisz:2008gg (), , effective mass  Hinchliffe:1996iu (), and its sisters Konar:2008ei (); Konar:2010ma (); Swain:2014dha (); Swain:2015qba (). They are useful for extracting the mass scale of new physics in a model independent way. While each method is more relevant for a particular outlook, the transverse mass variable and its extensions turned out to have the potential for precise measurement of the masses in short single-step decay chains involving invisible particles. Consequently, for our present analysis, we further develop and discuss this method in the next section.

The present work is inspired from another class of prescriptions known as the cusp method Han:2009ss (); Han:2012nm (); Han:2012nr (); Christensen:2014yya (), which can determine both the parent and invisible particle (or DM) masses simultaneously from single-step decay chains. However, the primary assumption is that both the parents are produced from an on-shell heavy resonance, whose mass is known already. This method uses the measured momenta of visible particles and constructs possible variables, among which the invariant mass and magnitude of the individual transverse momenta of the visible particle can exhibit non-smooth structure or cusp and end point in their distributions. The location of this cusp and end point are function of the unknown masses, so the parent and daughter mass can be determined simultaneously from these two measurements. It is argued that the longitudinal boost invariant variables, such as the individual transverse momenta and the invariant mass of visible particles, are minimally affected by the unknown transverse boost, which are sufficient to pin down both the unknown masses. This analysis was extended further in the context of the linear collider (Christensen:2014yya ().

In this paper, we propose a complementary procedure to measure both the parent mass and daughter mass simultaneously using the variable. We also assume that the heavy resonance mass is known, and this information is embedded in the minimization of resulting in a new constrained variable dubbed . Further discussion on this variable is in Sec. II with formal definitions and chronological motivations. Emboldened by the interesting characteristic of this variable which develops a kink in the distribution end-point maxima corresponding to the correct value of the unknown invisible mass, we propose a simultaneous measurement of both the masses by identifying the kink position. We also put forward a desirable way of recasting the kink by utilizing all available data, giving a robust outcome irrespective of other realistic effects. This variable proved to be useful in all mass range, including the region known as the ‘large mass gap’ in the kinematic cusp method where the cusp may not be very sharp. A unique event reconstruction can also be realized using .

As a (1+3)-dimensional variable, has the capability to use all components of three momenta. Thus, the utilization of additional mass constraints is possible if such information is available. This brings additional advantages, contrary to the transverse mass variables, which by construction are not similarly capable. Here we argue that the newly developed variable equips one to consider additional constraint from on-shell mass resonance, which in turn produces suitable kinematic constraints to get a new kink solution in antler topology. Importantly, this contribution towards the kink is not merely symbolic but is substantive, enabling better mass measurements.

This paper is organized as follows. In Sec. II, we motivate the antler production topology at the hadron collider describing all available constraints associated with such topology. In this context, we also open up discussion on the transverse mass variable and its -dimensional generalized sisters in the family. Following that, we introduce our new variable . We further discuss the effect of an additional heavy resonance constraint, describing the basic features and benefits of this variable. Sec. III is assigned to describing the kink solution coming from the distribution end point as a function of the trial invisible mass which is yet unknown. Further comparison is made with the corresponding maximum achievable in the variable. Realizing that the number of events at the end point is very limited for , we formulated an efficient way in Sec. IV to reconstruct the kink by recasting all available data. In Sec. V, we show that can be used for a unique event reconstruction, and the reconstructed momenta are well correlated with the true momenta of the invisible particle. Finally, in Sec. VI we summarize our main results and conclude.

Ii Antler topology and constrained variable with

Antler topology, realized in different Standard Model (SM) processes and a variety of new physics models, is very common and widely explored. The SM Higgs boson decaying semi-invisibly through the W-boson,333Note that except from the fact that probably signify most familiar SM antler channel, this off-shell production of is not pursued further in present analysis. , or via lepton, , are some of the significant channels in the context of the SM Higgs search at the LHC. Similarly, in several BSM theories, the search strategy relies on antler production topology. Some of these include the heavy Higgs of supersymmetry (SUSY) decaying to the Z-boson and lightest supersymmetric particle (LSP) via neutralinos,  Djouadi:2005gj () and the SUSY extended decaying to the lepton and LSP via the slepton,  Baumgart:2006pa (); Cvetic:1997wu (). Similarly, in a universal extra-dimensional model, second excitation states can decay to first excitation states,  Cheng:2002ej (); Datta:2005zs (). The semi-invisible decay of doubly charged exotic scalars in many BSM scenarios can produce SM particles via W pairs,  Bambhaniya:2013yca (). Moreover, the heavy Higgs or heavy can also decay semi-invisibly to SM particles via pairs, . In addition, antler topology can also be realized at the linear collider as fixed c.m. energy is equivalent to the heavy resonance produced at its rest frame before pair production and subsequent decay (for example, see Christensen:2014yya ()).

Before starting our analysis, let us describe the basic setup and the notation. The representative diagram for antler topology is shown in Fig. 1, where a parity-even444Parity is pertinent only for the BSM processes having stable invisible exotic particles in the final state. heavy resonance particle (grandparent) with mass decays to two parity-odd particles and , each of which subsequently decays to the Standard Model particle () and an invisible or dark matter particle (). We assign the momenta to visible and DM particles in each side of the decay chain as and , respectively. Moreover, we denote the masses of parents () and invisible daughters () as and , respectively. The primary motivation of this analysis is to determine these unknown parameters. Though we have shown a generic antler topology in Fig. 1, in this analysis we are interested in the symmetric antler process motivated by the above examples. The symmetric antler includes same parent ( = ) and same daughter ( = ) particles, or at least their masses are same, and .

Figure 1: Archetype of antler topology where , a heavy resonance particle with mass produced at hadron collider, decays to two daughter particles and through two-body decay, each of which subsequently decays to produce SM visible particle () and an invisible or dark matter particle (). Momenta of these visible and invisible particles are assigned as and , respectively. and are masses of parent and invisible particle . This topology can be considered for SM Higgs production with subsequent decays into SM visible and massless neutrinos as invisible particles. On the other hand, in a BSM scenario, (a parity-even state) can decay to produce (parity-odd states) and .

We would like to address this topology using the mass-constraining variable for the subsystem represented by the gray shaded region shown in Fig. 1. Let us start with the existing and popular transverse mass variable before moving into the generalization and finally extending to our new variable. is defined to have the potential to measure the masses of the BSM particles both in short or long decay chains, although its dominance and significance is mostly grounded in its capability to handle the former case. The classic definition555Here, “” and “” in stands for the transverse projection and two parent particles, respectively, in the topology under examination. Reference Barr:2011xt () generalized and unified the concept of mass variables and set a preferred nomenclature according to the order of operations to rewrite the same variable as within a general family. Notably, Barr:2011xt () also demonstrated the fact that the transverse projection can be done using not one, but three completely different schemes. of is given by the larger value between two transverse masses constructed from both sides of the decay chain and minimized over unknown invisible momenta satisfying the constraints of that event. Mathematically,


with the usual definition of the transverse mass for each decay chain,


and are -dimensional transverse energy-momenta corresponding to the visible and the invisible decay products in the decay chain, respectively. Note that, in the definition of , the minimization is done over all possible partitions of and the maximization of within the bracket ensures a closer shot towards the parent mass . By this definition, , calculated for each event, must be smaller than or equal to .

The maximum quantity for any of these mass variables (such as, , or ) over the available data set is


Now should provide a very close estimate of . Moreover, in the scenario where the invisible particle mass is a priori unknown, e.g. dark matter models, would still offer a useful correlation with the trial invisible mass , a daughter mass hypothesis used as an input for the calculation of . One can possess only this partial information on the unknown parent and daughter masses unless, under some special circumstances, this correlation curve generates a kink feature exactly at the correct mass point. We will have a slightly elaborate discussion about the kink feature in Sec. III.

Now to motivate the -dimensional generalization of previous definitions as in Eq. 1, one readily notes that the is not utilizing longitudinal components of the momenta and, thus, the available mass-shell constraints for a given topology. is thus constructed Barr:2011xt () out of the -dimensional momenta by removing all “” in the definition of Eqs. 1-3 (except that of the total missing transverse momentum constraint under the curly bracket, since the longitudinal part is not available in the context of the hadron collider). Now one can apply the on-shell mass constraints in the minimization of , and, depending on the constraints applied, different constrained classes of the variable (e.g. , and ) can be constructed; details about these variables can be found in Ref. Cho:2014naa (). Using similar notation, one can readily come up with the first two types of variables available from the subsystem considered in Fig. 1. Here, is the -dimensional generalization of with the equality of the parent mass constraint applied in the minimization,


with the -dimensional mass from each decay chain as


The corresponding variable is simply perceived once the last constraint inside bracket is absent, just like the transverse mass case in Eq. 1. It is straightforward to show Cho:2014naa ()


Also note that, in our example, there is one visible particle per decay chain in the final state. Hence, the and other variables always come from a balanced configuration irrespective of the choice of trial invisible mass. So once again the maximum (or the maxima of other variables as in Eq. 7 and Eq. 8) can only give a constraint between parent and invisible particle mass.

Now, by following the steps before the subsystem as in Fig. 1, one realizes that the parents () are actually originated from a heavy resonance (). In a BSM scenario, even-parity can directly decay to SM observable particles and hence the mass, , can in principle be measured. Here, before we move further, we assume that in our topology only this heavy resonance mass is known. We are now in a position to develop a variable using this mass constraint, so that666One can also consider an additional constraint using the equality of parent mass in . Although this would further constrain the allowed invisible momentum space, it would finally choose the same minima. Hence, our arguments with this present example and analysis remain the same.


where -dimensional invariant mass is as in Eq. 6. Additionally, the dependence on the unknown trial invisible mass is shown explicitly. With this additional constraint, one expects a squeezed phase space affecting this new variable from that of ; furthermore, we will soon realize that this effect is a little more far-reaching. Before we gradually move to demonstrate that, let us open the discussion with the consequences of this new variable in the invisible momenta space.

The additional heavy resonance mass-shell constraint in the minimization (last condition inside bracket of Eq. 9) constrains the invisible particle momenta, such that, the invariant mass of the parents is confined to a narrow resonance of mass . This phenomenon is true for each event. In Fig. 2, the effect of this constraint is demonstrated for one event with an example where the trial invisible particle mass is considered smaller than the yet unknown true mass . The region represented by the light temperature map color gradient is the maximum between two transverse masses , as in Eq. 1 before executing the minimization. This is shown with respect to the invisible momenta components, and , by taking care of the missing transverse momenta constraints. Now the minimum of this quantity, which is nothing but the , is the minimum point in the color map displayed by the filled circle , and different contour lines are shown by dashed curves. Moving to our -dimensional new variable with the heavy resonance mass-shell constraint in Eq. 9, once again after doing the similar exercise we get the solid contour curves superimposed in the same plot. Of course, we are no longer showing the color gradient as done in the transverse mass case. Transformation of dashed contour lines into corresponding solid ones (same color represents the same value of that contours) within the same region qualitatively indicate the effect of this additional mass constraint. Minimum of these solid contours represents the , displayed by circle plus in the same figure. Note that the longitudinal momenta components for the invisible pairs are eliminated in this demonstration by minimizing with the mass-shell constraint.

Figure 2: The effect of the heavy resonance mass-shell constraint is demonstrated using the mass variables and considering one antler event in the invisible momentum component space. The region represented by the color gradient is the maximum among two transverse masses coming from two decay chains. Corresponding contours are shown with dashed lines. Following Eq. 1, the minimization of this quantity, the , is represented by the filled circle . In the same plot, the solid lines (of same colors) are delineating the corresponding contours for the -dimensional new variable with heavy resonance mass-shell constraint as in Eq. 9. Note that only the contour lines are shown in this case, not the color gradient as in the transverse mass case. The minimum of these solid contours is represented by the , displayed by circle plus in the same figure. The mass-shell constraint restricts the invisible momenta, making the region shrink, as depicted by the dashed and solid lines. The white dashed (solid) line represents the equality of transverse-mass (mass) of parents and this equality line is also moving towards higher values because of the constraint. The red star is the position of true transverse momenta of invisible particle. The mass spectrum we choose is (, , ) = (1000.0, 200.0, 100.0) in GeV and the trial invisible particle mass we took for this plot is 10.0 GeV


As noted, constraint restricts the invisible momenta making the region shrink as depicted by the dashed and solid contour (e.g. following blue lines correspond to 100 GeV), the dashed line contour does not satisfy the additional mass-shell constraint while solid contour does. The same is true for all other lines also. The represented values of and considered in this example are 75.1 and 98.5 GeV, respectively, for trial mass at 10 GeV which is smaller than the true invisible mass 100 GeV. The corresponding true mass of parents and heavy resonance are 200.0 and 1000.0 GeV, respectively. The white dashed (solid) line represents the equality of the transverse-mass (mass) of parents and this equality line has moved towards higher value because of the constraint. As a result, one naturally expects event by event. The red star is the position of the true transverse momenta of the invisible particle. Clearly, constraint brings the minima, , closer to the true momenta, and this can improve any effort to reconstruct the invisible momenta. This feature will be further considered and discussed in Sec. V.

Figure 3: Normalized distributions of the mass variables are delineated using a toy model of antler topology with parents mass at 200 GeV. Both the (dark) and (green) distributions, considering the invisible particle mass at its true value (100 GeV), produce the end point at the correct parent mass. However, the heavy resonance constraint gives a higher value, resulting a larger number of events at the end point.

One more remarkable feature emerges at this point which will be capitalized in the next section. We already noted the event wise upward shift of values under the constraint. The next natural question in this context concerns the maximum value achievable by this mass variable and how it is related to the trial missing particle mass? The experimentally measured maxima can deviate (downward) from the theoretical maximum of the mass variable depending upon the accessible number of events, and more importantly, the abundance of events towards the end point of the distribution. We postpone this issue for the time being and consider it again in Sec. IV. Now coming back to our variable, it should not be surprising that at the true value of the invisible particle mass (i.e. when ), the maximum value of the constraint variable coincides with that of the variable without this constraint, . This is because both of these variables are derived for the same topology. On the other hand, at all other trial mass values, not only is the individual (event-by-event) constraint quantity larger, but also the maximum of that constraint mass is the larger value. To write in a compact form,


While this point is further discussed in the following section as a means of measuring the unknown masses, here we illustrate it with one example distribution for the aforementioned variables, considering a toy process with the antler topology as shown in Fig. 1. For demonstration purposes, we choose a mass spectrum with in GeV, which is a relatively difficult region for the kinematic cusp method Han:2009ss (); Han:2012nm () known as the “large mass gap” region, where the cusp may not be very sharp, leading to large errors in the mass determination. The variable can be effective for mass determination both in the large mass gap region as well as in other regions of phase space. In Fig. 3, we have compared the normalized distributions for and at the true mass of the invisible particle. Both the constrained (green histogram) and unconstrained (dark histogram) distributions share the same end point precisely at the parent mass, as argued earlier. However, from the distribution, one should also note the movement of the events towards the higher value under the heavy resonance constraint and, thus, expect a larger number of events at the end point.

Iii Mass measurements with kink

Kink in mass measurement techniques is a widely acclaimed feature, first shown in the context of the variable. Let us continue from the brief discussion below Eq. 1 where the distribution maximum is observed to offer a useful relation correlating the parent mass with the trial value of unknown invisible mass . It is shown Cho:2007qv (); Cho:2007dh () that simultaneously both parent mass and daughter mass can be determined by identifying a kink (continuous but not differentiable) in this correlation curve, where the true mass point resides. It is worthy of attention that the has two different functional forms before and after this kink, and they share the same value at the true mass point. This behavior stems from the fact that the visible system invariant mass of any (or both) decay chain(s) have to have a range of values; hence, there should be at least two visible particles per decay chain. Consequently, experimentally simpler single-step decay chain topology is deprived of such advantage. The above feature was shown where the system does not have any recoil from initial state radiation (ISR) or upstream transverse momenta (UTM). But the presence of ISR is inevitable during the production at any hadron collider. It is subsequently revealed Barr:2007hy (); Gripaios:2007is (); Burns:2008va () that kink can also arise from topology having a single-step decay chain on both sides, but there should be recoil to the system which may come from ISR or UTM. Both scenarios can naturally arise in subsystem context from a longer decay chain. However, sizable kink resolution only comes from the events with very high recoil , essentially with very low statistics.

In the last section, we define the constrained variable using the heavy resonance on-shell constraint in the minimization of . Analogous to the previous discussion,777At this point, we would like to make it clear that the effect of ISR/UTM is not considered in this present analysis. This study shows a new kink solution due to the kinematic constraint coming from on-shell mass resonance in antler events. If one consider such events associated with ISR, that may marginally contribute strengthening the case over already strong kink solution as demonstrated. the maximum of this variable also exhibits different dependence at either side from true mass, as a function of trial invisible particle mass . Following the Eq. 10, one can obtain the kink structure exactly at the true mass. However, the source for the appearance of this kink is attributed to the heavy resonance mass-shell constraint in the minimization.888One can argue that the heavy resonance constraint works in the same spirit of ‘relative’ constraint as defined in class of variables Cho:2014naa (). In fact, in a non-antler scenario the variables under usual relative constraints extend/shift their distribution end-point value over and above the end point where this relative constraint is absent. Similar to our case, this can happen when trial mass deviates from true the invisible particle mass. Hence, one expects formation or consolidation of similar kink structure. Although there is no analytic formula in support of the above empirical observation, we verified it by checking the slope numerically before and after the true mass point. The presence of this kink is also authenticated by various mass spectrums. Also note that unlike , the constrained variable cannot increase forever with the increase of the trial invisible particle mass , owing to the additional heavy resonance mass-shell constraint. can maximally reach up to half of the resonant mass and after that it would be unphysical. We have not studied the effects of ISR or UTM on this kink solution, but one expects that the presence of those extra transverse momenta will sharpen the kink structure, leaving these realistic studies for future work.

Figure 4: Upper plot depicts the behavior of the upper end points for both the constrained and unconstrained variables with respect to the trial invisible particle mass . The blue dashed line portrays , the red dashed line and the red thin dotted lines intersect at the value of true masses. This plot clearly illustrates that because of the on-shell constraint, attains a larger value, even bigger than the corresponding once is different from the true mass . The most compelling observation about this plot is the appearance of a kink exactly at the true mass point for , which can be used solely for measuring both and simultaneously. The lower plot describes the difference - with respect to the trial invisible particle mass . As expected, the difference between both the end points is zero at the true invisible particle mass.

To demonstrate this behavior in a more quantitative sense, we once again consider the toy process with the antler topology with the aforementioned mass spectrum. The top plot of Fig. 4 depicts the dependence of both as well as the constrained with respect to the trial invisible particle mass . The red thin dotted lines showing true mass lines intersect at the true mass point, in GeV. This plot clearly illustrates that because of the on-shell constraint, attains a larger value, even bigger than the corresponding once is different from the true mass . However, both of these maximum quantities attain the same value precisely at the true mass. The most compelling observation about this plot is the appearance of a kink exactly at this point for , which can be used for measuring both masses and simultaneously. The bottom plot describes the variation of difference between two maximums i.e. - with respect to the trial invisible particle mass . As expected, this difference between both the end points should ideally be zero at the true invisible particle mass.

Iv More aspects of kink measurement

Figure 5: Density of events as a percentage of total data contributing to distributions as a function of the trial invisible mass . On the left, the reference figure is shown for the variable which does not include heavy resonance constraint. The similar figure on the right is for the constrained variable , where constraint refers to the mass-shell constraint. The color coding represents the percentage of events per 2 GeV bin in . Since this distribution reaches a maximum for every trial value of , above which there are no events, this disallowed range kept as white. This upper end point in each plot represents the maximum curve shown in Fig. 4, last section. The presence of the kink can be clearly seen from the figure, and it is solely because of the on-shell heavy resonance constraint. But the reconstruction of the kink can be challenging due to the much smaller number of events at the endpoint, specifically when away from the kink. Also, it is interesting to note the changes in event density due to the application of an additional constraint. Evidently, a significant number of events shifted towards the end point at the true mass, as can be observed in the figure.

One of the significant challenges with most of the mass variables is the detection of the distribution end point, which can reach to the theoretical maximum only after using a large amount of data with significant statistics. The problem comes from the fact that negligible amount of events typically contribute towards the distribution end point. is also not an exception, forming a tail in the distribution towards its maximum value.999On the contrary, at the true invisible mass, produces a sharper end point as demonstrated in figure 3.

This feature is clarified in Fig. 5 where the density of events is displayed as a percentage of total data contributing to the distributions. As a function of the trial invisible mass , the left plot shows the reference density for the distribution which can reach up to a maximum, above which there are no events and it remains white. For the same data set, the right plot shows the density of events for the constrained variable . The color coding represents the percentage of events per 2 GeV bin in . These upper end points are equivalent to the maximum curve in Fig. 4, in the last section. Compared to the left figure, developed a clear kink solely because of the on-shell constraint of the heavy resonance. One notices that a tiny fraction of events is actually contributing at the end point, specifically when away from the kink position. Also, it is interesting to note the changes in event density due to the additional constraint. At the true mass (kink), a significant number of events shifted towards the end point, as is also observed in Fig. 3. This demonstration is also generated considering the toy process with the antler topology with the aforementioned mass spectrum = in GeV.

We pointed out and discussed the difficulty with determining the end points, which is in no way a shortcoming for this variable only. Fortunately, in this present case, the ability to simultaneously identify both and provides a solution for effectively pointing out the kink using all the events, not just relying on the events at the maximum.

Figure 6: The effectiveness of the variable as defined in Eq. 11 is in identifying the minimum and thus measuring the true mass of the invisible daughter. The function as a percentage of the event fraction clearly shows a sharp minimum at . So by identifying the minimum of , one can measure the invisible particle mass accurately. The red band shows the error accounting only for the statistical uncertainty.

We have already discussed in Sec. II that the additional constraint pushes the towards the higher value compared to , such that, as long as the trial invisible mass is unequal to the true mass , there can be enough events generating a larger than . Moreover, it is clear from Eq. 10 that the coincides with at the true invisible mass. This enables us to define a dimensionless variable pointing out the position of kink, in a way that was originally proposed in Konar:2009wn (). For a given , one counts all the events having value larger than the corresponding to get the fraction,


Here is the event index with total number . is the number of events in which for any given , satisfied by the Heaviside step function,


It is easy to follow from Eq. 10 that the quantity should ideally be zero at the true mass since both and share the same maximum value at that point. However, on both sides away from this point, substantial events contribute above the ; hence, poses a sharp minimum at the true mass point. Considering other realistic effects, such as backgrounds, mass width, and experimental errors, can lift the minimum from zero. These effects are not considered in this present analysis. However, it is safe to assume that the position of the functional minimum can be correctly identified to get the true invisible mass. The advantage of using is that it does not rely on some isolated event at the end point but rather it relies on a significant number of events distributed on a band in a two-dimensional plane between and which contribute to establish this minimum.

In our example, theoretical prediction of the function (as a fraction of total events) is shown in Fig. 6. The red band is the error accounting only for the statistical uncertainty. One can clearly identify the minimum and justify the relation


to measure the invisible particle mass accurately. Hence, it is straightforward to measure both the parent and daughter masses simultaneously.

V Reconstruction capability of events

Figure 7: Capability of reconstructing missing daughters momenta is demonstrated using constrained variable. Left figure is displaying a normalised distribution of with , hence parameterize the deviation from true momenta for the transverse part, With ”t” either x or y-component of momenta. This reconstruction of momenta is done from the minimisation of the with the true mass of the invisible particle as input. The reconstructed invisible momenta is unique and very well correlated with true momenta as the distribution of has a sharp peak at zero. In a process where the invisible particle mass is unknown, can be used for invisible particle mass determination and then event reconstruction using . Similarly, right figure displays a normalised distribution of with for more troublesome longitudinal momentum. Once again, reconstruction is unique and well correlated with true longitudinal momenta of invisible particle.

In this section, we are inclined to explore the event reconstruction capability coming from the constrained mass variable , typically once the invisible particle mass is determined, as in last section. Event reconstruction is extremely important in the case of spin, polarization and the coupling determination of new physics as well as SM processes with the Higgs and top. However, it is almost impossible to determine them exactly for a scenario involving multiple invisible particles in a hadron collider, especially for a topology with a short decay chain. Attempts have been made to reconstruct events using the transverse mass variable  Cho:2008tj (); Guadagnoli:2013xia (); Park:2011uz () known as the -assisted on-shell (MAOS) method in which the transverse momenta of the invisible particle are determined from the minimization of , and the longitudinal components are determined by solving mass-shell the constraints of parent and daughter. In Ref. Swain:2014dha (); Swain:2015qba (), it is shown that and its constrained sisters and , can also be used for event reconstruction especially in antler topology, where constraints refer to missing transverse momenta and available mass-shell constraints in an event. The reconstructed momenta of the invisible particles are derived at the extremum of and the constrained variables.

In this section we reconstruct all components of the invisible particle momenta from minimization of the -dimensional variable with the true mass of the invisible particle as input. The capability of reconstructing the missing daughters momenta is demonstrated in Fig. 7 using this constrained variable. The left plot displays a normalized distribution of the deviation (or error) in this reconstructed quantity from that of the true value, in the case of transverse part of the invisible momenta. This deviation is parametrized using a ratio defined as


where the subscript ”t” refers to the transverse (x or y) component of momenta. The reconstructed invisible momenta are proved to be unique and very well correlated with the true momenta, as the observed distribution has a sharp peak at its true value (i.e. zero deviation). Similarly, the right figure displays a normalized distribution of the corresponding variable for longitudinal momentum, and once again one gets a unique reconstruction well correlated with the true longitudinal momenta of the invisible particle. In a process where the invisible particle mass is unknown, can be used for invisible particle mass determination, and then events can be reconstructed using .

Vi Summary and conclusions

Looking forward to another breakthrough in the second phase of its journey, the LHC is discovering credible hints for new physics. Many of the popular BSM models are extended with massive exotic dark matter particles. As a result, common signatures, coming from such models at a high-energy collider, typically are detectable SM particles along with a pair of invisible particles. They are too complex to measure all the unknown masses or to fully reconstruct those events at the large hadron collider.

In this paper, we study one such class of events produced through antler topology. They are common in SM Higgs and several BSM scenarios, where the heavy resonance is produced before semi-invisible decay into visible decay products together with invisibles which can either be SM neutrinos or some exotic dark matter particles. Our objective is to determine all the unknown masses, including the dark matter particles produced from the heavy resonance. We consider a new constrained variable extending the -dimensional mass variable , by implementing additional heavy grandparent mass-shell constraint in the minimization.

This new variable contains several interesting features. We demonstrate how this variable acquires an event wise higher value owing to this constraint. In particular, we show how this variable moves closer to the unknown parent mass. In addition, the calculated invisible momenta at this minimum can provide a close estimate of the true momenta of the invisible particles for such events. Both these characteristic features are highlighted and exploited further to sharpen the measurements.

Another striking feature comes out once we analyze the distribution maxima of this new variable as a function of the trial values of yet unknown dark matter particle mass. This is constructed in an analogy with the popular study of , which gives a useful correlation curve relating the parents mass with the invisible particle mass. But now, under mass-shell constraint, develops a new kink solution over the correlation curve exactly at the value where this trial mass coincides with the true mass. Hence, this opened another new avenue that produces a kink feature to measure both masses simultaneously.

To handle the sparseness of events towards the distribution end point, we analyze with an experimentally feasible observable by utilizing both constrained and unconstrained variables. This observable does not rely on isolated events at the end point, but instead uses a significant amount of available data to pinpoint the unknown invisible particle mass from the sharp minimum.

Our method provides a complementary procedure to earlier antler studies and is applicable to any mass region. We demonstrate our analysis in the large mass gap region, considered as a difficult region for the kinematic cusp method. In this region, the cusps of many variables are not very sharp, which makes the mass determination more prone to error. But the present method can be used safely for better accuracy. We also investigate the event reconstruction capability of , and we reconstruct the unknown invisible particle momenta at the constrained minimization. The reconstructed momenta are found to be unique and well correlated with the true invisible momenta.

This work was funded by the Physical Research Laboratory (PRL), Department of Space (DoS), India. PK also gratefully acknowledges IACS and IISER Kolkata for hospitality.


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