1 Introduction

UT-HET 010

TU-820

June, 2008


Testing the Littlest Higgs Model with T-parity

at the Large Hadron Collider

Shigeki Matsumoto, Takeo Moroi, and Kazuhiro Tobe


Department of Physics, University of Toyama, Toyama 930-8555, Japan


Department of Physics, Tohoku University, Sendai 980-8578, Japan


Department of Physics, Nagoya University, Nagoya 464-8602, Japan


In the framework of the littlest Higgs model with T-parity (LHT), we study the production processes of T-even () and T-odd () partners of the top quark at the Large Hadron Collider (LHC). We show that the signal events can be distinguished from the standard-model backgrounds, and that information about mass and mixing parameters of the top partners can be measured with relatively good accuracies. With the measurements of these parameters, we show that a non-trivial test of the LHT can be performed. We also discuss a possibility to reconstruct the thermal relic density of the lightest T-odd particle using the LHC results, and show that the scenario where becomes dark matter may be checked.

1 Introduction

The hierarchy problem in the standard model (SM) is expected to give a clue to explore physics beyond the SM. This problem is essentially related to quadratically divergent corrections to the Higgs boson mass, and it strongly suggests the existence of new physics at the TeV scale. At the new physics scale, the problem is expected to be resolved due to the appearance of a new symmetry which controls the Higgs boson mass. With this philosophy, a lot of scenarios have been proposed so far. The most famous example is the supersymmety (SUSY), by which quadratically divergent corrections to the Higgs boson mass are completely cancelled. Another example is the Gauge-Higgs unification, by which the gauge invariance in higher dimensional space-time protects the Higgs potential from any ultraviolet (UV) divergent corrections.

In this article, we consider the third possibility, so-called the little Higgs (LH) scenario [1], in which the Higgs boson mass is controlled by a global symmetry. In this scenario, the Higgs boson is regarded as a pseudo Nambu-Goldstone boson arising from the spontaneous breaking of a symmetry. Due to the symmetry imposed, new particles such as heavy gauge bosons and top-partners are necessarily introduced, and main quadratically divergent corrections to the Higgs boson mass vanish at one-loop level due to contributions of these particles. Unlike the SUSY scenario, the cancellation of the quadratic divergence is achieved only at one-loop level, thus the LH model needs a UV completion at some higher scale. However, due to the cancellation at one-loop level, the fine-tuning of the Higgs boson mass is avoided even if the cutoff scale of the LH model is around 10 TeV. As a result, the LH model solves the little hierarchy problem [2].

Unfortunately, the original LH model is severely constrained by electroweak precision measurements due to direct couplings among a new heavy gauge boson and SM particles [3]. In order to resolve the problem, the implementation of the symmetry called T-parity to the model has been proposed [4, 5, 6]. Under the parity, almost all new particles are T-odd, while the SM particles are T-even#1#1#1One important exception is the top-partner , which is a T-even new particle as shown in the next section.. Thanks to the symmetry, dangerous interactions stated above are prohibited [7]. Furthermore, the lightest T-odd particle (LTP) becomes stable, which is electrically and color neutral, and has a mass of (100) GeV in many little Higgs models with T-parity [4]. Therefore, these models provide a good candidate for dark matter [8]#2#2#2For UV completion of T-parity models, see [9]..

In this article, we study signatures of the littlest Higgs model with T-parity (LHT) [5, 6] at the Large Hadron Collider (LHC), which is expected to explore various new-physics models [10, 11]. The LHT is the simplest model realizing the LH scenario with the T-parity, and considered to be an attractive reference model. Since the LHC is a hadron collider, new colored particles have an important role to explore physics beyond the SM. As shown in the next section, top-partners are necessarily introduced in the LH models, which are responsible for the cancellation of quadratically divergent corrections to the Higgs boson mass from top loop diagrams. Furthermore, masses of these partners are expected to be less than 1 TeV, and the partners will be copiously produced at the LHC [12]. Therefore, we consider the productions of the top partners at the LHC with a realistic simulation study, and show that these signatures are clearly distinguishable from SM backgrounds. Furthermore, we find that it is also possible to test the LHT by investigating a non-trivial relation among the signatures. We also consider how accurately model parameters of the LHT are determined, and discuss its implication to the property of the LTP dark matter such as how precisely the relic abundance of the dark matter is estimated with the LHC data.

This paper is organized as follows. In the next section, we briefly review the littlest Higgs model with T-parity paying particular attention to the gauge-Higgs and top sectors of the model. We also present representative points used in our simulation study. Signatures of the LHT at the LHC are shown in Sec. 3, especially focusing on the pair production of T-even top partner, the single production of T-even top partner, and the pair production of T-odd partner. The test of the LHT is discussed in Sec. 4, where we investigate a non-trivial relation among the signatures obtained in the previous section. We also discuss the implication of the result to the LTP dark matter phenomenology. Sec. 5 is devoted to summary.

2 Model

In this section, we briefly review the littlest Higgs model with T-parity focusing on gauge-Higgs and top sectors of the model. (For general reviews of little Higgs models and their phenomenological aspects, see [13, 14].) We also present a few representative points used in our simulation study at the end of this section.

2.1 Gauge-Higgs sector

The littlest Higgs model with T-parity is based on a non-linear sigma model describing an SU(5)/SO(5) symmetry breaking. The non-linear sigma field is given as

(2.1)

where TeV is the vacuum expectation value of the breaking. The Nambu-Goldstone (NG) boson matrix and the direction of the breaking are

(2.2)

Here, we omit the would-be NG fields in the matrix. An [SU(2)U(1)] subgroup in the SU(5) global symmetry is gauged, which is broken down to the diagonal subgroup identified with the SM gauge group SU(2)U(1). Due to the presence of the gauge interactions and Yukawa interactions introduced in the next subsection, the SU(5) global symmetry is not exact, and particles in the matrix become pseudo NG bosons. Fourteen (= 24 10) NG bosons are decomposed into representations under the electroweak gauge group. The first two representations are real, and become longitudinal components of heavy gauge bosons when the [SU(2)U(1)] is broken down to the SM gauge group. The other scalars and are a complex doublet identified with the SM Higgs field ( in Eq. (2.2)) and a complex triplet Higgs field ( in Eq. (2.2)), respectively.

The kinetic term of the field is given as

(2.3)

where () is the () gauge field and () is the () gauge coupling constant. With the Pauli matrix , the generator and the hyper-charge are given as

(2.4)
(2.5)

It turns out that the Lagrangian in Eq. (2.3) is invariant under the T-parity,

(2.6)

where .

This model contains four kinds of gauge fields. The linear combinations and correspond to the SM gauge bosons for the SU(2) and U(1) symmetries. The other linear combinations and are additional gauge bosons, which acquire masses of through the SU(5)/SO(5) symmetry breaking. After the electroweak symmetry breaking with , the neutral components of and are mixed with each other and form mass eigenstates and ,

(2.7)

The mixing angle is given as

(2.8)

where , , , and . Since the mixing angle is considerably suppressed, is dominantly composed of . Masses of gauge bosons are

(2.9)
(2.10)
(2.11)
(2.12)
(2.13)

As expected from the definitions of , , and , the new heavy gauge bosons behave as T-odd particles, while SM gauge bosons are T-even.

A potential term for and fields is radiatively generated as [1, 8]

(2.14)

Main contributions to come from logarithmic divergent corrections at 1-loop level and quadratically divergent corrections at 2-loop level. As a result, is expected to be smaller than . The triplet Higgs mass term, on the other hand, receives quadratically divergent corrections at 1-loop level, and therefore is proportional to . The quartic coupling is determined by the 1-loop effective potential from gauge and top sectors. Since both and depend on parameters at the cutoff scale , we treat them as free parameters in this paper. The mass of the triplet Higgs boson is given by , where is the mass of the SM Higgs boson. The triplet Higgs boson is T-odd, while the SM Higgs is T-even.

Gauge-Higgs sector of the LHT is composed of the kinetic term of field in Eq. (2.3) and the potential term in Eq. (2.14) in addition to appropriate kinetic terms of gauge fields , and gluon . It can be seen that the heavy photon is considerably lighter than other T-odd particles. Since the stability of is guaranteed by the conservation of T-parity, it becomes a good candidate for dark matter.

2.2 Top sector

To implement T-parity, two SU(2) doublets and and one singlet are introduced for each SM fermion. Furthermore, two vector-like singlets and are also introduced in the top sector in order to cancel large radiative corrections to the Higgs mass term. The quantum numbers of the particles in the top sector under the [SU(2) U(1)] gauge symmetry are shown in Table 1. All particles are triplets under the SM SU(3) (color) symmetry.

Table 1: Quantum number for for particles in the top sector.

With these particles, Yukawa interactions which are invariant under gauge symmetries and T-parity turn out to be

(2.15)

where , , and . The indices run from 1 to 3, while . The coupling constant is introduced to generate the top Yukawa coupling and gives the vector-like mass of the singlet . Under T-parity, and transform as and , thus T-parity eigenstates are given as

(2.16)

In terms of the eigenstates, mass terms in Eq. (2.15) are written as

(2.17)

T-even states and form the following mass eigenstates

(2.18)

Mixing angles , and mass eigenvalues , are given as

(2.19)

where , , , and with being . The quark is identified with the SM top quark, and is its T-even heavy partner. On the other hand, the T-odd fermions and form a Dirac fermion, , whose mass is given by . The remaining T-odd quark acquires mass by introducing an additional SO(5) multiplet transforming nonlinearly under the SU(5) symmetry. Therefore, the mass term of the quark does not depend on and . In this paper, we assume that the quark is heavy enough compared to other top partners, and that it is irrelevant for the direct production at the LHC experiment. (For the phenomenology of the quark, see [15].) Finally, it is worth notifying that the T-odd partner of top quark () does not participate in the cancellation of quadratically divergent corrections to the Higgs mass term. The cancellation is achieved by only loop diagrams involving and quarks.

2.3 Representative points

In this paper, we focus on productions at the LHC. For this purpose, we need to choose representative points to perform a numerical simulation. In order to find attractive points, we consider those consistent with electroweak precision measurements and the WMAP experiment for dark matter relics#3#3#3We consider the WMAP constraint only for choosing a representative point. In fact, the model does not have to satisfy the constraint, because, for instance, dark matter may be composed of other particles such as the axion..

We consider a -function to choose representative points;

(2.20)

where , , and are experimental result, theoretical prediction, and the error of the observation for observable . We consider following eight observables; boson mass ( 80.4120.042 GeV), weak mixing angle ( 0.231530.00016), leptonic width of the boson ( 83.9850.086 MeV) [16], fine structure constant at the pole ( 128.9500.048), top quark mass ( 172.72.9 GeV) [17], boson mass ( 91.18760.0021 GeV), Fermi constant ( (1.166370.00001)10 GeV) [18], and relic abundance of dark matter ( 0.1190.009) [19]. On the other hand, theoretical predictions of these observables depend on seven model parameters; , , , , , , and . (For the detailed expressions of the theoretical predictions, see [7, 8]). In order to obtain the constraint on vs.  plane, we minimize the function in Eq. (2.20) with respect to parameters , , , , and . In other words, we integrate out these parameters from the probability function .

Figure 1: Constraints to the littlest Higgs model with T-parity on vs.  plane at 99 % confidence level. The degree of fine-tuning to the quadratic coupling of the Higgs field is also shown as light shaded regions. Cross marks 1,2, and 3 are representative points for our simulation study.

The result is shown in Fig. 1, where the constraints on and at 99% confidence level ( 11.34) are depicted. The region is not favored due to electroweak precision measurements, because a large mixing angle between and is predicted in this region, which leads to a significant contribution to the custodial symmetry breaking. The region GeV, which corresponds to , is not attractive because the pair annihilation of into gauge-boson pair is kinematically forbidden. Here, we should comment on other parameters integrated out from the probability function. It can be easily seen that , , , and are almost fixed due to the precise measurements of these observables. Furthermore, once (, ) is fixed, is also fixed by the WMAP observation, because the annihilation cross section of dark matter is sensitive to . Here and hereafter, at each point, we use values of these parameters which minimize the -function. The degree of fine-tuning to set the Higgs mass on the electroweak scale is also shown in the figure. As mentioned in the previous subsections, the quadratic coupling of the Higgs field is generated radiatively. One of main contributions comes from the logarithmic divergent correction of a top-loop diagram, which yields [20]

(2.21)

where is the cutoff scale of the model. We used the ratio % to estimate the degree of fine-tuning. It can be seen that too large and are not attractive from the view point of the fine-tuning.

Point 1 Point 2 Point 3
(GeV) 570 600 570
1.0 1.1 1.4
0.20 0.16 0.11
(GeV) 145 131 145
(GeV) 80.1 85.4 80.1
(GeV) 570 660 798
(GeV) 772 840 914
(pb) 1.26 0.54 0.17
(pb) 0.21 0.13 0.07
(pb) 0.29 0.15 0.05
(pb) 0.14 0.07 0.02
50.8 % 50.8 % 53.3 %
21.1 % 21.8 % 23.6 %
15.8 % 17.4 % 19.1 %
12.3 % 10.0 % 4.03 %
Table 2: Representative points used in our simulation study.

Representative points used in our simulation study are shown in Fig. 1 and their details can be found in Table 2. Masses of and , cross sections for pair and single productions, and branching ratios of decay are also shown in each representative point. Note that the quark decays into the stable and the top quark with almost 100% branching ratio.

3 Signals from the LHT Events

Now, we consider the and production processes and their signals at the LHC. At the LHC, there are two types of production processes, pair production and single production processes, both of which are important. Thus, in the following, we discuss these processes separately. In addition, we also discuss the pair production.

3.1 pair production

First, we discuss the pair production process. Once produced, decays as , , , and . Branching ratios for individual decay modes depend on the underlying parameters. However, in most of the cases, becomes larger than 0.5, and many of decay into . Thus, in the experimental situation, the analysis using the decay mode is statistically preferred. In such a case, the quark production events become irreducible background. We will propose a set of kinematical cuts suitable for the elimination of background.

For the production process, the most dangerous background is the production which has larger cross section than the production#4#4#4We use the leading order calculation of the production cross section which is pb. . Thus, we need to develop kinematical cuts to suppress the background. We propose to use the fact that the jets in the signal events are likely to be very energetic because they are from the decay of heavy particles (i.e., or ). Consequently, the signal events are expected to have large , which is defined by the sum of transverse momenta of high objects and missing transverse momentum :

(3.1)

In our study, only the jets with are included into the high objects in order to reduce the contamination of QCD activities. We expect that the number of background events can be significantly reduced once we require that be large enough; in the following, we will see that this is indeed the case.

Once the backgrounds are reduced, the production events are reconstructed relatively easily. Here, we concentrate on the dominant decay mode . Then, the signal events are primarily from the process , followed by and . In particular, in order to constrain the mass of , we use the process in which one of the -boson decays hadronically while the other decays leptonically. At the parton level, the final state consists of two -jets, two quark jets from , one charged lepton and one neutrino from . Thus, the signal events are characterized by

  • Several energetic jets,

  • One isolated lepton,

  • Missing (due to the neutrino emission).

Using the fact that, in the signal events, the missing momentum is due to the neutrino emission, we reconstruct two systems, which we call -system and -system; here, the -system (-system) consists of high objects which are expected to be from or whose decay is followed by the leptonic (hadronic) decay of the -boson. To determine - and -systems, we first assume that all the missing is carried away by the neutrino. With this assumption, the neutrino momentum (in particular, the -component of ) is calculated, requiring . Then, we define -system as the charged lepton, reconstructed neutrino, and one of the three leading jets, while -system is the rest of the high objects. Since there is a two-fold ambiguity in reconstructing the neutrino momentum, there exist six possibilities in classifying high- objects into - and -systems. Using the fact that - and -systems have the same invariant mass in the ideal case, we choose one of the six combinations with which is minimized, where and are invariant masses of - and -systems, respectively. The distributions of the invariant masses of - and -systems are expected to provide information about the mass.

In order to demonstrate how well our procedure works, we generate the events for the processes and (as well as those for and ) with . The parton-level events are generated by using the MadGraph/MadEvent packages [21], which utilizes the HELAS package [22]. Then, Pythia package [23] is used for the hadronization processes and the detector effects are studied by using the PGS4 package [24]. In order to study the pair production process followed by the decay processes mentioned above, we require that the events should satisfy the following properties:

  • Three or more jets with , and only one isolated charged lepton.

In addition, we adopt the following kinematical cuts:

  • (with being the transverse momentum of the charged lepton),

  • ,

  • .

Notice that the third cut is to eliminate combinatorial backgrounds. We found that, after imposing these kinematical cuts, events from the and production processes are completely eliminated. Then, we calculate the distributions of . The results are shown in Fig. 2. As one can see, the distributions have distinguishable peaks at around . In addition, backgrounds are well below the signal. Thus, from the distribution of , we will be able to study the properties of .

Figure 2: Distribution of for the Points 1 3 (from the top to the bottom) with . The shaded histograms are the background distribution, while the solid ones are for signal background.

One important observable from the distribution of is the mass of ; once we obtain the peak of the distribution, it will provide us an important information about . To see the accuracy of the determination of , we consider the bin . Then, we calculate the number of events in the bin as a function of the center value with the width being fixed. The peak of the distribution is determined by which maximizes the number of events in the bin. We applied this procedure for (with ). Results for a set of signal and background events are shown in Table 3. With repeating the Monte-Carlo (MC) analysis with independent sets of signal samples, we found that the difference between the position of the peak and the input value of is typically or smaller. Thus, we expect a relatively accurate measurement of . In discussing the test of the LHT model at the LHC, we quote and as the uncertainty of and discuss the implication of the measurement of .

Point 1 Point 2 Point 3
Table 3: Peak of the distribution for , , , and .

3.2 Single production of

As well as the pair production, the single production processes and have sizable cross sections at the LHC. (Here, denotes light quark jets.) Such processes were discussed in [25] in the framework of the original littlest Higgs model without the T-parity, which pointed out that the discovery of may be possible by using this process. (See also [26].) Here, we reconsider the single production process for the test of the LHT model.

So far, we have discussed that the information about the mass of can be obtained by studying the pair production. Concerning the property of , another important parameter is the mixing angle , which determines the interaction between and weak bosons (i.e., and ). Importantly, the cross sections for the processes and are strongly dependent on . In particular, since these processes are dominated by the -channel -boson exchange diagram (with the use of - or -quark in the initial-state protons), the cross sections are approximately proportional to . Thus, if the cross sections of the single production processes are measured, it provides an information about the mixing angle . Although and have different cross section, their event shapes are very similar (if we neglect the charges of high objects). In the following, we consider how we can measure the total cross section .

As we have already discussed, once produced, dominantly decays into and . Thus, if we consider the leptonic decay of , there exist two energetic quarks and one charged lepton (as well as neutrino) at the parton level in the final state. Since the mass of is relatively large, the -jet is expected to be very energetic in this case. Thus, if we limit ourselves to the cases with the leptonic decay of , the single production events are characterized by

  • Two (or more) jets, one of which is very energetic (due to the -jet),

  • One isolated lepton,

  • Missing (due to the neutrino emission).

As we will see, the cross section of the background events are relatively large, so it is necessary to find a useful cut to eliminate the backgrounds as much as possible.

One of the possible cuts is to use the invariant mass of the “” system. In the signal event, the dominant source of the missing transverse momentum is the neutrino emission by the decay of . Thus, as we have discussed in the study of the pair production process, we can reconstruct the momentum of neutrino (and hence that of ). Then, we can calculate the invariant mass of the system. In such a study, we presume that the highest jet is the -jet because, at least at the parton level, the transverse momentum of the -quark from the decay of is much larger than that of the extra quark. Then, since we expect that the mass of is well understood by the study of pair production process, as discussed in the previous subsection, we only use the events with relevant value of the invariant mass to improve the signal-to-background ratio.

To estimate how well we can determine the cross section of the single production process, we generate the signal and background events for . In [25], it was pointed out that the most serious backgrounds are from production process as well as from the single production of the top-quark. Thus, in our study, we take account of these backgrounds.

Once the event samples are generated, we require the following event shape:

  • The number of isolated lepton is 1, the number of jets (with ) is 2.

In the next step, as in the case of the pair production, we reconstruct the momentum of the neutrino assuming that the transverse momentum of the neutrino is given by the observed missing . In reconstructing the neutrino momentum , there exists two-fold ambiguity; we denote the reconstructed neutrino momenta (). For each reconstructed momentum, we calculate the invariant mass of the system:

(3.2)

postulating that the highest jet corresponds to the jet. Even though one of is with the wrong , we found that, in the signal event, the typical difference between and are relatively small compared to that in the background events. Thus, we reject the events unless is small enough.

We also comment on another useful cut to eliminate the background. In the background events, the highest jet is likely to be from the overlapping of several hadronic objects from different partons if the is required to be very large. In our analysis, the cone algorithm (with ) is used to identify isolated jets. Then, if several partons from the decay of top quark or -boson are emitted in almost the same direction, hadronized objects from those partons are grouped into a single jet, which may be identified as the -originated jet in the present analysis. One of the method to reject such a background is to use the jet-mass variable, which is the invariant mass of the jet constructed from all the (observed) energy and momentum that are contained in the jet. The jet-mass of such a jet is likely to be much larger than that of the -jet. As we will show, the number of background from the production process is significantly reduced if the jet mass is required to be small enough.

Now, we show the results of our MC analysis. In our analysis, we use the following kinematical cuts:

  • , ,

  • , and , with being the invariant mass of total jets,

  • , with being the jet mass of the leading jet,

  • .

In Fig. 3, we plot the distribution of the “averaged” invariant mass of the system:

(3.3)

As one can see, the distribution from the signal events is peaked at around , while the background distribution is rather flat. In addition, at around , the number of signal events becomes significantly larger than that of background in particular when the parameter is relatively large. In such a case, the number of the single production events can be extracted from the distribution by using, for example, the side-band method#5#5#5It should be also possible to constrain the mass of from the peak of the distribution of . In this paper, we will not discuss such a possibility..

Figure 3: Distribution of for the Points 1 3 (from the top to the bottom). The shaded histograms are the background distribution, while the solid ones are for signal background.

In Table 4, with the data for the Point 2, we show the number of events in the event region, which we define , and those in the sidebands, and , after imposing the kinematical cuts mentioned above. Assuming that the numbers of signal and background events in the signal region are determined by using the sideband events, and that the cross section for the single production process can be obtained from the number of events in the signal region, the single production cross section may be determined with the uncertainty of . (The uncertainty here is statistical only.)

Using the result of the determination with the pair production process, the information about the cross section can be converted to that of the mixing angle . If the uncertainties in the theoretical calculation of the cross sections are under control, we obtain a constraint on . Since the cross section for the single production process is proportional to , is determined with the accuracy of if the cross section is determined with the accuracy of #6#6#6The cross section also depend on the mass of . Thus, the constraint on the cross section should provide a constraint on the vs.  plane. In our discussion, for simplicity, we only consider the dependence of the cross section by using the fact that the mass of is expected to be determined from the pair production process.. In the next section, we discuss the implication of the determination of at this level in testing the LHT model.

Lower Sideband Event Region Upper Sideband
Signal Signal Signal
II-0 313 21706 13509 522 12585 8609 116 7810 5362
II-0, 1 108 3366 376 234 2352 363 44 1747 237
II-0, 1, 2 45 428 53 144 446 76 14 440 86
II-0, 1, 2, 3 30 30 47 114 27 50 8 21 69
II-0, 1, 2, 3, 4 21 12 18 84 11 12 2 3 16
Table 4: Number of the signal events/ background/single top background events in the event region () as well as in the lower and upper sidebands ( and , respectively). Point 2, where , is used.

Before closing this subsection, we comment on the uncertainties which we have neglected so far. As we have mentioned, the single production process occurs by using the or in the sea quark of the initial-state proton. Thus, for the theoretical calculation of the cross sections, it is necessary to understand the parton distribution functions for the and quarks (as well as those of lighter quarks). Information about the parton distributions of the -quark may be obtained by using the single top (and anti-top) productions. As we have seen, significant amount of single top productions occur at the LHC (which has been seen to be one of the dominant backgrounds to the single production process). Since the single top production also occurs by using the quark in proton, information about the parton distribution function of will be obtained by studying the single top production process. In this paper, we do not go into the detail of such study, but we just assume that the parton distribution function of will become available with small uncertainty once the LHC experiment will start. We also note here that it is also important to understand the efficiency to accept the single production events (as well as the background events) after the cuts, whose uncertainties have been neglected in our discussion.

3.3 pair production

For the study of the LHT model at the LHC, it is also relevant to consider the -odd top partner, , and the lightest -odd particle, . For the study of -odd particles, it is important to consider the pair production process, which was discussed in [27, 28]. Here, we reconsider the importance of this process for the test of the LHT model.

At the LHC, is pair produced via , then decays as . Since is undetectable, the production events always result in missing events and hence the direct measurements of the masses of and are difficult.

One powerful method to study and is the so-called analysis [29], combined with the hemisphere analysis [30]. If the and systems are somehow reconstructed, one can constrain and from the distribution of the so-called variable. For the event followed by and , the variable is defined as

(3.4)

where the transverse mass is defined as

(3.5)

with being the postulated mass of