Kinematic Selection Criteria in the Search for Resonances

Abstract

In this paper, we discuss some general features of two-body decay kinematics and derive a new set of equations describing boundaries of the physical region in decays. These features via a graphical representation and implementation of selection criteria provide an efficient tool for the background suppression in experimental searches for conventional and exotic (pentaquark, etc.) resonance states.

1. Introduction

In the experimental search for a resonance production in particle collisions, momenta of secondary particles are combined to form the momentum of a resonance candidate. In high-energy reactions with high multiplicity of final states, this method of resonance reconstruction may cause a large combinatorial background which makes searches for new narrow states in invariant-mass distributions to be rather difficult. This was the case with the top-quark discovery, searches for new charmonium states or pentaquark searches. For reconstruction of the signal with a sufficient statistical significance, any additional physical information about the resonance and its decay properties (lifetime, the mass of secondary particles, etc.) helps considerably to reduce the background. For instance, the presence in an event a well-separated secondary vertex allows us to reconstruct Λ,𝐾⁰_𝑠 and mesons from the 𝐷-meson family with a very low background. One such example is the study of meson resonances in the 𝐾⁰_𝑠𝐾⁰_𝑠 final state in 𝑒𝑝 deep inelastic scattering at HERA [1]. Moreover, if masses of decay products are known and significantly differ from each other, this information also has to be used to reduce the background. This paper is a discussion of these issues.

Below only two-body decay modes are considered, and for each final state particle a set of kinematic equations describing boundaries of the physical region is given. Proposed selection criteria are based on the features of each physical region [2] and proved to be the efficient tool for experimental reconstruction both the conventional as well as exotic resonances [3, 4].

2. Two-Particle Decay

Let us consider a two-particle decay, 𝑅→𝐚+𝐛, of a resonance 𝑅 with a mass 𝑀_𝑅 and a momentum ^→𝑃_𝑅 in the laboratory frame. Masses of the decay products are denoted by 𝑚_𝐚 and 𝑚_𝐛. At the rest frame of 𝑅 the particles, 𝐚 and 𝐛 are flying in opposite directions with the momentum [5, 6]: 𝑃^∗=12𝑀_𝑅𝑀²_𝑅−𝑚²_𝐚−𝑚²_𝐛²−4𝑚²_𝐚𝑚²_𝐛^1/2. (2.1) In the laboratory frame, the absolute momenta 𝑝_𝐚 and 𝑝_𝐛 of the particles 𝐚 and 𝐛 depend on the relative orientation of the rest frame vectors ^→𝑝^∗_𝐚 and ^→𝑝^∗_𝐛 with respect to the boost vector. We shall consider only Lorentz boosts along the momentum ^→𝑃_𝑅 and denote by 𝜃^∗_𝐚 the polar angle between ^→𝑝^∗_𝐚 and the direction given by ^→𝑃_𝑅. In that case, 𝜃^∗_𝐛=𝜋−𝜃^∗_𝐚.

The energy and momentum components in both frames are related via [5, 6]: 𝐸_𝐚(𝐛)=𝛾𝐸^∗_𝐚(𝐛)−^→𝜂⋅^→𝑝^∗_𝐚(𝐛),_→𝑝_𝐚(𝐛)=^→𝑝^∗_𝐚(𝐛)+^→𝜂^→𝜂⋅^→𝑝^∗_𝐚(𝐛)𝛾+1−𝐸^∗_𝐚(𝐛). (2.2)(2.3) For a boost along ^→𝑃_𝑅, the boost parameters are 𝐸𝛾=^𝑅𝑀_𝑅,^→𝜂=−^→𝑃_𝑅𝑀_𝑅, (2.4) and therefore 𝑝_𝐚(𝐛)=1𝑀_𝑅𝐸_𝑅𝐸^∗_𝐚(𝐛)+𝑃_𝑅𝑃^∗cos𝜃^∗_𝐚(𝐛)²−𝑚²_𝐚(𝐛)𝑀²_𝑅. (2.5) If in an experiment there is no possibility to determine a particle type corresponding to a given charged track, then only information about the particle momentum (2.5) is used. In the opposite case, when the particle type can be identified, for instance, by ionization and time of flight, one may incorporate this information and in addition use (2.2). Below, these cases are considered separately.

2.1. Particles without Identification

The boundaries of the physical regions of the particle 𝐚 on the (𝑃_𝑅,𝑝_𝐚) plane are easy to obtain with the use of (2.5). For a given 𝑃_𝑅, 𝑝_𝐚 reaches the upper limit at 𝜃^∗_𝐚=0: 𝑝⁺_𝐚=1𝑀_𝑅𝑃_𝑅𝐸^∗_𝐚+𝐸_𝑅𝑃^∗, (2.6) when the lower limit at 𝜃^∗_𝐚=𝜋: 𝑝⁻_𝐚=1𝑀_𝑅𝑃_𝑅𝐸^∗_𝐚−𝐸_𝑅𝑃^∗. (2.7) The equations similar to (2.6)-(2.7) are valid for the particle 𝐛 too. At the resonance rest frame, (2.7) demonstrates an interesting feature of the momentum of the particle flying backward. With increasing 𝑃_𝑅, the momentum 𝑝_𝐚 (𝑝_𝐛) first decreases, at 𝑃_𝑅=𝑀_𝑅𝑃^∗/𝑚_𝐚(𝐛) it reaches the zero value and only at lager 𝑃_𝑅 starts to increase. When plotted on the momentum (𝑃_𝑅,𝑝_𝐚) plane, (2.6)-(2.7) select a band-like physical region (𝑚-band). For secondary particles with equal masses, 𝑚_𝐚=𝑚_𝐛, these 𝑚-bands are fully overlapping. However, if for instance, 𝑚_𝐚>𝑚_𝐛, the 𝑚-bands overlap only partially or even separate out at 𝑃_𝑅≥𝑃_𝑅^(𝑠)=2𝑀_𝑅𝑃^∗𝐸^∗_𝐚−𝐸^∗_𝐛²−4(𝑃^∗)². (2.8) The last equation follows from the condition 𝑝⁻_𝐚≥𝑝⁺_𝐛. In (2.8), the expression under the square root is positive only if 𝑃^∗<𝑚²_𝐚−𝑚²_𝐛8𝑚²_𝐚+𝑚²_𝐛. (2.9) Thus, for 𝑚_𝐚≫𝑚_𝐛 the physical regions of the particles 𝐚 and 𝐛 do not overlap if 𝑚_𝐚≥√8𝑃^∗ (2.10) and 𝑃_𝑅>𝑃_𝑅^(𝑠).

We shall now consider a few particular applications of (2.6)–(2.10). According to the PDG [7] mass values, only resonances with mass close to the threshold value 𝑚_𝐚+𝑚_𝐛 satisfy the condition (2.10). Some of them are listed in Table 1, where 𝑃^∗ and 𝑃_𝑅^(𝑠) values were calculated with use of (2.1) and (2.8), respectively.

Table 1 

The physical regions of the particles a and b do not overlap if 𝑚_𝑎≥√8𝑃^∗ and 𝑃_𝑅>𝑃_𝑅^(𝑠). Here 𝑀_𝑅 is the resonance mass, and 𝑚_𝑎 is the mass of the heaviest product, 𝑃^∗ is the rest frame momentum of the resonance 𝑅 decay products, and 𝑃_𝑅^(𝑠) is the threshold momentum of the 𝑚-bands separation (8).

Figures 1(a), 1(b), and 1(c) shows 𝑚-bands of decay products for some resonances listed in Table 1. For all of them always 𝑝_𝐚>𝑝_𝐛 (2.11) at 𝑃_𝑅>𝑃_𝑅^(𝑠), and 𝑚_𝐚 significantly larger 𝑚_𝐛. We will refer to the momentum condition (2.11) as a 𝑚-selector. Thus, for resonances of the type in Table 1, fulfillment of the condition (2.11) allows an assignment to the particle 𝐚 of the mass 𝑚_𝐚, that is, we identify the particle 𝐚. The 𝑚-selector (2.11) is a powerful condition for the background suppression.

(a)

(b)

(c)

(d)

Figure 1 

Phase space 𝑚-bands of the particles 𝐚 and 𝐛 in 𝑅→𝐚+𝐛 decays as a function of the resonance momentum 𝑃𝑅. The dashed line on (c) corresponds to the momentum cut 𝑝𝜋=0.15 GeV (see Section 3). The dashed line on (d) corresponds to (2.12).

In the two-body decay modes, one meets with a higher incidence the situation shown in Figure 1(d). For these decays the conditions (2.9)-(2.10) are not fulfilled, and at any 𝑃_𝑅 the phase space bands remain overlapping. Nevertheless, if 𝑚_𝐚>𝑚_𝐛, one can demand fulfillment of (2.11) for the particle 𝐚 in order to assign it the mass 𝑚_𝐚. In course of the resonance search, the condition (2.11) rejects not only a significant part of the background, but also some fraction of the signal combinations. To estimate the efficiency of the 𝑚-selector we proceed in the following way.

On (𝑃_𝑅,𝑝_𝐚) plane, the condition (2.11) is valid up to the line defined by the equation: 𝑝_𝐚cos𝜃^∗_𝐚,𝑃_𝑅=𝑝_𝐛cos𝜃^∗_𝐛,𝑃_𝑅. (2.12) For 𝐾^∗→𝐾𝜋 decays the last equality is shown in Figure 1(d) by the dashed line. The true 𝐾^∗ is a combination of a kaon from the region above the line (2.12) with a pion below it and vice versa. From (2.12) and (2.5), we find the variation of 𝜃^∗_𝐚=𝜋−𝜃^∗_𝐛 along the line (2.12): cos𝜃^∗_𝐚𝑃=−^𝑅𝐸_𝑅⋅𝐸^∗_𝐚−𝐸^∗_𝐛2𝑃^∗. (2.13) At the rest frame of 𝑅, (2.11) is equivalent to the exclusion of the region 𝜃^∗_𝐚>𝜃^∗_𝐚. For unpolarized particles the distribution of cos𝜃^∗_𝐚 is uniform, and we define the efficiency of the 𝑚-selector (2.11) as 𝜃Eff=1−cos^∗_𝐚2⋅100%. (2.14) With the definition (2.14), we get Eff=100%, when the conditions (2.9)-(2.10) are fulfilled, and Eff=50% for decays with 𝑚_𝐚=𝑚_𝐛(for real data Eff<50%, see discussion in Section 3). As another example, in Figure 4 by the dashed line is shown the evolution of Eff with 𝑃_𝑅 in 𝐾^∗→𝐾𝜋 decays. Eff grows because with the increase of 𝑃_𝑅 the overlap of 𝑚-bands decreases.

In the limit large 𝑃_𝑅, the expression for the efficiency can be simplified: 1Eff=2𝑚1+^2𝐚−𝑚²_𝐛𝑀²_𝑅−𝑚_𝐚−𝑚_𝐛²𝑀²_𝑅−𝑚_𝐚+𝑚_𝐛²≃1⋅100%2⋅𝑀²_𝑅𝑀²_𝑅−𝑚²_𝐚⋅100%. (2.15)(2.16) The last equation is a good approximation only if 𝑚_𝐚≫𝑚_𝐛. Equations (2.15)-(2.16) confirm the result we have already seen in Figure 4(a), where Eff is independent of 𝑃_𝑅 at large 𝑃_𝑅. On the other hand, for fixed values of 𝑚_𝐚 and 𝑚_𝐛, Eff decreases with increasing 𝑀_𝑅, the mass of the resonance candidate. Thus, the higher the invariant mass of a two-particle combination, the more strongly (2.11) suppresses that part of the mass spectrum.

2.2. Identified Particles

Accounting for the particle masses transforms (2.6)-(2.7) into 𝐸⁺_𝐚=1𝑀_𝑅𝐸_𝑅𝐸^∗_𝐚+𝑃_𝑅𝑃^∗,𝐸⁻_𝐚=1𝑀_𝑅𝐸_𝑅𝐸^∗_𝐚−𝑃_𝑅𝑃^∗ (2.17) and at low 𝑃_𝑅 leads to a “repulsion” between the phase space 𝐸-bands on the energy (𝑃_𝑅,𝐸_𝐚) plane (Figure 2). For all resonances with 𝑚_𝐚≫𝑚_𝐛 (see Table 1): 𝐸_𝐚>𝐸_𝐛 (2.18) independent of 𝑃_𝑅. This is not always true for the background combinations. Therefore, if applied, the condition (2.18) suppresses the background even more strongly than (2.11). We will refer to the energy condition (2.18) as an 𝐸-selector.

(a)

(b)

(c)

(d)

Figure 2 

Phase space 𝐸-bands in 𝑅→𝐚+𝐛 decays as a function of the resonance momentum 𝑃𝑅. The dashed line on (d) corresponds to (2.20).

As in the previous section, if the masses of the secondary particles do not differ significantly, at 𝑃_𝑅 greater than 𝑃_𝑅=𝑀_𝑅𝐸^∗_𝐚−𝐸^∗_𝐛4(𝑃^∗)²−𝐸^∗_𝐚−𝐸^∗_𝐛², (2.19) the 𝐸-bands start overlapping in the way shown in Figure 2(d). The loss of signal combinations by demanding (2.18) for resonance candidates we estimate again with (2.14). In the case under consideration, cos𝜃^∗_𝐚 is a solution of the equation: 𝐸_𝐚cos𝜃^∗_𝐚,𝑃_𝑅=𝐸_𝐛cos𝜃^∗_𝐛,𝑃_𝑅. (2.20) Thus, cos𝜃^∗_𝐚𝐸=−^𝑅𝑃_𝑅⋅𝐸^∗_𝐚−𝐸^∗_𝐛2𝑃^∗. (2.21)

In the limit large 𝑃_𝑅, with (2.21) we again recover (2.15)-(2.16). The evolution of Eff in 𝐾^∗→𝐾𝜋 decays is shown in Figure 4(a) by the full line. Eff=100% at 𝑃_𝑅<𝑃_𝑅 and drops down up to the value (2.15), Eff≃71%, with an increase in 𝑃_𝑅.

3. Remark about 𝐷^∗ Reconstruction

Charged particles are tracked in a tracking detector (TD). The resolution of the transverse momentum of a track traversing the TD is parametrized by 𝜎(𝑝_𝑇)/𝑝_𝑇=𝐴⋅𝑝_𝑇⊕𝐵⊕𝐶/𝑝_𝑇, with 𝑝_𝑇 being the track transverse momentum (in GeV). The coefficients 𝐴,𝐵, and 𝐶 characterize the resolution of the TD. Usually, to increase the momentum resolution, only tracks with 𝑝_𝑇>0.12–0.15 GeV are selected. That cut, as shown in Figure 1(c), makes impossible the reconstruction of 𝐷^∗ mesons with momenta lower than 1.8–2.0 GeV. The 𝑚-band of 𝜋 mesons is very narrow and grows rather slowly with 𝑃_𝑅. Tracks with momenta greater than 𝑝⁺_𝜋(𝑃_𝐷∗) belong to the background. This property can be used to suppress the background in the distribution on the mass difference, Δ𝑀=𝑀(𝐾𝜋𝜋_𝑠)−𝑀(𝐾𝜋), by imposing on the momentum of the soft pion 𝜋_𝑠, the following cut, 𝑝_𝜋𝑠<𝑝₀+𝑝⁺_𝜋𝑃_𝐷∗, (3.1) where 𝑝₀=0.0–0.3 GeV is a some shift from the pion 𝑚-band. In the decay 𝐷^∗→𝐷⁰𝜋, the rest frame momentum is small, 𝑃^∗=0.038 GeV, and the 𝐷^∗ momentum can be estimated with (2.6) by means of the reconstructed 𝐷⁰ momentum: 𝑃_𝐷∗=𝑀_𝐷∗𝑀_𝐷0⋅𝑃_𝐷0. (3.2) Thus, from (3.2) and (2.6) one get with (3.1): 𝑝_𝜋𝑠<𝑝₀+𝑚_𝜋+𝑃^∗𝑀_𝐷0⋅𝑃_𝐷0. (3.3) Instead of (3.3), it is possible to apply another, less-strict cut: 𝑝_𝜋𝑠<𝑝⁺_𝜋𝑃_max≃𝑚_𝜋+𝑃^∗𝑀_𝐷∗⋅𝑃_max. (3.4) Here, 𝑃_max is the right-hand edge of the momentum range of the 𝐷^∗ candidates, 𝑃_𝐷∗<𝑃_max.

4. Pentaquark States

Now we apply the results of the previous sections to new resonances predicted [8, 9] by Diakonov, Petrov, and Polyakov in the framework of the chiral soliton model and detected both in the formation type [4, 10–18] and in the production type [3] experiments. This is so-called five-quark states (pentaquarks). For a review on pentaquark phenomenology, see [19].

4.1. Θ(1530)→𝑁(939)+𝐾(498)

With the mass value predicted for 𝜃⁺, 𝑀_𝜃=1.530 GeV, and the masses (in calculations were used mass values averaged over the isomultiplet) of decay products, 𝑚_𝑁=0.939 GeV and 𝑚_𝐾=0.498 GeV, the condition (2.9) is not satisfied. That implies the overlap of the 𝑚-bands at all 𝑃_𝜃 (Figure 3(a)) as well as the overlap of the 𝐸-bands at 𝑃_𝜃>𝑃_𝜃=2.15 GeV (Figure 3(b)). The overlapping is not strong and both selectors (2.11) and (2.18) work with high efficiency (see Figure 4). The 𝑚- and 𝐸-selectors [2] were successfully applied in searches for the 𝜃⁺ in the production-type [3, 20, 21] and the formation-type [4, 17] experiments.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3 

Phase space 𝑚-bands ((a), (c), and (e)) and 𝐸-bands ((b), (d), and (f)) in decays of the pentaquark states Θ(1530) and Ξ3/2(2070). The dashed lines correspond to (2.12) and (2.20).

(a)

(b)

(c)

(d)

Figure 4 

Efficiency of the selection rule 𝑝𝐚>𝑝𝐛 (dashed line) and 𝐸𝐚>𝐸𝐛 (full line) at different 𝑃𝑅 in decays of 𝐾∗(892), Θ(1530), and Ξ3/2(2070) resonances.

The situation with the 𝜃⁺ particle searches is rather complicated and confusing, especially when similar experiments have different conclusions on a signal existence. To be more specific, let us remind that two similar collider experiments at DESY (Hamburg) [3, 23] and two similar fixed-target experiments at IHEP (Protvino) [4, 24] do not agree on the existence of an exotic state with the mass close to 1530 MeV, which may be interpreted as a strange pentaquark state 𝜃(1530) [8]. While the ZEUS [3] and SVD-2 [4] experiments claim the observation of a narrow peak near 1530 MeV in the decay mode 𝜃⁺→𝑝𝐾⁰_𝑆, the H1 and SPHINX collaborations do not see it. The inconsistency of the experimental situation with searches of 𝜃⁺ baryon has risen even more, after the recent public polemic on results obtained with the CLAS detector at Thomas Jefferson National Accelerator Facility [25, 26]. All this discussion is beyond the scope of the present paper and can be found elsewhere [27, 28].

4.2. Ξ_3/2(2070)→Ξ(1318)+𝜋(139)

A baryonic state Ξ is a weakly decaying particle. In a cascade decay, the vertex of the decay Ξ→Λ𝜋 is separated from the primary vertex, similarly to Λ→𝑝𝜋⁻ decays. This property is used to suppress combinations which do not result from a Ξ decay and helps to reconstruct the Ξ candidate and its invariant mass.

In spite of the large mass asymmetry, 𝑚_Ξ≫𝑚_𝜋, the conditions (2.9)-(2.10) are not fulfilled in Ξ_3/2 decays, and the pictures of the 𝑚- and 𝐸-bands shown in Figures 3(c) and 3(d) are very similar to those in Θ⁺ decays. At low momenta, the efficiency of the 𝐸-selector is 100%. The 𝐸-bands start overlapping at 𝑃_Ξ>1.96 GeV. Thus, the efficiency of the 𝐸-selector is no worse than 84% (Figure 4).

4.3. Ξ_3/2(2070)→Σ(1385)+𝐾(498)

The Σ(1385) decays strongly and exclusively at the primary event vertex. The decay modes Σ^±0→Λ𝜋^±0 are dominant. Therefore, after the reconstruction of Λ, the use of the 𝑚- or 𝐸-selector allows reconstruction of the Σ(1385) with high efficiency (see Figures 1(b) and 2(b)).

Figures 3(e) and 3(f) show the 𝑚- and 𝐸-bands in Ξ_3/2(2070) decays. The 𝑚-bands slightly overlap at low 𝑃_Ξ and diverge at 𝑃_Ξ>7 GeV. The 𝐸-bands are totally separated. Thus, in that decay mode, the 𝑚- and 𝐸-selectors work with 100% efficiency at high 𝑃_Ξ momenta (Figure 4).

4.4. Ξ_3/2(2070) or Ξ_3/2(1860)?

The NA49 collaboration provided an evidence for the existence in the decay mode Ξ⁻𝜋⁻ a narrow baryonic state with the mass of 1.862±0.002 GeV [22]. This state is considered as a candidate for the exotic pentaquark state Ξ_3/2. The reported mass value is much lower as predicted in [8]. The last developments in the theory of pentaquark states did not exclude the lower mass for the Ξ_3/2 [9]. In Figure 5 are shown the 𝑚-bands and the efficiency of both selectors. They are much the similar to those in Figures 3(e) and 4(d). The picture of the 𝐸-bands is also similar to Figure 3(f). Thus, the 𝐸-selector will not suppress the signal but will suppress the background at higher masses.

(a)

(b)

Figure 5 

The 𝑚-bands and the efficiency of the 𝑚- and 𝐸-selectors for the Ξ3/2→Ξ+𝜋 decay with the Ξ3/2 mass reported by the NA49 collaboration [22].

We have to note that there are arguments [29] that the result of the NA49 collaboration perhaps is inconsistent with data collected over the past decades.

5. Conclusions

Kinematics of the two-body decay, 𝑅→𝐚+𝐛, has been analyzed in terms of equations describing boundaries of the physical region in decays and the graphical phase space 𝑚- and 𝐸-bands. The efficiency of the 𝑚- and 𝐸-selectors was estimated with the use of (2.14). On the basis of many examples, in particular, the exotic antidecuplet baryons (pentaquark states), it has been demonstrated that for 𝑚_𝐚>𝑚_𝐛 the selection rules 𝑝_𝐚>𝑝_𝐛 and 𝐸_𝐚>𝐸_𝐛 can be with a high efficiency applied for the signal reconstruction from many resonances.

Dedication

This work is dedicated to the memory of Professor Pavel F. Ermolov (30.06.1932-14.05.2008), a prominent physicist, the organizer of Department of Experimental High Energy Physics, SINP MSU, the main “driving force” of the SVD experiment and a wonderful person.

Acknowledgments

The author is indebted to P. Bussey and S. Chekanov for comments and grateful to other colleagues from the ZEUS collaboration for useful discussions. Special thanks are due to P. F. Ermolov and A. Kubarovsky for discussions of preliminary results of the SVD collaboration.