The analysis of reactions within reggeon exchanges.
1. Fit and results.
The novel point of this analysis is a direct use of reggeon exchange technique for the description of the reactions at large energies of the initial pion. This approach allows us to describe simultaneously distributions over (invariant mass of two mesons) and (momentum transfer squared to nucleons). Making use of this technique, the following resonances (as well as corresponding bare states), produced in the reaction are studied: , ( in PDG notation), , , , , , , , . Adding data for the reactions , , and , , , we have performed simultaneous -matrix fit of two-meson spectra in all these reactions. The results of combined fits to the above-listed isoscalar -states and to isovector ones, , , , are presented.
PACS numbers: 11.25.Hf, 123.1K
The study of the mass spectrum of hadrons and their properties is
the key point for the understanding of colour particle interactions
at large distances. But even the meson sector, though less
complicated than the baryon one, is far from being
completely understood. We mean that
(i) there is no sufficient information about states above 2 GeV,
(ii) certain quark–antiquark states below 2 GeV (e.g. states) are still missing,
(iii) there is no clear understanding of the glueball spectrum (although strong candidates in the and sectors exist, we have no definite information about the sector),
(iv) some analyses reported the observation of other exotics (e.g. hybrid) states,
(v) in the scalar sector not only the properties but also the existence of states like , , ( in PDG notation) is under discussion.
So, there is indeed a strong demand for new data which can help us to identify the meson states in a more definite way. However, the situation is only partly connected with the lack of data. In the lower mass region there is a lot of data taken from the proton–antiproton annihilation at rest (Crystal Barrel, Obelix), from the interaction (L3), from the proton–proton central collisions (WA102), from decay (Mark III, BES), from - and -meson decays (Focus, D0, BaBar, Belle, Cleo C) and from reactions with high energy pion beams (GAMS, VES, E852). Most of these data are of high statistics, thus allowing us to determine resonance properties with a high accuracy (though, let us emphasize, in the reactions polarized-target data are lacking).
Nevertheless, in many cases there are significant contradictions between analyses performed by different groups. The ambiguities originate from two circumstances.
First, in the discussed sectors the analyses of data taken from a single experiment cannot provide us with a unique solution. A unique solution can be obtained only from the combined analysis of a large set of data taken in different experiments.
Second, there are some simplifications inherent in many analyses. The unitarity was neglected frequently even when the amplitudes were close to the unitarity limit. A striking example is that up to now there is no proper -matrix parametrization of the and waves which are considered by many physicists as mostly understood ones. As to multipartical final states, only a few analyses have ever considered the contributions of triangle or box singularities to the measured cross sections. However, these contributions can simulate the resonant behavior of the studied distributions, especially in the threshold region (for more detail, see  and references therein).
In the analysis of meson spectra in high energy reactions , many results are related to the decomposition of the cross sections into natural and unnatural amplitudes that is based on certain models developed for the two-pion production at small momenta transferred, (e.g., see [2, 3, 4]). However, as was discussed by the cited authors, a direct application of these methods at large momenta transferred to the analysis of data may lead to a wrong result. In addition, the data were discussed mostly in terms of -channel particle exchange, though without proper analysis of the -channel exchange amplitudes.
A decade ago our group performed a combined analysis of data on proton–antiproton annihilation at rest into three pseudoscalar mesons, together with the data on two-meson -waves extracted form the , , and reactions [5, 6, 7]. The analysis has been carried out in the framework of the -matrix approach which preserves unitarity and analyticity of the amplitude in the two-meson physical region. Although the two-meson data extracted from the reaction at small momentum transfer appeared to be highly compatible with those found in proton–antiproton annihilation, we have faced a set of problems, describing the data at large momentum transfer. As we have seen now, the problems were owing to the use of partial wave decomposition which was performed by the E852 Collaboration and showed a huge signal at 1300 MeV in the -wave.
The strategy of our present approach is as follows. The analysis of a large set of experimental data on proton–antiproton annihilation at rest is carried out together with the analysis of the data based on the -channel reggeized exchanges. For the reactions, the data at small and large momentum transfers are included. Here, as the first step, we perform the analysis in the framework of the -matrix parametrization for all fitting channels (-matrix approach insures the unitarity and analyticity in the physical region). At the next stage, we plan to use the method for two-meson amplitudes satisfying these requirements in the whole complex plane.
In this paper, we present the method for the analysis of the interactions based on the -channel reggeized exchanges supplemented by a study of the proton–antiproton annihilation at rest. The method is applied to a combined analysis of the data taken by E852 at small and large momentum transfers and Crystal Barrel data on the proton–antiproton annihilation at rest into three neutral pseudoscalar mesons. The even waves, which contributed to this set of data, are parametrized within the -matrix approach. To check a strong -wave signal around 1300 MeV, which has been reported by E852 Collaboration from the analysis of data at large momentum transfers, is a subject of a particular interest in the present analysis.
We present the results of the new -matrix analysis of two-meson spectra in the scalar, , and tensor, , sectors: these sectors need a particular attention because just here we meet with the low-lying glueballs, and . The situation with the tensor glueball is rather transparent allowing us to make a definite conclusion about the gluonium structure of , while the status of the broad state requires a special discussion: this state is nearly flavour-blind but the corresponding pole of the amplitude dives deeply into the complex- plane. It is definitely seen only in the analysis of a large number of different reactions in broad intervals of mass spectra (for example, see  and references therein).
So, here we consider the following reactions:
(i) at high energies of initial pion and small and large momenta transferred to nucleon, and
(ii) , , in liquid and gaseous — the data on these reactions give us the most reliable information about scalar and tensor sectors.
As was stressed above, the novel point of the performed -matrix analysis is the use of reggeon exchange technique for the description of at high energies that allows us to analyze the two-meson invariant mass spectra and nucleon momentum transfer distributions simultaneously.
The paper is organized as follows.
In Section 2, we consider meson–nucleon collisions at high energies and present formulas for peripheral two-meson production amplitudes in terms of reggeon exchanges. Amplitudes for the description of low-energy three-meson production in the -matrix approach are given in Section 3. The fitting procedure is described in Section 4. In Conclusion we summarize the results. Technical aspects of the fitting procedure are discussed in .
2 Meson–Nucleon Collisions at High Energies: Peripheral Two-Meson Production in Terms of Reggeon Exchanges
The two-meson production reactions , , , at high energies and small momentum transfers to the nucleon are used for obtaining the -wave amplitudes , , , at because, as commonly believed, the exchange dominates this wave at such momentum transferred. At larger momentum transfers, , we observe definitely a change of the regime in the -wave production — a significant contribution of other reggeons is possible (-exchange, daughter- and daughter- exchanges). Nevertheless, the study of the two-meson production processes at looks promising, for at such momentum transfers the contribution of the broad resonance (the scalar glueball ) vanishes. Therefore, the production of other resonances (such as the and ) appears practically without background – this is important for finding out their characteristics as well as a mechanism of their production.
What we know about the reactions , , , allows us to suggest that a consistent analysis of the peripheral two-meson production in terms of reggeon exchanges may be a good tool for studying meson resonances. Note that investigation of two-meson scattering amplitudes by means of the reggeon exchange expansion of the peripheral two-meson production amplitudes was proposed long ago  but was not used because of the lack of data until now.
The -matrix amplitude of the peripheral production of two mesons with total angular momentum reads:
This formula is illustrated by Fig. 1 for the production of , , , systems. Here the factor stands for the reggeon–nucleon vertex, and is the spin operator; is the reggeon propagator depending on the total energy squared of colliding particles, , and the momentum transfer squared , while the factor is related to the block of two-meson production; , and is the phase space matrix . In the reactions , , , , the factor describes transitions , , , : in this way the block is associated with the prompt meson production, and is the -matrix factor for meson rescattering (of the type of , , , and so on). The prompt-production block for transition (where , , , , , …) is parameterized with singular (pole) and smooth terms [5, 7, 10]:
The pole singular term, , determines the bare state: here is the bare state production vertex while the parameters and are the coupling and the mass of the bare state – they are the same as in the partial wave transition amplitudes , , , , , . The smooth term stands for the background production of mesons. The , , , are free parameters of the fitting procedure, while the characteristics of resonances are determined by poles of the -matrix amplitude (remind that the position of poles is given by zeros of the amplitude denominator, ).
Below we explain in detail the method of analysis of meson spectra using as an example the reactions , , , , .
2.1 Reggeon exchange technique and the -matrix analysis of meson spectra in the waves , , , , in high energy reactions
Here we present the technique of the analysis of high-energy reaction , with the production of mesons in the , , , , states at small and moderate momenta transferred to the nucleon.
The following points are to be emphasized:
(1) The technique can be used for performing the -matrix analysis not only for and wave, as in [5, 7, 10], but simultaneously in , , waves as well.
(2) We use the reggeon exchange technique for the description of the -dependence in all analyzed amplitudes. This allows us to perform a partial wave decomposition of the produced meson states directly on the basis of the measured cross sections without using the published moment expansions (which were done under some simplifying assumptions – it is discussed below in more detail).
(3) The mass interval of the analyzed spectra is extended up to 2500 MeV thus overlapping with the mass region studied in reactions (in flight) .
We discuss in detail the reactions at incident pion momenta 20–50 GeV/c, such as measured in [12, 13, 14, 15, 16, 17]: (i) , (ii) , (iii) , (iv) . At these energies, the mesons in the states , , , , are produced via -channel exchange by reggeized mesons belonging to the leading and daughter , and trajectories.
But, first, let us present notations used below.
2.1.1 Cross sections for the reactions , ,
We consider the process of the Fig. 1 type, that is, interaction at large momenta of the incoming pion with the production of a two-meson system with a large momentum in the beam direction. This is a peripheral production of two mesons.
The cross section is defined as follows:
where is the pion momentum in the c.m. frame of the incoming hadrons. Taking into account that invariant variables and are inherent in the meson peripheral amplitude, we rewrite the phase space in a more convenient form:
Momentum is calculated in the c.m. frame of the outgoing mesons: in this system one has and , while . We have:
The cross section can be expressed in terms of the spherical functions:
The coefficients , , are subjects of study in the determination of meson resonances.
Before describing the results of analysis based on the reggeon exchange technique, let us comment methods used in other approaches.
2.1.2 The CERN-Munich approach
The CERN-Munich model  was developed for the analysis of the data on the reaction. It is based partly on the absorption model but mainly on phenomenological observations. The amplitude squared is written as
and additional assumptions are made:
1) The helicity-1 amplitudes are equal for natural and unnatural exchanges ;
2) The ratio of the and the amplitudes is a polynomial over the mass of the two-pion system which does not depend on up to the total normalization, .
Then, in , the amplitude squared was rewritten using density matrices , , as follows:
Using this amplitude for the cross section, the fitting to the moments has been carried out.
The CERN–Munich approach cannot be applied to large , it does not work for many other final states either.
2.1.3 GAMS, VES, and BNL approaches
The functions are denoted as , the functions are defined as and the functions as . The equality of the helicity-1 amplitudes with natural and unnatural exchanges is not assumed in these approaches.
However, the discussed approaches are not free from other assumptions like the coherence of some amplitudes or the dominance of the one-pion exchange. In reality the interference of the amplitudes being determined by -channel exchanges of different particles leads to a more complicated picture than that given by (9), this latter may lead (especially at large ) to a misidentification of quantum numbers for the produced resonances.
For example, in  the S-wave appears in an unnatural set of amplitudes only. Natural exchanges have moments with m=1,2,3…. However, the a1-exchange is a natural one, therefore it contributes into the S-wave and does not interfere with unnatural exchanges – in this point the moment expansion  does not coincide with formula with reggeon exchanges.
2.2 The -channel exchanges of pion trajectories in the reaction
Consider now in more detail the production amplitude for the system with and and show the way of its generalization for higher .
2.2.1 Amplitude with leading and daughter pion trajectory exchanges
The amplitude with -channel pion trajectory exchanges can be written as follows:
The summation is carried out over the leading and daughter trajectories. Here is the transition amplitude for meson block in Fig. 1, is the reggeon– coupling and is the reggeon propagator:
where the slope parameters are given in (GeV/c) units. The normalization parameter is of the order of 2–20 GeV. To eliminate the poles at we introduce Gamma-functions in the reggeon propagators (recall that at ).
The angular momentum operators are constructed of momenta and which are orthogonal to the momentum of the two-pion system :
The coefficient normalizes the angular momentum operators, so that the unitarity condition appears in a simple form (for details see Appendix A).
2.2.2 The -channel -exchange
The -exchanges dominate the spin flip amplitudes, and the amplitudes with are here suppressed, see (6). However, their contributions are visible in the differential cross sections and should be taken into account. The effects appear owing to the interference in the two-meson production amplitude because of the reggeized exchange in the -channel. The corresponding amplitude is written as:
where is the meson block of the amplitude related to the -reggeized -channel transition, is the reggeon– vertex, is the reggeon propagator, and is the polarization tensor for the state. Let us remind that is the momentum of the outgoing nucleon.
The particles are located on the pion trajectories and are described by a similar reggeized propagator. But in the meson block, the state exchange leads to vertices different from those in the -exchange, so it is convenient to single out these contributions. Therefore, we use for the propagator given by (11) but with eliminated -contribution:
Taking into account that
In the large momentum limit of the initial pion, the second term in (20) is always small and can be neglected, while the convolution of with the momenta of the meson block results in the term . Hence, the amplitude for -exchange can be rewritten as follows:
A resonance with spin and fixed parity can be produced owing to the -exchange with three angular momenta , and , so we have:
2.3 Amplitudes with -trajectory exchanges
Here we present formulae for for leading and daughter -trajectories and leading -trajectory.
2.3.1 Amplitudes with -trajectory exchanges
The amplitude with -channel -exchanges is a sum of leading and daughter trajectories:
where is the reggeon–NN coupling and the reggeon propagator has the form:
As previously, the normalization parameter is of the order of 2–20 GeV, and the Gamma-functions in the reggeon propagators are introduced in order to eliminate the poles at .
For the nucleon–reggeon vertex we use two-component spinors in the infinite momentum frame, and , so the vertex reads where is the unit vector directed along the nucleon momentum in the c.m. frame of colliding particles.
At fixed partial wave , the channel (-1) is characterized by two angular momenta , therefore we have two amplitudes for each :
where the polarisation vector ; the GLF-vectors  defined in the c.m. system of the colliding particles as follows:
The products of and operators can be expressed through vectors and :