Exclusive diffractive production of \pi^{+}\pi^{-}\pi^{+}\pi^{-}via the intermediate \sigma\sigma and \rho\rho states in proton-proton collisions within tensor pomeron approach

# Exclusive diffractive production of π+π−π+π−via the intermediate σσ and ρρ states in proton-proton collisions within tensor pomeron approach

Piotr Lebiedowicz Institute of Nuclear Physics PAN, PL-31-342 Kraków, Poland    Otto Nachtmann Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany    Antoni Szczurek 111also at University of Rzeszów, PL-35-959 Rzeszów, Poland. Institute of Nuclear Physics PAN, PL-31-342 Kraków, Poland
###### Abstract

We present first predictions of the cross sections and differential distributions for the central exclusive reaction being studied at RHIC and LHC. The amplitudes for the processes are formulated in terms of the tensor pomeron and tensor reggeon exchanges with the vertices respecting the standard crossing and charge-conjugation relations of Quantum Field Theory. The and contributions to the final state are considered focussing on their specificities. The correct inclusion of the pomeron spin structure seems crucial for the considered sequential mechanisms in particular for the contribution which is treated here for the first time. The mechanism considered gives a significant contribution to the reaction. We adjust parameters of our model to the CERN-ISR experimental data and present several predictions for the STAR, ALICE, ATLAS and CMS experiments. A measurable cross section of order of a few is obtained including the experimental cuts relevant for the LHC experiments. We show the influence of the experimental cuts on the integrated cross section and on various differential distributions.

###### pacs:
12.40.Nn,13.60.Le,14.40.Be

## I Introduction

Last years there was a renewed interest in exclusive production of two meson pairs (mostly pairs) at high energies related to planned experiments at RHIC Adamczyk:2014ofa (), Tevatron Aaltonen:2015uva (); Albrow_Project_new (), and LHC Schicker:2014aoa (); Staszewski:2011bg (); CMS:2015diy (). From the experimental point of view the exclusive processes are important in the context of resonance production, in particular, in searches for glueballs. The experimental data on central exclusive production measured at the energies of the ISR, RHIC, Tevatron, and the LHC collider all show visible structures in the invariant mass. It is found that the pattern of these structures seems to depend on experiment. But, as we advocated in Ref. Lebiedowicz:2016ioh (), this dependence could be due to the cuts used in a particular experiment (usually these cuts are different for different experiments).

So far theoretical studies concentrated on two-pion continuum production. Some time ago two of us have formulated a Regge-type model with parameters fixed from phenomenological analysis of total and elastic and scattering Lebiedowicz:2009pj (). The model was extended to include also absorption effects due to proton-proton interaction Lebiedowicz:2011nb (); Lebiedowicz:2011tp (). In Ref. Lebiedowicz:2011nb () the exclusive reaction constitutes an irreducible background to the scalar meson production. These model studies were extended also to production Lebiedowicz:2011tp (). For a related work on the exclusive reaction , see Lebiedowicz:2010yb (). A revised view of the absorption effects including the nonperturbative interactions was presented very recently Lebiedowicz:2015eka (). Such an approach gives correct order of magnitude cross sections, however, does not include resonance contributions which interfere with the continuum.

It was known for a long time that the frequently used vector-pomeron model has problems considering a field theory point of view. Taken literally it gives opposite signs for and total cross sections. A way out of these problems was already shown in Nachtmann:1991ua () where the pomeron was described as a coherent superposition of exchanges with spin 2 + 4 + 6 + … . The same idea is realised in the tensor-pomeron model formulated in Ewerz:2013kda (). In this model pomeron exchange can effectively be treated as the exchange of a rank-2 symmetric tensor. The corresponding couplings of the tensorial object to proton and pion were worked out. In Ref. Lebiedowicz:2013ika () the model was applied to the diffractive production of several scalar and pseudoscalar mesons in the reaction . In Bolz:2014mya () an extensive study of the photoproduction reaction in the framework of the tensor-pomeron model was presented. The resonant () and non-resonant (Drell-Söding) photon-pomeron/reggeon production in collisions was studied in Lebiedowicz:2014bea (). The exclusive diffractive production of continuum together with the dominant scalar , , and tensor resonances was studied by us very recently in Ref. Lebiedowicz:2016ioh ().

The past program of central production of pairs of mesons was concentrated on the discussion of mesonic resonances. The low-energy program of studying meson excitations can be repeated at the LHC, where we expect dominance of one production mechanism only, two-pomeron exchange.

The identification of glueballs can be very difficult. The partial wave analyses of future experimental data of the STAR, ALICE, ATLAS and CMS Collaborations could be used in this context. Also the studies of different decay channels in central exclusive production would be very valuable. One of the possibilities is the reaction being analysed by the ATLAS, CMS and ALICE Collaborations at the LHC. Identification of the glueball-like states in this channel requires calculation/estimation of the four-pion background from other sources.

Pairs of (giving four pions) can be produced also in photon-hadron interactions in a so-called double scattering mechanism. In Ref. Goncalves:2016ybl () double vector meson production in photon-photon and photon-hadron interactions in // collisions was studied. In heavy ion collisions the double scattering mechanism is very important KlusekGawenda:2013dka (); Goncalves:2016ybl (). In proton-proton collisions, for instance at the center-of-mass energy of  TeV, total cross sections for the double meson production, taking into account the and double scattering mechanisms, were estimated Goncalves:2016ybl () to be of 182  and 4 , respectively.

In the present paper we wish to concentrate on the four charged pion continuum which is a background for future studies of diffractively produced resonances. We shall present a first evaluation of the four-pion continuum in the framework of the tensor pomeron model consistent with general rules of Quantum Field Theory. Here we shall give explicit expressions for the amplitudes of and production with the and decaying to . We shall discuss their specificity and relevance for the reaction. In the Appendix A we present the formulas of the double-pomeron exchange mechanism for the exclusive production of scalar resonances decaying into and/or pairs.

## Ii Exclusive diffractive production of four pions

In the present paper we consider the processes from the diagrams shown in Fig. 1,

 pp→ppσσ→ppπ+π−π+π−,pp→ppρρ→ppπ+π−π+π−. (1)

That is, we treat effectively the processes (1) as arising from processes, the central diffractive production of two scalar mesons and two vector mesons in proton-proton collisions. To calculate the total cross section for the reactions one has to calculate the 8-dimensional phase-space integral numerically Lebiedowicz:2009pj () 222In the integration over four-body phase space the transverse momenta of the produced particles (, , , ), the azimuthal angles of the outgoing protons (, ) and the rapidity of the produced mesons (, ) were chosen as integration variables over the phase space.. Some modifications of the reaction are needed to simulate the reaction with in the final state. For example, one has to include in addition a smearing of the and masses due to their instabilities. Then, the general cross-section formula can be written approximately as

 σ2→6=∫max{mX3}2mπ∫max{mX4}2mπσ2→4(...,mX3,mX4)fM(mX3)fM(mX4)dmX3dmX4. (2)

Here we use for the calculation of the decay processes with the spectral function

 fM(mXi)=(1−4m2πm2Xi)n/22πm2MΓM,tot(m2Xi−m2M)2+m2MΓ2M,totNI, (3)

where . In (3) , for meson and , for meson, respectively. The quantity smoothly decreases the spectral function when approaching the threshold, .

### ii.1 pp→ppσσ

Here we discuss the exclusive production of pairs in proton-proton collisions,

 p(pa,λa)+p(pb,λb)→p(p1,λ1)+σ(p3)+σ(p4)+p(p2,λ2), (4)

where , and , denote the four-momenta and helicities of the protons and denote the four-momenta of the mesons, respectively.

The diagram for the production with an intermediate meson is shown in Fig. 1 (left diagram). The amplitude for this process can be written as the following sum:

 M(σ−exchange)pp→ppσσ = M(PP→σσ)+M(Pf2R→σσ)+M(f2RP→σσ)+M(f2Rf2R→σσ). (5)

For instance, the -exchange amplitude can be written as

 M(PP→σσ)=M(^t)λaλb→λ1λ2σσ+M(^u)λaλb→λ1λ2σσ (6)

with the - and -channel amplitudes

 M(^t)λaλb→λ1λ2σσ=(−i)¯u(p1,λ1)iΓ(Ppp)μ1ν1(p1,pa)u(pa,λa)iΔ(P)μ1ν1,α1β1(s13,t1)iΓ(Pσσ)α1β1(pt,−p3)iΔ(σ)(pt)×iΓ(Pσσ)α2β2(p4,pt)iΔ(P)α2β2,μ2ν2(s24,t2)¯u(p2,λ2)iΓ(Ppp)μ2ν2(p2,pb)u(pb,λb), (7)
 M(^u)λaλb→λ1λ2σσ=(−i)¯u(p1,λ1)iΓ(Ppp)μ1ν1(p1,pa)u(pa,λa)iΔ(P)μ1ν1,α1β1(s14,t1)iΓ(Pσσ)α1β1(p4,pu)iΔ(σ)(pu)×iΓ(Pσσ)α2β2(pu,−p3)iΔ(P)α2β2,μ2ν2(s23,t2)¯u(p2,λ2)iΓ(Ppp)μ2ν2(p2,pb)u(pb,λb), (8)

where , , , , . Here and denote the effective propagator and proton vertex function, respectively, for the tensorial pomeron. The effective propagators and vertex functions for the tensorial pomeron/reggeon exchanges respect the standard crossing and charge-conjugation relations of Quantum Field Theory. For the explicit expressions of these terms see Sect. 3 of Ewerz:2013kda (). We assume that has the same form as (see (3.45) of Ewerz:2013kda ()) but with the coupling constant instead of the one . The scalar meson propagator is taken as in (4.7) and (4.8) of Lebiedowicz:2016ioh () with the running (energy-dependent) width. In a similar way the , and amplitudes can be written. For the vertex our ansatz is as for in (3.53) of Ewerz:2013kda () but with replaced by .

In the high-energy small-angle approximation we can write the amplitude (5) as

 M(σ−exchange)λaλb→λ1λ2σσ≃2(p1+pa)μ1(p1+pa)ν1δλ1λaF1(t1)FM(t1)×{Vμ1ν1(s13,t1,pt,−p3)Δ(σ)(pt)Vμ2ν2(s24,t2,pt,p4)[^Fσ(p2t)]2+Vμ1ν1(s14,t1,pu,p4)Δ(σ)(pu)Vμ2ν2(s23,t2,pu,−p3)[^Fσ(p2u)]2}×2(p2+pb)μ2(p2+pb)ν2δλ2λbF1(t2)FM(t2). (9)

The function has the form ( GeV)

 Vμν(s,t,k2,k1)=(k1+k2)μ(k1+k2)ν14s[3βPNNgPσσ(−isα′P)αP(t)−1+12M20gf2Rppgf2Rσσ(−isα′f2R)αf2R(t)−1], (10)

where = 1.87 GeV and = 11.04 from (6.53) and (6.55) of Ewerz:2013kda (), respectively.

If the meson has substantial gluon content or some component its coupling to and may be larger than for the pion. To illustrate effects of this possibility we take in the calculation two sets of the coupling constants

 setA:gPσσ=2βPππ,gf2Rσσ=gf2Rππ, (11) setB:gPσσ=4βPππ,gf2Rσσ=2gf2Rππ, (12)

where = 1.76 GeV and = 9.30 from (7.15) and (7.16) of Ewerz:2013kda (), respectively.

The form of the off-shell meson form factor in (9) and of the analogous form factor for mesons (see the next section) is unknown. We write generically for these form factors which we normalize to unity at the on-shell point, , and parametrise here in two ways:

 ^FM(k2)=exp⎛⎝k2−m2MΛ2off,E⎞⎠, (13) ^FM(k2)=Λ2off,Mp−m2MΛ2off,Mp−k2,Λoff,Mp>mM. (14)

The cut-off parameters for the exponential form or for the monopole form of the form factors can be adjusted to experimental data.

A factor due to the identity of the two mesons in the final state has to be taken into account in the phase-space integration in (2).

### ii.2 pp→ppρρ

Here we focus on exclusive production of in proton-proton collisions, see Fig. 1 (right diagram),

 p(pa,λa)+p(pb,λb)→p(p1,λ1)+ρ(p3,λ3)+ρ(p4,λ4)+p(p2,λ2), (15)

where and denote the four-momenta and helicities of the mesons, respectively. We write the amplitude as

 Mλaλb→λ1λ2ρρ=(ϵ(ρ)ρ3(λ3))∗(ϵ(ρ)ρ4(λ4))∗Mρ3ρ4λaλb→λ1λ2ρρ, (16)

where are the polarisation vectors of the meson.

Then, with the expressions for the propagators, vertices, and form factors, from Ewerz:2013kda () can be written in the high-energy approximation as

 M(ρ−exchange)ρ3ρ4λaλb→λ1λ2ρρ≃2(p1+pa)μ1(p1+pa)ν1δλ1λaF1(t1)FM(t1)×{Vρ3ρ1μ1ν1(s13,t1,pt,p3)Δ(ρ)ρ1ρ2(pt)Vρ4ρ2μ2ν2(s24,t2,−pt,p4)[^Fρ(p2t)]2+Vρ4ρ1μ1ν1(s14,t1,−pu,p4)Δ(ρ)ρ1ρ2(pu)Vρ3ρ2μ2ν2(s23,t2,pu,p3)[^Fρ(p2u)]2}×2(p2+pb)μ2(p2+pb)ν2δλ2λbF1(t2)FM(t2), (17)

 Vμνκλ(s,t,k2,k1)=2Γ(0)μνκλ(k1,k2)14s[3βPNNaPρρ(−isα′P)αP(t)−1+1M0gf2Rppaf2Rρρ(−isα′f2R)αf2R(t)−1]−Γ(2)μνκλ(k1,k2)14s[3βPNNbPρρ(−isα′P)αP(t)−1+1M0gf2Rppbf2Rρρ(−isα′f2R)αf2R(t)−1]. (18)

The explicit tensorial functions , = 0, 2, are given in Ref. Ewerz:2013kda (), see formulae (3.18) and (3.19), respectively. In our calculations the parameter set A of coupling constants and from Lebiedowicz:2014bea () was used, see Eq. (2.15) there.

We consider in (15) unpolarised protons in the initial state and no observation of polarizations in the final state. In the following we are mostly interested in the invariant mass distributions of the system and in distributions of the parent mesons. Therefore, we have to insert in (2) the cross section summed over the meson polarizations. The spin sum for a meson of momentum and squared mass is

 ∑λ=0,±1ϵ(ρ)μ(λ)(ϵ(ρ)ν(λ))∗=−gμν+kμkνm2X. (19)

But the terms do not contribute since we have the relations

 p3ρ3Mρ3ρ4λaλb→λ1λ2ρρ=0,p4ρ4Mρ3ρ4λaλb→λ1λ2ρρ=0. (20)

These follow from the properties of in (18); see (3.21) of Ewerz:2013kda ().

Taking also into account the statistical factor due to the identity of the two mesons we get for the amplitudes squared (to be inserted in in (2))

 (21)

So far we have treated the exchanged mesonic object for production as spin-1 particle. However, we should take into account the fact that the exchanged intermediate object is not a simple meson but may correspond to a whole family of daughter exchanges, that is, the reggeization of the intermediate meson is necessary. For related works, where this effect was included in practical calculations, see e.g. Cisek:2011vt (); Lebiedowicz:2013ika (). The “reggeization” of the amplitude given in Eq. (17) is included here for , only approximately, by replacing the propagator both in the - and -channel amplitudes by

 Δ(ρ)ρ1ρ2(p)→Δ(ρ)ρ1ρ2(p)(s34s0)αρ(p2)−1, (22)

where we take and with the momentum transfer .

To give the full physical amplitudes we should add absorptive corrections to the Born amplitudes (9) and (17) for the and reactions, respectively. For the details how to include the -rescattering corrections in the eikonal approximation for the four-body reaction see Sect. 3.3 of Lebiedowicz:2014bea ().

## Iii Preliminary results

In this section we wish to present first results for the and processes corresponding to the diagrams shown in Fig. 1.

We start from a discussion of the invariant mass distribution. In Fig. 2 we compare the - and -contributions to the CERN-ISR data Breakstone:1993ku () at  GeV. Here, the four pions are restricted to lie in the rapidity region and the cut 333The Feynman- variable was defined as in the center-of-mass frame with the longitudinal momentum of the outgoing proton. is imposed on the outgoing protons. In Ref. Breakstone:1993ku () five contributions to the four-pion spectrum were identified. A phase-space term with total angular momentum , two terms with and , and two terms with and . In the following we will compare the phase-space term with our result, the terms with our result. The theoretical results correspond to the calculation including absorptive corrections related to the nonperturbative interaction in the initial and final state. The ratio of full and Born cross sections (the gap survival factor) is approximately . In our calculation both the and the , , exchanges were included. At the ISR energy the exchanges, including their interference terms with the one, give about to the total cross section.

In the left panel of Fig. 2 we compare our contribution assuming the coupling constants (12) with the (, phase space) ISR data (marked as full data points). We present results for two different forms of off-shell meson form factor, the exponential type (13),  GeV, and the monopole type (14),  GeV, see the black lower line and the red upper line, respectively. There is quite a good agreement between our result with a monopole form factor and the (, phase space) data. Note that this implies that the set B of and couplings, which are larger than the corresponding pion couplings, seems to be preferred.

In the right panel of Fig. 2 we compare our result for the contribution to corresponding ISR data 444Here we plotted the sum of the experimental cross section for the and terms (see Figs. 3c and 3e in Breakstone:1993ku (), respectively) and the corresponding error is approximated as .. In the calculation of the contribution we take into account the intermediate meson reggeization. The reggeization leads to an extra strong damping of the large cross section. The effect of reggeization is expected only when the separation in rapidity between the two produced resonances is large. We will return to this issue in the further part of this section. We note that our model is able to give a qualitative account of the ISR data for  GeV within the large experimental errors.

The total experimental data (marked as open data points in the left panel of Fig. 2) are also shown for comparison. In Ref. Breakstone:1993ku () an integrated (total) cross section of 46  at  GeV was estimated. There are other processes besides the ones of (1) contributing to the final state, such as resonance production shown in the diagrams in Fig. 6 of Appendix A. Thus, the and the contributions considered here should not be expected to fit the ISR data precisely. In addition, the ISR data includes also a large ( and ) component (see Figs. 3b and 3d of Breakstone:1993ku ()) with an enhancement in the term which was interpreted there as a state. Also the () term indicates a signal of state; see Fig. 3e of Breakstone:1993ku (). Therefore, a consistent model for the resonance and continuum contributions, including the interference effects between them, would be required to better describe the ISR data. We leave this interesting problem for future studies.

In Fig. 3 we show our preliminary four-pion invariant mass distributions for experimental cuts relevant for the RHIC and LHC experiments. In the calculation of the and the contributions the pomeron and exchanges were included. Imposing limitations on pion (pseudo)rapidities, e.g. , and going to higher energies strongly reduces the role of subleading exchanges. The gap survival factors estimated within the eikonal approximation are 0.30, 0.21, 0.23 for = 0.2, 7, 13 TeV, respectively. In the case of contribution we use two sets of the coupling constants; standard (11) and enhanced ones (12).

The correlation in rapidity of pion pairs (e.g., means where the pions are produced from a meson decay) for both the and the contributions is displayed in Fig. 4 for  GeV. For the contribution, see the top panel, we observe a strong correlation . For the case the distribution extends over a much broader range of which is due to the exchange of the spin-1 particle. However, as discussed in the section devoted to the formalism we may include, at least approximately, the effect of reggeization of the intermediate meson. In the left and right bottom panels we show the results without and with the meson reggeization, respectively. As shown in the right panel this effect becomes crucial when the separation in rapidity between the two mesons increases. After the reggeization is performed the two-dimensional distribution looks very similar as for the case. The reggeization effect discussed here is also closely related to the damping of the four-pion invariant mass distribution discussed already in Fig. 2.

In Table 1 we have collected integrated cross sections in with different experimental cuts for the exclusive production including only the contributions shown in Fig. 1. The collected results were obtained in the calculations with the tensor pomeron and reggeon exchanges. In the calculations the off-shell-meson form factor (13) with  GeV was used. No absorption effects were included in the quoted numbers. The full cross section can be obtained by multiplying the Born cross section by the corresponding gap survival factor . These factors depend on the kinematic cuts and are 0.40 (ISR), 0.46 (STAR, lower ), 0.30 (STAR, higher ), 0.21 ( = 7 TeV), 0.19 ( = 13 TeV), 0.23 ( = 13 TeV, with cuts on ).

The cross sections for the final state found here are more than three orders of magnitude larger than the cross sections for the and double scattering mechanisms considered recently in Goncalves:2016ybl ().

Finally, in Fig. 5, we discuss some observables which are very sensitive to the absorptive corrections. Quite a different pattern can be seen for the Born case and for the case with absorption included. The absorptive corrections lead to significant modification of the shape of the distribution ( is the azimuthal angle between the vectors of the outgoing protons) and lead to an increase of the cross section for the proton four-momentum transfer = = at large . This effect could be verified in future experiments when both protons are measured which should be possible for ATLAS-ALFA Staszewski:2011bg () and CMS-TOTEM.

## Iv Conclusions

In the present paper we have presented first estimates of the contributions with the intermediate , resonance pairs to the reaction which is being analyzed experimentally by the STAR, ALICE, CMS, and ATLAS Collaborations. The results were obtained within a model where the pomeron and reggeon are treated as effective tensor exchanges. The results for processes with the exchange of heavy (compared to pion) mesons strongly depend on the details of the hadronic form factors. By comparing the theoretical results and the cross sections found in the CERN-ISR experiment Breakstone:1993ku () we fixed the parameters of the off-shell meson form factor and the and couplings. The corresponding values of parameters can be verified by future experimental results obtained at RHIC and LHC.

We have made estimates of the integrated cross sections for different experimental situations as well as shown several differential distributions. The pion-pair rapidities of the two mesons are strongly correlated (). This is due to a strong interference effect between the - and -channel amplitudes. For the contribution the situation is the following. If we take for the propagator of the exchanged in Fig. 1, right panel, the standard particle propagator the correlation is very weak. But for a reggeized propagator we get again a strong correlation, similar to that found in the case; see Fig. 4. This is understandable since the reggeization suppresses contributions where the two produced mesons have large subsystem energies, i.e. where there is a large rapidity distance between the two mesons.

We have found in this paper that the diffractive mechanism in proton-proton collisions considered by us leads to the cross section for the final state more than three orders of magnitude larger than the corresponding cross section for and double scattering photon-pomeron (pomeron-photon) mechanisms considered recently in Goncalves:2016ybl ().

Closely related to the reaction studied by us here is the production in ultra-peripheral nucleus-nucleus collisions. A phenomenological study of the reaction mechanism of was performed in Ref. KlusekGawenda:2013dka (). The application of our methods, based on the tensor-pomeron concept, to collisions involving nuclei is an interesting problem which goes, however, beyond the scope of the present work.

To summarize: we have given a consistent treatment of the production via two scalar mesons and two vector mesons in an effective field-theoretic approach. A measurable cross section of order of a few was obtained for the process which should give experimentalists interesting challenges to check and explore it.

## Appendix A Four-pion production through f0→σσ and f0→ρρ mechanisms

Here we discuss the diffractive production of the scalar , , and resonances decaying at least potentially into the final state. We present relevant formulas for the resonance contributions that could be used in future analyses. At present a precise calculation of the resonance contributions to the four-pion channel is not possible as some details of the relevant decays are not well understood.

The production and decay properties of the scalar mesons in the channel, such as the and states, have been investigated extensively in central diffractive production by the WA102 Collaboration at  GeV Barberis:1997ve (); Barberis:1999wn (); Barberis:2000em () and in and annihilations by the Crystal Barrel Collaboration Abele:2001js (). In central production, see Fig. 3 of Barberis:1999wn (), there is a very clear signal from in the spectra, especially in the channel, and some evidence of the broad resonance in the channel. In Fig. 1 of Ref. Barberis:2000em () the wave from the channel in four different intervals (each of ) was shown. A peak below  MeV was clearly seen which can be interpreted as the interference effect of the state, the state and the broad background. In principle, also the contributions from the and are not excluded. In Table 1 of Barberis:2000em () the percentage of each resonance in three intervals of the so-called “glueball filter variable” () was shown. The idea being that for small differences in the transverse momentum vectors between the two exchanged “particles” an enhancement in the production of glueballs relative to states may occur. The dependence and the distributions presented there are similar to what was found in the analysis of the channel Barberis:1999cq (). In Refs. Kirk:1999df (); Kirk:2000ws () it was shown that also the state has a similar behaviour in the azimuthal angle and in the variable as the state. That is, all the undisputed states are observed to be suppressed at small , but the glueball candidates , , together with the enigmatic , survive. It was shown in Barberis:2000em () that the and have similar and dependences. The fact that and states have different and dependences confirms that these are not simply dependent phenomena 555Some essential discrepancy for and states in the different decay channels was discussed, e.g., in Refs. Klempt:2007cp (); Ochs:2013gi (). In Ref. Janowski:2014ppa (), by using a three-flavor chiral effective approach, the authors found that is predominantly the gluonic state and the decay channel is strongly suppressed.. This is also true for the states, where the state has different dependences compared to the and states Barberis:1999cq (). We wish to emphasize that in Lebiedowicz:2013ika () we obtained a good description of the WA102 experimental distributions Barberis:1998ax (); Barberis:1999cq (); Barberis:1999zh () for the scalar and pseudoscalar mesons within the framework of the tensor pomeron approach. The and effects can be understood as being due to the fact that in general more than one pomeron-pomeron-meson coupling structure is possible Lebiedowicz:2013ika (). The behaviour of the tensor state was discussed recently in Lebiedowicz:2016ioh (); see Figs. 4 and 5 there.

In the following we present our analytic expressions for the diagrams of Fig. 6 for fusion only. The extension to include also , and fusion is straightforward.

#### a.0.1 pp→pp(f0→σσ)

Here we consider the amplitude for the reaction (4) through an -channel scalar resonance ; see Fig. 6 (left diagram). Here stands, for one of the , , states.

In the high-energy small-angle approximation we can write this amplitude as

 (23)

where , , , , , , and