Central exclusive diffractive production of K^{+}K^{-}K^{+}K^{-}via the intermediate \phi\phi state in proton-proton collisions

# Central exclusive diffractive production of K+k−K+k−via the intermediate ϕϕ state in proton-proton collisions

Piotr Lebiedowicz Institute of Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland    Otto Nachtmann Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany    Antoni Szczurek 111Also at Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, PL-35310 Rzeszów, Poland. Institute of Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland
###### Abstract

We present a study of the exclusive reaction at high energies. We consider diffractive mechanisms with the intermediate state with its decay into the system. We include the -channel exchanges and the -channel exchange mechanism. This state is a candidate for a tensor glueball. We discuss the possibility to use the process in identifying the odderon exchange. An upper limit for the coupling is extracted from the WA102 experimental data. The amplitudes for the processes are formulated within the tensor-pomeron and vector-odderon approach. We adjust parameters of our model to the WA102 data and present several predictions for the ALICE, ATLAS, CMS and LHCb experiments. The distributions in four-kaon invariant mass, in rapidity distance between the two mesons, in a special “glueball filter variable”, and in proton-proton relative azimuthal angle are presented. A measurable cross section of order of a few nb is obtained including the experimental cuts relevant for the LHC experiments. The distribution in rapidity difference of both -mesons could shed light on the coupling, not known at present. We discuss the possible role of the , , and resonances observed in the channel in radiative decays of .

## I Introduction

Diffractive studies are one of the important parts of the physics programme for the RHIC and LHC experiments. A particularly interesting class are central-exclusive-production (CEP) processes, where all centrally produced particles are detected; see Sect. 5 of N.Cartiglia:2015gve (). In the last years there was a renewed interest in exclusive production of pairs at high energies related to successful experiments by the CDF Aaltonen:2015uva () and the CMS Khachatryan:2017xsi () Collaborations. These measurements are important in the context of resonance production, in particular, in searches for glueballs. The experimental data on central exclusive production measured at Fermilab and CERN all show visible structures in the invariant mass. As we discussed in Ref. Lebiedowicz:2016ioh () the pattern of these structures has mainly resonant origin and is very sensitive to the cuts used in a particular experiment (usually these cuts are different for different experiments). In the CDF and CMS experiments only large rapidity gaps around the centrally produced dimeson system are checked but the forward and backward going (anti)protons are not detected. Preliminary results of similar CEP studies have been presented by the ALICE Schicker:2012nn () and LHCb McNulty:2016sor () Collaborations at the LHC. Although such results will have diffractive nature, further efforts are needed to ensure their exclusivity. Ongoing and planned experiments at RHIC (see e.g. Sikora:2018cyk ()) and future experiments at the LHC will be able to detect all particles produced in central exclusive processes, including the forward and backward going protons. Feasibility studies for the process with tagging of the scattered protons as carried out for the ATLAS and ALFA detectors are shown in Staszewski:2011bg (). Similar possibilities exist using the CMS and TOTEM detectors; see, e.g., Albrow:2014lrm ().

It was known for a long time that the frequently used vector-pomeron model has problems from the point of view of field theory. Taken literally it gives opposite signs for and total cross sections. A way how to solve these problems was discussed 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. In Ewerz:2016onn () it was shown that the tensor-pomeron model is consistent with the experimental data on the helicity structure of proton-proton elastic scattering at  GeV and small from the STAR experiment Adamczyk:2012kn (). In Ref. Lebiedowicz:2013ika () the tensor-pomeron 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 central exclusive diffractive production of continuum together with the dominant scalar , , and tensor resonances was studied by us in Lebiedowicz:2016ioh (). The meson production associated with a very forward/backward system in the and processes was discussed in Lebiedowicz:2016ryp (). Also the central exclusive production via the intermediate and states in collisions was considered in Lebiedowicz:2016zka (). In Lebiedowicz:2018sdt () the reaction was studied. Recently, in Lebiedowicz:2018eui (), the exclusive diffractive production of the in the continuum and via the dominant scalar , , , and tensor , resonances, as well as the photoproduction contributions, were discussed in detail. In Lebiedowicz:2019por () a possibility to extract the couplings from the analysis of pion angular distributions in the Collins-Soper system of reference was studied.

The identification of glueballs in the reaction, being analysed by the STAR, ALICE, ATLAS, CMS and LHCb Collaborations, can be rather difficult as the dipion spectrum is dominated by the states and mixing of the pure glueball states with nearby mesons is possible. The partial wave analyses of future experimental data could be used in this context. Studies of different decay channels in central exclusive production would be very valuable. One of the promising reactions is with both mesons decaying into the channel.

The advantage of this process for experimental studies is the following. The is a narrow resonance and it can be easily identified in the spectra. On the other hand, non- backgrounds in these spectra should have a broad distribution. However, identification of possible glueball-like states in this channel requires calculation/estimation both of resonant and continuum processes. It is known from the WA102 analysis of various channels that the so-called “glueball-filter variable” () Close:1997pj (), defined by the difference of the transverse momentum vectors of the outgoing protons, can be used to select out known states from non- candidates. It was observed by the WA102 Collaboration (see, e.g., Barberis:1996iq (); Barberis:1997ve (); Barberis:1998ax (); Barberis:1999cq (); Barberis:2000em (), Kirk:2000ws (); Kirk:2014nwa ()) that all the undisputed states are suppressed at small in contrast to glueball candidates. It is therefore interesting to make a similar study of the dependence for the system decaying into in central collisions at the LHC.

Structures in the invariant-mass spectrum were observed by several experiments. Broad structures around 2.3 GeV were reported in the inclusive reaction Booth:1985kv (); Booth:1985kr (), in the exclusive Etkin:1985se (); Etkin:1987rj () and Aston:1989gx (); Aston:1990wf () reactions, in central production Armstrong:1986ky (); Armstrong:1989hz (); Barberis:1998bq (), and in annihilations Evangelista:1998zg (). In the radiative decay an enhancement near  GeV with preferred was observed Bisello:1986pt (); Bai:1990hk (); Ablikim:2008ac (); Ablikim:2016hlu (). The last partial wave analysis Ablikim:2016hlu () shows that the state is significant, but a large contribution from the direct decay of , modeled by a phase space distribution of the system, was also found there. Also the scalar state , and two additional pseudoscalar states, and the were observed. Three tensor states, , and , observed previously in Etkin:1985se (); Etkin:1987rj (), were also observed in . It was concluded there that the tensor spectrum is dominated by the . The nature of these resonances is not understood at present and a tensor glueball has still not been clearly identified. According to lattice-QCD simulations, the lightest tensor glueball has a mass between 2.2 GeV and 2.4 GeV, see e.g. Morningstar:1997ff (); Morningstar:1999rf (); Hart:2001fp (); Loan:2005ff (); Gregory:2012hu (); Chen:2005mg (); Sun:2017ipk (). The and states are good candidates to be tensor glueballs. For an experimental work indicating a possible tensor glueball see Longacre:2004jn (). Also lattice-QCD predictions for the production rate of the pure gauge tensor glueball in radiative decays Yang:2013xba () are consistent with the large production rate of the in the Ablikim:2013hq (), Ablikim:2016hlu () and Ablikim:2018izx () channels.

In the present paper we wish to concentrate on CEP of four charged kaons via the intermediate state. Here we shall give explicit expressions for the amplitudes involving the pomeron-pomeron fusion to () through the continuum processes, due to the - and -channel reggeized -meson, photon, and odderon exchange, as well as through the -channel resonance reaction (). The pseudoscalar mesons having and can also be produced in pomeron-pomeron fusion and may contribute to our reaction if they decay to . Possible candidates are e.g. and which were observed in radiative decays of  Ablikim:2016hlu (). The same holds for scalar states with and , for example, the scalar meson. We will comment on the possible influence of these contributions for CEP of pairs. Some model parameters will be determined from the comparison to the WA102 experimental data Barberis:1998bq (); Barberis:2000em (). In order to give realistic predictions we shall include absorption effects calculated at the amplitude level and related to the nonperturbative interactions.

## Ii Exclusive diffractive production of four kaons

In the present paper we consider the process, CEP of four mesons, with the intermediate resonance pair,

 pp→ppϕϕ→ppK+K−K+K−. (1)

In Fig. 1 we show diagrams for this process which are expected to be the most important ones at high energies since they involve pomeron exchanges. Fig. 1 (a) shows the continuum process. In Fig. 1 (b) we have the process with intermediate production of an resonance

 pp→pp(f2→ϕϕ)→ppK+K−K+K−. (2)

In the place of the we can also have an - and an -type resonance. That is, we treat effectively the processes (1) - (2) as arising from the process, the central diffractive production of two vector mesons in proton-proton collisions.

The production of can also occur through diagrams of the type of Fig. 1 but with reggeons in the place of the pomerons. For example, in Fig. 1 (a) we can replace the pomerons by reggeons and the intermediate by a pomeron. In Fig. 1 (b) we can replace one or two pomerons by one or two reggeons. For high energies and central production such reggeon contributions are expected to be small and we shall not consider them in our present paper. We shall treat in detail the diagrams with pomeron exchange (Fig. 1) and diagrams involving odderon and also photon exchange; see Figs. 2 and 3 below.

A resonance produced in pomeron-pomeron fusion must have and charge conjugation , but it may have various spin and parity quantum numbers. See e.g. the discussion in Appendix A of Lebiedowicz:2013ika ().

In Table 1 we have listed intermediate resonances that can contribute to the reaction (2) and to other processes with two vector mesons in the final state. It must be noted that the scalar state , and three pseudoscalar states, , and , which were observed in the process Ablikim:2016hlu (), are only listed in PDG Tanabashi:2018oca () and are not included in the summary tables. Clearly these states need confirmation.

To calculate the total cross section for the reactions one has to calculate the 8-dimensional phase-space integral 222In the integration over four-body phase space the transverse momenta of the produced particles (, , , ), the azimuthal angles of the outgoing protons (, ) and the rapidities of the produced mesons (, ) were chosen as integration variables over the phase space. numerically Lebiedowicz:2009pj (). Some modifications of the reaction are needed to simulate the reaction with in the final state. For example, since the is an unstable particle one has to include a smearing of the masses due to their resonance distribution. Then, the general cross-section formula can be written approximately as

 σ2→6 = [B(ϕ(1020)→K+K−)]2 (3) ×∫max{mX3}2mK∫max{mX4}2mKσ2→4(...,mX3,mX4)fϕ(mX3)fϕ(mX4)dmX3dmX4

with the branching fraction Tanabashi:2018oca (). We use for the calculation of the decay process the spectral function

 fϕ(mXi)=Cϕ(1−4m2Km2Xi)3/22πm2ϕΓϕ(m2Xi−m2ϕ)2+m2ϕΓ2ϕ, (4)

where , is the total width of the resonance, its mass, and is found from the condition

 ∫∞2mKfϕ(mXi)dmXi=1. (5)

The quantity smoothly decreases the spectral function when approaching the threshold, , and takes into account the angular momentum of the state.

To include experimental cuts on charged kaons we perform the decays of mesons isotropically 333This is true for unpolarised ’s. In principle our model also makes predictions for the polarisation of the ’s and the anisotropies of the resulting decay distributions. Once a good event generator for our reaction is available all these effects should be included. in the rest frames and then use relativistic transformations to the overall center-of-mass frame.

In principle, there are other processes contributing to the final state, for example, direct continuum production and processes with resonances:

 pp→ppK+K−K+K−, (6) pp→ppf0,2K+K−→ppK+K−K+K−, (7) pp→ppf0,2f0,2→ppK+K−K+K−, (8) pp→pp(f2→f0f0)→ppK+K−K+K−. (9)

Here stands for one of the scalar or tensor mesons decaying to . It should be noted that a complete theoretical model of the process should include interference effects of the processes (1), (2), (6) to (9). However, such a detailed study of the reaction will only be necessary once high-energy experimental data for the purely exclusive measurements will be available. We leave this interesting problem for future studies. The GenEx Monte Carlo generator Kycia:2014hea (); Kycia:2017ota () could be used in this context. We refer the reader to Ref. Kycia:2017iij () where a first calculation of four-pion continuum production in the reaction with the help of the GenEx code was performed.

## Iii pp→ppϕϕ

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

 p(pa,λa)+p(pb,λb)→p(p1,λ1)+ϕ(p3,λ3)+ϕ(p4,λ4)+p(p2,λ2), (10)

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

The amplitude for the reaction (10) can be written as

 Mλaλb→λ1λ2ϕϕ=(ϵ(ϕ)ρ3(λ3))∗(ϵ(ϕ)ρ4(λ4))∗Mρ3ρ4λaλb→λ1λ2ϕϕ, (11)

where are the polarisation vectors of the meson.

We consider here unpolarised protons in the initial state and no observation of polarisations in the final state. Therefore, we have to insert in (3) the cross section summed over the meson polarisations. The spin sum for a meson of momentum and squared mass is

 ∑λ=0,±1ϵ(ϕ)μ(λ)(ϵ(ϕ)ν(λ))∗=−gμν+kμkνm2X. (12)

But in our model the terms do not contribute to the cross section since we have the relations

 p3ρ3Mρ3ρ4λaλb→λ1λ2ϕϕ=0,p4ρ4Mρ3ρ4λaλb→λ1λ2ϕϕ=0, (13)

which will be shown below in Secs. III.1 and III.2.

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 (3))

 (14)

To give the full physical amplitude for the reaction we should include absorptive corrections to the Born amplitudes discussed below. 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.1 ϕ-meson exchange mechanism

The diagram for the production with an intermediate -meson exchange is shown in Fig. 1 (a). The Born-level amplitude can be written as the sum

 M(ϕ−exchange)ρ3ρ4λaλb→λ1λ2ϕϕ=M(^t)ρ3ρ4λaλb→λ1λ2ϕϕ+M(^u)ρ3ρ4λaλb→λ1λ2ϕϕ (15)

with the - and -channel amplitudes:

 M(^t)ρ3ρ4=(−i)¯u(p1,λ1)iΓ(Ppp)μ1ν1(p1,pa)u(pa,λa)iΔ(P)μ1ν1,α1β1(s13,t1)×iΓ(Pϕϕ)ρ1ρ3α1β1(^pt,−p3)iΔ(ϕ)ρ1ρ2(^pt)iΓ(Pϕϕ)ρ4ρ2α2β2(p4,^pt)×iΔ(P)α2β2,μ2ν2(s24,t2)¯u(p2,λ2)iΓ(Ppp)μ2ν2(p2,pb)u(pb,λb), (16)
 M(^u)ρ3ρ4=(−i)¯u(p1,λ1)iΓ(Ppp)μ1ν1(p1,pa)u(pa,λa)iΔ(P)μ1ν1,α1β1(s14,t1)×iΓ(Pϕϕ)ρ4ρ1α1β1(p4,^pu)iΔ(ϕ)ρ1ρ2(^pu)iΓ(Pϕϕ)ρ2ρ3α2β2(^pu,−p3)×iΔ(P)α2β2,μ2ν2(s23,t2)¯u(p2,λ2)iΓ(Ppp)μ2ν2(p2,pb)u(pb,λb), (17)

where , , , , . Here and denote the effective propagator and proton vertex function, respectively, for the tensorial pomeron. The corresponding expressions, as given in Sect. 3 of Ewerz:2013kda (), are as follows

 iΔ(P)μν,κλ(s,t) = 14s(gμκgνλ+gμλgνκ−12gμνgκλ)(−isα′P)αP(t)−1, (18) iΓ(Ppp)μν(p′,p) = −i3βPNNF1(t){12[γμ(p′+p)ν+γν(p′+p)μ]−14gμν(p/′+p/)}, (19)

where  GeV. For extensive discussions of the properties of these terms we refer to Ewerz:2013kda (). Here the pomeron trajectory is assumed to be of standard linear form, see e.g. Donnachie:1992ny (); Donnachie:2002en (),

 αP(t)=αP(0)+α′Pt,αP(0)=1.0808,α′P=0.25GeV−2. (20)

Our ansatz for the vertex follows the one for the in (3.47) of Ewerz:2013kda () with the replacements and . This was already used in Sect. IV B of Lebiedowicz:2018eui ().

In the hadronic vertices we should take into account form factors since the hadrons are extended objects. The form factors in (19) and in the vertex are chosen here as the electromagnetic form factors only for simplicity,

 F1(t)=4m2p−2.79t(4m2p−t)(1−t/m2D)2, (21) FM(t)=11−t/Λ20; (22)

see Eqs. (3.29) and (3.34) of Ewerz:2013kda (), respectively. In (21) is the proton mass and  GeV is the dipole mass squared. As we discussed in Fig. 6 of Lebiedowicz:2018eui () we should take in (22)  GeV instead of  GeV used for the vertex in Ewerz:2013kda ().

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

 M(ϕ−exchange)ρ3ρ4λaλb→λ1λ2ϕϕ=2(p1+pa)μ1(p1+pa)ν1δλ1λaF1(t1)FM(t1)×{Vρ3ρ1μ1ν1(s13,t1,^pt,p3)Δ(ϕ)ρ1ρ2(^pt)Vρ4ρ2μ2ν2(s24,t2,−^pt,p4)[^Fϕ(^p2t)]2+Vρ4ρ1μ1ν1(s14,t1,−^pu,p4)Δ(ϕ)ρ1ρ2(^pu)Vρ3ρ2μ2ν2(s23,t2,^pu,p3)[^Fϕ(^p2u)]2}×2(p2+pb)μ2(p2+pb)ν2δλ2λbF1(t2)FM(t2), (23)

 (24)

with two rank-four tensor functions,

 Γ(0)μνκλ(k1,k2)=[(k1⋅k2)gμν−k2μk1ν][k1κk2λ+k2κk1λ−12(k1⋅k2)gκλ], (25) Γ(2)μνκλ(k1,k2)=(k1⋅k2)(gμκgνλ+gμλgνκ)+gμν(k1κk2λ+k2κk1λ) −k1νk2λgμκ−k1νk2κgμλ−k2μk1λgνκ−k2μk1κgνλ −[(k1⋅k2)gμν−k2μk1ν]gκλ; (26)

see Eqs. (3.18) and (3.19) of Ewerz:2013kda (). In Lebiedowicz:2018eui () we have fixed the coupling parameters of the tensor pomeron to the meson based on the HERA experimental data for the reaction Derrick:1996af (); Breitweg:1999jy (). We take the coupling constants  GeV and  GeV from Table II of Lebiedowicz:2018eui () (see also Sect. IV B there). The relations (13) are now easily checked from (23) - (III.1) using the properties of the above tensorial functions; see (3.21) of Ewerz:2013kda (). We can then make in (23) the following replacement for the -meson propagator

 Δ(ϕ)ρ1ρ2(^p)→−gρ1ρ2Δ(ϕ)T(^p2), (27)

where we take the simple Breit-Wigner expression for .

The amplitude (23) contains a form factor taking into account the off-shell dependences of the intermediate -mesons. The form factor is normalized to unity at the on-shell point and parametrised here in the exponential form

 ^Fϕ(^p2)=exp⎛⎝^p2−m2ϕΛ2off,E⎞⎠, (28)

where the cut-off parameter could be adjusted to experimental data.

We should take into account the fact that the exchanged intermediate object is not a simple spin-1 particle ( meson) but may correspond to a Regge exchange, that is, the reggeization of the intermediate meson is necessary, see e.g. Lebiedowicz:2016zka (). The “reggeization” of the amplitude given in Eq. (23) is included here approximately, by replacing the -meson propagator both in the - and -channel amplitudes by Lebiedowicz:2016zka ()

 Δ(ϕ)ρ1ρ2(^p)→Δ(ϕ)ρ1ρ2(^p)(s34s0)αϕ(^p2)−1, (29)

where we take and with from Collins:1977 () and  GeV.

### iii.2 f2 resonance production

Now we consider the amplitude for the reaction (10) through the -channel -meson exchange as shown in Fig. 1 (b). The , and mesons could be considered as potential candidates; see Table 1.

The Born amplitude for the fusion is given by

 M(PP→f2→ϕϕ)ρ3ρ4λaλb→λ1λ2ϕϕ=(−i)¯u(p1,λ1)iΓ(Ppp)μ1ν1(p1,pa)u(pa,λa)iΔ(P)μ1ν1,α1β1(s1,t1)×iΓ(PPf2)α1β1,α2β2,ρσ(q1,q2)iΔ(f2)ρσ,αβ(p34)iΓ(f2ϕϕ)αβρ3ρ4(p3,p4)×iΔ(P)α2β2,μ2ν2(s2,t2)¯u(p2,λ2)iΓ(Ppp)μ2ν2(p2,pb)u(pb,λb), (30)

where , , , , , , and .

The vertex can be written as

A possible choice for the coupling terms is given in Appendix A of Lebiedowicz:2016ioh (). The corresponding coupling constants are not known and should be fitted to existing and future experimental data. In the following we shall, for the purpose of orientation, assume that only is unequal to zero.

In practical calculations we take the factorized form for the form factor

 ~F(PPf2)(q21,q22,p234)=~FM(q21)~FM(q22)F(PPf2)(p234) (32)

normalised to . We will further set

 ~FM(t)=11−t/~Λ20,~Λ20=1GeV2; (33) F(PPf2)(p234)=exp⎛⎝−(p234−m2f2)2Λ4f2⎞⎠,Λf2=1GeV. (34)

For the vertex we take the following ansatz:

 iΓ(f2ϕϕ)μνκλ(p3,p4) = i2M30g′f2ϕϕΓ(0)μνκλ(p3,p4)F′(f2ϕϕ)(p234) (35) −i1M