The \rho^{0} and Drell-Söding contributions to central exclusive production of \pi^{+}\pi^{-} pairsin proton-proton collisions at high energies

# The ρ0 and Drell-Söding contributions to central exclusive production of π+π− pairs in proton-proton collisions at high energies

Piotr Lebiedowicz,    Otto Nachtmann    and Antoni Szczurek111Also at Rzeszów University, PL-35-959 Rzeszów, Poland.
###### Abstract

We present a study of the central exclusive production via the photoproduction mechanism in nucleon-nucleon collisions. The photon-pomeron/reggeon and pomeron/reggeon-photon exchanges both for the resonance contribution and the Drell-Söding contribution are considered. The amplitudes for the processes are formulated in terms of vertices respecting the standard crossing and charge-conjugation relations of Quantum Field Theory. The coupling parameters of tensor pomeron and reggeon exchanges are fixed based on the H1 and ZEUS experimental data for the reaction. We present first predictions of this mechanism for the reaction being studied at COMPASS, RHIC, Tevatron, and LHC. We show the influence of the experimental cuts on the integrated cross section and on various differential distributions for outgoing particles. Distributions in rapidities and transverse momenta of outgoing protons and pions as well as correlations in azimuthal angle between them are presented. We compare the photoproduction contribution to distributions with double pomeron/reggeon two-pion continuum. We discuss whether the high-energy central production of mesons could be selected experimentally.

###### Keywords:
Phenomenological Models
Institute of Nuclear Physics PAN, PL-31-342 Kraków, PolandInstitut für Theoretische Physik, Universität Heidelberg,
Philosophenweg 16, D-69120 Heidelberg, Germany

## 1 Introduction

There is a growing interest in understanding the mechanism of exclusive resonance and continuum production in the nucleon-nucleon collisions. This is closely related to ongoing experimental studies of the COMPASS Austregesilo:2013yxa (), STAR Turnau_DIS2014 (); Adamczyk:2014ofa (), CDF Albrow:2013mva (); Albrow_Project_new (), ALICE Schicker:2012nn (); Schicker:2014aoa (), ATLAS Staszewski:2011bg () and CMS CMS_private_com (); Osterberg:2014mta () collaborations.

Some time ago two of us proposed a simple Regge-like model for the continuum based on the exchange of two pomerons/reggeons Lebiedowicz:2009pj (). For further work see Lebiedowicz:2011nb (). These model studies were extended also to production Lebiedowicz:2011tp (). Predictions for experiments at different energies have been presented also in Chapter 2 of Lebiedowicz:thesis () in order to make precise comparison between calculations and experimental data. In addition to the continuum one has to include also two-pion resonances. Production of scalar and pseudoscalar resonances was studied by us very recently Lebiedowicz:2013ika () in the context of the theoretical concept of tensor pomeron proposed in Ref. Ewerz:2013kda (). In the present paper we focus on exclusive production of the resonance followed by the decay . Due to its quantum numbers this resonance state cannot be produced by pomeron-pomeron fusion. The exchanges contributing are photon-pomeron/reggeon and reggeon-pomeron/reggeon.

The process has been discussed recently Bolz:2014mya (); Sauter_LowX () within the model for tensor-pomeron and vector-odderon Ewerz:2013kda (). It was known for a long time that the shape of the in photoproduction is skewed. An explanation was given by Söding following a suggestion by Drell Drell:1960zz (); Drell:1961zz (); Soding:1965nh (); see also Szczurek:2004xe (). The skewing is due to the interference of continuum production with the production through the meson. Usually there are problems of gauge invariance when adding these two contributions to the production and restoration of the gauge invariance is to some extent arbitrary. The authors of Ref. Bolz:2014mya () obtained a gauge-invariant version of the Drell-Söding mechanism which produces the skewing of the -meson shape. The reaction would be helpful to test the model and its parameters against available data.

In the literature exist some phenomenological models of light vector meson photoproduction in the reaction, e.g. the color dipole approach Armesto:2014sma (); Santos:2014vwa (), and the pQCD -factorization approach Schafer:2007mm (); Cisek:2010jk (); Cisek:2011vt (); Cisek:2014ala (). In the latter case the authors also consider absorption effects due to strong proton-proton interactions.

In this paper we focus on the four-body reaction with the pion pair produced by photon-pomeron/reggeon fusion. For the resonance production we consider the diagrams shown in Fig. 1. In these diagrams all vertices and propagators will be taken here according to Ref. Ewerz:2013kda (). The diagrams to be considered for the dominant non-resonant (Drell-Söding) contribution are shown in Fig. 2. In the following we collect formulae for the amplitudes for the (or ) reaction within the tensor pomeron model Ewerz:2013kda (). We expect that the central exclusive photoproduction and its subsequent decay are the main source of -wave in the channel in contrast to even waves populated in double-pomeron/reggeon processes.

A complete calculation for central exclusive production in collisions at high energies clearly must take into account more diagrams than those of Figs. 1 and 2. For instance, in these figures we could replace the virtual photon by the reggeon. As already mentioned, we can have double pomeron/reggeon exchange leading to in the continuum or to the resonance decaying to . We shall come back to all these processes in a further publication. Here we concentrate on our first “building block” for this program, the processes of Figs. 1 and 2. Due to the photon propagators occurring in these diagrams we expect these processes to be most important when at least one of the protons is undergoing only a very small momentum transfer.

There is also a non-central diffractive production via the bremsstrahlung-type mechanism. But the mesons originating from such a mechanism are expected to be produced very forward or very backward in analogy to the -bremsstrahlung considered in Ref. Cisek:2011vt (). Similar processes were discussed at high energies also for the exclusive meson Lebiedowicz:2013vya (), and Lebiedowicz:2013xlb () production.

Our paper is organised as follows. In section 2 we discuss the reaction. Turning to the reaction we give in section 3 analytic expressions for the non-resonant (Drell-Söding) and the resonant (through the meson) amplitudes. In section 4 we present numerical results for total and differential cross sections and discuss interference effects between the two contributions. Moreover, we will present our prediction for the two-pion invariant mass distribution at LHC energy of 7 TeV in proton-proton collisions, which is currently under analysis by the ALICE and CMS collaborations.

Closely related to the reaction studied by us here are the reactions of central production in ultra-peripheral nucleon-nucleus and nucleus-nucleus collisions, and . For the latter process high-energy data exist from the STAR Adler:2002sc (); Abelev:2007nb (); Abelev:2008ew (); Agakishiev:2011me () and the ALICE Nystrand:2014vra () collaborations. For theoretical reviews treating such collisions see for instance Baur:2001jj (); Bertulani:2005ru (); Baltz:2007kq (). The application of our methods, based on the tensor-pomeron concept, to collisions involving nuclei is an interesting problem which, however, goes beyond the scope of the present work.

## 2 Photoproduction of ρ0 meson

The amplitude for the reaction shown in Fig. 3 includes not only pomeron (), but also reggeon (, ) exchanges.

First, we write down the amplitude for the reaction via the tensor-pomeron exchange as follows

 \Braketρ0(pρ,λρ),p(p2,λ2)Tγ(q,λγ),p(pb,λb)≡ Mλγλb→λρλ2=(−i)(ϵ(ρ)μ)∗iΓ(IPρρ)μναβ(pρ,q)iΔ(ρ)νκ(q)iΓ(γ→ρ)κσ(q)ϵ(γ)σ ×iΔ(IP)αβ,δη(s,t)¯u(p2,λ2)iΓ(IPpp)δη(p2,pb)u(pb,λb), (1)

where , and , denote the four-momenta and helicities of the ingoing and outgoing protons, and are the polarisation vectors for photon and meson with the four-momenta , and helicities , , respectively. We use standard kinematic variables

 s=W2γp=(pb+q)2=(p2+pρ)2, t=(p2−pb)2=(pρ−q)2. (2)

The vertex is given in Ewerz:2013kda () by formula (3.47). The propagator of the tensor-pomeron exchange is written as (see (3.10) of Ewerz:2013kda ()):

 (3)

and fulfils the following relations

 Δ(IP)μν,κλ(s,t)=Δ(IP)νμ,κλ(s,t)=Δ(IP)μν,λκ(s,t)=Δ(IP)κλ,μν(s,t), gμνΔ(IP)μν,κλ(s,t)=0,gκλΔ(IP)μν,κλ(s,t)=0. (4)

Here the pomeron trajectory is assumed to be of standard linear form with intercept slightly above 1:

 αIP(t)=αIP(0)+α′IPt,αIP(0)=1.0808,α′IP=0.25GeV−2. (5)

The corresponding coupling of tensor pomeron to protons (antiprotons) including a vertex form-factor, taken here to be the Dirac electromagnetic form factor of the proton for simplicity (see Section 3.2 of Donnachie:2002en ()), is written as (see (3.43) of Ewerz:2013kda ()):

 iΓ(IPpp)μν(p′,p)=iΓ(IP¯p¯p)μν(p′,p) =−i3βIPNNF1((p′−p)2){12[γμ(p′+p)ν+γν(p′+p)μ]−14gμν(p/′+p/)}, (6)

where and  GeV. A sufficiently good representation of the Dirac form factor is given by the dipole formula

 F1(t)=4m2p−2.79t(4m2p−t)(1−t/m2D)2, (7)

where is the proton mass and  GeV is the dipole mass squared.

For the reggeon exchange a similar form of the propagator and the effective vertex is assumed, see (3.12) and (3.49) of Ewerz:2013kda (), with the Regge parameters:

 αIR+(t)=αIR+(0)+α′IR+t,αIR+(0)=0.5475,α′IR+=0.9GeV−2. (8)

The vertex is obtained from (6) replacing by with  GeV and .

In the high-energy small-angle approximation we get, using (D.19) in Appendix D of Lebiedowicz:2013ika (),

 Mλγλb→λρλ2(s,t)≅ iem2ργρΔ(ρ)T(0)(ϵ(ρ)μ)∗ϵ(γ)νVμνκλ(s,t,q,pρ) (9) ×2(p2+pb)κ(p2+pb)λδλ2λbF1(t)FM(t).

Here , with the meson mass taken from PDG Agashe:2014kda (). The function has the form

 Vμνκλ(s,t,q,pρ)=14s ×{2Γ(0)μνκλ(pρ,−q)[3βIPNNaIPρρ(−isα′IP)αIP(t)−1+M−10gf2IRppaf2IRρρ(−isα′IR+)αIR+(t)−1] −Γ(2)μνκλ(pρ,−q)[3βIPNNbIPρρ(−isα′IP)αIP(t)−1+M−10gf2IRppbf2IRρρ(−isα′IR+)αIR+(t)−1]},

where the explicit tensorial functions , = 0, 2, are given in Ref. Ewerz:2013kda (), formulae (3.18) and (3.19), respectively. In (9) is the pion electromagnetic form factor in a parametrization valid for ,

 FM(t)=11−t/Λ20, (11)

where  GeV; see e.g. (3.22) of Donnachie:2002en () and (3.34) of Ewerz:2013kda ().

Fig. 4 (left panel) shows the integrated cross section for the reaction, calculated from (9), as function of the center-of-mass energy together with the experimental data. We have checked that the cross section from the -exchange is about three orders of magnitude smaller than from the -exchange. This is due to the fact, that the - coupling is much smaller than the - coupling and ; see (3.50) and (3.52) of Ewerz:2013kda () . Thus, in the present calculations we have neglected the -exchange contribution. As shown in Ewerz:2013kda () the and coupling constants and are expected to approximately fulfil the relations: 222The coupling constants of the leading trajectories in Eq. (LABEL:rhop_tot_opt_aux2) have been estimated from the parametrization of total cross sections for and scattering assuming for . The theoretical formulae of total cross sections are discussed in Section 7.1 of Ewerz:2013kda ().

 2m2ρaIPρρ+bIPρρ=4βIPππ=7.04GeV−1, (12) 2m2ρaf2IRρρ+bf2IRρρ=M−10gf2IRππ=9.30GeV−1; (13)

see (7.27) and (7.28) of Ewerz:2013kda (). In our calculations two parameter sets of coupling constants are used:

 setA: aIPρρ=0.7GeV−3,af2IRρρ=0GeV−3, (14) bIPρρ=6.2GeV−1,bf2IRρρ=9.3GeV−1, setB: aIPρρ=af2IRρρ=0GeV−3,bIPρρ=7.04GeV−1,bf2IRρρ=9.3GeV−1. (15)

At low energies there are other processes contributing, such as meson exchanges (e.g. , , ), the  bremsstrahlung, baryonic resonances decaying into the channel etc. Thus, the Regge terms should not be expected to fit the low-energy data precisely. We refer the reader to Friman:1995qm (); Laget:2000gj (); Oh:2003gm (); Oh:2003aw (); Riek:2008ct (); Obukhovsky:2009th () for reviews and details concerning the photoproduction mechanism at low energies. We see from Fig. 4, left panel, that our model calculation describes the total cross section for fairly well for energies  GeV.

In the right panel of Fig. 4 we show the differential cross section for elastic photoproduction. The calculations, performed for  GeV, are compared with ZEUS data Breitweg:1997ed (); Breitweg:1999jy (). We can see that, according to our calculation, the amplitude for longitudinal meson polarisation is negligible and vanishes at . The sum of the contributions describes the data well up to  GeV.

In Ref. Bolz:2014mya () detailed model calculations for the reaction have been presented using the approach to soft scattering from Ewerz:2013kda (). The diagrams considered in Bolz:2014mya () include production, as in our Fig. 3, but also a number of other processes; see Fig. 1 of Bolz:2014mya (). Our results here for production are in agreement with those from Bolz:2014mya (); see Figs. 3 and 4 there. This gives a very valuable cross check of the programs used for the calculations in Bolz:2014mya () and in our present paper. We shall now go on and calculate with the same methods amplitudes and cross sections corresponding to the diagrams of Figs. 1 and 2.

## 3 The central exclusive two-pion production

We shall study exclusive production of in proton-proton collisions at high energies

 p(pa,λa)+p(pb,λb)→p(p1,λ1)+π+(p3)+π−(p4)+p(p2,λ2), (16)

where , and , denote the four-momenta and helicities of the protons, and denote the four-momenta of the charged pions, respectively. In the following we will calculate the contributions from the diagrams of Figs. 1 and 2 to the process (16), that is, the photon-pomeron and photon- reggeon exchange contributions. These processes are expected to be the dominant ones for highly peripheral -collisions. Experimentally such collision could be selected by requiring only a very small deflection angle for at least one of the outgoing protons.

The kinematic variables for reaction (16) are

 p34=p3+p4,q1=pa−p1,q2=pb−p2, s=(pa+pb)2=(p1+p2+p34)2,M2ππ=p234, s1=(pa+q2)2=(p1+p34)2,s2=(pb+q1)2=(p2+p34)2, t1=q21,t2=q22; (17)

see also Appendix D of Lebiedowicz:2013ika ().

The Born amplitude for exclusive photoproduction of can be written as the following sum:

 MBornpp→ppπ+π−=M(γIP)+M(IPγ)+M(γf2IR)+M(f2IRγ). (18)

If we want to treat -collisions (Tevatron) we must be careful since then there is no symmetry any more for the amplitude under , , etc. Using the charge conjugation () properties of the , and exchanges we get for scattering from (18)

 MBornp¯p→p¯pπ+π−=M(γIP)−M(IPγ)+M(γf2IR)−M(f2IRγ). (19)

Note that these sign changes are automatically obtained using the Feynman rules for the tensor pomeron and exchanges but have to be implemented by hand for the vectorial pomeron and exchanges.

### 3.1 ρ0-resonance contribution

The “bare” amplitude (excluding rescattering effects) for the -exchange, see diagram (a) in Fig. 1, can be written in terms of our building blocks as follows:

 M(γIP)λaλb→λ1λ2π+π−=(−i)¯u(p1,λ1)iΓ(γpp)μ(p1,pa)u(pa,λa) ×iΔ(γ)μσ(q1)iΓ(γ→ρ)σν(q1)iΔ(ρ)νρ1(q1)iΔ(ρ)ρ2κ(p34)iΓ(ρππ)κ(p3,p4) ×iΓ(IPρρ)ρ2ρ1αβ(p34,q1)iΔ(IP)αβ,δη(s2,t2)¯u(p2,λ2)iΓ(IPpp)δη(p2,pb)u(pb,λb). (20)

All propagators and vertices used in (20) are defined in section 3 of Ewerz:2013kda (); see also Appendix B of Bolz:2014mya (). For the -exchange the amplitude has the same structure with , and . In a similar way we can write down the and amplitudes.

For simplicity, in the following we shall consider the amplitude (20) in the high-energy small-angle limit; see Appendix D of Lebiedowicz:2013ika (). Including both, pomeron and exchanges, we obtain in this way

 M(γIP+γf2IR)λaλb→λ1λ2π+π−≃ie(p1+pa)μF1(t1)δλ1λa ×em2ργρ1t1Δ(ρ)μρ1(q1)Δ(ρ)ρ2κ(p34)gρππ2(p3−p4)κ~F(ρ)(q21)~F(ρ)(p234) ×Vρ2ρ1αβ(s2,t2,q1,p34)FM(t2)2(p2+pb)α(p2+pb)βF1(t2)δλ2λb. (21)

The function is as defined in (LABEL:rhop_tot_opt_aux2) and includes two tensorial functions for . From (3.21) and (3.22) of Ewerz:2013kda () we have

 qρ11Γ(i)ρ2ρ1αβ(p34,−q1)=0,pρ234Γ(i)ρ2ρ1αβ(p34,−q1)=0, gαβΓ(i)ρ2ρ1αβ(p34,−q1)=0,i=0,2. (22)

Thus, the terms proportional to in cannot contribute. The same is true for the terms proportional to in . Thus, only the transverse part of the propagator , as defined in (4.1)-(4.4) of Ewerz:2013kda (), contributes in (21). As is easily seen, the same holds for (20). The decay vertex for is well known (e.g. see (3.35) of Ewerz:2013kda ()) and the relevant coupling constant is . We emphasize that in the work Melikhov:2003hs () the authors found a very good description of the line shape from the reaction - up to  GeV - without a form factor in the vertex.

In the diagram of Fig. 1 at the pomeron-- vertex the incoming is always off shell, the outgoing also may be away from the nominal “mass shell” . As suggested in Bolz:2014mya (), see (B.82) there, we insert, therefore, in the vertex extra form factors. A convenient form, given in (B.85) of Bolz:2014mya () is

 ~F(ρ)(k2)=[1+k2(k2−m2ρ)Λ4ρ]−nρ. (23)

The form factor (23) has the property: which is consistent with the traditional, phenomenologically successful, vector-meson-dominance model.

### 3.2 Drell-Söding contribution

The -exchange amplitudes given by the diagrams shown in Fig. 2, can be written on the Born level for the tensor-pomeron exchange as follows

 M(a)λaλb→λ1λ2π+π−=(−i)¯u(p1,λ1)iΓ(γpp)μ(p1,pa)u(pa,λa)iΔ(γ)μν(q1)iΓ(γππ)ν(pt,−p3) ×iΔ(π)(pt)iΓ(IPππ)αβ(p4,pt)iΔ(IP)αβ,δη(s2,t2)¯u(p2,λ2)iΓ(IPpp)δη(p2,pb)u(pb,λb), (24) M(b)λaλb→λ1λ2π+π−=(−i)¯u(p1,λ1)iΓ(γpp)μ(p1,pa)u(pa,λa)iΔ(γ)μν(q1)iΓ(γππ)ν(p4,pu) ×iΔ(π)(pu)iΓ(IPππ)αβ(pu,−p3)iΔ(IP)αβ,δη(s2,t2)¯u(p2,λ2)iΓ(IPpp)δη(p2,pb)u(pb,λb), (25) M(c)λaλb→λ1λ2π+π−=(−i)¯u(p1,λ1)iΓ(γpp)μ(p1,pa)u(pa,λa)iΔ(γ)μν(q1) ×iΓ(IPγππ)ν,αβ(q1,p4,−p3)iΔ(IP)αβ,δη(s2,t2)¯u(p2,λ2)iΓ(IPpp)δη(p2,pb)u(pb,λb). (26)

Above we have introduced and . In order to assure gauge invariance and “proper” cancellations among the three terms (24) to (26) we have introduced, somewhat arbitrarily, one common energy dependence for the pomeron propagator in all three diagrams instead of naively , , and , respectively, where . See also the discussion of this point in section 2.5 of Bolz:2014mya () where a justification for this procedure is given.

For the -exchange we have the same structure as for the above amplitudes with , , . In a similar way we can write the and amplitudes.

Starting with the coupling, see Eq. (7.3) of Ewerz:2013kda (), and making a minimal substitution there gives the couplings involving pions, photons and the pomeron (see (B.66) to (B.71) of Bolz:2014mya ()). We have the following vertex, see Eq.(3.45) of Ewerz:2013kda () and (B.69) of Bolz:2014mya (),

 iΓ(IPππ)αβ(k′,k)=−i2βIPππ[(k′+k)α(k′+k)β−14gαβ(k′+k)2]FM((k′−k)2), (27)

where  GeV. For the standard electromagnetic vertex we have

 iΓ(γππ)ν(k′,k)=ie(k′+k)νFM((k′−k)2). (28)

Finally, there is the contact term, see (B.71) of Bolz:2014mya (),

 iΓ(IPγππ)ν,αβ(q,k′,k)= −ie2βIPππ[2gαν(k′+k)β+2gβν(k′+k)α−gαβ(k′+k)ν] (29) ×FM(q2)FM((k′−q−k)2).

Compared to (B.67) and (B.71) of Bolz:2014mya () we have introduced in both, (28) and (29), an extra form factor (11). For an on shell photon with this form factor is equal to 1 and, thus, would not make any difference for the calculations of photoproduction in Bolz:2014mya (). With the normal pion propagator the above ansatz for the vertices guarantees gauge invariance of the continuum contribution.

In the high-energy approximation we can write for tensor-pomeron exchange

 M(a)λaλb→λ1λ2π+π−≃ie2(p1+pa)μδλ1λaF1(t1)FM(t1)1t1(pt−p3)μ1p2t−m2π ×2βIPππ(p4+pt)α(p4+pt)β14s2(−is2α′IP)αIP(t2)−1 ×3βIPNN2(p2+pb)α(p2+pb)βδλ2λbF1(t2)FM(t2), (30) M(b)λaλb→λ1λ2π+π−≃ie2(p1+pa)μδλ1λaF1(t1)FM(t1)1t1(p4+pu)μ1p2u−m2π ×2βIPππ(pu−p3)α(pu−p3)β14s2(−is2α′IP)αIP(t2)−1 ×3βIPNN2(p2+pb)α(p2+pb)βδλ2λbF1(t2)FM(t2), (31) M(c)λaλb→λ1λ2π+π−≃−ie2(p1+pa)νδλ1λa1t1F1(t1)FM(t1) ×2βIPππ[2gαν(p4−p3)β+2gβν(p4−p3)α]14s2(−is2α′IP)αIP(t2)−1 ×3βIPNN2(p2+pb)α(p2+pb