Exclusive central diffractive production of scalar, pseudoscalar and vector mesons

We discuss exclusive central diffractive production of scalar ($f_{0}(980)$, $f_{0}(1370)$, $f_{0}(1500)$), pseudoscalar ($\eta$, $\eta'(958)$), and vector ($\rho^{0}$) mesons in proton-proton collisions. The amplitudes are formulated in terms of effective vertices required to respect standard rules of Quantum Field Theory and propagators for the exchanged pomeron and reggeons. Different pomeron-pomeron-meson tensorial (vectorial) coupling structures are possible in general. In most cases two lowest orbital angular momentum - spin couplings are necessary to describe experimental differential distributions. For the $f_{0}(980)$ and $\eta$ production the reggeon-pomeron, pomeron-reggeon, and reggeon-reggeon exchanges are included in addition, which seems to be necessary at relatively low energies. The theoretical results are compared with the WA102 experimental data, in order to determine the model parameters. For the $\rho^{0}$ production the photon-pomeron and pomeron-photon exchanges are considered. The coupling parameters of tensor pomeron and/or reggeon are fixed from the H1 and ZEUS experimental data of the $\gamma p \to \rho^{0} p$ reaction. We present first predictions of this mechanism for $pp \to pp \pi^{+} \pi^{-}$ reaction being studied at COMPASS, RHIC, Tevatron, and LHC. Correlation in azimuthal angle between outgoing protons and distribution in pion rapidities at $\sqrt{s} = 7$TeV are presented. We show that high-energy central production of mesons could provide crucial information on the spin structure of the soft pomeron.


Introduction
There is a growing interest in understanding the mechanism of exclusive meson production both at low and high energies, for a review see [1]. In Ref. [2] we performed application of the "tensorial" pomeron model [3] for the exclusive production of scalar (J PC = 0 ++ ) and pseudoscalar (J PC = 0 −+ ) mesons in the pp → ppM reaction and we compared results of our calculations with WA102 experimental data. At high energies the dominant contribution to this reaction comes from pomeronpomeron (or pomeron-reggeon) fusion. While it is clear that the effective Pomeron must be a colour singlet, its spin structure and couplings to hadrons are not finally established. Although, it is also commonly assumed in the literature that the pomeron has a "vectorial" nature [4] there are strong hints from the Quantum Field Theory that it should have rather "tensorial" properties [3]. a e-mail: Piotr. Lebiedowicz@ifj Left: Diagram of the central exclusive I G = 0 + meson production by a fusion of pomeron/reggeonpomeron/reggeon. Right: Diagram of the central exclusive ρ 0 meson production via pomeron/reggeon-photon (or photon-pomeron/reggeon) exchanges and its subsequent decay into π + π − pair.
Indeed, tests for the helicity structure of the pomeron have been devised in [5] for diffractive contributions to electron-proton scattering, that is, for virtual-photon-proton reactions. For central meson production in proton-proton collisions such tests were discussed in [6][7][8]. The general structure of helicity amplitudes of the simple Regge behaviour was also considered in Ref. [9,10].
In Ref. [2] we discussed differences between results of the "tensorial pomeron" and "vectorial pomeron" models of central exclusive production for the scalar and pseudoscalar mesons. In order to calculate these contributions we must know the effective I P propagator, the I Ppp and I PI PM vertices, see Fig. 1 (left). The formulae for corresponding propagators and vertices are presented and discussed in detail in Refs. [2,3]. In [2] we gave the values of the lowest orbital angular momentum l and of the corresponding total spin S , which can lead to the production of M in the fictitious fusion of two tensorial or vectorial "pomeron particles". In most cases one has to add coherently amplitudes for two couplings with different orbital angular momentum and spin of two "pomeron particles". The corresponding coupling constants are not known and have been fitted to existing experimental data.
We focus on exclusive central production of ρ 0 resonance in the context of theoretical concept of tensor pomeron proposed in Ref. [3]. Due to its quantum numbers this resonance state can be produced only by photon-pomeron or photon-reggeon mechanisms. The diagrams to be considered are shown in Fig. 1 (right). In the amplitude for the γp → ρ 0 p subprocess we included both pomeron and f 2I R exchanges. All effective vertices and propagators have been taken here from Ref. [3]. All the details will be presented elsewhere [11]. Needless to say that at lower energies there is also an important ρ 0 → π + π − background from the non-central diffractive processes. 1 Two of us proposed some time ago a simple Regge-like model for the π + π − and K + K − continuum based on the exchange of two pomerons/reggeons [1,[16][17][18].

Results
The theoretical results are compared with the WA102 experimental data in order to determine the model parameters. At low energy the ππ-fusion contribution [25] dominates while at high energies the dominant contribution comes from pomeron-pomeron (I PI P) fusion. For pseudoscalar meson production we can expect large contributions from ωω exchanges due to the large coupling of the ω meson to the nucleon. At higher subsystem energies squared s 13 and s 23 the meson exchanges are corrected to obtain the high energy behaviour appropriate for ω-reggeon exchanges. This seems to be the case for the η meson production where we have included also exchanges of subleading trajectories [2] which improves the agreement with experimental data.
In Fig.2 we show the distribution in azimuthal angle φ pp between outgoing protons for central exclusive meson production by the fusion of two tensor (solid line) or two vector (long-dashed line) pomerons at √ s = 29.1 GeV. The strengths of the (l, S ) couplings were adjusted to roughly reproduce the WA102 data from [27]. The tensorial pomeron with the (l, S ) = (0, 0) coupling alone already describes the azimuthal angular correlation for f 0 (1370) meson reasonable well. The vectorial pomeron with the (l, S ) = (0, 0) term alone is disfavoured here. The preference of the f 0 (1370) for the φ pp ≈ π domain in contrast to the enigmatic f 0 (980) and f 0 (1500) scalars has been observed by the WA102 Collaboration [28]. The contribution of the (0, 0) component alone is not able to describe the azimuthal angular dependence for these states. We observe a large interference of two (l, S ) components in the amplitude. For production of pseudoscalar mesons in both models of pomeron the theoretical distributions are somewhat skewed due to phase space angular dependence. The contribution of the (1, 1) component alone in the tensorial pomeron model is not able to describe the WA102 data [27]. Fig. 3 (left panel) shows the integrated cross section for the γp → ρ 0 p reaction as a function of center-of-mass energy together with the experimental data. In our calculations we taken default values for the parameters of the propagators and vertices [3,11]. We show distributions in the azimuthal angle between outgoing protons (center panel) and in the pion rapidity (right panel) at √ s = 7 TeV. For the φ pp distribution, the effect of deviation from a constant is due to interference of pomeron-photon and photon-pomeron amplitudes. Similar effect was discussed first in Ref. [29] for the exclusive production of J/ψ meson. These correlations are quite different than those for double-pomeron mechanism [1,17] and could be therefore used at least to partial separation of these two mechanisms. 2 The rapidities of the two pions are strongly correlated. The f 2I R exchange included in the amplitude contributes at backward/forward pion rapidities and its contribution is non-negligible even at the LHC energies.

Conclusions
We have analysed proton-proton collisions with the exclusive central production of scalar and pseudoscalar mesons. We have presented the predictions of two different models of the soft pomeron. The first one is the commonly used model with vectorial pomeron which is, however, difficult to be supported from a theoretical point of view. The second one is a recently proposed model of tensorial pomeron [3], which, in our opinion, has better theoretical foundations. We have performed calculations of several differential distributions. We wish to emphasize that the tensorial pomeron can, at least, equally well describe experimental data on the exclusive meson production discussed here as the less theoretically justified vectorial pomeron frequently used in the literature. The models contain only a few free coupling parameters to be determined by experiment. The existing low-energy experimental data do not allow to clearly distinguish between the two models as the presence of subleading reggeon exchanges is at low energies very probable for many reactions. Production of η ′ meson seems to be less affected by contributions from subleading exchanges. Pseudoscalar meson production could be of particular interest for testing the nature of the soft pomeron since there the distribution in the azimuthal angle φ pp between the two outgoing protons may contain, for the tensorial pomeron, a term which is not possible for the vectorial pomeron, see [2]. It would clearly be interesting to extend the studies of central meson production in diffractive processes to higher energies. For the resonances decaying e.g. into the ππ channel an interference of the resonance signals with the two-pion continuum has to be included in addition. This requires a consistent model of the resonances and the non-resonant background.
We have made first estimates of the contribution of exclusive ρ 0 production to the pp → ppπ + π − reaction. We have shown some differential distributions in pion rapidities as well as some observables related to final state protons. We have obtained that the ρ 0 contribution constitutes about 10% of the double-pomeron/reggeon contribution calculated in a simple Regge-like model [1,16,17]. We expect that the exclusive ρ 0 photoproduction and its subsequent decay are the main source of P-wave in the π + π − channel in contrast to even waves populated by the double-pomeron processes. Different