# Exclusive and hadroproduction as a probe of the QCD Odderon^{1}^{1}1Dedicated to the memory of Leszek Łukaszuk, co-father of the odderon, who recently passed away.

###### Abstract

We study the exclusive production of or in and collisions, where the meson emerges from the pomeron–odderon and the pomeron–photon fusion. We estimate the cross sections for these processes for the kinematical conditions of the Tevatron and of the LHC.

## 1 Introduction

The new analysis of exprimental data on the exclusive hadroproduction processes by the CDF collaboration [1] shows that these types of processes can be objects of detailed study at the Tevatron and in the near future at the LHC. Up to now, the most intensively studied exclusive hadroproduction processes include the dijet or the production in the central rapidity region and the Higgs meson production [2], see Fig. 1.

Here, we discuss the exclusive hadroproduction of and mesons, i.e.

(1) |

The main motivation of our recent study [3] of the process (1) is that the production of a charmonium , with the quantum numbers , occurs as the result of a pomeron-odderon or pomeron-photon fusion. Such studies can thus probe the dynamics of the odderon [4], i.e. the counterpart with negative charge parity of the pomeron. Odderon escapes experimental verification and until now has remained a mystery, although various ways to detect it through its interference with a pomeron mediated amplitude [5] have been recently proposed (for a review see [6]).

## 2 The scattering amplitude

In the lowest order of perturbative QCD, the pomeron and the odderon are described by the exchange of two and three non-interacting gluons, respectively. The lowest order contribution to the hadroproduction

(2) |

is illustrated by diagrams of Fig. 2, from which the diagrams (a,b) describe the pomeron-odderon fusion and (c,d) the photon-pomeron fusion. The momenta of particles are parametrized by the Sudakov decompositions

(3) |

with ,

(4) |

which lead to the mass-shell condition for the vector meson,
,

The scattering amplitude written within the -factorization approach is a convolution in transverse momenta of channel fields. For instance, the contribution of Fig. 2a reads:

(5) | |||

where and are the impact factors describing the coupling of the pomeron and the odderon to scattered hadrons, respectively, whereas is the effective -meson production vertex.

The proton impact factors are non-perturbative objects and we describe them within the Fukugita-Kwieciński eikonal model [7]. For the pomeron exchange the impact factor of the proton is the product

(6) |

of the impact factor of a quark

(7) |

and the phenomenological form-factor describing the proton internal structure

(8) |

which vanishes when any of , as required by colour gauge invariance. The function is chosen in the form

(9) |

with a phenomenological constant chosen to be half of the meson mass, MeV. The analogous impact-factor for the odderon exchange reads

(10) |

where

(11) |

and the form-factor has a form

(12) |

The derivation of the effective production vertex of , , in Eq. (5) is one of the main results of our study. The charmonium is treated in the non-relativistic approximation and it is described by the vertex

(13) |

with the coupling constant related to the electronic width of the decay

(14) |

The effective vertex is described by the sum of the contributions of the diagrams in Fig. 3 which has the form

(15) | |||

In the numerical analysis we set and .

The analogous formula which describes the photon-pomeron fusion in Fig. 2c has the form

(16) | |||

where is the phenomenological form-factor of the photon coupling to the proton chosen as . The pomeron impact factor is given by Eq. (6) and is the corresponding effective vertex expressed through in Eq. (2)

(17) |

The phases of the scattering amplitudes describing the two mechanisms of -meson production differ by the factor . Consequently, they do not interfere and they contribute to the cross section as a sum of two independent contributions.

## 3 Estimates for the cross sections

We analyse the contributions of pomeron-odderon fusion and the photon-pomeron fusion separately. Denoting and we calculate the differential cross sections with respect to the rapidity , the squared momentum transfers in the two -channels, , and the azimuthal angle between and

(18) |

and the partially integrated cross sections

(19) |

with for the -fusion and , for the -fusion, and we set GeV. This leads to the naive predictions shown in the Table 1.

odderon | photon | odderon | photon | |
---|---|---|---|---|

20 nb | 1.6 nb | 36 pb | 1.1 pb | |

11 nb | 2.3 nb | 21 pb | 1.7 pb |

More realistic cross-sections are obtained by taking into account phenomenological improvements, such as related to the BFKL evolution (which is very important for the pomeron exchange and which may be omitted for the odderon exchange [3]), the effects of soft rescatterings of hadrons, and the precise determination of the value of the model parameter in the impact factors. For that we write the corrected cross-sections in the form

(20) |

where are the cross sections given by (19) at . The BFKL evolution for pomeron exchange is taken into account by inclusion of the enhancement factor, which for the central production (i.e. for the rapidity ) has the form

(21) |

Here, is the maximal fraction of incoming hadron momenta exchanged in the channels (or the initial condition for the BFKL evolution) and it is set . The effective pomeron intercept is determined by HERA data and it equals () for the () production [8].

The gap surviving factor for the exclusive production via the pomeron-odderon fusion is fixed by the results of Durham two channel eikonal model [9]: for the exclusive production at the Tevatron and for LHC. In the case of production from the photon-pomeron fusion, [10].

The available estimates of the effective strong coupling constant in the Fukugita–Kwieciński model yield results with rather large spread: from [7], through determined from the HERA data [3] to determined from data on elastic and scattering [11]. This led us to introduce three scenarios which differ by the values of and of . In the optimistic scenario we use a large value of the coupling, , combined with the gap survival factors obtained in the Durham two-channel eikonal model. We believe that the best estimates should follow from the central scenario defined by , and Durham group estimates () at the Tevatron (LHC). The pessimistic scenario is defined by and .

Table 2 shows our predictions for the phenomenologically improved cross sections in all three scenarios. Their magnitudes justify our hope that the process (1) is a subject of experimental study in the near future at the Tevatron and at the LHC [12]. The encouraging feature of our results is due to the fact, that the measurement of the dependence of the cross section partially permits filtering out the contributions and to uncover the ones.

odderon | photon | odderon | photon | |
---|---|---|---|---|

Tevatron | 0.3–1.3–5 nb | 0.8–5–9 nb | 0.7–4–15 pb | 0.8–5–9 pb |

LHC | 0.3–0.9–4 nb | 2.4–15–27 nb | 1.7–5–21 pb | 5–31–55 pb |

## Acknowledgments

I acknowledge common research and discussions with A. Bzdak, J-R. Cudell and L.Motyka. This work is partly supported by the Polish (MEiN) research grant 1 P03B 028 28 and by the Fonds National de la Recherche Scientifique (FNRS, Belgium). I acknowledge a warm hospitality at Ecole Polytechnique and at LPT-Orsay.

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