Production of χc meson pairs with additional emission

I. Babiarz1,∗, W. Schäfer1, and A. Szczurek1,2 1Institute of Nuclear Physics PAS, Krakow, Poland 2Faculty of Mathematics and Natural Sciences, University of Rzeszow, Poland Abstract. We discus mechanism of double χc production with large rapidity separation. The first order pertubative correction to gg→ χcχc process includes additional emission of gluon among χc pair. We have considered real gluon distribution as well as virtual correction to the production process of χc pairs. Results for two scalar χc0χc0 and two axial mesons χc1χc1 are shown.


Introduction
The production process of χ c pairs with extra emission of a gluon is the first order perturbative correction to the Born result for χ c pair production, see Fig. 1.  Figure 1. Diagrams for production χ c pairs. In the left panel there is a leading order process and in the right panel there is pair production associated with extra gluon emission.
The cross section for the elementary process gg → χ c χ c , at gluon gluon center mass energy squaredŝ, shown in the first diagram in Fig. 1, can be written as follows: where the squared amplitude is: |A| 2 =ŝ 2 N 2 c − 1 I 1 (q 1 )I 2 (q 2 ), Here q 1 and q 2 are transverse momenta of outgoing χ c . For scalar particle or axial particle T µν takes the form given in Ref. [1], and n + n − = 1. * e-mail: izabela.babiarz@ifj.edu.pl The cross section for the inclusive production of χ C pair: where terms dσ (1) (2 → 2) and dσ(2 → 3) correspond to high order mechanisms. The cross section dσ(2 → 3) is calculated as for a process with large rapidity distance between two χ c(J) and an extra gluon emission among them. The cross-section dσ (1) (2 → 2) includes the virtual corrections to the first order in α S Y in the BFKL formalism.
2 Real-gluon contribution to the inclusive gg → χ c χ c X process The cross section for gg → χ c gχ c process is: where y denotes rapidity of the gluon. In the impact factor representation the amplitude for the gg → χ c gχ c reads: where K r is derived from real radiative correction to gluon gluon scattering amplitude in the BFKL formalism, obtained using effective Lipatov vertex [2]. The squared amplitude is independent of y, thus one can integrate it out and receive The cross section has a singularity when the transverse momentum of the produced gluon q 3⊥ vanishes. Because of that fact let us introduce a parameter µ and a regulator function: The factor F suppresion ,which is included in the amplitude, provides that the cross section is integrable in the region of small q 3⊥ . Note, that the singularity appears in the back-to-back kinematical situation. 3)]

Virtual corrections
The virtual corrections in the first order in α s Y means that there are two diagrams in which the two gluons are in octet state and give a ln(ŝ) contribution [2]. It leads to a replacement of the gluon propagator by: 1 Subsequently expressing Q by q ⊥ − q 1⊥ and doing some calculation one can obtain an expression free off singularity, also using the same parameter µ: Then expanding the exponential function and decomposing into two terms the formula for the diffrential cross section for dσ(2 → 2), the radiative correction can be written: K In order to omit the singularity, there we also use the parameter µ and as before y veto in purpose to guarantee numerical compatibility with real correction. Here y veto is applied in the gluon propagator by replacing Y by Y − 2y veto . After summation of all terms in the first order correction in α s Y the singularity disappears and the single meson transverse momentum distribution is indeed infrared finite.  Figure 3. Differential cross section as a function of χ c1 transverse momentum when the second produced particle is χ c1 . The dotted-dashed (red) line is for leading order result.
In the Fig. 3 one can notice some effects caused when we include the virtual correction as well as the real gluon distriubtion at the parton level. Adding the virtual corrections cause reduction of the cross section, while including extra gluon emission cause enhancement. It will be interesting to see these effects for hadronic cross sections.