^{1}

^{*}

^{2}

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^{2}

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ahmadovazar@yahoo.com

coskun@ktu.edu.tr

oguzhan_deu@hotmail.com

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In this study, we have considered the contribution of the higher-twist (HT) effects of the subprocesses to inclusive pion pair production cross section in the high energy proton-antiproton collisions by using various pion distribution amplitudes (DAs) within the frozen coupling constant approach and compared them with the leading-twist contributions. The feature of the HT effects may help the theoretical interpretation of the future PANDA experiment. The dependencies of the HT contribution on the transverse momentum

It is well known that quantum chromodynamics (QCD) is the fundamental theory of strong interactions. QCD describes the strong interactions between quarks and gluons, also the structure and dynamics of hadrons at the amplitude level.

The hadronic distribution amplitude (DA) in terms of internal structure degrees of freedoms is important in QCD process predictions. Parton DAs are important ingredients in applying QCD to hard exclusive processes via the factorization theorem

The main difficulty in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale

From Eq.

The dependence of the DA on the factorization scale

In Refs.

Precision experimental studies of meson pair production in proton-antiproton collisions at low energies are proposed in the experiment named PANDA

The physical information of the inclusive pion pair production can be obtained efficiently in the pQCD and it is, hence, possible to compare directly with the experimental data. The corresponding hard-scattering subprocesses occur via three different mechanisms. The first one is the direct production of charged pion pairs which are produced directly at the hard-scattering subprocess (see Fig.

QCD Feynman diagrams of the partonic process

QCD Feynman diagrams of the partonic process

The rest of the paper is organized as follows. In Sec.

The inclusive production of charged pion pair with large transverse momenta

In principle, all measurable quantities in QCD should be invariant under any choice of renormalization scale and scheme. It is clear that the use of different scales and schemes may lead to different theoretical predictions. Therefore, the constructive mathematical tool for defining QCD is a choice of the renormalization scale which makes scheme independent results at all fixed order in running coupling constant

The parton-level differential cross sections for the direct pion pair production are obtained as

The corresponding differential cross sections of the subprocesses are defined for these cases as

The longitudinal momentum fractions of partons are defined in this form

For the calculation of the HT cross sections in the case of direct pion pair production, we assume in Eq.

It is an important task to compare the HT corrections with LT contributions and to extract the HT corrections to the pion pair production cross section.

For the LT cross section for the production of pion pairs, we take the next subprocesses in which the final particles are fragmented to pion pairs as

The corresponding differential cross section of the LT subprocesses are written as

The standard approach to distribution amplitudes, which is due to Brodsky and Lepage

It would be important and interesting to compare the proton-antiproton annihilation process

Let us now discuss in detail the numerical predictions of the HT and LT cross sections of the pion pair production process

HT contribution to charged pion pair production

The sum of HT and LT contribution to charged pion pair production

Ratio

HT contribution to charged pion pair production

The sum of HT and LT contribution to charged pion pair production

Ratio

HT contribution to charged pion pair production

The sum of HT and LT contribution to charged pion pair production

Ratio

Ratio HT cross sections

HT

HT

HT

Ratio of HT to LT contributions as a function of the variable

First, we compare the HT and LT cross sections obtained within holographic QCD and pQCD. In Figs.

In Fig.

Through Figs.

For the region

One can also observe that the HT cross section in the region

In Figs.

In this study, the HT contributions, which are included in the direct and semidirect productions of the hard scattering process, to large-

The HT cross section obtained in our study should be observable at a hadron collider. Also, the feature of HT effects can help theoretical interpretations of the future PANDA experimental data for the direct inclusive pion pair production cross section in the proton-antiproton collisions. As a result, it can be indicated that the HT processes for large-

A.I.A. is grateful for the financial support by the Science Development Foundation under the President of the Republic of Azerbaijan - Grant No. EIF/MQM/Elm-Tehsil-1-2016-1(26)-71/11/1 and Baku State University Grant No. “50+50” (2018–2019). The authors are grateful to S. V. Mikhailov for useful discussions.