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Semihard QCD Processes beyond the Collinear Approximation

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Abstract

The main features and characteristics of the \({{k}_{{\text{T}}}}\)-factorization QCD approach, which is based on BFKL-like noncollinear gluon evolution equations, are outlined. One of the advantages of this approach is connected with a convenience of including a significant part of higher order perturbative QCD corrections. The explicit expression and solutions of some parton distribution evolution equations in the leading logarithmic approximation are presented. Frequently used in calculations gluon and quark distribution functions, which depend on the transverse momenta of the initial interacting particles, are shown. Various techniques to calculate the off-shell amplitudes of parton subprocesses are discussed. The application of the \({{k}_{{\text{T}}}}\)-factorization approach to some QCD processes that are intensively studied in modern experiments at the LHC is considered, in particular, to processes of the single and double production of heavy quarks, as well as the inclusive and associated (correlated with one or two hadronic jets) production of direct photons (or heavy gauge boson) in proton collisions at high energies. A comparison of the obtained results with predictions made using the standard (collinear) approach in the next-to-leading order of perturbative QCD is presented. Basic capabilities and features of modern Monte-Carlo event generators PEGASUS, CASCADE and KATIE are also discussed.

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Notes

  1. As a rule, the value \({{\mu }^{2}}\) is taken as these scales, which thereby eliminates contributions proportional to \({{\ln {{\mu }^{2}}} \mathord{\left/ {\vphantom {{\ln {{\mu }^{2}}} {\mu _{{\text{R}}}^{2}}}} \right. \kern-0em} {\mu _{{\text{R}}}^{2}}}\) and \({{\ln {{\mu }^{2}}} \mathord{\left/ {\vphantom {{\ln {{\mu }^{2}}} {\mu _{{\text{F}}}^{2}}}} \right. \kern-0em} {\mu _{{\text{F}}}^{2}}}\), which arise when calculating corrections of the higher orders to the parton cross sections calculated in the leading order of perturbative QCD.

  2. The initial conditions for quark and gluon distributions are not calculated within the perturbative QCD. They can be determined, e.g., from experiments on deep inelastic ep-scattering.

  3. Within the leading logarithmic approximation, only terms proportional to \({{\alpha _{s}^{n}{{{\ln }}^{n}}s} \mathord{\left/ {\vphantom {{\alpha _{s}^{n}{{{\ln }}^{n}}s} {\Lambda _{{{\text{QCD}}}}^{2}}}} \right. \kern-0em} {\Lambda _{{{\text{QCD}}}}^{2}}}\) are considered, while within the next-to-leading approximation, also terms of order \({{\alpha _{s}^{{n + 1}}{{{\ln }}^{n}}s} \mathord{\left/ {\vphantom {{\alpha _{s}^{{n + 1}}{{{\ln }}^{n}}s} {\Lambda _{{{\text{QCD}}}}^{2}}}} \right. \kern-0em} {\Lambda _{{{\text{QCD}}}}^{2}}}\).

  4. In QCD, a Pomeron is a bound state of two reggeized gluons and determines the total scattering cross section of a particle. The odderon, which is responsible for the difference between the scattering cross sections of a particle and an antiparticle, is a bound state of three reggeized gluons.

  5. A method for calculating the TMD quark and gluon distributions in the proton using a numerical solution of the DGLAP evolution equations was proposed within the so-called parton branching approach (PB) [97, 98] (see also [99, 100]).

  6. A detailed description of the corresponding calculations is given in [118].

  7. https://theory.sinp.msu.ru/doku.php/pegasus/news.

  8. Various methods for calculating the gluon density (in the leading and next-to-leading orders of perturbative QCD) from experimental data for the functions \({{F}_{2}}(x,{{Q}^{2}})\), \({{F}_{{\text{L}}}}(x,{{Q}^{2}})\) and the logarithmic derivative \({{\partial {{F}_{2}}(x,{{Q}^{2}})} \mathord{\left/ {\vphantom {{\partial {{F}_{2}}(x,{{Q}^{2}})} {\partial \ln {{Q}^{2}}}}} \right. \kern-0em} {\partial \ln {{Q}^{2}}}}\) have been proposed, e.g., in [175–179].

  9. A similar approach can be used to study the TMD quark distribution function in the proton using Drell–Yan processes [190].

  10.  An essential part of the mathematical calculations may be the expansion of the amplitude into a suitable set of basis vectors (orthogonal amplitude method). This method was proposed in [196] and was widely used previously (see, e.g., [197–201]). A rigorous justification of the method is given in [197]. Using the orthogonal amplitude method can significantly reduce the volume and time of computer calculations.

  11.  The processes of associative production of direct photons and heavy quark jets at the Tevatron collider within the \({{k}_{{\text{T}}}}\)-factorization approach were considered in [207, 208]. In addition, the processes of inclusive production of direct photons in \(ep\) and \(pp\) collisions were studied [208, 210], as well as the processes of production of photon pairs [211] (see also [154]).

  12.  In contrast to the usual collinear perturbative QCD, where the kinematics (and the very presence) of jets is determined only by the corresponding amplitude of parton scattering.

  13. See also calculations [46].

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ACKNOWLEDGMENTS

The authors are grateful to A.V. Kotikov, G.I. Lykasov, and H. Jung for numerous discussions and assistance in the work.

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This work was supported by ongoing institutional funding. No additional grants to conduct or direct this particular research were obtained.

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  1. Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992, Moscow, Russia

    A. V. Lipatov & M. A. Malyshev

  2. Lebedev Physical Institute, Russian Academy of Sciences, 119991, Moscow, Russia

    S. P. Baranov

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  1. A. V. Lipatov
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Lipatov, A.V., Baranov, S.P. & Malyshev, M.A. Semihard QCD Processes beyond the Collinear Approximation. Phys. Part. Nuclei 55, 256–307 (2024). https://doi.org/10.1134/S1063779624020047

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