Iranian Journal of Physics Research

Analytical solution of the evolution equations of the parton distribution functions in the small-x region through the Kramers-Moyal expansion of the master equation of Markov processes

Document Type : Original Article

Author

Naeimeh Olanj

Bu-Ali Sina University, Hamedan

https://doi.org/10.47176/ijpr.22.2.81308
Abstract
Recently, we generated the evolution equations of the parton distribution functions (PDF) usually used in the hadrons phenomenology using the stochastic modeling of the non-equilibrium statistical mechanics in the momentum space. The evolution equations obtained from stochastic modeling are the same as the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations, but are obtained by a more simplistic mathematical procedure based on the non-equilibrium statistical mechanics and the theory of Markov processes. In this paper, we analytically solve the parton evolution equation for the non-singlet quark distribution function in the small-x (the longitudinal momentum fraction) region through the Kramers-Moyal expansion of the master equation. Finally, we compare the cutoff dependent non-singlet quark distribution function obtained from the analytical solution by considering the strong ordering and the angular ordering constraints with the ordinary non-singlet quark distribution function produced by the MMHT2014 group. In general, we show that our results at the small x and moderate Q2 (the energy scale) are in good agreement with the results of the MMHT2014 group.

Keywords

parton distribution function (PDF)
stochastic modeling
master equation
Markov process
Kramers-Moyal expansion
Subjects
  1. L A Harland-Lang, et al., Eur. J. C 75, 5 (2015) 1.

2.       V N Gribov and L N Lipatov, Yad. Fiz. 15 (1972) 781.

  1. A D Martin, M G Ryskin, and G Watt, Phys. J. C 66 (2010) 163.

4.       G C Nayak, Phys. Part. Nucl. 43 (2012) 742.

5.       L Bellantuono, R Bellotti, and F Buccella, J. Stat. Mech.: Theory Exp. 2019, 7 (2019) 073302.

  1. N Olanj, E Moradi, and M Modarres, Physica A 551 (2020) 124585.
  2. M Alimohammadi and N Olanj, Physica A 389 (2010) 1549.

8.       L Mankiewicz, A Saalfeld, and T Weigl, Phys. Lett. B 393 (1997) 175.

  1. N N K Borah, D K Choudhury, and P K Sahariah, High Energy Phys. (2013) 1.
  2. G Alvarez and I Kondrashuk, Phys. Commun. 4 (2020) 075004.
  3. M Mottaghizadeh, F Taghavi Shahri, and P Eslami, Lett. B 773 (2017) 375.
  4. G Alvarez, et al., High Energy Phys. (2016).
  5. L E Reichl and A Modern, “Course in Statistical Physics” Wiley (2009).
  6. H A Kramers, 7, 4 (1940) 284.
  7. J E Moyal, R. Stat. Soc., B: Stat. 11, 2 (1949) 150.
  8. K Golec-Biernat and A M Stasto, Lett. B 781, (2018) 633.
  9. N Olanj and M Modarres, Phys. J. C 79 (2019) 615.
  1. L A Harland-Lang, et al., Eur. J. C 75, 5 (2015) 1.

2.       V N Gribov and L N Lipatov, Yad. Fiz. 15 (1972) 781.

  1. A D Martin, M G Ryskin, and G Watt, Phys. J. C 66 (2010) 163.

4.       G C Nayak, Phys. Part. Nucl. 43 (2012) 742.

5.       L Bellantuono, R Bellotti, and F Buccella, J. Stat. Mech.: Theory Exp. 2019, 7 (2019) 073302.

  1. N Olanj, E Moradi, and M Modarres, Physica A 551 (2020) 124585.
  2. M Alimohammadi and N Olanj, Physica A 389 (2010) 1549.

8.       L Mankiewicz, A Saalfeld, and T Weigl, Phys. Lett. B 393 (1997) 175.

  1. N N K Borah, D K Choudhury, and P K Sahariah, High Energy Phys. (2013) 1.
  2. G Alvarez and I Kondrashuk, Phys. Commun. 4 (2020) 075004.
  3. M Mottaghizadeh, F Taghavi Shahri, and P Eslami, Lett. B 773 (2017) 375.
  4. G Alvarez, et al., High Energy Phys. (2016).
  5. L E Reichl and A Modern, “Course in Statistical Physics” Wiley (2009).
  6. H A Kramers, 7, 4 (1940) 284.
  7. J E Moyal, R. Stat. Soc., B: Stat. 11, 2 (1949) 150.
  8. K Golec-Biernat and A M Stasto, Lett. B 781, (2018) 633.
  9. N Olanj and M Modarres, Phys. J. C 79 (2019) 615.

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