Application of a computer algebra systems to the calculation of the \(\pi\pi\)-scattering amplitude

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Abstract

The aim of this work is to develop a set of programs for calculation the scattering amplitudes of the elementary particles, as well as automating the calculation of amplitudes using the appropriate computer algebra systems (Mathematica, Form, Cadabra). The paper considers the process of pion-pion scattering in the framework of the effective Nambu-Iona-Lasinio model with two quark flavours. The Package-X for Mathematica is used to calculate the scattering amplitude (starting with the calculation of Feynman diagrams and ending with the calculation of Feynman integrals in the one-loop approximation). The loop integrals are calculated in general kinematics in Package-X using the Feynman parametrization technique. A simple check of the program is made: for the case with zero temperature, the scattering lengths \(a_0 = 0.147\) and \(a_2 = -0.0475\) are calculated and the total cross section is constructed. The results are compared with other models as well as with experimental data.

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Introduction The heavy ion collision experiment is an instrument for the study of the matter properties under critical conditions. The modern experiment is a multi- stage process, which includes the event selection, the event reconstruction (the reconstruction of the primary particles) and the simulation of the collision process. The simulation is made on the base of the chosen model and the final result has to reproduce the real data. The fulfil of such analysis or simulation among other things requires a good understanding and a strict description of the final state particle interaction. The information about the particles properties and their interactions can be extracted from the probabilities of the processes occurring during their © Kalinovsky Y. L., Friesen A. V., Rogozhina E. D., Golyatkina L. I., 2020 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ collision. The interaction probability is associated with the cross section of the given reaction and the phase volume, which is uniquely determined by the laws of conservation of energy-momentum, i.e., by the kinematics of the reaction. From the theoretical point of view, the cross section is defined by the scattering amplitude, which is described in the framework of the model under consideration. The model can include description of the quantum mechanical properties of the particles, describe the type of the interaction, take into account the matter properties or the quark structure of the colliding particles, etc. That is why to obtain the scattering amplitude is not the trivial task both from theoretical and computing point of view. This paper is dedicated to the calculation of

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About the authors

Yuriy L. Kalinovsky

Joint Institute for Nuclear Research; Dubna State University

Author for correspondence.
Email: kalinov@jinr.ru

Doctor of Physical and Mathematical Sciences, senior researcher

6, Joliot-Curie St., Dubna, Moscow region, 141980, Russian Federation; 19, Universitetskaya St., Dubna, Moscow Region, 141982, Russian Federation

Alexandra V. Friesen

Joint Institute for Nuclear Research

Email: avfriesen@theor.jinr.ru

Candidate of Physical and Mathematical Sciences, researcher of Joint Institute for Nuclear Research

6, Joliot-Curie St., Dubna, Moscow region, 141980, Russian Federation

Elizaveta D. Rogozhina

Joint Institute for Nuclear Research; Dubna State University

Email: liorinoff@mail.ru

Student of Dubna State University; Senior laboratory assistant of Joint Institute for Nuclear Research

6, Joliot-Curie St., Dubna, Moscow region, 141980, Russian Federation; 19, Universitetskaya St., Dubna, Moscow Region, 141982, Russian Federation

Lyubov’ I. Golyatkina

Joint Institute for Nuclear Research; Dubna State University

Email: lubovgolyatkina@mail.ru

Student of Dubna State University; Senior laboratory assistant of Joint Institute for Nuclear Research

6, Joliot-Curie St., Dubna, Moscow region, 141980, Russian Federation; 19, Universitetskaya St., Dubna, Moscow Region, 141982, Russian Federation

References

  1. S. P. Klevansky, “The Nambu—Jona-Lasinio model of quantum chromodynamics,” Reviews of Modern Physics, vol. 64, pp. 649–708, 3 Jul. 1992. doi: 10.1103/RevModPhys.64.649.
  2. Wei-jie Fu and Yu-xin Liu, “Mesonic excitations and pi–pi scattering lengths at finite temperature in the two-flavor Polyakov–Nambu–Jona- Lasinio model,” 2009. arXiv: 0904.2914 [hep-ph].
  3. V. Jos. (1989). “FORM (symbolic manipulation system),” [Online]. Available: https://www.nikhef.nl/~form/.
  4. P. Kasper. (2020). “Cadabra,” [Online]. Available: https://cadabra. science/.
  5. R. Mertig, M. Bohm, and A. Denner, “FEYN CALC: Computer algebraic calculation of Feynman amplitudes,” Computer Physics Communications, vol. 64, pp. 345–359, 1991. doi: 10.1016/0010-4655(91)90130-D.
  6. T. Hahn and M. Perez-Victoria, “Automatized one loop calculations in four-dimensions and D-dimensions,” Computer Physics Communications, vol. 118, pp. 153–165, 1999. doi: 10.1016/S0010-4655(98)00173-8. arXiv: hep-ph/9807565.
  7. T. Binoth, J.-P. Guillet, G. Heinrich, E. Pilon, and T. Reiter, “Golem95: A Numerical program to calculate one-loop tensor integrals with up to six external legs,” Computer Physics Communications, vol. 180, pp. 2317–2330, 2009. doi: 10.1016/j.cpc.2009.06.024. arXiv: 0810.0992 [hep-ph].
  8. G. Passarino and M. Veltman, “One-loop corrections for e+e− annihilation into µ+µ− in the Weinberg model,” Nuclear Physics B, vol. 160, no. 1, pp. 151–207, 1979. doi: 10.1016/0550-3213(79)90234-7.
  9. H. H. Patel, “Package-X: A Mathematica package for the analytic calculation of one-loop integrals,” Computer Physics Communications, vol. 197, pp. 276–290, Dec. 2015. doi: 10.1016/j.cpc.2015.08.017.
  10. D. Ebert, Y. L. Kalinovsky, L. Munchow, and M. K. Volkov, “Mesons and diquarks in a NJL model at finite temperature and chemical potential,” International Journal of Modern Physics A, vol. 8, pp. 1295–1312, 1993. doi: 10.1142/S0217751X93000539.
  11. M. L. Goldberger and S. B. Treiman, “Decay of the pi meson,” Physical Review, vol. 110, pp. 1178–1184, 1958. doi: 10.1103/PhysRev.110.1178.
  12. E.Quack, P. Zhuang, Y. L. Kalinovsky, S. P. Klevansky, and J.Hufner, “ππ scattering lengths at finite temperature,” Physics Letters B, vol. 348, pp. 1–6, 1995.
  13. S. Narison, “Techniques of dimensional regularization and the two-point functions of QCD and QED,” Physics Reports, vol. 84, no. 4, pp. 263–399, 1982. doi: 10.1016/0370-1573(82)90023-0.
  14. J. D. Bjorken and S. D. Drell, Relativistic quantum mechanics, ser. International series in pure and applied physics. New York, NY: McGraw-Hill, 1964.
  15. H. J. Schulze, “Pion pion scattering lengths in the SU(2) Nambu-Jona-Lasinio model,” Journal of Physics G, vol. 21, pp. 185–191, 1995. doi: 10.1088/0954-3899/21/2/006.
  16. V. Srinivasan et al., “π−π+ → π−π+ interactions below 0.7 GeV from π−p→π−π+n data at 5 GeV/c,” Physical Review D, vol. 12, pp. 681–692, 3 Aug. 1975. doi: 10.1103/PhysRevD.12.681.
  17. S. D. Protopopescu, M. Alston-Garnjost, A. Barbaro-Galtieri, S. M. Flatté, J. H. Friedman, T. A. Lasinski, G. R. Lynch, M. S. Rabin, and F. T. Solmitz, “ππ Partial-Wave Analysis from Reactions π+p → π+π−∆++ and π+p → K+K−∆++ at 7.1 GeV/c,” PPhysical Review D, vol. 7, pp. 1279–1309, 5 Mar. 1973. doi: 10.1103/PhysRevD.7.1279.
  18. Y. L. Kalinovsky, V. D. Toneev, and A. V. Friesen, “Phase diagram of baryon matter in the SU(2) Nambu – Jona-Lasinio model with a Polyakov loop,” Physics-Uspekhi, vol. 59, no. 4, pp. 367–382, 2016. doi: 10.3367/UFNe.0186.201604b.0387.
  19. S. R. Cotanch and P. Maris, “QCD based quark description of pi pi scattering up to the sigma and rho region,” Physical Review D, vol. 66, p. 116 010, 2002. DOI: 10. 1103 / PhysRevD. 66. 116010. arXiv: hep-ph/0210151.
  20. V. Bernard, U. G. Meissner, A. Blin, and B. Hiller, “Four point functions in quark flavor dynamics: Meson meson scattering,” Physics Letters B, vol. 253, pp. 443–450, 1991. doi: 10.1016/0370-2693(91)91749-L.

Copyright (c) 2020 Kalinovsky Y.L., Friesen A.V., Rogozhina E.D., Golyatkina L.I.

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