Abstract
We study the eikonal approximation of the total cross section for the scattering of two unpolarized particles and obtain an approximate formula in the case where the eikonal function χ(b) is moderately small, |χ(b)| ≲ 0.1. We show that the total cross section is given by a series of improper integrals of the Born amplitude AB. The advantage of this representation compared with standard eikonal formulas is that these integrals contain no rapidly oscillating Bessel functions. We prove two theorems that allow relating the large-b asymptotic behavior of χ(b) to analytic properties of the Born amplitude and give several examples of applying these theorems. To check the effectiveness of the main formula, we use it to calculate the total cross section numerically for a selection of specific expressions for AB, choosing only Born amplitudes that result in moderately small eikonal functions and lead to the correct asymptotic behavior of χ(b). The numerical calculations show that if only the first three nonzero terms in it are taken into account, this formula approximates the total cross section with a relative error of O(10−5).
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References
G. Molière, “Theorie der Streuung schneller geladener Teilchen I: Einzelstreuung am abgeschirmten Coulomb-Feld,” Z. Naturforsch. A, 2, 133–145 (1947); “Theorie der Streuung schneller geladener Teilchen II: Mehrfachund Vielfachstreuung,” Z. Naturforsch. A, 3, 78–97 (1948).
R. J. Glauber, “High-energy collision theory,” in: Lectures in Theoretical Physics (University of Colorado, Boulder, 1958, W. E. Brittin and L. G. Dunham, eds.), Vol. 1, Interscience, New York (1959), pp. 315–414.
H. Cheng and T. T. Wu, “High-energy elastic scattering in quantum electrodynamics,” Phys. Rev. Lett., 22, 666–669 (1969); “Impact factor and exponentiation in high-energy scattering processes,” Phys. Rev., 186, 1611–1618 (1969).
A. A. Logunov and A. N. Tavkhelidze, “Quasi-optical approach in quantum field theory,” Nuovo Cimento, 29, 380–389 (1963).
P. D. B. Collins, An Introduction to Regge Theory and High Energy Physics, Cambridge Univ. Press, Cambridge (1977).
S. K. Lucas, “Evaluating infinite integrals involving products of Bessel functions of arbitrary order,” J. Comput. Appl. Math., 64, 269–282 (1995).
J. Van Deun and R. Cools, “Integrating products of Bessel functions with an additional exponential or rational factor,” Comp. Phys. Commun., 178, 578–590 (2008).
A. V. Kisselev, “Approximate formulas for moderately small eikonal amplitudes,” Theor. Math. Phys., 188, 1197–1209 (2016).
G. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1, The Hypergeometric Function. Legendre Functions, McGraw-Hill, New York (1953).
M. Abramowitz and I. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Natl. Bur. Stds. Appl. Math. Ser., Vol. 55), U.S. Government Printing Office, Washington, D. C. (1964).
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Cambridge Univ. Press, Cambridge (1927).
F. G. Mehler, “Über die Vertheilung der statischen Elektricität in einem von zwei Kugelkalotten begrenzten Körper,” J. für die Reine und Angewandte Math., 1868, 134–150 (1868); “Notiz über die Dirichlet’schen Integralausdrücke für die Kugelfunktion P n(cos θ) und über eine analoge Integralform für die Zylinderfunktion J(x),” Math. Ann., 5, 141–144 (1872).
G. Szegő, Orthogonal Polynomials, Amer. Math. Soc., Providence, R. I. (1975).
A. V. Kisselev, “Ramanujan’s master theorem and two formulas for zero-order Hankel transform,” arXiv: 1801.06390v1 [math.CA] (2018).
H. Bateman and A. Erdelyi, Higher Trancendental Functions, Vol. 2, Bessel Functions, Parabolic Cylinder Functions, Orthogonal Polynomial, McGraw-Hill, New York (1953).
C. L. Frenzen and R. Wong, “A note on asymptotic evaluation of some Hankel transforms,” Math. Comp., 45, 537–548 (1985).
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge (1944).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series: Special Functions [in Russian], Nauka, Moscow (1983).
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Fizmatgiz, Moscow (1963); English transl.: Tables of Integrals, Series, and Products, Acad. Press, San Diego, Calif. (2000).
H. Bateman and A. Erdélyi, Higher transcendental functions, Vol. 3, Elliptic and Modular Functions. Lame and Mathieu Functions, McGraw-Hill, New York (1955).
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 201, No. 1, pp. 84–104, October, 2019.
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Kisselev, A.V. Approximate Formula for the Total Cross Section for a Moderately Small Eikonal Function. Theor Math Phys 201, 1484–1502 (2019). https://doi.org/10.1134/S0040577919100064
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DOI: https://doi.org/10.1134/S0040577919100064