---

Prog. Theor. Phys. Vol. 36 No. 4 (1966) pp. 745-760

Impact Parameter Formalism. IV

— The Non-Zero Spin Case —

Toshimi Adachi, Tsuneyuki Kotani^* and Minoru Takeda^**

The Tokyo Metropolitan Technical College, Shinagawa, Tokyo
^*Institute of Mathematical Sciences, Adyar, Madras, India
and
^**Department of Physics, Tokai University, Hiratsuka, Kanagawa

(Received May 24, 1966)

Abstract:

An extension of our impact parameter formalism to the elastic scattering of particles with spin is proposed; this is valid for arbitrary physical energies and scattering angles. The helicity representation of the full scattering amplitude and the Jacob-Wick partial wave expansion are used. Our impact parameter amplitude is related to both the full scattering and the partial wave amplitudes without introducing any approxmation. A function α_ν(β ) is defined by
α_ν(β) = \int₀¹y dy J_ν(βy)T(y),
where we assume y^1/2 T(y) ∈ L²(0,1) and Re ν> -½ or an integer ν. The function β-^νa_ν (β) is an even entire function of exponential type 1 and β ^1/2α_ν(β) ∈ L²(0,∞). It satisfies the relation
α_ν+2m (β) = \int₀^∞ αdαφ _ν(α, β) α_ν+2m (α),
where m is a positive integer and the entire function φ_ν (α,β) is defined by
φ_ν(α, β) = \int₀¹y dy J_ν(αy)J_ν (βy).
If we choose T(y) to be a helicity amplitude, ν an integer related with helicities of the initial and final states, and y = sin (θ/2), θ being the scattering angle, then α_ν(β) is our impact parameter amplitude.

URL : http://ptp.ipap.jp/link?PTP/36/745/
DOI : 10.1143/PTP.36.745

[ Full Text PDF : FREE ACCESS (742K) ] Citation: Plain Text BiBTeX EndNote(RIS) RSS

---

References:

1. W. N. Cottingham and R. F. Peierls, Phys. Rev. 137 (1965), B147[APS].
2. T. Adachi and T. Kotani, Prog. Theor. Phys. Suppl. Extra Number (1965), 316[PTP].
3. T. Adachi, Prog. Theor. Phys. 35 (1966), 463[PTP].
4. T. Adachi and T. Kotani, Prog. Theor. Phys. 35 (1966), 485[PTP].
5. I. Mannelli, A. Bigi, R. Carrara, M. Wahlig and L. Sodickson, Phys. Rev. Lett. 14 (1965), 408[APS].
   A. V. Stirling, P. Sonderegger, J. Kirz, P. Falk-Vairant, O. Guisan, C. Bruneton, P. Borgeaud, M. Yvert, J. P. Guillaud, C. Caverzasio and B. Amblard, Phys. Rev. Lett. 14 (1965), 763[APS].
6. V. Barger and M. Ebel, Phys. Rev. 138 (1965), B1148[APS].
7. N. Byers and C. N. Yang, Phys. Rev. 142 (1966), 976[APS].
8. M. Jacob and G. C. Wick, Ann. of Phys. 7 (1959), 404[CrossRef].
9. K. Gottfried and J. D. Jackson, Nuovo Cim. 34 (1964), 735.
10. L. Durand, III, and Y. T. Chiu, Phys. Rev. 139 (1965), B646[APS].
11. M. Luming and E. Predazzi, Nuovo Cim. A 42 (1966), 878.
12. M. E. Rose, Elementary Theory of Angular Momentum (John Wiley and Sons, New York, 1957), Chap. IV.
13. R. P. Boas, Jr., Entire Functions (Academic Press, New York, 1954), p. 8.
14. K. R. Unni, Bulletin American Math. Soc. 71 (1965), 511; Matscience Symposia on Theoretical Physics (Plenum Press, New York, 1966), ed. by Alladi Ramakrishnan, Vol. 6, in press.
15. G. Källen, Matscience Symposia on Theoretical Physics (Plenum Press, New York, 1966), Vol. 6, in press; and private communication.
16. T. Kotani, Lecture Note on An Impact Parameter Formalism (Institute of Mathematical Science, Madras, India, 1966), Appendix A.
17. Y. Yamamoto, Prog. Theor. Phys. 35 (1966), 806[PTP].
18. M. Ida, Prog. Theor. Phys. 35 (1966), 1074[PTP].
19. T. M. MacRobert, Proc. Roy. Soc. Edinburgh 51 (1931), 116.
20. J. E. Wilkins, Trans. Amer. Math. Soc. 64 (1948), 359.
21. G. Szegö, Orthogonal Polynomials (American Methematical Society Colloquium Publications, New York, 1939), Vol. 23, p. 195.
22. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, 1948), 2nd edition.

Citing Article(s) :

1. Progress of Theoretical Physics Vol. 37 No. 6 (1967) pp. 1211-1227 :

   Impact-Parameter Representation for Proton-Proton Helicity Amplitudes at High Energies

   Reijiro Kubo

2. Progress of Theoretical Physics Vol. 39 No. 2 (1968) pp. 430-452 :

   An Impact Parameter Representation of the Scattering Problem^*

   Toshimi Adachi and Tsuneyuki Kotani

3. Progress of Theoretical Physics Vol. 39 No. 3 (1968) pp. 785-816 :

   An Impact Parameter Representation of the Scattering Problem^*

   Toshimi Adachi and Tsuneyuki Kotani

4. Progress of Theoretical Physics Vol. 39 No. 4 (1968) pp. 1006-1019 :

   An Impact-Parameter Representation of Pion-Nucleon Scattering Amplitudes at High Energies

   Reijiro Kubo

---

[PTP HOME] [BROWSE] [SEARCH] [SUBSCRIPTION] [SUBMISSION] [REGISTRATION]

Copyright © Progress of Theoretical Physics 2013 All rights reserved.