- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review D

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

New approach to the singularities of Feynman amplitudes in the zero-mass limit

T. Kinoshita

A. Ukawa*

  • Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14853

  • Department of Physics, University of Tokyo, Tokyo 113, Japan

  • *Present address: Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14853.
PDF

Phys. Rev. D 13, 1573 – Published 15 March, 1976

DOI: https://doi.org/10.1103/PhysRevD.13.1573

Abstract

We present an improved method for studying the behavior of Feynman amplitudes in the limit in which some or all of the masses of the internal lines vanish. Feynman amplitudes develop singularities in such a limit for certain configurations of external momenta. Our method enables us to determine these configurations, be they Euclidean or Minkowskian, on or off the mass shell. They are worked out explicitly in a few relatively simple cases which include most of the physically interesting singularities. Power-counting rules for these singularities are briefly discussed.

View figure in article

References (59)

  1. R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The Analytic S-Matrix (Cambridge Univ. Press, London, 1966) N. Nakanishi, Graph Theory and Feynman Integrals (Gordon and Breach, New York, 1971)
  2. T. Kinoshita, J. Math. Phys. 3, 650 (1962) in Lectures in Theoretical Physics, edited by W. E. Brittin (Interscience, New York, 1962), p. 121
  3. T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964)
  4. D. R. Yennie, S. C. Frautschi, and H. Suura, Ann. Phys. (N.Y.) 13, 379 (1961) S. Weinberg, Phys. Rev. 140, B516 (1965) G. Grammer, Jr. and D. R. Yennie, Phys. Rev. D 8, 4332 (1973) P. Cvitanović and T. Kinoshita, [40] V. Chung, Phys. Rev. 140, B1110 (1965) T. W. B. Kibble, ibid. 173, 1527 (1968) ibid.174, 1882 (1968) L. Faddeev and P. Kulish, Teor. Mat. Fiz. 4, 153 (1970) [Theor. Math. Phys. 4, 745 (1970)] D. Zwanziger, Phys. Rev. Lett. 30, 934 (1973) Phys. Rev. D 7, 1082 (1973)
  5. S. Weinberg, [4] Phys. Rev. Lett. 31, 494 (1973) D. J. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973)
  6. A. Casher, J. Kogut, and L. Susskind, Phys. Rev. Lett. 31, 792 (1973) Phys. Rev. D 10, 732 (1974)
  7. K. G. Wilson, Phys. Rev. D 10, 2445 (1974) J. B. Kogut and L. Susskind, ibid. 10, 3468 (1974) R. Balian, J. M. Drouffe, and C. Itzykson, ibid. 10, 3376 (1974) S. Blaha, ibid. 10, 4268 (1974) R. A. Brandt and Ng Wing-Chiu, Phys. Rev. Lett. 33, 1640 (1974)
  8. T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975)
  9. S. Weinberg, Phys. Rev. 118, 838 (1960)
  10. D. A. Slavnov, Teor. Mat. Fiz. 17, 169 (1973) [Theor. Math. Phys. 17, 1067 (1973)
  11. J. P. Fink, J. Math. Phys. 9, 1389 (1968)
  12. M. J. Westwater, Fortschr. Phys. 17, 1 (1969)
  13. M. C. Bergère and Y. P. Lam, Commun. Math. Phys. 39, 1 (1974) Freie Universität, Berlin, Report No. HEP 74/9 (unpublished)
  14. K. Pohlmeyer, DESY Report No. DESY 74/36 (unpublished) H. Trute, DESY Report No. 74/44 (unpublished)
  15. E. Speer, Generalized Feynman Amplitudes (Princeton Univ. Press, Princeton, N. J., 1969)
  16. D. Ruelle, Nuovo Cimento 19, 356 (1961) K. Symanzik, Commun. Math. Phys. 23, 49 (1971)
  17. Pohlmeyer's [14]
  18. R. J. Eden et al. in [1] C. Lovelace, Nucl. Phys. B95, 12 (1975) H. Cheng and T. T. Wu Proceedings of the 1971 International Symposium on Electron and Photon Interactions at High Energies, edited by N. B. Mistry (Laboratory of Nuclear Studies, Cornell Univ., Ithaca, N. Y., 1972)
  19. T. Appelquist and J. R. Primack, Phys. Rev. D 1, 1144 (1970) ibid.4, 2454 (1971)
  20. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781 (1972) ibid.15, 1218 (1972) [Sov. J. Nucl. Phys. 15, 438 (1972) Sov. J. Nucl. Phys. 15, 675 (1972)]
  21. J. J. Carazzone, E. R. Poggio, and H. R. Quinn, Phys. Rev. D 11, 2286 (1975) ibid.12, 3368(E) (1975) J. M. Cornwall and G. Tiktopoulos, Phys. Rev. Lett. 35, 338 (1975)
  22. M. Gell-Mann and F. Low, Phys. Rev. 95, 1300 (1954) N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience, New York, 1959) L. V. Ovsiannikov, Dokl. Akad. Nauk SSSR 109, 1121 (1956)
  23. G. 't Hooft, Nucl. Phys. B61, 455 (1973) S. Weinberg, Phys. Rev. D 8, 3497 (1973)
  24. K. Symanzik, Commun. Math. Phys. 18, 227 (1970) Springer Tracts in Modern Physics, edited by G. Höhler (Springer, Berlin, 1971), Vol. 57, p. 222 C. G. Callan, Jr., Phys. Rev. D 2, 1541 (1970)
  25. D. J. Gross and F. Wilczek, Phys. Rev. Lett. 26, 1343 (1973) H. D. Politzer, ibid. 26, 1346 (1973) K. Symanzik, DESY Report No. T-73/3, 1973 (unpublished) Commun. Math. Phys. 34, 7 (1973)
  26. K. Symanzik, Commun. Math. Phys. 23, 49 (1971) C. G. Callan, Jr., Phys. Rev. D 5, 3202 (1972)
  27. K. G. Wilson, Phys. Rev. 179, 1499 (1969) W. Zimmermann, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. Deser (MIT Press, Cambridge, Mass., 1971) Ann. Phys. (N.Y.) 77, 536 (1973) ibid.77, 570 (1973)
  28. N. Christ, B. Hasslacher, and A. Mueller, Phys. Rev. D 6, 3543 (1972) H.-J. Thun, Nuovo Cimento 26A, 329 (1975) A. Mueller, Phys. Rev. D 9, 963 (1974)
  29. P. Cvitanović and T. Kinoshita, Phys. Rev. D 10, 3978 (1974)

  30. Omitted endnote

  31. Omitted endnote

  32. Omitted endnote

  33. Omitted endnote

  34. [29]

  35. Omitted endnote

  36. [29]

  37. Omitted endnote

  38. Omitted endnote

  39. P. V. Landshoff, Phys. Rev. D 10, 1024 (1974) P. Cvitanović, ibid. 10, 338 (1974) S. J. Brodsky and G. Farrar, ibid. 11, 1309 (1975) I. G. Halliday, J. Huskins, and C. T. Sachrajda, Nucl. Phys. B83, 189 (1974)
  40. P. Cvitanović and T. Kinoshita, Phys. Rev. D 10, 3991 (1974)
  41. N. Nakanishi, [1] P. Cvitanović and T. Kinoshita, [40]

  42. Omitted endnote

  43. Omitted endnote

  44. Omitted endnote

  45. Omitted endnote

  46. Omitted endnote

  47. Omitted endnote

  48. Omitted endnote

  49. [44]
  50. [48]
  51. [46]

  52. Omitted endnote

  53. Omitted endnote

  54. [48]

  55. Omitted endnote

  56. [16]

  57. Omitted endnote

  58. K. Symanzik, [16,25]
  59. T. Kinoshita and A. Ukawa (unpublished)

Outline

Information

Phys. Rev. D 13, 1573– Published 15 March, 1976
  • Received 6 November 1975

© 1976 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation