Gell-Mann–Low equation: Relation between the vacuum polarization coefficients up to the eighth order

B. P. Nigam

doi:10.1103/PhysRevD.60.025006

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Gell-Mann–Low equation: Relation between the vacuum polarization coefficients up to the eighth order

B. P. Nigam

Department of Physics & Astronomy, Box 871504, Arizona State University, Tempe, Arizona 85287-1504

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Phys. Rev. D 60, 025006 – Published 23 June, 1999

DOI: https://doi.org/10.1103/PhysRevD.60.025006

Abstract

Gorishny, Kataev, Larin, and Surguladze have obtained the fourth-order Gell-Mann–Low function $ψ_{4}$ and the Callan-Symanzik function $β_{4} .$ In earlier papers, we used their result to establish relationships between the fifth-, sixth-, and seventh-order vacuum polarization coefficients $b_{5},$ $b_{6},$ and $b_{7}$ and the fifth-, sixth-, and seventh-order Gell-Mann–Low functions $ψ_{5},$ $ψ_{6},$ and $ψ_{7} .$ In this paper, we summarize the results of the previous papers and extend the calculations to include the relationship between the eighth-order $b_{8}$ and $ψ_{8}$ using the Gell-Mann–Low equation and other relations for the Gell-Mann–Low function.

References (14)

S. G. Gorishny, A. L. Kataev, S. A. Larin, and L. R. Surguladze, in Quark 90, Proceedings of the International Seminar, Telavi, Georgia, U.S.S.R., edited by V. A. Matveev, V. A. Rubakov, A. N. Tavkhelidze, and P. G. Tinyakov (World Scientific, Singapore, 1991), pp. 194–206; Phys. Lett. B 256, 81 (1991).
L. R. Surguladze and M. A. Samuel, Phys. Rev. Lett. 66, 560 (1991); M. A. Samuel and L. R. Surguladze, Phys. Rev. D 44, 1602 (1991).
B. P. Nigam and R. Acharya, Phys. Rev. D 47, 1726 (1993).
M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954).
D. J. Broadhurst, A. L. Kataev, and V. Tarasov, Phys. Lett. B 298, 445 (1993).
B. P. Nigam, Phys. Rev. D 50, 5446 (1994); J. Phys. G 20, L95 (1994).
B. P. Nigam and R. Acharya, J. Phys. G 19, L95 (1993).
R. Acharya and B. P. Nigam, J. Phys. G 18, L135 (1992).
B. P. Nigam, Nuovo Cimento A 109, 479 (1996); Phys. Rev. D 54, 5377 (1996).
B. P. Nigam, Phys. Rev. D 56, 8115 (1997).
C. Callan, Phys. Rev. D 2, 1541 (1970); K. Symanzik, Commun. Math. Phys. 18, 227 (1970); S. Weinberg, Phys. Rev. 118, 838 (1960).
E. DeRafael and J. Rosner, Ann. Phys. (N.Y.) 82, 369 (1974); ibid.J. Rosner, 44, 11 (1967).
B. P. Nigam, Phys. Rev. D 53, 4698 (1996). In Ref. [6], $β_{3}$ is incorrectly printed with the opposite sign, and in $β_{4}$ the first term is incorrectly printed with the opposite sign.
S. L. Adler, Phys. Rev. D 5, 302 (1972).

[c1] S. G. Gorishny, A. L. Kataev, S. A. Larin, and L. R. Surguladze, in Quark 90, Proceedings of the International Seminar, Telavi, Georgia, U.S.S.R., edited by V. A. Matveev, V. A. Rubakov, A. N. Tavkhelidze, and P. G. Tinyakov (World Scientific, Singapore, 1991), pp. 194–206; Phys. Lett. B 256, 81 (1991).

[c2] L. R. Surguladze and M. A. Samuel, Phys. Rev. Lett. 66, 560 (1991); M. A. Samuel and L. R. Surguladze, Phys. Rev. D 44, 1602 (1991).

[c3] B. P. Nigam and R. Acharya, Phys. Rev. D 47, 1726 (1993).

[c4] M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 (1954).

[c5] D. J. Broadhurst, A. L. Kataev, and V. Tarasov, Phys. Lett. B 298, 445 (1993).

[c6] B. P. Nigam, Phys. Rev. D 50, 5446 (1994); J. Phys. G 20, L95 (1994).

[c7] B. P. Nigam and R. Acharya, J. Phys. G 19, L95 (1993).

[c8] R. Acharya and B. P. Nigam, J. Phys. G 18, L135 (1992).

[c9] B. P. Nigam, Nuovo Cimento A 109, 479 (1996); Phys. Rev. D 54, 5377 (1996).

[c10] B. P. Nigam, Phys. Rev. D 56, 8115 (1997).

[c11] C. Callan, Phys. Rev. D 2, 1541 (1970); K. Symanzik, Commun. Math. Phys. 18, 227 (1970); S. Weinberg, Phys. Rev. 118, 838 (1960).

[c12] E. DeRafael and J. Rosner, Ann. Phys. (N.Y.) 82, 369 (1974); ibid.J. Rosner, 44, 11 (1967).

[c13] B. P. Nigam, Phys. Rev. D 53, 4698 (1996). In Ref. [6], $β_{3}$ is incorrectly printed with the opposite sign, and in $β_{4}$ the first term is incorrectly printed with the opposite sign.

[c14] S. L. Adler, Phys. Rev. D 5, 302 (1972).

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Gell-Mann–Low equation: Relation between the vacuum polarization coefficients up to the eighth order

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