Renormalizable interactions with only one coupling constant

David N. Levin

doi:10.1103/PhysRevD.13.832

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Renormalizable interactions with only one coupling constant

David N. Levin

Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637
Department of Physics, University of California, Los Angeles, California 90024

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Phys. Rev. D 13, 832 – Published 15 February, 1976

DOI: https://doi.org/10.1103/PhysRevD.13.832

Abstract

A search is begun for renormalizable Lagrangians in which the bare interaction vertices of several different fields are all related to the value of a single coupling constant. Such models have great predictive power. A study is made of the five $φ^{4}$ -type interactions of two real scalar fields. We search for renormalizable ways of imposing four relations on the five bare coupling constants. Specifically, we look for all renormalizable models in which the five bare coupling constants lie on a line in coupling-constant space, which is linearly, quadratically, or cubically parameterized by a single coupling constant. It is shown that the only theories of this type either have decoupled $φ_{1}$ , $φ_{2}$ interactions or have the O(2)-invariant form, which is known to be renormalizable. Therefore, in this example all renormalizable interactions with a single coupling constant are symmetric under a group of field transformations; no "new" renormalizable Lagrangians are found.

References (15)

D. N. Levin and G. Tiktopoulos, Phys. Rev. D 12, 415 (1975) J. M. Cornwall, D. N. Levin, and G. Tiktopoulos, ibid. 10, 1145 (1974)
H. Georgi and S. L. Glashow, Phys. Rev. D 7, 2457 (1973)
S. Coleman, in Properties of the Fundamental Interactions, proceedings of the 1971 International Summer School of Physics "Ettore Majorana", edited by A. Zichichi (Editrice Compositori, Bologna, 1973), p. 359
J. Schwinger, Ann. Phys. (N.Y.) 2, 407 (1957) M. GellMann and M. Lévy, Nuovo Cimento 16, 705 (1960)
B. W. Lee, Nucl. Phys. B9, 649 (1969) J.-L. Gervais and B. W. Lee, ibid. B12, 627 (1969)
J. Wess and B. Zumino, Phys. Lett. 49B, 52 (1974)
R. E. Cutkosky, in Particle Symmetries, edited by M. Chrétien and S. Deser (Gordon and Breach, New York, 1966)
G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972)
Omitted endnote
G. 't Hooft, Nucl. Phys. B62, 444 (1973)
Omitted endnote
S. Nandi (unpublished)
L. P. Eisenhart, An Introduction to Differential Geometry (Princeton University Press, Princeton, N. J., 1940)
[1]
P. Nath and R. Arnowitt, Phys. Lett. 56B, 177 (1975)

[c1] D. N. Levin and G. Tiktopoulos, Phys. Rev. D 12, 415 (1975) J. M. Cornwall, D. N. Levin, and G. Tiktopoulos, ibid. 10, 1145 (1974)

[c2] H. Georgi and S. L. Glashow, Phys. Rev. D 7, 2457 (1973)

[c3] S. Coleman, in Properties of the Fundamental Interactions, proceedings of the 1971 International Summer School of Physics "Ettore Majorana", edited by A. Zichichi (Editrice Compositori, Bologna, 1973), p. 359

[c4] J. Schwinger, Ann. Phys. (N.Y.) 2, 407 (1957) M. GellMann and M. Lévy, Nuovo Cimento 16, 705 (1960)

[c5] B. W. Lee, Nucl. Phys. B9, 649 (1969) J.-L. Gervais and B. W. Lee, ibid. B12, 627 (1969)

[c6] J. Wess and B. Zumino, Phys. Lett. 49B, 52 (1974)

[c7] R. E. Cutkosky, in Particle Symmetries, edited by M. Chrétien and S. Deser (Gordon and Breach, New York, 1966)

[c8] G. 't Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972)

[c9] Omitted endnote

[c10] G. 't Hooft, Nucl. Phys. B62, 444 (1973)

[c11] Omitted endnote

[c12] S. Nandi (unpublished)

[c13] L. P. Eisenhart, An Introduction to Differential Geometry (Princeton University Press, Princeton, N. J., 1940)

[c14] [1]

[c15] P. Nath and R. Arnowitt, Phys. Lett. 56B, 177 (1975)

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Renormalizable interactions with only one coupling constant

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