Nonlinear Realizations of SU(3)×SU(3) and the Symmetry-Breaking Meson Lagrangian

S. P. Rosen; A. McDonald

doi:10.1103/PhysRevD.4.1833

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Nonlinear Realizations of SU(3)×SU(3) and the Symmetry-Breaking Meson Lagrangian

S. P. Rosen and A. McDonald

Physics Department, Purdue University, Lafayette, Indiana 47907

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Phys. Rev. D 4, 1833 – Published 15 September, 1971

DOI: https://doi.org/10.1103/PhysRevD.4.1833

Abstract

We show how an octet of pseudoscalar mesons which transform nonlinearly under SU(3) × SU(3) can be converted to the linear realizations ( $3, \bar{3}$ ) and ( $\bar{3}, 3$ ). Our results are manifestly covariant with respect to redefinitions of the meson field. We use them to construct an effective meson Lagrangian in which all the symmetry breaking occurs in the mass term and belongs to a single representation { $(p, \bar{p}) + (\bar{p}, p)$ }. From an analysis of $π - π$ scattering lengths, we find it unlikely that the symmetry-breaking interaction belongs either to the self-adjoint sequence (8, 8), (27, 27),..., or to the triangular one $p = 6, 10, 15, \dots$ . There are, however, many other representations, including { $(3, \bar{3}) + (\bar{3}, 3)$ }, which cannot be ruled out. We also examine weak currents and find that the $f_{-}$ form factor in $K_{l 3}$ decay gives rise to a serious problem in the symmetry breaking.

References (14)

S. P. Rosen, Phys. Rev. D 1, 3392 (1970) Phys. Rev. 188, 2542 (1969)
M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960) M. Lévy, ibid. 52, 23 (1967)
S. Weinberg, Phys. Rev. 166, 1568 (1968)
M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968)
J. A. Cronin, Phys. Rev. 161, 1483 (1967) L. Turner, Nucl. Phys. B11, 355 (1969)
A. J. Macfarlane, A. Sudbery, and P. Weisz, Proc. Roy. Soc. (London) A314, 217 (1970)
D. Carlstone, S. P. Rosen, and S. Pakvasa, Phys. Rev. 174, 1877 (1968)
S. P. Rosen, J. Math. Phys. 12, 673 (1971)
S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177, 2239 (1969)
S. P. Rosen, J. Math. Phys. 5, 289 (1964)
S. Coleman, in Hadrons and Their Interactions, edited by A. Zichichi (Academic, New York, 1967), p. 39
F. T. Meiere and M. Sugawara, Phys. Rev. 153, 1702 (1967)
D. Haidt et al., Phys. Rev. D 3, 10 (1971)
A. J. Macfarlane and P. H. Weisz, Lett. Nuovo Cimento 2, 25 (1969)

[c1] S. P. Rosen, Phys. Rev. D 1, 3392 (1970) Phys. Rev. 188, 2542 (1969)

[c2] M. Gell-Mann and M. Lévy, Nuovo Cimento 16, 705 (1960) M. Lévy, ibid. 52, 23 (1967)

[c3] S. Weinberg, Phys. Rev. 166, 1568 (1968)

[c4] M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968)

[c5] J. A. Cronin, Phys. Rev. 161, 1483 (1967) L. Turner, Nucl. Phys. B11, 355 (1969)

[c6] A. J. Macfarlane, A. Sudbery, and P. Weisz, Proc. Roy. Soc. (London) A314, 217 (1970)

[c7] D. Carlstone, S. P. Rosen, and S. Pakvasa, Phys. Rev. 174, 1877 (1968)

[c8] S. P. Rosen, J. Math. Phys. 12, 673 (1971)

[c9] S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177, 2239 (1969)

[c10] S. P. Rosen, J. Math. Phys. 5, 289 (1964)

[c11] S. Coleman, in Hadrons and Their Interactions, edited by A. Zichichi (Academic, New York, 1967), p. 39

[c12] F. T. Meiere and M. Sugawara, Phys. Rev. 153, 1702 (1967)

[c13] D. Haidt et al., Phys. Rev. D 3, 10 (1971)

[c14] A. J. Macfarlane and P. H. Weisz, Lett. Nuovo Cimento 2, 25 (1969)

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Nonlinear Realizations of SU(3)×SU(3) and the Symmetry-Breaking Meson Lagrangian

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