<span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow><mo>−</mo><mrow><msup><mrow><mi>η</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow><mo>−</mo><mrow><msup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math></span> Mixing in Broken Chiral Symmetry

Ranabir Dutt; Shalom Eliezer

doi:10.1103/PhysRevD.4.180

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$π^{0} - η^{0} - X^{0}$ Mixing in Broken Chiral Symmetry

Ranabir Dutt^* and Shalom Eliezer

Department of Physics, Technion-Israel Institute of Technology, Haifa, Israel

^*Postdoctoral research fellow from the Department of Physics and Astrophysics, University of Delhi, Delhi-7, India.

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Phys. Rev. D 4, 180 – Published 1 July, 1971

DOI: https://doi.org/10.1103/PhysRevD.4.180

Abstract

The mixing of $π^{0}$ , $η$ , and $X (958)$ mesons has been investigated within the framework of the broken-chiral-symmetry model of Gell-Mann, Oakes, and Renner. Considering $SU (2)$ symmetry breaking of the Hamiltonian due to the nonelectromagnetic isospin-violation term (so-called $u_{3}$ ) and imposing the smoothness assumption on certain three-point vertex functions, we calculate the mixing angles and the symmetry-breaking parameters in terms of the known masses of the pseudoscalar mesons and one input variable, $F = \frac{f_{K}}{f_{π}}$ . We obtain two sets of solutions (e.g., $η - X$ mixing angle $ω \approx {10.5}^{\circ} or 19^{\circ}$ for $F = 1$ ) and predict negligible mixing between $π^{0} - X$ compared to $π^{0} - η$ . The effect of the vacuum symmetry breaking on these mixing parameters has been studied as well. In the case of an $SU (3)$ -invariant vacuum, we reproduce separately the known results about $η - X$ and $π^{0} - η$ mixings and obtain the corrections to the Gell-Mann-Okubo mass formulas.

References (30)

R. H. Dalitz and D. G. Sutherland, Nuovo Cimento 37, 1777 (1965) ibid.38, 1945(E) (1965)
S. Coleman, S. L. Glashow, and D. J. Kleitman, Phys. Rev. 135, B779 (1964)
G. Alexander, H. J. Lipkin, and F. Scheck, Phys. Rev. Letters 17, 412 (1966)
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Riazuddin and A. Q. Sarker, Phys. Rev. Letters 20, 1455 (1968)
L. Schülke, Phys. Rev. Letters 22, 626 (1969)
R. S. Willey, Phys. Rev. 183, 1397 (1969)
S. L. Glashow, R. Jackiw, and S. S. Shei, Phys. Rev. 187, 1916 (1969)
Y. Y. Lee, Nuovo Cimento 64A, 474 (1969)
L. K. Pande, Phys. Rev. Letters 25, 777 (1970)
S. Okubo and V. S. Mathur, Phys. Rev. D 1, 2046 (1970) V. S. Mathur, S. Okubo, and J. Subba Rao, ibid.1 2058 (1970)
H. Osborn and D. J. Wallace, Nucl. Phys. B20, 23 (1970)
S. Okubo and B. Sakita, Phys. Rev. Letters 11, 50 (1963)
R. H. Dalitz and F. von Hippel, Phys. Letters 10, 153 (1964)
S. Matsuda, S. Oneda, and P. Desai, Phys. Rev. 178, 2129 (1969)
S. Oneda and S. Matsuda, Phys. Rev. D 1, 944 (1970)
S. Oneda, and H. Umezawa, and S. Matsuda, Phys. Rev. Letters 25, 71 (1970)
M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968) S. L. Glashow and S. Weinberg, Phys. Rev. Letters 20, 224 (1968) S. L. Glashow, in Hadrons and Their Interactions, edited by A. Zichichi (Academic, New York, 1968), pp. 83-140
R. Gatto, G. Sartori, and M. Tonin, Lett. Nuovo Cimento 1, 1 (1969) R. Gatto, in Springer Tracts in Modern Physics, edited by G. Höhler (Springer, Berlin, 1970), Vol. 53
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R. J. Oakes, Phys. Letters 29B, 683 (1969)
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R. Dutt and S. Eliezer, Technion report (unpublished)
R. Dashen, Phys. Rev. 183, 1245 (1969)
M. Gell-Mann, Phys. Rev. 125, 1067 (1962)
[17]
P. R. Auvil and N. G. Deshpande, Phys. Rev. 183, 1463 (1969)
H. J. Schnitzer and S. Weinberg, Phys. Rev. 164, 1828 (1967)
Particle Data Group, Phys. Letters 33B, 1 (1970)
[6] ([8,11,27])

[c1] R. H. Dalitz and D. G. Sutherland, Nuovo Cimento 37, 1777 (1965) ibid.38, 1945(E) (1965)

[c2] S. Coleman, S. L. Glashow, and D. J. Kleitman, Phys. Rev. 135, B779 (1964)

[c3] G. Alexander, H. J. Lipkin, and F. Scheck, Phys. Rev. Letters 17, 412 (1966)

[c4] A. J. Macfarlane and R. H. Socolow, Phys. Rev. 144, 1194 (1966)

[c5] Riazuddin and A. Q. Sarker, Phys. Rev. Letters 20, 1455 (1968)

[c6] L. Schülke, Phys. Rev. Letters 22, 626 (1969)

[c7] R. S. Willey, Phys. Rev. 183, 1397 (1969)

[c8] S. L. Glashow, R. Jackiw, and S. S. Shei, Phys. Rev. 187, 1916 (1969)

[c9] Y. Y. Lee, Nuovo Cimento 64A, 474 (1969)

[c10] L. K. Pande, Phys. Rev. Letters 25, 777 (1970)

[c11] S. Okubo and V. S. Mathur, Phys. Rev. D 1, 2046 (1970) V. S. Mathur, S. Okubo, and J. Subba Rao, ibid.1 2058 (1970)

[c12] H. Osborn and D. J. Wallace, Nucl. Phys. B20, 23 (1970)

[c13] S. Okubo and B. Sakita, Phys. Rev. Letters 11, 50 (1963)

[c14] R. H. Dalitz and F. von Hippel, Phys. Letters 10, 153 (1964)

[c15] S. Matsuda, S. Oneda, and P. Desai, Phys. Rev. 178, 2129 (1969)

[c16] S. Oneda and S. Matsuda, Phys. Rev. D 1, 944 (1970)

[c17] S. Oneda, and H. Umezawa, and S. Matsuda, Phys. Rev. Letters 25, 71 (1970)

[c18] M. Gell-Mann, R. J. Oakes, and B. Renner, Phys. Rev. 175, 2195 (1968) S. L. Glashow and S. Weinberg, Phys. Rev. Letters 20, 224 (1968) S. L. Glashow, in Hadrons and Their Interactions, edited by A. Zichichi (Academic, New York, 1968), pp. 83-140

[c19] R. Gatto, G. Sartori, and M. Tonin, Lett. Nuovo Cimento 1, 1 (1969) R. Gatto, in Springer Tracts in Modern Physics, edited by G. Höhler (Springer, Berlin, 1970), Vol. 53

[c20] N. Cabibbo and L. Maiani, Phys. Letters 28B, 131 (1968) Phys. Rev. D 1, 707 (1970)

[c21] R. J. Oakes, Phys. Letters 29B, 683 (1969)

[c22] A. Goyal, L. F. Li, and G. Segrè, Phys. Rev. D 2, 2373 (1970)

[c23] R. Dutt and S. Eliezer, Technion report (unpublished)

[c24] R. Dashen, Phys. Rev. 183, 1245 (1969)

[c25] M. Gell-Mann, Phys. Rev. 125, 1067 (1962)

[c26] [17]

[c27] P. R. Auvil and N. G. Deshpande, Phys. Rev. 183, 1463 (1969)

[c28] H. J. Schnitzer and S. Weinberg, Phys. Rev. 164, 1828 (1967)

[c29] Particle Data Group, Phys. Letters 33B, 1 (1970)

[c30] [6] ([8,11,27])

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$π^{0} - η^{0} - X^{0}$ Mixing in Broken Chiral Symmetry

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π0−η0−X0 Mixing in Broken Chiral Symmetry

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References (30)

$π^{0} - η^{0} - X^{0}$ Mixing in Broken Chiral Symmetry