Static solutions in the vacuum sector of the Skyrme model

J. Bagger; W. Goldstein; M. Soldate

doi:10.1103/PhysRevD.31.2600

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Static solutions in the vacuum sector of the Skyrme model

J. Bagger, W. Goldstein, and M. Soldate

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305

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Phys. Rev. D 31, 2600 – Published 15 May, 1985

DOI: https://doi.org/10.1103/PhysRevD.31.2600

Abstract

We use topological methods to prove the existence of a nontrivial static solution in the vacuum sector of the gauged Skyrme model. We then search for a purely strong-interaction saddle point by computing the interaction energy of a Skyrmion–anti-Skyrmion pair.

References (18)

C. H. Taubes, Commun. Math. Phys. 86, 257 (1982); ibid. 86, 299 (1982).
N. S. Manton, Phys. Rev. D 28, 2019 (1983); F. R. Klinkhamer and N. S. Manton, ibid. 30, 2227 (1984).
T. H. R. Skyrme, Proc. R. Soc. London A 260, 127 (1961).
G. 't Hooft, Nucl. Phys. B 72, 461 (1974); ibid. B 75, 461 (1974).
E. Witten, Nucl. Phys. B 160, 57 (1979).
The Wess-Zumino term should be added when $N_{F}$ >= 3. However, it does not affect the energy of static configurations and we shall ignore it in this paper.
E. Witten, Nucl. Phys. B 223, 433 (1983).
A. P. Balachandran, V. P. Nair, S. G. Rajeev and A. Stern, Phys. Rev. Lett. 49, 1124 (1982).
G. Adkins, C. Nappi and E. Witten, Nucl. Phys. B 228, 552 (1983).
H. Walliser and G. Eckart, Nucl. Phys. A 429, 514 (1984); A. Hayashi, G. Eckart, G. Holzwarth and H. Walliser, Phys. Lett. 147 B, 5 (1984); J. Breit and C. Nappi, Phys. Rev. Lett. 53, 889 (1984); M. Mattis and M. Karliner, Phys. Rev. D (to be published).
C. Callan and E. Witten, Nucl. Phys. B 239, 161 (1984).
See, for example, L. A. Ljusternick, The Topology of the Calculus of Variations in the Large (American Mathematical Society, Providence, 1966).
D. Finkelstein and J. Rubinstein, J. Math. Phys. 9, 1762 (1968).
A larger set of configurations is given by rescaling $U_{S}$ ( x vec ) in Eq. (12): $U_{S S¯}$ ( x vec )= $U_{S}$ ( λ x vec -λ x ${vec}_{1}$ )B ( β vec ) $U_{S}^{dag}$ ( λ x vec - λ x ${vec}_{2}$ ) $B^{dag}$ ( β vec ). We have checked that the minimax procedure still fails to converge when working within this larger configuration space.
A. Angelopoulos et al., CERN Report No. SP/84-47, 1984 (unpublished).
K. Fujii, S. Otsuki, and F. Toyoda, Kyushu Report No. KYUSHU-84-HE-5 (unpublished).
We have used the values of a and $F_{pi}$ found in Ref. 9.
When $N_{F}$ = 3, the strange-quark mass should help stabilize a saddle point in the additional directions in configuration space.

[c1] C. H. Taubes, Commun. Math. Phys. 86, 257 (1982); ibid. 86, 299 (1982).

[c2] N. S. Manton, Phys. Rev. D 28, 2019 (1983); F. R. Klinkhamer and N. S. Manton, ibid. 30, 2227 (1984).

[c3] T. H. R. Skyrme, Proc. R. Soc. London A 260, 127 (1961).

[c4] G. 't Hooft, Nucl. Phys. B 72, 461 (1974); ibid. B 75, 461 (1974).

[c5] E. Witten, Nucl. Phys. B 160, 57 (1979).

[c6] The Wess-Zumino term should be added when $N_{F}$ >= 3. However, it does not affect the energy of static configurations and we shall ignore it in this paper.

[c7] E. Witten, Nucl. Phys. B 223, 433 (1983).

[c8] A. P. Balachandran, V. P. Nair, S. G. Rajeev and A. Stern, Phys. Rev. Lett. 49, 1124 (1982).

[c9] G. Adkins, C. Nappi and E. Witten, Nucl. Phys. B 228, 552 (1983).

[c10] H. Walliser and G. Eckart, Nucl. Phys. A 429, 514 (1984); A. Hayashi, G. Eckart, G. Holzwarth and H. Walliser, Phys. Lett. 147 B, 5 (1984); J. Breit and C. Nappi, Phys. Rev. Lett. 53, 889 (1984); M. Mattis and M. Karliner, Phys. Rev. D (to be published).

[c11] C. Callan and E. Witten, Nucl. Phys. B 239, 161 (1984).

[c12] See, for example, L. A. Ljusternick, The Topology of the Calculus of Variations in the Large (American Mathematical Society, Providence, 1966).

[c13] D. Finkelstein and J. Rubinstein, J. Math. Phys. 9, 1762 (1968).

[c14] A larger set of configurations is given by rescaling $U_{S}$ ( x vec ) in Eq. (12): $U_{S S¯}$ ( x vec )= $U_{S}$ ( λ x vec -λ x ${vec}_{1}$ )B ( β vec ) $U_{S}^{dag}$ ( λ x vec - λ x ${vec}_{2}$ ) $B^{dag}$ ( β vec ). We have checked that the minimax procedure still fails to converge when working within this larger configuration space.

[c15] A. Angelopoulos et al., CERN Report No. SP/84-47, 1984 (unpublished).

[c16] K. Fujii, S. Otsuki, and F. Toyoda, Kyushu Report No. KYUSHU-84-HE-5 (unpublished).

[c17] We have used the values of a and $F_{pi}$ found in Ref. 9.

[c18] When $N_{F}$ = 3, the strange-quark mass should help stabilize a saddle point in the additional directions in configuration space.

Physical Review D

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Static solutions in the vacuum sector of the Skyrme model

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