Abstract
We study static solutions of the standard Skyrme model with a pion mass. Using approximately 105 pseudo-random initial configurations made of single Skyrmions in the non-symmetrized product Ansatz and an automatic detection of repeated solutions, we find 409 local energy minimizers (Skyrmions) of the model with baryon numbers 1 through 16, of which 383 are new. In particular, we find new solutions for baryon numbers 5, 8, 9, 10, 11, 12, 13, 14, 15, and 16. Our results for the number of solutions per baryon number suggest that this number could grow either polynomially or exponentially. We identify new families of solutions: sheets of Skyrmions in synchronized and antisynchronized hexagonal layers (which we call graphene); chains of 2- and 3-tori; chain-like solutions containing a hinge and many clustered Skyrmions. Contrary to common lore, only the B = 12 global energy minimizer is made of alpha particles or some chunk of a cubic crystal, whereas the B = 9, 11, 14, 15 minimizers contain the B = 7 icosahedrally symmetric Skyrmion as a component. The B = 10, 13, 16 are symmetric graphene-like solutions. We find B = 5 and B = 8 minimizers with numerically indistinguishable energies. The B = 8 candidates are the chain of two cubes, which is a chunk of the cubic Skyrme crystal and the fullerene-type ball found originally by the rational map approximation. The B = 5 global minimizer is either the well-known D2d symmetric fullerene or a new C2v symmetric solution. Finally, our findings show a large number of solutions have no discrete symmetries or just one symmetry, contrary to the common lore that Skyrmions are highly symmetric configurations.
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References
T.H.R. Skyrme, A nonlinear field theory, Proc. Roy. Soc. Lond. A 260 (1961) 127.
T.H.R. Skyrme, A Unified Field Theory of Mesons and Baryons, Nucl. Phys. 31 (1962) 556 [INSPIRE].
G.S. Adkins, C.R. Nappi and E. Witten, Static Properties of Nucleons in the Skyrme Model, Nucl. Phys. B 228 (1983) 552 [INSPIRE].
I. Zahed and G.E. Brown, The Skyrme Model, Phys. Rept. 142 (1986) 1 [INSPIRE].
V.B. Kopeliovich and B.E. Stern, Exotic Skyrmions, JETP Lett. 45 (1987) 203 [INSPIRE].
N.S. Manton, Is the B = 2 Skyrmion Axially Symmetric?, Phys. Lett. B 192 (1987) 177 [INSPIRE].
J.J.M. Verbaarschot, Axial Symmetry of Bound Baryon Number Two Solution of the Skyrme Model, Phys. Lett. B 195 (1987) 235 [INSPIRE].
E. Braaten, S. Townsend and L. Carson, Novel Structure of Static Multi-Soliton Solutions in the Skyrme Model, Phys. Lett. B 235 (1990) 147 [INSPIRE].
R.A. Battye and P.M. Sutcliffe, Symmetric skyrmions, Phys. Rev. Lett. 79 (1997) 363 [hep-th/9702089] [INSPIRE].
R. Battye and P. Sutcliffe, Skyrmions and the pion mass, Nucl. Phys. B 705 (2005) 384 [hep-ph/0410157] [INSPIRE].
C.J. Houghton, N.S. Manton and P.M. Sutcliffe, Rational maps, monopoles and Skyrmions, Nucl. Phys. B 510 (1998) 507 [hep-th/9705151] [INSPIRE].
S. Krusch, Finkelstein-Rubinstein constraints for the Skyrme model with pion masses, Proc. Roy. Soc. Lond. A 462 (2006) 2001 [hep-th/0509094] [INSPIRE].
R. Battye and P. Sutcliffe, Skyrmions with massive pions, Phys. Rev. C 73 (2006) 055205 [hep-th/0602220] [INSPIRE].
R. Battye, N.S. Manton and P. Sutcliffe, Skyrmions and the alpha-particle model of nuclei, Proc. Roy. Soc. Lond. A 463 (2007) 261 [hep-th/0605284] [INSPIRE].
D.T.J. Feist, P.H.C. Lau and N.S. Manton, Skyrmions up to Baryon Number 108, Phys. Rev. D 87 (2013) 085034 [arXiv:1210.1712] [INSPIRE].
P.H.C. Lau and N.S. Manton, States of Carbon-12 in the Skyrme Model, Phys. Rev. Lett. 113 (2014) 232503 [arXiv:1408.6680] [INSPIRE].
C.J. Halcrow, C. King and N.S. Manton, A dynamical α-cluster model of 16O, Phys. Rev. C 95 (2017) 031303 [arXiv:1608.05048] [INSPIRE].
N.S. Manton, Lightly Bound Skyrmions, Tetrahedra and Magic Numbers, arXiv:1707.04073 [INSPIRE].
C.J. Halcrow, C. King and N.S. Manton, Oxygen-16 Spectrum from Tetrahedral Vibrations and their Rotational Excitations, Int. J. Mod. Phys. E 28 (2019) 1950026 [arXiv:1902.09424] [INSPIRE].
N.S. Manton, Evidence for Tetrahedral Structure of Calcium-40, Int. J. Mod. Phys. E 29 (2020) 2050018 [arXiv:2002.08744] [INSPIRE].
N.S. Manton, Skyrmions, Tetrahedra and Magic Numbers, Quart. J. Math. Oxford Ser. 72 (2021) 735.
D. Finkelstein and J. Rubinstein, Connection between spin, statistics, and kinks, J. Math. Phys. 9 (1968) 1762 [INSPIRE].
S. Bjarke Gudnason and C. Halcrow, Vibrational modes of Skyrmions, Phys. Rev. D 98 (2018) 125010 [arXiv:1811.00562] [INSPIRE].
G.S. Adkins and C.R. Nappi, The Skyrme Model with Pion Masses, Nucl. Phys. B 233 (1984) 109 [INSPIRE].
O.V. Manko, N.S. Manton and S.W. Wood, Light nuclei as quantized skyrmions, Phys. Rev. C 76 (2007) 055203 [arXiv:0707.0868] [INSPIRE].
D. Harland, Topological energy bounds for the Skyrme and Faddeev models with massive pions, Phys. Lett. B 728 (2014) 518 [arXiv:1311.2403] [INSPIRE].
C. Adam and A. Wereszczynski, Topological energy bounds in generalized Skyrme models, Phys. Rev. D 89 (2014) 065010 [arXiv:1311.2939] [INSPIRE].
C. Adam, M. Haberichter and A. Wereszczynski, The volume of a soliton, Phys. Lett. B 754 (2016) 18 [arXiv:1511.01104] [INSPIRE].
S.B. Gudnason and J.M. Speight, Realistic classical binding energies in the ω-Skyrme model, JHEP 07 (2020) 184 [arXiv:2004.12862] [INSPIRE].
D. Harland and R.S. Ward, Chains of Skyrmions, JHEP 12 (2008) 093 [arXiv:0807.3870] [INSPIRE].
R.A. Battye and P.M. Sutcliffe, A Skyrme lattice with hexagonal symmetry, Phys. Lett. B 416 (1998) 385 [hep-th/9709221] [INSPIRE].
J. Silva Lobo and R.S. Ward, Skyrmion Multi-Walls, J. Phys. A 42 (2009) 482001 [arXiv:0910.5457] [INSPIRE].
R.M. Battye and P.M. Sutcliffe, Solitonic fullerenes, Phys. Rev. Lett. 86 (2001) 3989 [hep-th/0012215] [INSPIRE].
R.A. Battye and P.M. Sutcliffe, Skyrmions, fullerenes and rational maps, Rev. Math. Phys. 14 (2002) 29 [hep-th/0103026] [INSPIRE].
N.S. Manton and B.M.A.G. Piette, Understanding skyrmions using rational maps, Prog. Math. 201 (2001) 469 [hep-th/0008110] [INSPIRE].
S. Bolognesi, Magnetic Bags and Black Holes, Nucl. Phys. B 845 (2011) 324 [arXiv:1005.4642] [INSPIRE].
C.J. Halcrow, Vibrational quantisation of the B = 7 Skyrmion, Nucl. Phys. B 904 (2016) 106 [arXiv:1511.00682] [INSPIRE].
J. Cork and C. Halcrow, ADHM skyrmions, Nonlinearity 35 (2022) 3944 [arXiv:2110.15190] [INSPIRE].
P. Sutcliffe, Skyrmions, instantons and holography, JHEP 08 (2010) 019 [arXiv:1003.0023] [INSPIRE].
P. Sutcliffe, Skyrmions in a truncated BPS theory, JHEP 04 (2011) 045 [arXiv:1101.2402] [INSPIRE].
C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A Skyrme-type proposal for baryonic matter, Phys. Lett. B 691 (2010) 105 [arXiv:1001.4544] [INSPIRE].
C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A BPS Skyrme model and baryons at large Nc, Phys. Rev. D 82 (2010) 085015 [arXiv:1007.1567] [INSPIRE].
M. Gillard, D. Harland and M. Speight, Skyrmions with low binding energies, Nucl. Phys. B 895 (2015) 272 [arXiv:1501.05455] [INSPIRE].
S.B. Gudnason, Loosening up the Skyrme model, Phys. Rev. D 93 (2016) 065048 [arXiv:1601.05024] [INSPIRE].
S.B. Gudnason, Exploring the generalized loosely bound Skyrme model, Phys. Rev. D 98 (2018) 096018 [arXiv:1805.10898] [INSPIRE].
C. Naya and P. Sutcliffe, Skyrmions and clustering in light nuclei, Phys. Rev. Lett. 121 (2018) 232002 [arXiv:1811.02064] [INSPIRE].
M. Gillard, D. Harland, E. Kirk, B. Maybee and M. Speight, A point particle model of lightly bound skyrmions, Nucl. Phys. B 917 (2017) 286 [arXiv:1612.05481] [INSPIRE].
N.S. Manton, Classical Skyrmions: Static Solutions and Dynamics, Math. Methods Appl. Sci. 35 (2012) 1188 [arXiv:1106.1298] [INSPIRE].
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Gudnason, S.B., Halcrow, C. A Smörgåsbord of Skyrmions. J. High Energ. Phys. 2022, 117 (2022). https://doi.org/10.1007/JHEP08(2022)117
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DOI: https://doi.org/10.1007/JHEP08(2022)117