Abstract
Construction of a supersymmetric extension of the Skyrme term was a long-standing problem because of the auxiliary field problem; that is, the auxiliary field may propagate and cannot be eliminated, and the problem of having fourth-order time derivative terms. In this paper, we construct for the first time a supersymmetric extension of the Skyrme term in four spacetime dimensions, in the manifestly supersymmetric superfield formalism that does not suffer from the auxiliary field problem. Chiral symmetry breaking in supersymmetric theories results not only in Nambu-Goldstone (NG) bosons (pions) but also in the same number of quasi-NG bosons so that the low-energy theory is described by an SL(N, \( \mathrm{\mathbb{C}} \))-valued matrix field instead of SU(N) for NG bosons. The solution of auxiliary fields is trivial on the canonical branch of the auxiliary field equation, in which case our model results in a fourth-order derivative term that is not the Skyrme term. For the case of SL(2, \( \mathrm{\mathbb{C}} \)), we find explicitly a nontrivial solution to the algebraic auxiliary field equations that we call a non-canonical branch, which when substituted back into the Lagrangian gives a Skyrme-like model. If we restrict to a submanifold, where quasi-NG bosons are turned off, which is tantamount to restricting the Skyrme field to SU(2), then the fourth-order derivative term reduces exactly to the standard Skyrme term. Our model is the first example of a nontrivial auxiliary field solution in a multi-component model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T.H.R. Skyrme, A Unified Field Theory of Mesons and Baryons, Nucl. Phys. 31 (1962) 556 [INSPIRE].
T.H.R. Skyrme, A Nonlinear field theory, Proc. Roy. Soc. Lond. A 260 (1961) 127 [INSPIRE].
E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
E. Witten, Current Algebra, Baryons and Quark Confinement, Nucl. Phys. B 223 (1983) 433 [INSPIRE].
R.A. Battye and P.M. Sutcliffe, Symmetric skyrmions, Phys. Rev. Lett. 79 (1997) 363 [hep-th/9702089] [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].
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].
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 and N.S. Manton, A Skyrme model approach to the spin-orbit force, JHEP 01 (2015) 016 [arXiv:1410.0880] [INSPIRE].
M. Karliner, C. King and N.S. Manton, Electron Scattering Intensities and Patterson Functions of Skyrmions, arXiv:1510.00280 [INSPIRE].
M. Haberichter, P.H.C. Lau and N.S. Manton, Electromagnetic Transition Strengths for Light Nuclei in the Skyrme model, arXiv:1510.08811 [INSPIRE].
L.D. Faddeev, Some Comments on the Many Dimensional Solitons, Lett. Math. Phys. 1 (1976) 289 [INSPIRE].
N.S. Manton and P.J. Ruback, Skyrmions in Flat Space and Curved Space, Phys. Lett. B 181 (1986) 137 [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-N c , Phys. Rev. D 82 (2010) 085015 [arXiv:1007.1567] [INSPIRE].
C. Adam, C. Naya, J. Sanchez-Guillen, R. Vazquez and A. Wereszczynski, The Skyrme model in the BPS limit, arXiv:1511.05160 [INSPIRE].
E.B. Bogomolny, Stability of Classical Solutions, Sov. J. Nucl. Phys. 24 (1976) 449 [Yad. Fiz. 24 (1976) 861] [INSPIRE].
E.A. Bergshoeff, R.I. Nepomechie and H.J. Schnitzer, Supersymmetric Skyrmions in Four-dimensions, Nucl. Phys. B 249 (1985) 93 [INSPIRE].
B. Zumino, Supersymmetry and Kähler Manifolds, Phys. Lett. B 87 (1979) 203 [INSPIRE].
L. Freyhult, The Supersymmetric extension of the Faddeev model, Nucl. Phys. B 681 (2004) 65 [hep-th/0310261] [INSPIRE].
L.D. Faddeev and A.J. Niemi, Knots and particles, Nature 387 (1997) 58 [hep-th/9610193] [INSPIRE].
J.M. Queiruga, Skyrme-like models and supersymmetry in 3+1 dimensions, Phys. Rev. D 92 (2015) 105012 [arXiv:1508.06692] [INSPIRE].
J.M. Queiruga, private communication.
A. Karlhede, U. Lindström, M. Roček and G. Theodoridis, Supersymmetric Nonlinear Maxwell Theories and the String Effective Action, Nucl. Phys. B 294 (1987) 498 [INSPIRE].
I.L. Buchbinder, S. Kuzenko and Z. Yarevskaya, Supersymmetric effective potential: Superfield approach, Nucl. Phys. B 411 (1994) 665 [INSPIRE].
I.L. Buchbinder, S.M. Kuzenko and A. Yu. Petrov, Superfield chiral effective potential, Phys. Lett. B 321 (1994) 372 [INSPIRE].
S.J. Gates Jr., Why auxiliary fields matter: The Strange case of the 4-D, N = 1 supersymmetric QCD effective action, Phys. Lett. B 365 (1996) 132 [hep-th/9508153] [INSPIRE].
S.J. Gates Jr., Why auxiliary fields matter: The strange case of the 4-D, N = 1 supersymmetric QCD effective action. 2., Nucl. Phys. B 485 (1997) 145 [hep-th/9606109] [INSPIRE].
M. Nitta, A Note on supersymmetric WZW term in four dimensions, Mod. Phys. Lett. A 15 (2000) 2327 [hep-th/0101166] [INSPIRE].
C. Adam, J.M. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, N=1 supersymmetric extension of the baby Skyrme model, Phys. Rev. D 84 (2011) 025008 [arXiv:1105.1168] [INSPIRE].
C. Adam, J.M. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, BPS bounds in supersymmetric extensions of K field theories, Phys. Rev. D 86 (2012) 105009 [arXiv:1209.6060] [INSPIRE].
J. Khoury, J.-L. Lehners and B. Ovrut, Supersymmetric P(X,ϕ) and the Ghost Condensate, Phys. Rev. D 83 (2011) 125031 [arXiv:1012.3748] [INSPIRE].
C. Adam, J.M. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, Extended Supersymmetry and BPS solutions in baby Skyrme models, JHEP 05 (2013) 108 [arXiv:1304.0774] [INSPIRE].
M. Nitta and S. Sasaki, BPS States in Supersymmetric Chiral Models with Higher Derivative Terms, Phys. Rev. D 90 (2014) 105001 [arXiv:1406.7647] [INSPIRE].
M. Nitta and S. Sasaki, Higher Derivative Corrections to Manifestly Supersymmetric Nonlinear Realizations, Phys. Rev. D 90 (2014) 105002 [arXiv:1408.4210] [INSPIRE].
M. Nitta and S. Sasaki, Classifying BPS States in Supersymmetric Gauge Theories Coupled to Higher Derivative Chiral Models, Phys. Rev. D 91 (2015) 125025 [arXiv:1504.08123] [INSPIRE].
S. Bolognesi and W. Zakrzewski, Baby Skyrme Model, Near-BPS Approximations and Supersymmetric Extensions, Phys. Rev. D 91 (2015) 045034 [arXiv:1407.3140] [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
M. Bando, T. Kuramoto, T. Maskawa and S. Uehara, Structure of Nonlinear Realization in Supersymmetric Theories, Phys. Lett. B 138 (1984) 94 [INSPIRE].
M. Bando, T. Kuramoto, T. Maskawa and S. Uehara, Nonlinear Realization in Supersymmetric Theories, Prog. Theor. Phys. 72 (1984) 313 [INSPIRE].
M. Bando, T. Kuramoto, T. Maskawa and S. Uehara, Nonlinear Realization in Supersymmetric Theories. 2., Prog. Theor. Phys. 72 (1984) 1207 [INSPIRE].
K. Itoh, T. Kugo and H. Kunitomo, Supersymmetric Nonlinear Realization for Arbitrary Kählerian Coset Space G/H, Nucl. Phys. B 263 (1986) 295 [INSPIRE].
K. Itoh, T. Kugo and H. Kunitomo, Supersymmetric Nonlinear Lagrangians of Kählerian Coset Spaces G/H: G = E6, E7 and E8, Prog. Theor. Phys. 75 (1986) 386 [INSPIRE].
W. Lerche, On Goldstone Fields in Supersymmetric Theories, Nucl. Phys. B 238 (1984) 582 [INSPIRE].
G.M. Shore, Supersymmetric Higgs Mechanism With Nondoubled Goldstone Bosons, Nucl. Phys. B 248 (1984) 123 [INSPIRE].
A.C.W. Kotcheff and G.M. Shore, Kähler σ Models From Supersymmetric Gauge Theories, Int. J. Mod. Phys. A 4 (1989) 4391 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992), pg. 1–259.
S. Sasaki, M. Yamaguchi and D. Yokoyama, Supersymmetric DBI inflation, Phys. Lett. B 718 (2012) 1 [arXiv:1205.1353] [INSPIRE].
M. Roček and A.A. Tseytlin, Partial breaking of global D = 4 supersymmetry, constrained superfields and three-brane actions, Phys. Rev. D 59 (1999) 106001 [hep-th/9811232] [INSPIRE].
I. Antoniadis, E. Dudas and D.M. Ghilencea, Supersymmetric Models with Higher Dimensional Operators, JHEP 03 (2008) 045 [arXiv:0708.0383] [INSPIRE].
M. Gomes, J.R. Nascimento, A. Yu. Petrov and A.J. da Silva, On the effective potential in higher-derivative superfield theories, Phys. Lett. B 682 (2009) 229 [arXiv:0908.0900] [INSPIRE].
S.M. Kuzenko and S.J. Tyler, The one-loop effective potential of the Wess-Zumino model revisited, JHEP 09 (2014) 135 [arXiv:1407.5270] [INSPIRE].
G.M. Shore, Geometry of Supersymmetric σ Models, Nucl. Phys. B 320 (1989) 202 [INSPIRE].
G.M. Shore, Geometry of Supersymmetric σ Models. 2. Fermions, Connections and Currents, Nucl. Phys. B 334 (1990) 172 [INSPIRE].
K. Higashijima, M. Nitta, K. Ohta and N. Ohta, Low-energy theorems in N = 1 supersymmetric theory, Prog. Theor. Phys. 98 (1997) 1165 [hep-th/9706219] [INSPIRE].
M. Nitta, Moduli space of global symmetry in N = 1 supersymmetric theories and the quasiNambu-Goldstone bosons, Int. J. Mod. Phys. A 14 (1999) 2397 [hep-th/9805038] [INSPIRE].
M. Ostrogradsky, Memoires sur les equations differentielles relatives au probleme des isoperimetres, Mem. Ac. St. Petersbourg VI 4 (1850) 385.
M. Koehn, J.L. Lehners and B.A. Ovrut, Higher-Derivative Chiral Superfield Actions Coupled to N=1 Supergravity, Phys. Rev. D 86 (2012) 085019 [arXiv:1207.3798] [INSPIRE].
F. Farakos and A. Kehagias, Emerging Potentials in Higher-Derivative Gauged Chiral Models Coupled to N = 1 Supergravity, JHEP 11 (2012) 077 [arXiv:1207.4767] [INSPIRE].
C. Adam, C. Naya, J. Sanchez-Guillen and A. Wereszczynski, The gauged BPS baby Skyrme model, Phys. Rev. D 86 (2012) 045010 [arXiv:1205.1532] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1512.07557
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Gudnason, S.B., Nitta, M. & Sasaki, S. A supersymmetric Skyrme model. J. High Energ. Phys. 2016, 74 (2016). https://doi.org/10.1007/JHEP02(2016)074
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2016)074