A supersymmetric Skyrme model

Journal of High Energy Physics, Feb 2016

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.

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A supersymmetric Skyrme model

HJE A supersymmetric Skyrme model Sven Bjarke Gudnason 0 1 4 5 6 7 Muneto Nitta 0 1 2 5 6 7 Shin Sasaki 0 1 3 5 6 7 0 Keio University 1 Lanzhou 730000 , China 2 Department of Physics, and Research and Education Center for Natural Sciences 3 Department of Physics, Kitasato University 4 Institute of Modern Physics, Chinese Academy of Sciences 5 @X in the algebraic 6 F @X in the action and hence a 7 @(XF ) giving a term Construction of a supersymmetric extension of the Skyrme term was a longstanding problem because of the auxiliary propagate and cannot be eliminated, and the problem of having fourth-order time derivative terms. In this paper, we construct for the rst time a supersymmetric extension of the Skyrme term in four spacetime dimensions, in the manifestly supersymmetric super eld formalism that does not su er from the auxiliary eld 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 ,C)-valued matrix eld instead of SU(N ) for NG bosons. The solution of auxiliary elds is trivial on the canonical branch of the auxiliary model results in a fourth-order derivative term that is not the Skyrme term. For the case of SL(2,C), we nd explicitly a nontrivial solution to the algebraic auxiliary eld 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 o , which is tantamount to restricting the Skyrme eld to SU(2), then the fourthorder derivative term reduces exactly to the standard Skyrme term. Our model is the rst example of a nontrivial auxiliary eld solution in a multi-component model. Supersymmetric E ective Theories; Solitons Monopoles and Instantons - 2 3 4 5 1 2.1 2.2 3.1 3.2 3.3 3.4 1 Introduction The formalism General action Chiral symmetry breaking The supersymmetric Skyrme term Gauging the global symmetry Conclusion and discussion Introduction A fourth-order derivative term in the chiral Lagrangian Canonical branch The Dirichlet term Non-canonical branch: a supersymmetric Skyrme term The Skyrme model was rst introduced as a toy model describing baryons in a low-energy mesonic eld theory [ 1, 2 ]. Later it was shown to be the low-energy limit of large-Nc QCD [ 3, 4 ]. After this it gained popularity as a model of nuclei in the literature. It took, however, some time before the numerical calculations (and the computing power) could tackle solutions for higher baryon numbers. The breakthrough came with the introduction of the rational maps as an approximation to the real Skyrmion solution [5, 6]. These are very useful as initial guesses for numerical calculations. For vanishing pion mass, the fullerenes adequately described by the rational maps are believed to be the global minimizers of the Skyrmion energy functional. Once a pion mass of the order of the physical pion mass is introduced, the Skyrmions prefer to order themselves in a lattice of B = 4 cubes, which can be interpreted as a crystal of alpha particles [7]. Quite a few phenomenologically appealing results have been achieved in the Skyrme model; for recent works, see e.g. [8{11]. A withstanding problem of the Skyrme model, is that the binding energies naturally come out too large (by about an order of magnitude). For this reason, quite some work has been devoted to nding a BPS limit of the Skyrme model. The minimal (original) Skyrme model has a BPS bound [12], that, however, can be saturated only on the 3-sphere [13]. Recently, a di erent model has been suggested, called the BPS Skyrme model [14, 15], which has a BPS limit and many exact solutions have been found [16]. Naively, one may think that the BPS limits of the Skyrme model above are related to supersymmetry, as is the case for Abrikosov-Nielsen-Olesen vortices or for 't Hooft-Polyakov monopoles [17]. This is, however, not the case for the Skyrme model. { 1 { constraint j 1j2 + j 2j2 = 1, was put by hand for the chiral super elds 1;2.1 A more notorious problem called the auxiliary eld problem arises when considering higher-derivative models with manifest supersymmetry. The problem is that once derivatives act on the auxiliary eld, its equation of motion becomes dynamical instead of algebraic. This means that the auxiliary eld becomes propagating and cannot simply be eliminated. This problem is related to the above mentioned problem and in fact was encountered in both ref. [ 18 ] and [ 20 ]. Two situations occur. If the derivatives act on the auxiliary eld F as X@F , then the problem can be avoided by adding a total derivative of the form eld problem can be constructed. First examples of such constructions include refs. [24{29]. The manifestly supersymmetric term found in ref. [25, 26] o ers a manifestly supersymmetric class of higher-derivative theories | free (...truncated)


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Sven Bjarke Gudnason, Muneto Nitta, Shin Sasaki. A supersymmetric Skyrme model, Journal of High Energy Physics, 2016, pp. 74, Volume 2016, Issue 2, DOI: 10.1007/JHEP02(2016)074