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
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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)