  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,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,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.
