# Explicit construction of constrained instantons

• Published 1999

#### Abstract

Instantons have been prominent tools for the computation of nonperturbative effects in classically conformally invariant field theories including gauge theories since the pioneering achievements of Belavin et al. [1] and ’t Hooft [2]. In the presence of mass, including mass generation because of spontaneous symmetry breaking, instantons leading to a finite action do not exist as a consequence of a generalization of Derrick’s theorem [3]. However, as pointed out by Frishman and Yankielowicz [4] and Affleck [5], a finite action solution of the field theory in question can be obtained if a constraint is imposed on the theory restricting the scale of the instantons to be small compared to the inverse mass parameter. Since then, constrained instantons have enjoyed considerable attention [6], [7], [8], [9], [10], [11]. However, little consideration has been given to a systematic explicit analytic construction of constrained instantons. In the present paper, a detailed account is given of the explicit construction of constrained instantons in the context of the two models also considered in [5], viz. φ theory with a negative potential, and SU(2) Yang-MillsHiggs theory. The latter example is especially interesting because of its relevance for the standard model of electroweak unification. The constructions in the two models are carried out recursively in the mass parameters, following the pattern indicated in [5] and in such a way that the constrained instanton solutions at short distances do not contain singularities spoiling the finiteness of the actions, while their large-distance behavior is determined by the modified Bessel function K1, thus ensuring the exponential fall off familiar from massive field theories.

### Cite this paper

@inproceedings{Nielsen1999ExplicitCO, title={Explicit construction of constrained instantons}, author={Morten Nielsen and Nanna Kj{\ae}rsgaard Nielsen}, year={1999} }