ELLIPTIC FUNCTION METHODS IN PHI-4 THEORY AND YANG-MILLS THEORY

We present a systematic method for constructing elliptic solutions of the massive or massless φ4 and Yang-Mills (YM) theories. As input this method requires a known solution, say f(x), of the φ4 theory. Following Cervero, Jacobs, and Nohl (CJN) this known solution f(x) is multiplied by a suitable Jacobi elliptic function E(u(x)) to obtain a more general (elliptic) solution of the φ4 theory. A corresponding elliptic solution of the YM theory follows automatically from the 't Hooft-Corrigan Fairlie-Wilczek Ansatz. In our analysis E(u) can be any of the 12 Jacobi elliptic functions. The only problem is to find its argument u = u(x), which is coupled to f(x) by two partial differential equations. No general existence criteria for u(x) are established. We analyze in detail the CJN elliptic two-meron solution, with particular emphasis on the distribution of Euclidean topological charge. For arbitrary elliptic parameter k this solution represents two pointlike topological charges within a nonsingular cloud of topological charge. In the limit k → 0 the cloud vanishes and the point charges increase to half-unit strength (merons). For k → 1 the point charges vanish and the cloud becomes the instanton. This solution may describe the dissociation of an instanton into two merons. © 1979.