On finite energy monopole solutions in Weinberg–Salam model

Abstract

We study the problem of existence of finite energy monopole solutions in the Weinberg–Salam model starting with the most general ansatz for static axially-symmetric electroweak magnetic fields. The ansatz includes an explicit construction of field configurations with various topologies described by the monopole and Hopf charges. We introduce a unique [Formula: see text] gauge invariant definition for the electromagnetic field. It has been proved that the magnetic charge of any finite energy monopole solution must be screened at far distance. This implies nonexistence of finite energy monopole solutions with a nonzero total magnetic charge. In the case of a special axially-symmetric Dashen–Hasslacher–Neveu ansatz, we revise the structure of the sphaleron solution and show that sphaleron represents a nontrivial system of monopole and antimonopole with their centers located in one point. This is different from the known interpretation of the sphaleron as a monopole–antimonopole pair like Nambu’s “dumb-bell.” In general, the axially-symmetric magnetic field may admit a helical structure. We conjecture that such a solution exists and estimate an upper bound for its energy, [Formula: see text].