Revisiting the Wu-Yang Monopole: classical solutions and conformal invariance

Abstract

The Wu-Yang monopole for pure SU(2) Yang-Mills theory is revisited. New classical solutions with finite energy are found for a generalized Wu-Yang configuration. Our method relies on known asymptotic series solutions and explores the conformal invariance of the theory to set the scale for the monopole configurations. Furthermore, by breaking the scale invariance of the pure Yang-Mills theory by a term which simulates the coupling to quark fields, four different monopole configurations with an energy of 2.9 GeV and spatial extent of 1.2 fm, 1.4 fm, 2.4 fm and 2.6 fm are obtained. These configurations may play a role in the transition to the plasma phase.

Non-existence theorems and solutions of the Wu-Yang monopole equation

“…Now we introduce a new function f = g + 1. This deviates from Marinho et al [42] (2010) who in our notation use g = f + 1 to obtain the same results. This new function allows us to rewrite L into the more compelling form…”

“…contains arbitrary powers of r and has the free parameter b 1 =: b. b has units of energy and so sets the energy scale of the B(r) solution. The first 10 coefficients of A(r) and B(r) are given in Marinho et al [42]. We do not agree with their assertion of signs(neither do Rosen [43] or Actor [41]).…”

“…According to Marinho et al [42], these estimates of convergence changes little when using higher order coefficients. We find that neither of the power series converge for all r > 0, but each may come arbitrarily close by choosing √ a very small or b very large in some particular length units.…”

“…Here A(r) represents those solutions which have a Tayler series expansion at r = 0 and B(r) the solutions which have an expansion at r = ∞. These expansions have been considered in the literature, consider Rosen (1972) [43], Actor (1979) [41] and for a discussion Marinho et al (2009) [42]. We have supplemented the literature with a more detailed discussion of how to find the coefficients in Appendix A 4.…”

“…As such, the introduction of a discontinuity in the derivative does not break scale invariance, except if the scales R or ζ are taken to be universal. See Marinho et al [42] for a discussion of ( 24) in the context of QCD. They suggest that the delta function may be thought of as localized static quarks at radius R with ζ as the square of the quark wave function.…”