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We calculate a one loop effective action of SU (2) QCD in the presence of the monopole background, and find a possible connection between the resulting QCD effective action and a generalized Skyrme-Faddeev action of the non-linear sigma model. The result is obtained using the gauge-independent decomposotion of the gauge potential into the topological degrees which describes the non-Abelian monopoles and the local dynamical degrees of the potential, and integrating out all the dynamical degrees of QCD.

The one-loop effective action of QED obtained by Heisenberg and Euler and by Schwinger has been expressed by an asymptotic perturbative series which is divergent. In this Letter we present a nonperturbative but convergent series of the effective action. With the convergent series we establish the existence of the manifest electric-magnetic duality in the one-loop effective action of QED. DOI: 10.1103/PhysRevLett.86.1947 It has been well known that Maxwell's electrodynamics gets a quantum correction due to the electron loops. This quantum correction has first been studied by Heisenberg and Euler and by Schwinger a long time ago [1,2], and later by many others in detail [3,4]. The physics behind the quantum correction is also very well understood, and the various nonlinear effects arising from the quantum corrections (the pair production, the vacuum birefringence, the photon splitting, etc.) are being tested and confirmed by experiments [5,6].Unfortunately it is also very well known that the oneloop effective action of QED has been expressed only by a perturbative series which is divergent. For example, for a uniform magnetic field B, the Euler-Heisenberg effective action is given by [2,3]where m is the electron mass and B n is the Bernoulli number. Clearly the series (1) is an asymptotic series which is divergent [3,4]. This is not surprising. In fact, one could argue that the effective action, as a perturbative series, can be expressed only by a divergent asymptotic series [7,8]. This suggests that only a nonperturbative series could provide a convergent expression for the effective action. There have been many attempts to improve the convergence of the series with a Borel-Pade resummation. Although these attempts have made remarkable progress for various purposes, they have not produced a convergent series so far. The purpose of this Letter is to provide a nonperturbative but convergent series of the one-loop effective action of QED. Using a nonperturbative series expansion we prove that the one-loop effective action of QED can be expressed by