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Abstract
Force potential exerting between two classical static sources of pure non-abelian gauge theory in the Coulomb gauge is reconsidered at a periodic/twisted box of size $L^3$. Its perturbative behavior is examined by the short-distance expansion as well as by the derivative expansion. The latter expansion to one-loop order confirms the well-known change in the effective coupling constant at the Coulomb part as well as the Uehling potential while the former is given by the convolution of two Coulomb Green functions being non-singular at $\bm{x}=\bm{y}$. The effect of the twist comes in through its Green function of the sector.
Highlights
The force potential between two static classical sources is a classic object in quantum field theory since Yukawa
In the theory where the gauge principle is operating, the computation of this quantity at the Coulomb gauge is a most straightforward one as the Coulomb potential is present in the interaction Hamiltonian as its instantaneous part
Non-Abelian gauge theory formulated in a finite box has been exploited in several directions both for the periodic boundary condition and for the twisted boundary conditions [13,14,15,16,17,18,19], combining them with several approximations
Summary
Non-Abelian gauge theory formulated in a finite box has been exploited in several directions both for the periodic boundary condition (see, for example, [12]) and for the twisted boundary conditions [13,14,15,16,17,18,19], combining them with several approximations.. The goal of this paper is rather modest: we will reexamine the force potential of the non-Abelian gauge theory in the Coulomb gauge at a finite periodic as well as twisted box of size L3 and determine its form both in the derivative expansion and in the short-distance expansion to one-loop order in old-fashioned perturbation theory. We deal with the non-Abelian case to confirm the asymptotic freedom from the effective coupling constant and to obtain the Uehling potential (see, for example, [23]) at the derivative expansion to one-loop order. We briefly conclude our results in the bigger perspective
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