Casimir scaling hypothesis on the nonperturbative force in QCD vs . dual superconducting scenario of confinement ∗

From the SU(2) and SU(3) lattice QCD studies of the static potential between color charges in various dimensional representations, there arises the Casimir scaling hypothesis for the intermediate distance force [ 1, 2, 3]. This hypothesis tells that the ratio of forces associated with various dimensional representations of color charges, for a given color group, is determined by that of eigenvalues of the quadratic Casimir operator for each representation. It is expected that such a hypothesis is realized for the short distance force as described by one-gluon exchange, since the coupling is proportional to the quadratic Casimir operator. However, it is hard to imagine that this property is kept until intermediate distance where the nonperturbative effects set in. If the behavior of the ratio is governed exclusively by the group theoretical factor it should be manifest in arbitrary SU(N) gauge theory. Recent studies of k-strings in SU(4) and SU(6) lattice gauge theories, however, do not support Casimir scaling [ 4, 5]. Therefore, it seems natural to consider that there are nonperturbative dynamics, rather than Casimir factor, which has inspired the Casimir scaling hypothesis for the intermediate distance force in early lattice investigations. In this paper, we show that the dual superconducting picture of the nonperturbative QCD vacuum, as practically described by the dual Ginzburg-Landau (DGL) theory, provides us an understandable idea to explain the mechanism hidden behind the lattice data. In the dual superconducting vacuum, the color-electric flux is squeezed almost one dimensional tube, called the flux tube, due to the dual Meissner effect. For the quark and the antiquark system, then the flux tube is formed between the sources, leading to the linear potential, where its slope is identified as the string tension.