How to extract the dominant part of the Wilson loop average in higher representations

Abstract

In previous works, we have proposed a new formulation of Yang-Mills theory on the lattice so that the so-called restricted field obtained from the gauge-covariant decomposition plays the dominant role in quark confinement. This framework improves the Abelian projection in the gaugeindependent manner. For quarks in the fundamental representation, we have demonstrated some numerical evidences for the restricted field dominance in the string tension, which means that the string tension extracted from the restricted part of the Wilson loop reproduces the string tension extracted from the original Wilson loop. However, it is known that the restricted field dominance is not observed for the Wilson loop in higher representations if the restricted part of the Wilson loop is extracted by adopting the Abelian projection or the field decomposition naively in the same way as in the fundamental representation. In this paper, therefore, we focus on confinement of quarks in higher representations. By virtue of the non-Abelian Stokes theorem for the Wilson loop operator, we propose suitable gauge-invariant operators constructed from the restricted field to reproduce the correct behavior of the original Wilson loop averages for higher representations. Moreover, we perform lattice simulations to measure the static potential for quarks in higher representations using the proposed operators. We find that the proposed operators well reproduce the behavior of the original Wilson loop average, namely, the linear part of the static potential with the correct value of the string tension, which overcomes the problem that occurs in naively applying Abelian-projection to the Wilson loop operator for higher representations.

How to extract the "Abelian" part of double-winding Wilson loop

“…There is no reason why we consider that the Abelian operator must be defined by just replacing the gauge field with the diagonal component of the gauge field naively. Such Abelian operators are found firstly for Wilson loops in the adjoint representation of SU(2) [8], and then for Wilson loops in an arbitrary representation of an arbitrary gauge group [9]. The numerical simulations confirm this claim for Wilson loops in the adjoint representation of SU(2) [8,9] and in the adjoint representation and the sextet representation of SU(3) [9].…”

“…(3.9) is only the factor 2 sin 2 θ . Therefore, we guess from this that if the operator including the difference of the measure is modified as when J = 1 in the previous studies [8,9]. Here the symbol "∼" means that the both sides decrease exponentially with the area surrounded by C at approximately the same rate.…”

“…It is known that the string tensions extracted from the Wilson loops in higher representations cannot be reproduced when the Abelian projection is applied naively. To avoid this problem, in [9], "the highest weight part of the Abelian Wilson loop" is proposed and it is numerically checked that the string tensions of several higher representations are reproduced by using this operator. In the following, we introduce another operator that reproduces the string tension in higher representations and give the reason why this operator reproduces the correct string tension [11].…”

“…Such Abelian operators are found firstly for Wilson loops in the adjoint representation of SU(2) [8], and then for Wilson loops in an arbitrary representation of an arbitrary gauge group [9]. The numerical simulations confirm this claim for Wilson loops in the adjoint representation of SU(2) [8,9] and in the adjoint representation and the sextet representation of SU(3) [9]. Another operator whose behavior is not reproduced by the operator defined by just replacing the gauge field with the diagonal component of the gauge field is a double-winding Wilson loop in the SU(2) pure gauge theory [10].…”

How to extract the "Abelian" part of double-winding Wilson loop

“…It is known that the string tensions extracted from the Wilson loops in higher representations cannot be reproduced when the Abelian projection is applied naively. To avoid this problem, in [9], "the highest weight part of the Abelian Wilson loop" is proposed and it is numerically checked that the string tensions of several higher representations are reproduced by using this operator. In the following, we introduce another operator that reproduces the string tension in higher representations and give the reason why this operator reproduces the correct string tension [11].…”