Lattice study of area law for double-winding Wilson loops

We study the double-winding Wilson loops in the SU(N) Yang-Mills theory on the lattice. We discuss how the area law falloff of the double-winding Wilson loop average is modified by changing the enclosing contours C1 and C2 for various values of the number of color N. By using the strong coupling expansion, we evaluate the double-winding Wilson loop average in the lattice SU(N) Yang-Mills theory. Moreover, we compute the double-winding Wilson loop average by lattice Monte Carlo simulations for SU(2) and SU(3). We further discuss the results from the viewpoint of the Non-Abelian Stokes theorem in the higher representations.


Introduction
The Wilson loop is a gauge-invariant and important operator for the lattice study. By using a single winding Wilson loop, we investigate the static potential and the flux tube between quark and antiquark in the fundamental representation. We further investigate the evidence of the dual superconductivity such as the restricted field dominance and the magnetic monopole dominance for the string tension, and dual Meissner effect.
There still exist two promising mechanisms for quark confinement. One is the dual superconductivity [1] in which the magnetic monopole plays a dominant role for confinement. The other is the vortex picture in which the center vortex plays a relevant role for confinement [2]. Recently, Greensite and Hölwieser presented the testing method for the mechanism of confinement by using the double-winding Wilson loop which enclosing contours are in the same plain [3]. For the S U(2) case, they investigate the average of the double-winding Wilson loop made of Yang-Mills field, the center field extracted in the maximal center gauge, and the Abelian-projection field in the maximal Abelian gauge. They showed that the string tension for the minimum surface of the Wilson loop is the difference of area behavior in case of the center-projection field as well as the Yang-Mills field and the center-projection. In case of Abelian-projection field, the string tension is the sum of area behavior in the same way as the Abelian case. However, it must be examined whether replacing the Yang-Mills field with the Abelian-projected field in the Wilson loop operator leads to the correct result or not in view of the non-Abelian Stokes theorem. In this talk, we investigate the double-winding Wilson loops of SU(N) Yang-Mills theory on the lattice to know the correct behavior of the expectation values such as the gauge group dependence, the relation to N-ality, and the relation to the non-Abelian stokes theorem in view of the dual superconductivity.

Double-winding Wilson loop
We set up the double-winding Wilson loop on the lattice W(C) (see Figure 1). The contour C = C 1 ×C 2 winds once around a loop C 1 and once around a loop C 2 in the same direction, where the two coplanar loops C 1 and C 2 share one point in common. The loop C 2 lies entirely in the minimal area of the loop C 1 . The area S 1 represents the minimum area formed by C 1 , i.e., S 1 = L × L 2 + δL × (2L + L 2 + δL). The area S 2 represents the minimum areas formed by C 2 , i.e., S 2 = L 1 × L 2. For simplicity of the analysis, we sometimes use the case of δL = 0 (the center plane). The rightmost panel represents the double-winding Wilson loop with an identical contour (two identical loops, C 1 = C 2 , or δL = 0 and L 1 = L).
We investigate how the area law falloff of the double-winding Wilson loop average is modified by changing the enclosing contours C 1 and C 2 for various values of the number of color N in S U(N)-Yang-Mills theory. We first study the double-winding Wilson loop average by using the string coupling expansion. Then, we evaluate the double-winding Wilson loop average for S U(2) and S U(3) by using Monte-Carlo simulations to examine the result of the strong-coupling expansion and the calculation in the continuum theory [4].

Strong Coupling expansion
By using the strong coupling expansion [5], we evaluate the double-winding Wilson loop average in the lattice S U(N) Yang-Mills theory. For simplicity of calculation, we consider the δL = 0 case of Fig. 1. We adopt the Wilson standard action where U x,µ (∈ S U(N)) is a gauge link variable, and β = 2N/g 2 is the gauge coupling parameter. We consider the case β ≪ 1 (g ≫ 1) and the average of the double-winding Wilson loop average is  Figure 2 shows the examples which contribute to the expectation of the double-winding Wilson loop. The leading term of the expansion is given by the planner diagram that covers the minimal areas S 1 and S 2 .

SU(2) case :
The leading contribution is given by the leftmost diagram of Fig.2. The area S 2 is not fulfilled by plaquettes, and we have only contribution from area S 1 − S 2 . Thus we have the difference-of-area behavior: The right diagram of Fig.2 is the higher order term, which gives the sum-of-area behavior: where the coefficient q N c (S ) is given by As for U(1) case it should be noticed that the average of the left panel of Fig.2 vanishes in the U(1) case, and the leading term starts from the right panel. Therefore, we have the sum-of-area behavior. The leading term is given by the left panel of Fig.3.
The higher corrections are given by the center and rightmost diagrams of Fig.3 The area law is neither the neither difference-of-area behavior nor the sum-of-area behavior.

SU(4) case :
For N c = 4, the left and center diagrams in Fig.3 give the same contribution: SU(N c )Ṅ c ≥ 5 case : For N c ≥ 5, the leading diagram is interchanged, and the center diagram in Fig.3 is the leading term:

Numerical simulation
We perform the numerical simulation on the lattice by using the Wilson action. For S U(2) case, we generate 1000 configurations for 32 4 lattice with β = 2.6 by using the standard pseudo heat-bath method. For S U(3) case, we generate 1000 configurations for 24 4 lattice with β = 6.2 by using Cabibo-Marinari [8] and over-relaxation algorithms. In the measurement of the Wilson loop average, the gauge links are smeared by using the APE smearing method [9].

SU(3) case
We investigate the S U(3) case. Figure 5 shows Next we investigate the case of δL 0 (see left panel of Fig. 1). Figure 6 shows measurement of the double-winding Wilson loop average for various δL. As is the same with the case of δL = 0, W(C) deceases from W(C 1 ) (> 0) as L 1 increases. For large S 1 and S 2 , the Wilson loop average W(C) is negative and almost constant. Therefore, the double-winding Wilson loop average is independent of L 1 (S 2 ), and it follows the area law of the area enclosed by C 1 , i.e., S 1 .

Double-winding Wilson loop with an identical contour
We finally discuss the double-winding Wilson loop with an identical contour (C 1 = C 2 = C) (See the rightmost panel of Fig.1). This can be rewritten by using the Wilson loops in the irreducible representations [4] : Here the representation is specified by the Dynkin indices, e.g., the (anti)fundamental representation 3 * with the Dynkin index [0, 1], the sextet representation 6 with the Dynkin index [2,0]. If one assumes the Casimir scaling of the string tension, one can estimate the double-winding Wilson loop average, where σ F , σ R , C 2 (F), and C 2 (R) denote the string tension of the fundamental representation (F) and the representation R, and the quadratic Casimir operator of the fundamental representation and the representation R, respectively. In the calculation of the strong-coupling expansion, the Wilson loop average can be estimated for large area S . Therefore, the Wilson-loop average in the lower dimensional representation become dominant for large area S , that is, the first term in each eq(10) becomes dominant. These results are consistent with results in strong coupling expansion.
Next, we examine the relation eqs(10) and the numerical simulations. Figure 7 shows the Wilson loop averages of the single-winding Wilson loop for the fundamental representation (left panel) and the double-winding Wilson loop with identical contour (right panel) for the S U(3) case. The singlewinding-Wilson-loop average of the representation R is positive and it falls off monotonically as the area S increases. However, the double-winding Wilson loop average decreases as the area S increases, and changes the sign from positive to negative. As S further increases, the absolute value of the Wilson-loop average decreases to zero. These are consistent with eq(10b), because the second term W [2,0] (C) in eq(10b), is dominant for small S , and fall off quickly as S increases. For larger S , the dominant term is switched from the second one to the first one.

Summary and discussion
We have investigated the double-winding Wilson loop average for S U(N c ) Yang-Mills theory by using the strong coupling expansion and the lattice simulation. By using the strong coupling expansion, we obtain the difference-of-area behavior for the S U(2) case. For the S U(N) (N ≥ 3) case, however, the area law is the neither difference-of-area behavior nor sum-of-area behavior. By using numerical simulation, we have confirmed the result of the strong coupling expansion for the Wilson loop with large areas S 1 and S 2 . These results are consistent with the results from the continuum theory [4] .
We are further interested in the dual superconductivity in the higher dimensional representation of quarks. It has been pointed out that naively replacing the Yang-Mills field with the Abelian projected field cannot reproduces the correct result [3]. In order to confirm the dual superconductivity, we should go back to the non-Abelian Stokes theorem (NAST) for the higher-dimensional representation [10], and derive the Wilson-loop operator with the restricted ("Abelian") field that can reproduce the area law in the NAST. These studies will appear in near future work.