CHIBA-EP-249

B64 Lattice QCD

The double-winding Wilson loop operator $W(C_{1}\cup C_{2})$ is a trace of the path-ordered product of gauge link variables $U_{\ell}$ along a closed loop $C$ composed of two loops $C_{1}$ and $C_{2}$:

The double-winding Wilson loop is called coplanar if the two loops $C_{1}$ and $C_{2}$ lie in the same plane, while it is called shifted if the two loops $C_{1}$ and $C_{2}$ lie in planes parallel to the $x-t$ plane, but are displaced from one another in the transverse direction, e.g., $z$ by distance $R$, and are connected by lines running parallel to the $z$-axis to keep the gauge invariance. The coplanar case is regarded as a non-shifted $R=0$ limit of the shifted one. Note that the double-winding Wilson loop operators are defined in a gauge invariant manner, irrespective of shifted $R\not=0$ or coplanar $R=0$. See Fig.1. From the physical viewpoint, a double-winding Wilson loop operator represents the correlation between the two interacting quarkonia, namely, two interacting mesons which consist of a heavy quark and its antiquark. It can be used as a probe to study the interactions between two flux tubes connecting quark-antiquark pairs.

The preceding works GH15 ; MK17 ; KSK20 investigated the area ($S_{1}$ and $S_{2}$) dependence of the expectation value $\left<W(C_{1}\cup C_{2})\right>$ of a double-winding Wilson loop operator $W(C_{1}\cup C_{2})$ where $S_{1}$ and $S_{2}$ are respectively the minimal areas bounded by loops $C_{1}$ and $C_{2}$. In the lattice $SU(2)$ Yang-Mills theory, it has been first shown in GH15 that the coplanar double-winding Wilson loop average obeys the “difference-of-areas law” by using the strong coupling expansion and the numerical simulations on the lattice.

In the continuum $SU(N)$ Yang-Mills theory, subsequently, a double-winding, a triple-winding, and general multiple-winding Wilson loops are investigated in MK17 to show that a coplanar double-winding $SU(3)$ Wilson loop average follows a novel area law which is neither difference-of-areas law nor sum-of-areas law, and that “sum-of-areas law” is allowed for $SU(N)$ ($N\geqslant 4$), provided that the string tension obeys the Casimir scaling for quarks in the higher representations.

In the lattice $SU(N)$ Yang-Mills theory, moreover, it has been shown in KSK20 by using the strong coupling expansion and the numerical simulations on the lattice that the coplanar double-winding Wilson loop average has the $N$ dependent area law falloff: “max-of-areas law” for $N=3$ and “sum-of-areas law” for $N\geqslant 4$. Moreover, a shifted double-winding Wilson loop average as a function of the distance in a transverse direction has the long distance behavior which does not depend on the number of color $N$, while the short distance behavior depends on $N$.

In order to give better understanding on these results, in this paper, we investigate the area law falloff of the double-winding Wilson loops using the lattice gauge theory with the gauge group $Z_{N}$, which is the center group of the original color gauge group $SU(N)$. We show that the $N$ dependent area law falloffs of the double-winding $SU(N)$ Wilson loop averages can be reproduced by using the Abelian gauge theory with the corresponding center group $Z_{N}$. These results show that the center group as a subgroup of the original non-Abelian gauge group has the essential contribution for quark confinement. For obtaining the explicit expression of the double-winding Wilson loop average, we make use of the character expansion Creutz83 ; MM94 to rewrite the weight coming from the action and perform the $Z_{N}$ group integration.

Indeed, such dominance of the center group in quark confinement was shown long ago by Fröhlich in Frohlich79 which states that confinement in $Z_{N}$ lattice gauge theories implies confinement in $SU(N)$ lattice Higgs theories. According to Fröhlich’s original argument, we can extend the center group dominance to an arbitrary closed loop composed of any number of loops. Therefore, Fröhlich’s original result holds not only for the case of the single-winding Wilson loop average but also for the case of the double-winding and more general multiple-winding Wilson loop averages, beyond the ordinary single-winding Wilson loop average. Our calculations of the double-winding Wilson loop average to be performed in the approximation up to the leading order are consistent with this rigorous result, which reinforces the validity of our result of approximate calculations. In the $SU(N)$ theory, it is an involved task to evaluate the double-winding Wilson loop average, due to the non-Abelian nature of the gauge group $SU(N)$. Applying the Fröhlich’s argument, we can avoid this complexity to obtain the qualitative properties of the double-winding Wilson loop operator in the $SU(N)$ theory owing to the simplicity of the $Z_{N}$ theory.

In a Wilson loop operator $W(C):=\prod_{\ell\in C}{U}_{\ell}$, there is a single link variable ${U}_{\ell}$ on a link $\ell\in C$. For obtaining non-vanishing contributions after integration, there must be another link variable ${U}_{\ell}^{*}={U}_{\ell}^{N-1}$ for ${U}_{\ell}\ (\ell\in C)$ which is supplied from the character expansion of the weight ${e}^{S_{G}[U]}$. It should be remarked that for an Abelian group has a special property:

where $S$ is an arbitrary surface composed of a connected set of plaquettes and bounded by a loop $C$. From this special property of the Abelian group, a Wilson loop operator is converted to the product of $U_{p}$. Therefore, the leading contribution is given by the diagram in Fig.3 where the minimal area $S$ bounded by the loop $C$ is tiled by a planar set of plaquettes. For $N=2$, given in the left panel of Fig.3, especially, the link variable ${U}_{\ell}$ is real (${U}_{\ell}=\pm 1$) and hence no orientation.

We take a lattice $\Lambda$ sufficiently large so as to include these gauge-invariant operators and calculate the leading contribution. Therefore the leading contribution for the Wilson loop average $\langle W(C)\rangle$ is calculated as

where the partition function $Z_{S}$ for $\Lambda=S$ reads

Here we have used $\prod_{\ell^{\prime}\in C}{U}_{\ell^{\prime}}=\prod_{p^{\prime}\in S}{U}_{p^{\prime}}$ following from (12) in the second equality of (Center group dominance in quark confinement) and the integration formula obtained from (11):

Furthermore, we use the integration formula following from (11): for the integration on a link $\ell\in S\setminus C$,

and for the integration on a link $\ell\in C$,

Therefore, the non-zero contribution in the integration is given by terms with ${m}_{\ell}={n}_{\ell}=0$, namely terms without $U_{p}$:

where $|S|$ is the total number of plaquettes on $S$. This leading result does not have the dependence on the dimensionality $D$ and gives the exact result for $D=2$.

For $N=2$, ${U}_{\ell}$ is a real number taking the value ${U}_{\ell}=\pm 1$. Therefore, the double link variables ${({U}_{\ell})}^{2}\ (\ell\in C_{1}\cap C_{2})$ have no contribution due to ${({U}_{\ell})}^{2}=1$. Consequently, a double-winding Wilson loop operator $W(C_{1}\cup C_{2})$ reduces to a single-winding Wilson loop operator on the loop $C:=(C_{1}\cup C_{2})\setminus(C_{1}\cap C_{2})$. The leading contribution is given by the diagram in the left panel of Fig.4(a) where the area $S_{1}\setminus S_{2}$ is tiled by a planar set of plaquettes. Thus, the leading contribution for the $\langle W(C_{1}\cup C_{2})\rangle$ is calculated as the difference-of-area law:

For $N\geqslant 3$, the leading contribution is given by the diagram in the right panel of Fig.4 where the area $S_{1}\setminus S_{2}$ is tiled once and $S_{2}$ is tiled twice by a set of plaquettes which are supplied from the character expansion of the weight ${e}^{S_{G}[U]}$. Therefore, the leading contribution for $\langle W(C_{1}\cup C_{2})\rangle$ is calculated as

Here we have used $\prod_{\ell^{\prime}\in C_{1}\cup C_{2}}{U}_{\ell^{\prime}}=\prod_{p^{\prime}\in S_{2}}{U}_{p^{\prime}}^{2}\prod_{p^{\prime}\in S_{1}\setminus S_{2}}{U}_{p^{\prime}}$ following from (12) in the second equality, and (15) in the last equality. In a similar way, we obtain

In the similar way to the single-winding Wilson loop, the result of integration is given by

Especially, in the case of $N=3$, due to $c_{1}(\beta)=c_{2}(\beta)$, we find $\langle W(C_{1}\cup C_{2})\rangle\simeq{c_{1}(\beta)}^{|S_{1}|}$.

Summarizing the results,

For $N=2$, $\langle W(C_{1}\cup C_{2})\rangle$ obeys the difference-of-areas law.

For $N=3$, $\langle W(C_{1}\cup C_{2})\rangle$ obeys the max-of-areas law. This special result in the $N=3$ case is derived from the relation $c_{1}(\beta)=c_{2}(\beta)$, which holds only in the $N=3$ case and follows from the intrinsic property of the $Z_{3}$ group in the $Z_{N}$ theory: ${U}^{2}={U}^{*}$ for $U\in Z_{3}$.

For $N\geqslant 4$, the area law depends on $\beta$. In the strong coupling region $\beta\ll 1$, due to $c_{1}(\beta)\sim\mathcal{O}(\beta)$ and $c_{2}(\beta)\sim\mathcal{O}({\beta}^{2})$, $\langle W(C_{1}\cup C_{2})\rangle$ obeys the sum-of-areas law. While in the weak coupling region $\beta\gg 1$, due to $c_{1}(\beta)\sim c_{2}(\beta)$, $\langle W(C_{1}\cup C_{2})\rangle$ obeys the max-of-areas law.
This result reproduces the areas law falloff obtained by Kato, Shibata, and Kondo in KSK20 for the double-winding Wilson loop average in the lattice $SU(N)$ pure gauge model, except for the $N$-dependent coefficients reflecting the non-Abelian structure of the group $SU(N)$. The max-of-areas law for $N\geqslant 4$ in the weak coupling region is a new result, beyond the region $\beta\ll 1$ where the strong coupling expansion is effective. For the spacetime dimension $D>2$, it should be remarked that the higher order contributions become the same order as the leading contribution in the weak coupling region due to the asymptotic property $c_{n}(\beta)\sim 1$. Hence, the evaluation of the double-winding Wilson loop average only from the leading contribution is valid only in the strong coupling region $\beta\ll 1$ for $D>2$.

For the original $SU(N)$ theory, in the strong coupling expansion, the candidates of the leading contribution to the coplanar double-winding Wilson loop average are given by the two types of tiling patterns according to the $SU(N)$ group integration formula. Although the area $S_{1}\setminus S_{2}$ is tiled by the set of the single plaquettes, in one type of the tiling patterns the area $S_{2}$ is tiled by the set of $(N-2)$-fold plaquettes, while in another type the area $S_{2}$ is tiled by the set of the double plaquettes. See Fig.7 of KSK20 . For $N=2,3$, the former gives the leading contribution (The difference-of-areas law holds for $N=2$ and the max-of-areas law holds for $N=3$), while for $N\geqslant 4$, the latter gives the leading contribution (The sum-of-areas law holds for $N\geqslant 4$). This switching giving the leading contribution reproduces the difference between the cases $N=2,3$ and $N\geqslant 4$. Therefore, the specialness of the $N=3$ case comes from this switching between the two types of contributions. Note that the right figure in Fig.4 for the leading contribution in the case of $N\geqslant 3$ is $N$-independent in $Z_{N}$ gauge theory, in contrast to the $SU(N)$ gauge theory.

Additionally, taking the continuous group limit $N\to\infty$, $b_{n}(\beta)$ converges to the first-kind modified Bessel function $I_{n}(\beta)$. From the asymptotic form of the $I_{n}(\beta)$, $b_{n}(\beta)$ behaves as

and $c_{n}(\beta)$ have the $N\to\infty$ limit

Therefore, the areas law for $N\geqslant 4$ persists in the limit $N\to\infty$ i.e. the continuous group $U(1)$. This result suggests the double-winding Wilson loop average in the $U(N)$ lattice gauge model obeys the same area law as that in the $N\geqslant 4$ case of the $SU(N)$ lattice gauge model up to the leading contribution in accord with the center group dominance for the $U(N)$ gauge model, since the center group of $U(N)$ is $U(1)$.

First, the four sides of a cuboid with the area $2R(L_{2}+T)$, whose height is $R$ and top is $S_{2}$, are tiled by plaquettes. After this tiling, the integration is reduced to the case of the coplanar (non-shifted) version. Thus, this diagram gives the $R$-dependent result

where $C^{\prime}_{2}$ denotes a loop $C_{2}$ after taking the $R\to 0$ limit. Therefore, the total leading contribution for ${\langle W(C_{1}\cup C_{2})\rangle}_{R\neq 0}$ is composed of the $R$-independent contribution ${\langle W(C_{1}\cup C_{2})\rangle}_{R\neq 0}^{I}$ and $R$-dependent contribution ${\langle W(C_{1}\cup C_{2})\rangle}_{R\neq 0}^{II}$ :

This result reproduces the $R$-dependence of the shifted double-winding Wilson loop average obtained in KSK20 .

Moreover, we can evaluate the mass gap ${\Delta}(\beta)$ from a shifted double-winding Wilson loop average by considering the case of $S_{1}=S_{2}=1$ and $R\gg 1$:

according to the relation

where $C(\beta)$ is a $\beta$-dependent constant. Therefore, the non-zero mass gap ${\Delta}(\beta)$ is obtained from the result for a shifted double-winding Wilson loop average:

We use the inequality within the pure gauge theory, since we discuss only the pure gauge theory in this paper and the gauge-scalar model is the subject to be tackled in a subsequent paper. By examining the proof given in Frohlich79 , we find that the similar inequality holds also for the double-winding Wilson loop as

This inequality implies that the area law falloff of the double-winding Wilson loop average in the lattice non-Abelian gauge theory with the gauge group $G$ follows from that in the lattice Abelian gauge theory with the center gauge group $Z(G)$. In two and three dimensional spacetime, the area law falloff holds for any gauge coupling constant for the Abelian gauge theory, while in the four-dimensional spacetime the area law falloff must hold only in the strong coupling phase because the weak coupling region must be the deconfinement phase (free charge phase for the $Z_{N}$ group and the Coulomb phase for the $U(1)$ group). For the non-Abelian gauge theory in the four-dimensional spacetime, therefore, this equality can be used to study the area law falloff only in the strong gauge coupling region. Anyway, the results obtained by explicit calculations in this paper are consistent with this rigorous result.

Moreover, we evaluated the $N$-dependence of the area law falloff for the coplanar double-winding Wilson loop average ${\langle W(C_{1}\cup C_{2})\rangle}_{R=0}$ up to the leading contribution. We obtained the difference-of-areas law for $N=2$, the max-of-areas law for $N=3$, and discovered a new $\beta$-dependent result for $N\geqslant 4$ that the sum-of-areas law in the strong coupling region $\beta\ll 1$, and the max-of-areas law in the weak coupling region $\beta\gg 1$. This result reproduces the area law falloff in the lattice $SU(N)$ gauge model obtained in KSK20 . We also checked the continuous group limit $N\to\infty$, the area law of the double-winding Wilson loops for $N\geqslant 4$ persists in the lattice $U(1)$ gauge model. This result suggests that the coplanar double-winding Wilson loop average in the lattice $U(N)$ gauge model and the lattice $SU(N)\ (N\geqslant 4)$ gauge model obeys the same area law up to the leading contribution, since $U(1)$ is the center group of $U(N)$.

In addition, we investigated the shifted double-winding Wilson loop average ${\langle W(C_{1}\cup C_{2})\rangle}_{R\neq 0}$. The total leading contribution is composed of a $R$-independent contribution and a $R$-dependent one. Our result reproduces the $R$-dependent behavior of the shifted double-winding Wilson loop average in the lattice $SU(N)$ gauge model obtained in KSK20 . We also evaluated the mass gap from this leading result, and obtained the $\beta$-dependent non-zero mass gap $\Delta(\beta)$ for any finite value of $\beta$.

From the above results, we confirmed the center group dominance in reproducing the $N$-dependent area law falloffs of the double-winding Wilson loop averages in the lattice $SU(N)$ gauge theory, as suggested from the rigorous result of Fröhlich Frohlich79 . The result for the double-winding Wilson loop average for the pure gauge model presented in this paper is exact in the whole gauge coupling region only for $D=2$, while it is exact only in the strong gauge coupling region for $D>2$ which excludes the weak gauge coupling region for $D=4$.

In order to obtain the result valid even in the weak gauge coupling region for $D=4$, we investigate the lattice gauge-scalar model in a subsequent paper. This model allows the analytically connected region OS78 ; FS79 between the confinement region ($0\leqslant\beta\ll 1,K\ll 1$) and the Higgs region ($\beta\gg 1,K_{c}\leqslant K<\infty$) which includes the weak gauge coupling region (off the pure gauge region) even if the deconfinement phase exists in the weak gauge coupling region in the pure gauge theory for $D=4$. Therefore, the Abelian gauge dynamics can be relevant in understanding the confinement in the non-Abelian gauge theory for $D=4$. We will discuss the double-winding Wilson loop averages in the lattice $SU(N)$ gauge scalar theory from this viewpoint in a subsequent paper IK21b , by performing the cluster expansion adopted in OS78 ; FS79 to estimate the expectation value of the Wilson loop operator which is valid in a specific parameter region of analyticity.