Monopole and vortex content of a meron pair

We investigate the monopole and vortex content of a meron pair by calculating the points at which the transformation to the Laplacian Center Gauge is ill-defined and by studying the behavior of Wilson loops. These techniques reveal complementary aspects of the vortex and monopole structure, including the presence of closed monopole lines and closed vortex surfaces joining the two merons, and evidence for intersecting vortex surfaces at each meron.


Introduction
The QCD vacuum is characterized by two striking phenomena, the breaking of chiral symmetry and the confinement of color charge. Chiral symmetry breaking may be understood in terms of localized topological excitations of the gluon field and their associated quark zero-modes that produce a non-vanishing value of the chiral condensate. Classical instanton [1] solutions of the Yang Mills equations with topological charge Q = 1 and their quantum fluctuations provide a physical foundation for these topological excitations and thus a natural understanding of chiral symmetry breaking.
In contrast, the mechanism for confinement is not presently well understood, and various pictures have been investigated to try to explain it in terms of relevant structures in the QCD vacuum. Various point-like solutions to the Yang Mills equations, which fall off at large distances in all space-time dimensions, have been considered. Although Q=1 instanton solutions provide an understanding of chiral symmetry breaking, in the dilute gas and instanton liquid approximations they do not lead to confinement [2]. Merons, topological charge 1 2 solutions found by De Alfaro, Fubini and Furlan [3], are more strongly disordering objects than instantons and were proposed as a mechanism for confinement by Callan, Dashen and Gross [4]. Fractons, also solutions of the Yang Mills equations of motion with fractional topological charge, appear on the four-dimensional torus, T 4 , when twisted boundary conditions are imposed [5]. The possible relevance of these objects to confinement was pointed out in [6], and a scenario for confinement based on the fractional charge solution found in reference [7], was proposed by González-Arroyo and Martínez [8].
One and two-dimensional structures in the QCD vacuum have also been considered as mechanisms for confinement. In the dual superconductor picture [9], the condensation of monopoles in the QCD vacuum leads to confinement. Monopoles are one-dimensional curves in space-time that appear in QCD as defects in the abelian gauges proposed by 't Hooft [10]. The gauge is fixed up to the Cartan subgroup of the gauge group and monopoles appear at points in space where this gauge fixing is ill-defined, leaving a gauge freedom larger than the abelian subgroup. In the vortex theory [11], confinement is due to the condensation of vortices. Vortices are two-dimensional surfaces carrying flux in the center of the SU(N) group, which means that a Wilson loop intersecting the surface of the vortex takes the value of one of the elements of the center of the group. Classical vortex solutions to the SU(N) Yang Mills equations have been found numerically [12].
The mechanism for chiral symmetry breaking and the alternative descriptions of confinement are not mutually exclusive -rather they are highly interrelated. The fact that the intersection of two vortices has topological charge 1 2 [13][14][15] provides a provocative connection between chiral symmetry breaking and confinement and suggests that the confinement properties of charge 1 2 merons may also be understood in terms of the intersections of vortices. In addition, as elaborated below, monopole lines lie on vortex surfaces, so that both structures coexist and may be studied simultaneously. In this picture, a meron pair corresponds to the intersection of two closed vortex sheets containing closed monopole loops and provides the simplest system in which one could explore this structure quantitatively.
As the separation between the merons decreases to zero and they merge into an instanton, one would expect a vortex sheet and a monopole loop on it to shrink to a point at the center of the instanton [16,17]. A similar picture of the separation of an instanton into two fractionally charged objects connected by hedgehog world lines is given in reference [18].
In this article we investigate numerically the monopole and vortex content of a meron pair in SU(2) Yang Mills theory by calculating the points at which Laplacian Center Gauge fixing is ill-defined [19,20] and by calculating the behavior of Wilson loops. The monopole and vortex content of an isolated meron has already been studied analytically by Reinhardt and Tok [21] using Laplacian Center Gauge fixing and Wilson loops, and provides an essential foundation for the present work. Since their work, as well as that of others, has shown Laplacian Center Gauge fixing to be an imperfect tool, in this study we also explore the limitations of this tool as well as the physics of the QCD vacuum.
The outline of this letter is the following. In section 2 we describe the meron pairs that we study and in section 3 we use Wilson loops to explore their vortex content. Section 4 presents the monopole and vortex content of these configurations determined from Laplacian Center Gauge defects and section 5 summarizes our conclusions.
Merons [3] are solutions to the classical Yang-Mills equations of motion in four Euclidean dimensions, which can be written as where η aµν is the 't Hooft symbol. Using the conformal symmetry of the classical Yang-Mills action, it can be shown that in addition to a meron at the origin, there is a second meron at infinity, and these two merons may be mapped to arbitrary positions. The gauge field for the two merons [3] is This gauge field for the meron pair has infinite action density at points To avoid the problem of these singularities, we use the following expression [4] A a µ (x) = η aµν x ν Here, the singular meron fields for √ x 2 < r and √ x 2 > R are replaced by instanton caps, each containing topological charge 1 2 to agree with the topological charge carried by each meron. We study the monopole and vortex content of this configuration by putting the gauge field on a lattice of size N t ×N 3 s . For details of the procedure for putting the meron pair on the lattice and relaxing it to a solution of the field equations, see reference [22].
In this article, we analyze four meron pair configurations obtained on N t ×N 3 s lattices with N s = 16, 24 and N t = 2N s . We study configurations with different cap sizes, c, distances between merons, d, and sizes of the lattice, N s . We used a configuration with N s = 16, c = 4 and d = 10 (configuration I), and three configurations with N s = 24: one with c = 1 and d = 12 (configuration II), one with c = 5 and d = 12 (configuration III), and one with c = 1 and d = 16 (configuration IV). We have checked that the field strength from each of the lattice configurations has the essential properties described in reference [22] for the continuum field strength. We have also applied up to five cooling sweeps to the meron pair configurations in order to relax them close to lattice solutions, and checked that the monopole and vortex content for these meron pair configurations are independent of this cooling. Although we do not explicitly address Dirac zero modes in this work, note that the zero mode for a meron pair configuration has been calculated for a range of separations in reference [22] and displays two peaks at the positions of the merons.  figure 1A. We see that at short distance, the value of the Wilson loop goes from +1 towards the value −1, as for a single meron, and only changes this behavior at large distance where the contribution of the second meron starts to be significant, approaching the value +1 when the loop is bigger than the distance between the merons. We also see in figure 1A the effect of the cap size. The cap gives a characteristic size c to the meron, which is reflected in the distance one must go for the value of the Wilson loop to start to be approach −1 and thus enclose the vortex flux. Note     The results in the xt plane for the same configurations as in figure 1A, are shown in figure   1B. We see that again at short distances, the Wilson loop goes from +1 to −1 as the size of the loop increases, and as r exceeds half the separation between merons, the Wilson loop begins to approach +1, which it will reach when both merons are included. Again, the loop must be larger than the cap size, c, to enclose all the vortex flux. As before, the results for the other two planes, yt and zt, are the same, and the other configurations show analogous behavior reflecting the other cap sizes and separations.

Vortex content from Wilson loops
The conclusion from this study of Wilson loops in a meron pair is that, like an isolated meron, a meron in a pair behaves like a source or sink for flux in non-trivial elements of the center of the group for all six planes defined by the Cartesian axes, and the size of the source or sink is of order of the cap size, c. Thus, each meron corresponds to the intersection of orthogonal pairs of vortices.

Monopole and vortex content from Laplacian Center Gauge defects
In this section, we present the monopole and vortex content of the meron pair configurations described in the previous section, as inferred from the points at which gauge fixing to the Laplacian Center Gauge is ill-defined.
Fixing the gauge to Laplacian Center Gauge [19,20] involves the use of the two eigenvectors with lowest eigenvalues, ψ a 1 (n) and ψ a 2 (n), of the Laplacian operator, in the presence of a gauge field R ab (n, µ) in the adjoint representation of the gauge group.
The lowest eigenvector, ψ a 1 (n), is rotated to the (σ 3 ) direction in color space. This step fixes the gauge up to the abelian subgroup of the SU(2) group. The U(1) abelian freedom is fixed by imposing the additional condition that the ψ a 2 (n) eigenvector is rotated to lie in the positive (σ 1 , σ 3 ) half-plane. After these two steps, the gauge is completely fixed up to the center degrees of freedom.
Monopoles and vortices are found in Laplacian Center Gauge as defects of the gauge fixing procedure, which means we have to look at the points at which the gauge fixing prescription is ill-defined. The first step, rotation of the first eigenvector to the third direction in color space, is ill-defined if ψ a 1 (t, x, y, z) = 0. This defines lines in four-dimensional space and these lines are identified as monopole lines. The second step, rotation of the second eigenvector to the positive (σ 1 , σ 3 ) half-plane, is ill-defined at points at which the first and second eigenvectors are parallel. This condition defines surfaces in four dimensional space and these surfaces are identified as vortex sheets.
To fix to the Laplacian Center Gauge we use the algorithm presented in [23] to calculate the lowest eigenvectors of the Laplacian operator. We calculate the four eigenvectors with lowest eigenvalues, and find that the three lowest eigenvalues are degenerate. With two vectors chosen from these three, or from linear combinations of these three, we can fix the gauge to Laplacian Center Gauge. Note that because of the degeneracy in the lowest eigenvalues, the monopole and vortex content is ambiguously defined, and in this work we will consider all the different monopole and vortex patterns that may be obtained from the lowest eigenvectors.
Before considering the monopole and vortex content of our meron pair configurations, it is useful to review the monopole and vortex content of two limiting cases, an instanton and a single meron. The eigenfunctions of the lowest state of the Laplacian for these two cases are known analytically [21]. To obtain the locus of all the points which can be monopoles or vortices for our meron configurations, we calculate the determinant of these three vectors, I, II and III, at each lattice point. First, note that if any of the vectors is zero, the condition to find monopoles, the determinant is zero. Second, note that if there is a linear combination between them giving a zero vector, the condition to find vortices, the determinant is also zero. Finally, note that the determinant is independent under linear combinations of the three vectors.
Hence, all the points for our meron configurations that can be monopoles or vortices are determined by the condition that the determinant vanishes.
The result we obtain is the following. We find a region on the lattice in which the determinant is always positive and another one in which it is always negative, both regions separated by a three-dimensional volume in which the determinant vanishes and defines all positions that can be monopoles or vortices. We describe the shape of this volume by showing some of its two-dimensional sections. First, we show its temporal dependence.
Consider the determinant as a function of x and t, for values of y and z fixed to the values Figure 2: Figure A shows the action density S(t,x,y,z) for the meron pair with d = 16 and c = 1 (configuration IV) as a function of x and t, with y and z fixed to the values that maximize the action density. Figure B shows the absolute value of the discriminant of the three lowest Laplacian eigenvectors, D(t, x, y, z), to the 1/4 power as a function of x and t, for the same meron pair configuration and values of the y and z coordinates used in figure A. Figure C shows the absolute value of D(t, x, y, z) to the 1/4 power as a function of x and y for z fixed to the value that maximizes the action density and t fixed to the midpoint between the two merons.    surfaces joining the two merons that intersect the merons in spatial planes. We note that the high symmetry of the background field produces a highly atypical situation including, for example, the intersection of monopole loops and a high degeneracy of equivalent solutions. It is possible that the introduction of a small perturbation would not only remove the intersections and degeneracy, but also produce a more generic situation of intersecting vortices. If this is not the case, more powerful techniques will be required to fully analyze the vortex structure.
Finally, looking at the combination of the results we obtain for the meron pair from Wilson loops and Laplacian Center Gauge fixing, it is reasonable to conclude that in a pair as well as in isolation, a meron is a localized source of monopole trajectories and a localized object with topological charge 1 2 carrying center flux in six orthogonal space-space and space-time planes.