Type of dual superconductivity for the $SU(2)$ Yang--Mills theory

We investigate the type of dual superconductivity responsible for quark confinement. For this purpose, we solve the field equations of the $U(1)$ gauge-scalar model to obtain the static vortex solution in the whole range without restricting to the long-distance region. Then we use the resulting magnetic field of the vortex to fit the gauge-invariant chromoelectric field connecting a pair of quark and antiquark which was measured by numerical simulations for $SU(2)$ Yang--Mills theory on a lattice. This result improves the accuracy of the fitted value for the Ginzburg--Landau parameter to reconfirm the type I dual superconductivity for quark confinement which was claimed by preceding works based on a fitting using the Clem ansatz. Moreover, we calculate the Maxwell stress tensor to obtain the distribution of the force around the flux tube. This result suggests that the attractive force acts among chromoelectric flux tubes, in agreement with the type I dual superconductivity.


I. INTRODUCTION
In high energy physics, quark confinement is a longstanding problem to be solved in the framework of quantum field theories, especially quantum chromodynamics (QCD). The dual superconductivity picture [1] for the QCD vacuum is known as one of the most promising scenarios for quark confinement. For a review of the dual superconductivity picture, see, e.g., [2]. For this hypothesis to be realized, we must show the existence of some magnetic objects which can cause the dual Meissner effect. Then, the resulting chromofields are squeezed into the flux tube by the dual Meissner effect. This situation should be compared with the Abrikosov-Nielsen-Olesen (ANO) vortex [3] in the U (1) gauge-scalar model as a model describing the superconductor. In the context of the superconductor, in type II the repulsive force works among the vortices, while in type I the attractive force acts. The boundary of the type I and type II is called the Bogomol'nyi-Prasad-Sommerfield (BPS) limit and no forces work among the vortices. From the viewpoint of the dual superconductivity picture, the type of dual superconductor characterizes the vacuum of the Yang-Mills theory or QCD for quark confinement.
The type of dual superconductor has been investigated for a long time by fitting the chromoelectric flux obtained by lattice simulations to the magnetic field of the ANO vortex. The preceding studies [4] done in 1990's concluded that the vacuum of the Yang-Mills theory is of type II or the border of type I and type II as a dual * Electronic address: shogo.nishino@chiba-u.jp † Electronic address: kondok@faculty.chiba-u.jp ‡ Electronic address: Akihiro.shibata@kek.jp § Electronic address: skato@oyama-ct.ac.jp superconductor. In these studies, however, the fitting range was restricted to a long-distance region from the flux tube, where only the long-distance asymptotic form of the solution was used for the fitting. Recent studies [5][6][7], on the other hand, show that the vacua of the SU (2) and SU (3) Yang-Mills theories are the type I dual superconductor. In these works, the Clem ansatz [8] was used to incorporate also the short distance behavior of the flux tube. The Clem ansatz assumes an analytical form for the behavior of the complex scalar field (as the order parameter of a condensation of the Cooper pairs), which means that it still uses an approximation. In this work, we shall fit the chromoelectric flux tube to the magnetic field of the ANO vortex in the U (1) gaugescalar model without any approximations to examine the type of dual superconductor. Indeed, we determine the Ginzburg-Landau (GL) parameter by fitting the lattice data of the chromoelectric flux to the numerical solution of the ANO vortex for the whole range. The resulting value of the GL parameter reconfirms that the dual superconductivity of SU (2) Yang-Mills theory is of type I.
In addition, in order to estimate the force working among the flux tubes, we investigate the Maxwell stress force carried by a single vortex configuration. Recently, the Maxwell stress force distribution around the quarkantiquark pair was directly measured on a lattice via the gradient flow method [9]. Our results should be compared with theirs. For this purpose, we shall calculate the energy-momentum tensor originating from a single ANO vortex solution to obtain the distribution of the Maxwell stress force corresponding to the obtained value of the GL parameter. This paper is organized as follows. In Section II, we introduce the operator on a lattice for measuring the gaugeinvariant field strength generated by a pair of quark and antiquark and give the results of lattice measurements in [6]. In Section III, we give a brief review of the ANO vortex in the U (1) gauge-scalar model. Then, we discuss the type of superconductor characterized by the GL parameter. In Section IV, we explain a new method of fitting after giving a brief review of the fitting method based on the Clem ansatz adopted in the previous study [6] in order to compare our new result with the previous one. In Section V, we study the distribution of the force around the flux tube by considering the Maxwell stress tensor. In Section VI, we summarize our results.

II. OPERATOR ON A LATTICE TO MEASURE THE FLUX TUBE
In order to measure the chromofield strength F µν generated by a pair of a static quark and antiquark belonging to the fundamental representation of the gauge group G = SU (2), we use the gauge-invariant operator proposed by Giacomo, Maggiore, and Olejnik [10] using the Wilson loop operator W [U ] along a path C (L × T rectangular) with the Yang-Mills link variable U ∈ SU (2): where A ∈ su(2) stands for the gauge field of the continuum SU (2) Yang-Mills theory, which is related to the link variable U as U x,µ = exp (−igǫA µ (x)). Thus, the field strength generated by a pair of quark and antiquark F µν [U ] can be obtained by FIG.2 shows the measurements of the chromofield strength F µν [U ] at the midpoint of the qq pair for the 8 × 8 Wilson loop on the 24 4 lattice at β = 2.5 [6].
In the previous study [6], we used the new formulation of the lattice Yang-Mills theory by decomposing the gauge field U into V and X, U = XV , where V ∈ SU (2) is called the restricted link variable which has the same transformation law as the original link variable U under the gauge transformation, and X ∈ SU (2) is a remaining part called the remaining site variable which transforms in an adjoint way under the gauge transformation. The restricted like variable V plays a very important role for realizing the dual superconductor picture, since the dominant mode for quark confinement is extracted from it, for example, V induces naturally the magnetic current. See, e.g., [2] for more details. Therefore, we can define the operator ρ[V ] similar to (1) by replacing the full link variable U by the restricted link variable V : In the continuum limit ǫ → 0, ρ[V ] reduces to and therefore, we can define the chromofield strength generated by qq pair F µν [V ] for the restricted link variable V by FIG.3 shows the measurements of the restricted chromofield strength F µν [V ] in the same settings as F µν [U ] [6]. We observe that the z-component of the restricted chromoelectric field E z [V ] forms the uniform flux tube rather than E z [U ] [6,7], since the effect due to the static sources placed at a finite distance in E z [V ] is smaller than E z [U ]. Therefore, the restricted chromoelectric flux E z [V ] can be well approximated by the ANO vortex with an infinite length. Moreover, it was shown in the previous studies [6,7] that the type of dual superconductor determined only by the flux tube does not change whether we use E z [U ] or E z [V ]. By these reasons, we shall use the data of E z [V ] for fitting.
It should be noticed that we can define the magnetic current k µ induced by the chromofield F µν [V ] as with the lattice derivative ∇ ν so that the conservation law ∇ µ k µ = 0 holds [6,7]. Since the nontrivial component of the chromofield

A. The Abrikosov-Nielsen-Olesen vortex
In this subsection, we give a brief review of the U (1) gauge-scalar model with the Lagrangian density given by where λ is the coupling constant of the scalar selfinteraction, and v is the value of the magnitude |φ(x)| of the complex scalar field φ(x) at the vacuum |x| = ∞. The asterisk ( * ) denotes the complex conjugation. The field strength F µν of the U (1) gauge field A µ and the covariant derivative D µ φ of the scalar field φ are defined by where q is the charge of the scalar field φ(x). The Euler-Lagrange equations are given as where we define the electric current j µ by In order to describe the vortex solution, we introduce the cylindrical coordinate system (ρ, ϕ, z) for the spatial coordinates with unit vectors e ρ , e ϕ , and e z for the corresponding directions, and adopt a static and axisymmetric ansatz: (14) where n is an integer. Under this ansatz, the field equations (12) and (11) are cast into where a non-vanishing component j ϕ of the electric current is written as Moreover, the magnetic field B is given in the present ansatz by To determine the boundary conditions, let us consider the static energy E. The energy-momentum tensor T µν is obtained from the Lagrangian density (8) as Notice that this energy-momentum tensor is symmetric, i.e., T µν = T νµ . Then, the static energy E is obtained In what follows, we consider the energy per unit length of a vortex to avoid the divergence, since the energy density T 00 does not depend on z. The static energy E given by (20) is nonnegative E ≥ 0. The equality E = 0 holds if and only if are satisfied. Since the equation (21) is the solution of the field equations (15) and (16), we call it the vacuum solution. Therefore, we require the solution to satisfy the boundary conditions for ρ → ∞: so that the energy E does not diverge in the long-distance region ρ ≫ 1. Indeed, these boundary conditions describe that in the long-distance region, the scalar field φ(x) goes to its vacuum value |φ(∞)| = v and the gauge field A µ (x) becomes the pure gauge configuration.
In the limit ρ → 0, we assume so that the energy E does not have a short-distance divergence. Now we can clarify the meaning of the integer n by using the boundary conditions. Let us consider the magnetic flux Φ passing through the surface S bounded by a circle C with the center at the origin and the large radius ρ → ∞, which implies that the integer n corresponds to the quantization of the magnetic flux. By this reason, we call the integer n the topological charge, especially the winding number of a vortex. Motivated by the vacuum solution (21), we modify the ansatz for the gauge field A(ρ) as Moreover, in order to make the field equations dimensionless, we introduce the dimensionless variable: and redefine the profile functions as f (ρ) = f (R) and a(ρ) = a(R). Thus, the field equations (15), (16), and (17), are rewritten into where the prime ( ′ ) stands for the derivative with respect to R. The boundary conditions are also modified as We have simultaneously solved the field equations (27) The right panel of FIG.5 shows the dimensionless magnetic field b(R) corresponding to (32). Notice that the magnetic field b(R) has no short-distance divergences, which is supported by the boundary condition (23). This means that the boundary condition (23) implies the regularity of the magnetic field b(R) and the finiteness of the energy E for a short distance.

B. Type of the superconductor
In order to investigate the asymptotic forms of the profile functions in the long-distance region R ≫ 1, we introduce g and w in place of f and a as functions of R by with |g(R)|, |w(R)| ≪ 1 for R ≫ 1. Then, the field equations for g and w read The second equation (35) can be solved by using the modified Bessel function of the second kind K ν (x) as which behaves for R ≫ 1 as .
Inserting the asymptotic form (37) of w(R) into the first equation (34), we have the closed equation for g(R) The solution of this inhomogeneous equation is given by where the first term is the general solution of the homogeneous equation which is obtained by ignoring the right hand side of (39) and the second term is a particular solution of (39). In terms of the dimensionful variable ρ, g(R) behaves as which means that the damping factor of the scalar field must be distinguished by the value of λ/q. We can define two typical lengths δ and ξ by and the ratio by The length δ is called the penetration length (or depth), at which the magnitude of the magnetic field B z falls to 1/e ≃ 37% of its original value at the origin ρ = 0. The length ξ is called the coherent length because the magnitude of the scalar field |φ(x)| grows to 1 − 1/e ≃ 63% of its vacuum value v. See FIG.6. Taking into account the damping rates (or the masses) of the gauge and scalar fields, the mass of the gauge field m V = √ 2qv is larger than that of the scalar field m S = √ 2λv for κ < 1 √ 2 , while for κ > 1 √ 2 the opposite situation occurs. At the critical value κ = 1 √ 2 , the two masses m V and m S become equal: m V = m S . Therefore, the superconductor is classified by the value of the ratio κ as κ < 1 √ 2 : type I, κ = 1 √ 2 : BPS, κ > 1 √ 2 : type II.
(44) The ratio κ is called the Ginzburg-Landau (GL) parameter. The limit κ → ∞, which is realized by ξ → 0 or m S → ∞, is called the London limit.

IV. TYPE OF DUAL SUPERCONDUCTOR
To determine the type of dual superconductivity for SU (2) Yang-Mills theory, we simultaneously fit the chromoelectric field and the induced magnetic current obtained by the lattice simulation [6] (see FIG.3 and FIG.4) to the magnetic field and electric current of the n = 1 ANO vortex.
A. The previous study using the Clem ansatz In this subsection, we give a review of the approximated method of fitting with the Clem ansatz [8]. The previous studies [5][6][7] considered only the regression of the chromoelectric flux, however in this paper, we also take into account the regression of the induced magnetic current to compare with our new method. In the Clem ansatz adopted to the U (1) gauge-scalar model, the scalar profile function f (ρ) is assumed to be where ζ is a variational parameter for the core radius of the ANO vortex and ρ is the dimensionful variable ρ = R/(qv). For the profile function of the gauge field a(ρ), we introduce the new function w(ρ) by which satisfies the boundary condition a(ρ = 0) = 0. Then, the field equation (28) for the gauge field is now written as the differential equation for w: where we have defined a variable x := ρ 2 + ζ 2 . The solution is given by the modified Bessel function of the second kind K ν (z) as and hence Therefore, the magnetic field B(ρ) is given by where we have defined with the external flux Φ = 2πn/q. The electric current J(ρ) = J(ρ)e ϕ is also written as: In the present setting, the energy per unit length E can be calculated by restricting ourselves to the unit vortex with n = 1 as where we have introduced the parameter s = √ 2qvζ. Since the vortex solution is obtained by minimizing the energy with respect to the parameter s, or ζ, for a given GL parameter κ, the energy (53) must satisfy (54) Therefore, the GL parameter κ is given by In the previous study [6], we adopted the fitting only for the flux. In this paper, we adopt the fitting for the flux and current simultaneously. In what follows, we use values measured in the lattice unit, e.g., the distanceŷ = y/ǫ with a lattice spacing ǫ, the chromoelectric flux E z (ŷ) = F 34 [V ](ŷ) in (6), and the magnetic current k ϕ (ŷ) in (7). Then, we denote the set of data as (ŷ i , E z (ŷ i ), δE z (ŷ i )) for the chromoelectric field, and (ŷ j , k ϕ (ŷ j ), δk ϕ (ŷ j )) for the induced magnetic current, where δO represents the error of the measurement O.
To define the dimensionless regression functions, let us rescale the parameters β and ζ to be dimensionless by using the lattice spacing ǫ aŝ and hence the parameter α is rescaled aŝ We also rescale the magnetic field B and the electric current J asB Then, we can define the regression functions bŷ J(ρ;α,β,ζ) =αβρ ρ 2 +ζ 2 K 1 β ρ 2 +ζ 2 , (60) with the dimensionless variableρ := ρ/ǫ in the lattice unit. Then, the error functions of the regression with the weights are given by When we assume that these errors obey independent standard normal distributions, the parametersα,β, and ζ can be estimated by maximizing the log-likelihood function ℓ(α,β,ζ) for (61) and (62) defined by The GL parameter κ is determined according to (55) in terms of the estimated valuesβ ⋆ andζ ⋆ by The obtained values in the previous work [6], which can be achieved by ignoring the second term in (63) and restricting the fitting range to 2 ≤ρ ≤ 8, are given bŷ α ⋆ = 0.41 ± 0.44,β ⋆ = 0.77 ± 0.13,

B. The new method
In this subsection, we shall fit the chromoelectric flux and the magnetic current to the magnetic field and the electric current of the ANO vortex simultaneously without any approximations. The advantage of a new method could be that the value of the GL parameter κ is a direct fitting parameter unlike the case in the Clem ansatz.
Such a fitting can be done by using the regression functions B and J constructed by the solutions, f (R) and a(R), of the field equations (27) and (28) through the dimensionless magnetic field b(R) in (32) and the electric current j(R) in (31). However, there are difficulties to estimate the model parameters, when we flow the same procedure as in the previous subsection. When we construct the regression functions B and J from the numerical solutions f (R) and a(R) by solving the field equations (27) and (28), we also calculate the regression functions numerically. Indeed, it is necessary to numerically calculate the derivative in (32) separately, and this causes a large numerical error even if one obtains the solutions f (R) and a(R) with small errors. To avoid these difficulties, we reorganize the field equations to include both b(R) and j(R) as independent unknown functions by where we have decomposed the second order differential equation (28) for the gauge profile function a(R) into two independent first order differential equations (68) and (69) and one algebraic equation (70). We solve these coupled equations simultaneously. We impose the following boundary conditions for four unknown functions f (R), a(R), b(R), and j(R): From (31) and (32) , we obtain the regression functions with the dimensionless variational parametersη := qv 2 ǫ 2 τ := q 2 v 3 ǫ 3 in the lattice unit bŷ B(ρ;η,τ , κ) :=ηb(τρ; κ),Ĵ(ρ;η,τ , κ) :=ητ j(τρ; κ), whereρ := ρ/ǫ is the dimensionless variable, and κ is the GL parameter. By numerically solving (67)-(70) simultaneously and maximizing the log-likelihood function (63) with the regression functions (73) by varying the parametersη,τ , and κ, we estimate the model parametersη,τ , and κ. Note that since the coupled differential equations (67)-(70) with respect to R =τρ depends on only the GL parameter κ, the variation of the parametersτ andη does not deform the functions b(R) and j(R). Thus we obtain the results: η ⋆ = 0.0448 ± 0.0050,τ ⋆ = 0.508 ± 0.032, κ ⋆ = 0.565 ± 0.053, We further obtain the penetration δ and coherent ξ lengths defined in (42) by using the fitted values (74) and the value of the lattice spacing ǫ = 0.08320 fm at β = 2.5 for SU (2) [6], This new result shows that the vacuum of SU (2) Yang-Mills theory is of type I, κ = 0.565 ± 0.053 < 1/ √ 2 ≈ 0.707, which is consistent with the results based on the Clem ansatz (65) and (66) within errors. We find that the inclusion of the regression for the magnetic current (66) and (74) give small errors of the GL parameter κ than the excluded one (65). We also observe that the sums of squared residuals for both the flux and current in the new method become smaller than the fitting method based on the Clem ansatz. Therefore, the inclusion of the fitting for the magnetic current is important to improve the accuracy.
It should be noticed that we adopt the fitting range 0 ≤ρ ≤ 8 for the new method. We find that the effect restricting the fitting range is negligible in new method, since it appears in the order of 10 −5 . Therefore, we can use a whole data for the fitting in new method. This fact quite differs from the previous method based on the Clem ansatz. If we use the whole range 0 ≤ρ ≤ 8 to the previous method, the regression (63) gives the value of the GL parameter κ = 0.303 ± 0.07. On the other hand, for 3 ≤ρ ≤ 8, the GL parameter κ reads κ = 0.506±0.20. Thus, we can trust the fitted values obtained in the new method rather than the previous method.

V. DISTRIBUTION OF THE STRESS FORCE AROUND A VORTEX
In what follows, to clarify the difference between type I and II of dual superconductors in view of force among the chromoelectric fluxes, we consider the Maxwell stress tensor. The components of T µν defined in (19) are writ- where n is a normal vector perpendicular to the area element dS, and ∆S stands for the area of dS. See FIG.10. The left and mid panels show the situations for the ANO vortex, while the right panel shows the corresponding situation in the electromagnetism, where a pair of electric charges ±q is located at ∓∞ on the z-axis.
If we choose n to be equal to the normal vector pointing the ρ-direction, i.e., n = e ρ , the corresponding stress force F (ρ) reads Since T ρρ obeys (80) and (81), we observe that F (ρ) ·e ρ = T ρρ ∆S is always positive in type I, while always negative in type II. Therefore, we find that F (ρ) represents the attractive force for type I, while the repulsive force for type II. If we choose n as the unit vector for the ϕ-direction, n = e ϕ , the corresponding stress force F (ϕ) is written as The signature of F (ϕ) · e ϕ = T ϕϕ ∆S changes, since the signature of T ϕϕ flips at some critical value R = R * . This feature could be an artifact due to the infinite length of the ANO vortex and should be investigated in a more realistic situation. The other choice of n is to be parallel to the ANO vortex, i.e., n = e z . The corresponding stress force F (z) can be written as FIG .11 shows the distribution of the stress forces F (ρ) and F (z) in y − z plane. Therefore, F (z) represents the attractive force. Since T zz is always positive T zz > 0 due to (77), F (z) points the same direction regardless of the value of the GL parameter κ.
It should be noted that the situation of the type II superconductor is similar to the electromagnetism, see the mid and right panels of FIG.10.
Using the parameters obtained by fitting to the ANO vortex, we can reproduce the distribution of the Maxwell stress force around the flux tube, which is shown in FIG.12. This result indeed supports the type I dual superconductor for quark confinement.

VI. CONCLUSION
In this paper, we have studied the type of dual superconductivity for the SU (2) Yang-Mills theory by fitting the chromoelectric flux tube and the induced magnetic current obtained by lattice simulations to the magnetic field and the electric current produced by the ANO vortex in the U (1) gauge-scalar model.
We have reconfirmed that the vacuum of the SU (2) Yang-Mills theory is of type I as a dual superconductor with the GL parameter κ = 0.565 ± 0.053. This result of type I agrees with the preceding one [6] based on the Clem ansatz with κ = 0.38±0.23 within errors where only the regression of the chromoelectric flux was adopted. We  FIG. 12: The chromoelectric flux obtained in [6] and the distribution of the Maxwell stress forces F (ρ) and F (z) for the fitted value of the GL parameter κ = 0.565. We have taken the height of the cylinder as h = 8ǫ to correspond to the distance between the static sources. The red line (the thick line in the y − z plane) stands for the ANO vortex.
further obtained the result κ = 0.37 ± 0.20 by using the Clem ansatz including also the regression of the induced magnetic current, which shows the type I and is also consistent with our new method. We found that the new method proposed in this paper improves the accuracy of the fitting as seen from the error of the GL parameter, or the mean of squared residuals.
We also found that the approximated method based on the Clem ansatz is sensitive to the fitting range. This can be considered that the short-distance behavior of the Clem ansatz (45) is controlled by the parameter ζ of the core radius. Because the parameter ζ appears as the product with the mass parameter β, this causes the shortdistance arbitrariness or the large errors. In the new method, on the other hand, there is no short-distance arbitrarinesses, since the parameter of the core radius does not exist. Then, the effect of the fitting range is negligible: it appears in the order of 10 −5 . This fact suggests that our new method gives more reliable results than the previous one.
Moreover, we have obtained the distribution of the Maxwell stress force around the flux tube by using the obtained GL parameter. It was observed that there exists the attractive force among the chromoelectric flux tubes, which also supports the type I dual superconductor.