Type of dual superconductivity for the SU(2) Yang–Mills theory
Abstract
We investigate the type of dual superconductivity responsible for quark confinement. For this purpose, we solve the field equations of the U(1) gaugescalar model to obtain a single static vortex solution in the whole range without restricting to the longdistance region. Then we use the resulting magnetic field of the vortex to fit the gaugeinvariant 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 the 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.
1 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, 2, 3] 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., [4]. 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 [5, 6] in the U(1) gaugescalar model as a model describing the superconductor. In the context of the superconductor, in type II the repulsive force acts 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 act 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 [7, 8, 9, 10] 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 superconductor. In these studies, however, the fitting range was restricted to a longdistance region from a flux tube. The improved studies [11, 12] concluded that the vacuum of the Yang–Mills theory can be classified as weakly type I dual superconductor. Recent studies [13, 14, 15, 16] based on the standard framework of lattice gauge theory, and studies [17, 18] based on the new formulation [19, 20], on the other hand, show that the vacua of the SU(2) and SU(3) Yang–Mills theories are strictly type I dual superconductor. In these works [13, 14, 15, 16, 17, 18], the Clem ansatz [21] was used to incorporate also the short distance behavior of a 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 in 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 acting among the flux tubes, we investigate the Maxwell stress force carried by a single vortex configuration. Recently, the Maxwell stress force distribution around a quarkantiquark pair was directly measured on a lattice via the gradient flow method [22]. Our results should be compared with theirs. For this purpose, we shall calculate the energymomentum 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 Sect. 2, we introduce an operator to measure chromofields produced by a pair of quark and antiquark on a lattice. We review the results of lattice measurements in [17]. In Sect. 3, we give a brief review of the ANO vortex in the U(1) gaugescalar model. Then, we discuss the type of superconductor characterized by the GL parameter. In Sect. 4, 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 [17] in order to compare our new result with the previous one. In Sect. 5, we study the distribution of the force around a single flux tube by considering the Maxwell stress tensor. In Sect. 6, we summarize our results. In Appendix A, we explain the advantage of the operator which we propose based on the new formulation to measure the gaugeinvariant field strength on a lattice.
2 Operator on a lattice to measure the flux tube
In Appendix A, we demonstrate advantages of using \(\rho [V]\) constructed from the restricted link variable V based on the new formulation, in sharp contrast to the preceding operator \(\rho [U]\) defined in terms of the original link variable U based on the ordinary framework of lattice gauge theory: (i) The operator \(\rho [V]\) enables us to extract the nontrivial gaugeinvariant and Abelianlike field strength which is used to measure the chromoelectrix flux, in sharp contrast to the gaugecovariant nonAbelian field strength. (ii) The operator \(\rho [V]\) does not depend on the choice of the Schwinger lines \(L, L^\dagger \), namely, the shape of \(L, L^\dagger \) and the position z at which the Schwinger lines are inserted.
Recent study [25] suggests that the operator \(\rho [U]\) undergoes nontrivial renormalizations, which depend on the length and on the number of cusps in the Schwinger lines. The study [26] suggests that the extended smearing behaves like an effective renormalization of the operator \(\rho [U]\). For \(\rho [V]\), however, such renormalizations are not necessary since \(\rho [V]\) does not depend on the Schwinger lines. On the other hand, renormalization or smearing for the restricted Wilson loop operator and the probe should be taken into account. In the previous study [17], we used the hypercubic blocking (HYP) method [27] once for the link variables on the Wilson loop to reduce highenergy noises for both U and V. However, we find numerically that for the restricted field V, the measured expectation value hardly differs from the unsmeared case.
First of all, we observe that the zcomponent of the restricted chromoelectric field \(E_{z} [V]\) forms a uniform flux tube compared with a nonuniform one \(E_{z} [U]\) [17, 18], 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 [17, 18] that the type of dual superconductor determined only by the flux tube does not change irrespective of whether we use \(E_{z} [U]\) or \(E_{z} [V]\). By these reasons, we shall use the data of \(E_{z} [V]\) for fitting.
3 The gaugescalar model and type of superconductor
3.1 The Abrikosov–Nielsen–Olesen vortex
3.2 Type of the superconductor
4 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 [17] (see Figs. 3 and 4) to the magnetic field and electric current of the \(n=1\) ANO vortex.
4.1 The previous study using the Clem ansatz
4.2 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 the new method could be that the value of the GL parameter \(\kappa \) is a direct fitting parameter unlike the case in the Clem ansatz.
This new result shows that the vacuum of SU(2) Yang–Mills theory is of type I, \(\kappa = 0.565 \pm 0.053 < 1/\sqrt{2} \approx 0.707\), which is consistent with the results based on the Clem ansatz (67) and (68) within errors. We find that the inclusion of the regression for the magnetic current (68), (70), and (78) give small errors of the GL parameter \(\kappa \) than the excluded ones (67) and (69). 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.
5 Distribution of the stress force around a vortex
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.
Our analysis on the Maxwell stress tensor around an ANO vortex agrees with the result obtained by the preceding work [22, 28, 29, 30].
6 Conclusion

We have introduced the restricted field V to extract the dominant mode for quark confinement and define the induced magnetic current in a gaugeinvariant way.

We have solved the field equations of the ANO vortex in the U(1) gaugescalar model numerically without any approximations. The previous method is based on the Clem ansatz which assumes an analytic form of the complex scalar field without solving the field equations.

We have used the resulting magnetic field and the electric current to fit respectively the chromoelectric flux tube and the induced magnetic current obtained by lattice simulations. In the previous method, only the regression for the chromoelectric flux tube was considered.

We found that the result of type I agrees with [17] reproduced and supplemented by (67)–(70). In the new method, we determined the GL parameter with good accuracy.

We have investigated the sensitivity for the fitting range. We found that the inclusion of the short range modifies the value of the GL parameter \(\kappa \) to a smaller one under the Clem ansatz. For the new method, on the other hand, we found that the inclusion or exclusion of the short range does not effect the GL parameter \(\kappa \).

We also 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 in both methods. Therefore, the inclusion of the regression of the magnetic current is important.
Footnotes
 1.
We use the notation \(\rho [U]\) to indicate the average coming from the operators defined in terms of the original link variable U, since we define the similar operator defined in terms of the different variable later.
 2.
Here, we change the sign of \(T^{j k}\) defined in (20) by using the ambiguity of the overall sign of the Noether current in order to reproduce the conventional Maxwell stress tensor.
Notes
Acknowledgements
The authors would like to thank Hideo Suganuma for valuable discussions, especially suggestions on error estimations. They would like to express sincere thanks to Ryosuke Yanagihara, Takumi Iritani, Masakiyo Kitazawa, and Tetsuo Hatsuda for very helpful and illuminating discussions on the Maxwell stress tensor in the early stage of their investigations, on which a part of the result presented in section V is based. This work was supported by GrantinAid for Scientific Research, JSPS KAKENHI Grant Number (C) No. 15K05042 and No. 19K03840. S.N. thanks Nakamura Sekizenkai for a scholarship.
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