On the Landau Ginzburg theory of MAG projected SU(2) lattice gauge theory

Maximal Abelian gauge fixing and subsequent Abelian projection of SU(2) lattice gauge theory defines closed trajectories of magnetic monopoles. These trajectories can be interpreted in terms of an effective scalar field theory of the MAG monopoles using the worldline representation of the functional determinants. Employing the monopole worldlines detected in the numerical simulation, we show that a scalar bound state exists. The screening mass $m$ of this state properly scales towards the continuum limit. We find m ~ 1.3 $GeV when the string tension sigma = 440 MeV is used as reference scale.


Introduction.
Lattice calculations [1]- [7] performed in the so-called "Abelian gauges" [8,9] have provided evidence that a condensate of magnetic monopoles exist in the Yang-Mills vacuum. Consequently, the vacuum expels color-electric flux by virtues of the dual Meissner effect and produces confinement. In this sense, the vacuum represents a dual superconductor. On a phenomenological level a superconductor can be described by a Ginzburg Landau theory. There have been attempts to construct the pertinent dual Ginzburg Landau theory for the QCD vacuum or to extract it from lattice gauge simulations [10]- [13]. The difficulty seems to consist in the mapping of the monopole degrees of freedom to the one of the scalar Ginzburg Landau field.
The Ginzburg Landau theory describes a complex scalar field interacting with an electromagnetic gauge field. The (dual) Abelian electromagnetic field can be, in principle, be integrated out yielding an effective theory of a complex self-interacting scalar field. In this paper, we will extract the effective scalar field theory describing the dual superconductor of the QCD vacuum with the help of lattice gauge simulations. To this end we firstly determine the ensemble of magnetic monopole loops of the vacuum by performing a lattice calculation in the maximum Abelian gauge, performing the Abelian projection and identifying the magnetic monopole loops by the method of DeGrand and Toussaint [14]. The obtained ensemble of closed (magnetic monopole) trajectories is then described in terms of an effective scalar field theory by using the worldline formalism [15,16]. Our lattice simulations will show that a scalar anti-scalar bound state survives in the continuum limit.
The Ginzburg Landau theory of MAG monopole trajectories. The central idea of the present paper is that the theory of closed monopole trajectories arising from MAG projected SU(2) lattice gauge theory is equivalent to a theory of a charged scalar field. This scalar field theory designed to describe the monopole properties of the Abelian projected SU(2) Yang-Mills theory necessarily inherits the scaling laws from the underlying SU(2) Yang-Mills theory and, in particular, the property of asymptotic freedom by construction (provided that SU(2) monopole theory properly scales towards the continuum limit). It was pointed out by Zakharov [18] that the scalar field theory which emerges from SU(2) monopole loops is an interesting candidate for avoiding the so-called fine tuning problem, which is generic in (4-dimensional) scalar field theory equipped with the standard φ 4 potential. Since scalar field theories with local interactions of the scalar field possess an infra-red fixed point, the action term is presumably not a polynomial of finite order. One might argue that the increase of complexity due to the non-local interactions prohibit the access to such theories at a practical level (e.g., the numerical simulation). However, examples of scalar theories incorporating asymptotic freedom at the expense of the non-locality of the action have been treated in the literature [19]- [21].
In order to establish the equivalence between the theory of the monopole loops and the scalar field theory, we consider the general form of the partition function of a complex scalar field Here m is the usual mass term, and M(x) is an external source, which we will specify later. V (φ † φ) describes the interaction of the scalar field. The only restriction which we impose here is that we assume the potential term V (φ † φ) to admit a Taylor expansion so that (1) can be written as Using the proper-time representation of the functional determinant in (3), i.e., the emerging heat kernel can be interpreted as the time evolution operator of a point particle, for which the usual Feynman path integral representation holds (we refer to [22] for a recent review of the world line formalism) Here we have split off the integral over the zero-modes of the path integral, d 4 x 0 , where x 0 , the so-called loop center of mass, corresponds to the average position of the loop: In order to relate the functional integral over the world lines x(τ ) in (5) to the expectation values over loop clouds, we normalize it with respect to the free theory (M = 0) and introduce where Equation (6)  ]. These definitions lead us to the compact formula The world line representation of the interacting scalar field theory (1) is obtained by inserting (8) into (2). Thereby, the interaction of the scalar field V (φ † φ) gives rise to an effective interaction of the loops V (x(τ )), i.e., (9) is in principle straightforward.
In this paper, we propose to determine the closed monopole loops of the Yang-Mills vacuum using lattice gauge simulations (see below for details). Employing the above illustrated equivalence between a world line ensemble and a scalar field theory, the effective scalar field theory underlying the dual superconductor of the Yang-Mills vacuum can be in principle extracted as suggested in [18]. It was recently observed that Monte-Carlo calculations of the loop averages (such as those in (8) and (9)) are feasible [23,24].
In order to connect properties of the monopole loops with expectation values of the scalar field theory, we study different choices of the external current M(x) in the context of the scalar field theory (2,3) and in the context of the worldline formalism (9), respectively.
To shorten the presentation, we introduce the shorthand notation Firstly, we choose and insert this ansatz into (1) yielding On the other hand, it is clear from (9) and, hence, corresponds to the probability of finding a monopole or antimonopole (depending on the orientation of the trajectory) at x 0 . We will call this quantity monopole density. Comparing (13) and (12), one identifies In order to get a first insight into the propagators of the full interacting scalar theory (1), we investigate the particular choice of the source Inserting (15) into (1), taking the derivative with respect to the currents j 1 and j 2 , respectively, we obtain the connected Green function which in view of (13) can be interpreted as the correlation function of the monopole density. The long distance behavior of this correlation function determines the screening mass of the scalar anti-scalar excitation.   Taking the derivatives with respect to the currents j 1 and j 2 of the effective action (9) in the world line formulation yields the loop representation of the above correlation function: The latter equation has a simple interpretation: the correlation function C(x 0 − y 0 ) is obtained by taking the average over all closed loops which pass through points x 0 and y 0 , respectively (see figure 1). Once the monopole world lines are at our disposal, we are able to calculate the full propagator (16) of the corresponding scalar field theory without specifying the scalar interactions V (φ † φ).

Lattice results.
In order to determine the closed worldlines of the magnetic monopoles we performed simulations on a 12 3 × 24 lattice using the Wilson action. In order to express the size of the lattice spacing in physical units, we will make use of the 1-loop scaling relation where s 0 is the parameter expressing the string tension in units of the lattice spacing for β = 2.3. The scaling relation is assumed to reproduce the string tension within the scaling window β ∈ [2.1, 2.6]. Since we are using an asymmetric lattice, we have checked this assumption by calculating the quark anti-quark potential V (r) for the above lattice size. For this purpose, we calculated the Polyakov loop correlation function P (r) as function of the quark anti-quark distance r, i.e., For this task, we used the 2-level Lüscher-Weisz method [25]. The averages at level 1 were performed using 50 iterations while 10 evaluations were employed for the averages at level 2. 150 independent 2-level measurements were performed to yield the accuracy of the data shown in figure 1 (right panel). The raw data were fitted to the potential form Fit parameters have been the parameter s 0 in (18) and the offset of the potential for each β value. We finally obtained: This value for s 0 is in reasonable agreement with the known value 0.125 for symmetric lattices [26]. The agreement between the lattice data and the model potential (20) is very good (see figure 1). In the following, we will use a string tension σ = 440 MeV as reference scale.
The Maximal Abelian gauge (MAG) condition, i.e. {x},µ where U Ω µ (x) = Ω(x) U µ (x) Ω † (x+µ) are the gauge transformed link variables, is implemented by employing a standard iteration over-relaxation algorithm. We do not expect that our numerical procedure locates the global maximum of the non-linear functional (22). Choosing different sets of local maxima of (22) implies that different gauge conditions are implemented (see e.g. [27] for a more detailed study of the issue gauge fixing ambiguities). We stress that the properties of the monopoles corresponding to these different gauges might turn out to be different. Rather than pursuing a detailed study of the effects of these so-called Gribov ambiguities, the aim of the present paper is to show that a monopole anti-monopole bound state exists at least for a specific choice of the gauge (MAG, using an iteration over-relaxation algorithm to install the  gauge condition). Once the MAG is implemented, the SU(2) gauge theory is projected onto a compact U(1) gauge theory by the usual prescription

m=1.3(4) GeV
Once the compact U(1) gauge theory is at our disposal, we use the standard method of DeGrand and Toussaint [14] to extract the closed monopole trajectories.
In order to obtain the screening mass m of the scalar bound state, instead of (17) we consider the related correlation function where The correlation function C(t) is obtained from the monopole trajectories as follows: the number n t of monopoles is counted for a given time slice. The correlation function provides the disconnected counterpart of the Green function (24). The function can be well represented by the fit function where we used the fact that the disconnected correlation function asymptotically (t ≫ 1/m) approaches the monopole density squared.
After thermalization, we performed 100 measurements which were separated by 15 dummy sweeps to reduce the auto-correlations. For each β, we extract ma from a fit of (27) to the numerical data for C dis (t) (26). The figure 2 shows the normalized connected correlation function where the value of t (in physical units), t = a(β) N t , is obtained by using the scaling relation (18). We observe that the data points obtained from several β values fall on top of the same curve. This signals that the screening mass extrapolates to the continuum limit. Using a string tension σ = (440 MeV) 2 , we find a mass m ≈ 1.3 GeV, which is of order of the mass of the low lying glueballs.