A finite temperature investigation of dual superconductivity in the modified SO(3) lattice gauge theory

We study the SO(3) lattice gauge theory in 3+1 dimensions with the adjoint Wilson action modified by a $\mathbb{Z}_2$ monopole suppression term and by means of the Pisa disorder operator. We find evidence for a finite temperature deconfinement transition driven by the condensation of U(1) magnetic charges. A finite-size scaling test shows consistency with the critical exponents of the 3D Ising model.

non-perturbative investigations carried out for Wilson [6,7] as well as for Villain discretizations [8] showed a different behavior. For the adjoint cases no evidence for a finite temperature phase transition was found, whereas a bulk transition separating the strong from the weak coupling regions appeared for both kinds of discretizations.
The bulk transition was explained in terms of a condensation of lattice artifacts -Z 2 monopoles [8]. It was argued that the finite temperature transition could be overshadowed by the bulk one and Z 2 monopole suppression terms were thus proposed [9]. More recently the Villain mixed fundamental-adjoint SU(2) model with a monopole (and vortex) suppression has been reinvestigated [10,11,12] and first evidence for a deconfinement transition in the Ising 3D universality class, at least for strong coupling (N τ = 2), was given. Moreover, in the weak coupling region negative states of the Polyakov loop in the adjoint representation were found [10,13] and later linked to the non-trivial twist sectors of the theory [14], given that on the hypertorus T 4 the partition function of the SO(3) theory in the Villain formulation with complete Z 2 monopole suppression was shown to be equivalent to that of the SU (2) theory in the fundamental representation when summed over all twist sectors [15,16,17,18].
In a recent paper we have reinvestigated the SO(3) theory using the Wilson action and a Z 2 monopole suppression term with a "chemical potential" λ [19,20,21,22]. The phase diagram in the β A − λ plane was studied at zero and finite temperature monitoring the tunneling between twist sectors and its disappearance for strong enough Z 2 monopole suppression. For λ ≥ .85 we have found a strong indication for the existence of a finite temperature deconfinement transition although having restricted the simulation to a fixed (e.g. trivial) twist sector. The proposed phase diagram is redrawn in Fig.   1. Since no proper order parameter was available a determination of critical 0 0 loop L A and the averaged electric twist variable z (see [21]).
exponents was not intended, and there was no answer given to the question about the underlying confinement mechanism.
A disorder parameter related to Abelian monopole condensation in the dual superconductivity picture of confinement [23,24,25] has been devised by the Pisa group some time ago [26,27,28,29,30]. It is the vacuum expectation value of a magnetically charged operator µ shown to be different from zero in the confined phase, thus signaling dual superconductivity, and going to zero at the deconfining phase transition. Similar parameters have been constructed more recently by Fröhlich and Marchetti [31] as well as in the framework of the lattice Schrödinger functional [32,33] leading to analogous results. The main advantage of these parameters is that they can be applied also to full QCD, where the center symmetry is explicitly broken by the fermionic degrees of freedom and -as we shall show -to the adjoint pure gauge theory, where center symmetry becomes trivial. In this letter we will use the Pisa disorder operator in order to answer the questions raised above for the SO(3) lattice gauge theory. In particular we will check the critical exponents and whether the dual superconductor scenario applies also to this case.
We will study the SU(2) adjoint representation Wilson action modified by a Z 2 monopole suppression term where the product σ c = P ∈∂c sign(Tr The Pisa disorder operator [26,27,28,29,30] was shown to be a reliable order parameter for SU(2) and SU(3) gauge theories in the fundamental repre-sentation, with and without dynamical quarks, giving critical exponents in agreement with other order parameters. Moreover, it can give important informations about the mechanism which confines quarks into hadrons. The Pisa disorder operator is motivated by the dual superconductor scenario for the QCD vacuum [23,24,25], driven by the condensation of U(1) magnetic charges. The construction of the operator in the case of the modified SO (3) theory follows the same line as in the fundamental case, so we will avoid going into the details and we will refer to the original papers for further details [26,27,28,29,30].
The idea is to construct a magnetically charged operator µ which shifts the quantum field at a given time slice by a classical external field corresponding to a magnetic monopole. The U(1) subgroup of the gauge group which defines the magnetic charge is selected by an Abelian projection, usually fixed by diagonalizing an operator X in the adjoint representation. The disorder parameter is defined as where S M (t) denotes the Wilson action with the space-time plaquettes U i4 ( x, t) at a fixed time-slice t modified by an insertion of an external monopole field with Ω the gauge transformation which diagonalizes an operator X in the adjoint representation. T a denote the generators of the Cartan subalgebra and b the discretized transverse field generated at the lattice spatial point x by a magnetic monopole sitting at y. We decided to work with the completely random Abelian projection (RAP) [29,30] in which 5 we do not diagonalize any operator X: it can be thought as a kind of averaging over a continuous infinity of Abelian projections. It should be stressed that only the plaquette contribution to the action (1) is modified by the insertion of the monopole field and not the chemical potential term. From the definition of µ, making use of an iterated change of variables, it can be shown that the correlation function D(∆t) = μ( y, t + ∆t)µ( y, t) describes the creation of a monopole at ( y, t) and its propagation from t to t + ∆t [26,27,28,29,30]. As a consequence the magnetic charge is conserved and everything is consistent. Of course, the denominator in Eq. (2) is computed using standard periodic boundary conditions. We used links in the fundamental representation, so we implemented exactly the above condition, but an analogous con-dition holds also for link variables defined in the adjoint representation, i.e.
Since µ is the average of the exponential of a sum over the physical volume, it is affected by huge fluctuations which make it difficult to be measured in Monte Carlo simulations. A way out is to compute the derivative with respect to the coupling parameter β ≡ β A , which contains all the relevant information Eq. (5) tells us that in order to have µ = 0 in a confined phase, where the dual magnetic symmetry becomes broken, ρ should stay finite for β < β c in the thermodynamical limit (i.e. in the limit of spatial lattice extent N s → ∞), while a sharp negative peak for ρ occuring at β c and diverging for N s → ∞ should signal the phase transition associated with the restoration of the dual magnetic symmetry. Above the transition a sufficient condition for µ to vanish would be to have ρ → −∞ in the thermodynamical limit. In [26,27,28,29,30] for the SU(2) case it was argued using perturbation theory that ρ for β → ∞ reaches negative plateau values linearly scaling with N s . In [34] a more detailed numerical analysis has been performed for both SU(2) and SU(3) pure gauge theories, showing that, in the case where µ is magnetically charged, ρ diverges (to negative values) linearly in the weak coupling limit, and more and more rapidly as T → T c from above, where it diverges as N 1/ν s , thus proving that µ is strictly zero for every temperature T > T c , as follows from Eq. (5). Therefore, when simulating the theory at accessibly large β values and lattice sizes, a good criterion for µ being exactly zero above the transition temperature, is that ρ keeps diverging at least linearly with N s in a wide range of β values above the transition.

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We will now present our numerical results. First of all we studied the bulk transition (compare with Fig. 1) for some λ-values between 0 (no monopole suppression) and 0.8 (partial monopole suppression) by varying β A . We do this for finite temperature (N τ = 4) to investigate the interplay between the bulk transition and the finite temperature one. Fig. 2 Fig. 4 (b) shows the quality of finite-size scaling assuming the Ising model value for the critical index. Our results are in agreement with the finite-size scaling observed for the specific heat in the Villain action case for N τ = 2 [10,11,12].
We can conclude that our investigation of the modified lattice SO(3) gauge theory with Wilson action and Z 2 monopole suppression using the Pisa disorder operator has confirmed our previous results [21]. Although having restricted to the fixed trivial twist case we find a clear indication for the existence of a finite temperature transition decoupled from the notorious bulk transition. The critical behavior at such a transition reasonably agrees with the critical exponents of the 3D Ising model. The nature of the Pisa disorder parameter we used tells us that the transition is related to a condensation of U(1) magnetic charges for T < T c which disappears above the transition, proving that the dual superconductor scenario is a good model of confinement also for the adjoint theory. Of course, a final answer can be given only after all twist sectors will be taken into account simultaneously. This study is currently under way. Moreover, it would certainly be interesting to compute also the free energy of an (extended) center vortex in order to check how the vortex condensation mechanism works in this case.
We thank A. Di Giacomo, L. Del Debbio, and P. de Forcrand for valuable comments and discussions.