• topology
• fisica teorica
• double trace deformation
• high temperature Yang-Mills theory
• SU(N) Yang-Mills theory
• phase diagram

Yang-Mills theories are gauge theories based on non-abelian gauge groups. Many theories stem from them, for example Quantum Chromodynamics (QCD), the theory of the strong interactions, based on the SU(3) gauge group. When we study this theory at high energies (ultraviolet limit) we can use the perturbative expansion because the theory is weakly coupled. On the contrary, at low energies (infrared limit) the perturbative expansion cannot be used because the theory becomes strongly coupled. In this regime intrinsically non-perturbative phenomena such as confinement, chiral symmetry breaking and theta-dependence can be studied on the lattice, by means of Monte Carlo simulations. Consider SU(N) Yang-Mills theory discretized with R^3 x S^1 topology, i.e. with a compactified direction and periodic boundary conditions imposed on that direction. This framework is equivalent to study the system at finite temperature. The high temperature regime correspond to a small value of the compactified length and vice-versa. Starting from the low temperature regime, if we squeeze too much the compactified radius, the theory undergoes a phase transition (called deconfinement phase transition) in which center symmetry is spontaneously broken: the order parameter associated to this transition is the Polyakov loop. On the lattice, a center symmetry transformation acts on all the temporal links at a given time slice by multiplying them by an element of the center of the gauge group, that in the SU(N) case is Z_N. The Polyakov loop is the ordered product of the links along time direction at fixed space point. It is related to the free energy of a quark-antiquark pair, hence to confinement and its mean value is different from zero when center symmetry is spontaneously broken. Because of this phase transition we have two different phases that are not analytically connected to each other. For this reason it is not possible to study the high temperature regime (which is weakly coupled) and get information about the low temperature one (which is strongly coupled). In this framework M. Unsal and L. Yaffe proposed a deformed action for SU(N) Yang-Mills theory, with the purpose of restoring center symmetry even at small length of the compactified radius. The deformation is proportional to the sum of the square traces of the powers of the Polyakov loop up to integer part of N/2. In this way gauge configurations with different value of the mean of the (Polyakov loop)^n are explicitly suppressed. Moreover, the authors proposed a 1-loop effective potential for the new theory, whose minima will represent the vacua of the theory. Doing so it is possible to recover center symmetry even in the high temperature regime, after the deconfinement phase transition. This theory is interesting because if the deformation avoids the spontaneous breaking of center symmetry, it would be possible to use semiclassical methods in the high temperature regime and from that obtain information on intrinsically low temperature properties, such as confinement and topology. However it was not clear if this theory possesses the same non perturbative properties of the original Yang-Mills theory. For this reason, C. Bonati, M. Cardinali and M. D’Elia investigated the relations between the deformed theory and the undeformed one by means of numerical simulations. They focused on the SU(3) gauge group case and investigated the behavior of not-center-related physical observables as functions of the new parameter introduced in the theory, for example the topological susceptibility and other quantities related to the theta-dependence. For these quantities they found a good agreement between the two theories. Within the scenario described above we place our work. In fact it is quite remarkable that lattice results obtained for SU(3) for the deformed theory show that one recovers exactly the same theta-dependence as in the confined phase of the underformed theory. What one can do is then use the deformed theory as a tool to study theta-dependence and extending the results to larger SU(N) gauge groups.
This requires greater knowledge about the deformed theory: for example, understanding better what the 1-loop effective potential predicts and eventually comparing these results with numerical simulations. Furthermore for larger SU(N) gauge groups we have that the possible breaking patterns of the
center symmetry group Z_N are more complex, including also symmetry breaking of Z_N to its non-trivial subgroups, which are present if N is not a prime number. Firstly we study more in detail the phase diagram for the SU(3) case, where we do not have partial breaking patterns. We fix the bare coupling beta and study the phase transition by means of Monte Carlo simulations and we find that it is a first order one. Then we focus on the SU(4) gauge group. Apart from the fact that the space of trace deformations extends to two independent parameters, we have that the possible breaking patterns of the center symmetry group Z_4 includes also a partial Z_4 to Z_2 breaking which corresponds to a phase differing from both the standard confined and the deconfined phase of the undeformed theory.
We construct a phase diagram of the theory based on the minima of the 1-loop effective potential. Then we compare the results with a phase diagram obtained via numerical simulations. We find a disagreement between predictions of potential and numerical simulations. In particular we find that the center symmetry can be restored by disorder even in regions where the effective potential predicts symmetry breaking. Trying to understand the reason for this discrepancy we focus on locally averaged quantities and we find that the deformed and the undeformed theory are different from a dynamical point of view. We also discuss the new results concerning the theta-dependence. Finally we make preliminary progress regarding the SU(5) gauge group, comparing the phase diagram calculated via 1-loop effective potential with the one calculated by numerical simulations.