• double trace deformation
• high temperature Yang-Mills theory
• SU(3) Yang-Mills theory
• topology

Topological Properties of SU(3) Yang-Mills Theory With Double Trace Deformation} }Yang-Mills theory and QCD are characterised by the property of asymptotic freedom. When these theories are studied at high energies (ultraviolet limit) they are weakly coupled and it is possible to perform a perturbative expansion. On the contrary when they are studied at low energies (infrared limit) they become strongly coupled and perturbation theory is not reliable any more. Intrinsically non perturbative phenomena, such as confinement, chiral symmetry breaking and topology, can be studied on the lattice using Monte Carlo simulations. We consider $SU(3)$ Yang-Mills theory discretised on R^3xS ^1, i.e. with a compactified direction and periodic boundary condition imposed on that direction. This is equivalent to consider the system at finite temperature. In particular, if the length of the compactified direction decreases, the temperature increases, and vice versa. We are interested in considering the theory in the limit of small compactification radius (high temperatures). In such a limit it could be possible to use semiclassical methods to study intrinsically non-perturbative properties, such as the confinement mechanism or topology. However, if we squeeze the compactification radius too much the system undergoes a phase transition. In this phase transition centre symmetry is spontaneously broken and the Polyakov loop, the order parameter, acquires a non-zero value. A centre symmetry transformation consists in multiplying all the temporal links at a given time slice by an element of the centre of the gauge group, the centre of SU(3) being Z3. Polyakov loop is the ordered product of the links in the time direction and it is different from zero if centre symmetry is spontaneously broken. This phase transition is known in the literature as the deconfinement phase transition, since the correlator of the Polyakov loop is related to the free energy of a quark-antiquark pair and thus is related to the confining mechanism. Because of this phase transition the theory is composed of two different phases that are not analytically connected and so it is not possible to study the high temperature regime and obtain from that information about properties of the low temperature one.

In this framework M. Unsal and L. Yaffe proposed a deformed action for SU(3) Yang-Mills theory with the purpose of restoring centre symmetry even for small length of the compactified direction. Avoiding the phase transition it would be possible, in principle, to study non perturbative properties using semiclassical methods. The deformation is proportional to the square of the trace of the Polyakov loop and explicitly disadvantages gauge configurations with a different from zero value of the mean Polyakov loop, thus it is possible to restore centre symmetry even for small values of the compactification radius.

In our Master Thesis we study, using Monte Carlo simulations, the properties of this deformed theory, in order to understand if the phase in which centre symmetry is restored possesses the same characteristics of the undeformed, low temperature phase. First of all we concentrated on the computation of the mean value of Polyakov loop to check if centre symmetry is recovered in the high temperature regime. We observed that, for large enough values of the parameter coupled to the deformation, the mean value of the Polyakov loop is centred around zero and so centre symmetry is recovered. In order to see how centre symmetry is recovered we computed the adjoint Polyakov loop P^adj. This observable gives information on how the eigenvalues of the Polyakov loop are displaced. In the undeformed SU(3) YM theory P^adj is small and positive. When centre symmetry is recovered, in the deformed theory, we found that P^adj is negative and different from zero. The realisation of centre symmetry, even if unbroken in both cases, is qualitatively different in the undeformed, low temperature regime and in the deformed, high temperature one. Even though centre symmetry is realised in a slightly different way we ask ourselves if the non-perturbative properties of the two theories were equivalent or not.

To that purpose we decided to investigate the topological properties and the theta dependence of the deformed theory. We consider the Lagrangian of SU(3) YM theory with a non zero theta parameter. The theta term breaks explicitly parity and time reversal. Moreover there are quite stringent experimental bounds on it: |theta| < 10^-10. Nevertheless it enters in different aspects of the hadron phenomenology, e.g. the solution of the $U(1)_A$ problem. Monte Carlo simulations with non-zero theta in the SU(3) Lagrangian are not possible because we have an imaginary action (hence a non positive definite weight in the updating procedure), but it is possible to expand the free energy around theta = 0. The first two coefficients of such expansion are chi, the topological susceptibility, and b_2. We can relate both chi (T) and b_{2} to the momenta of the topological charge distribution computed at theta = 0. We have measured on the lattice this two observables using the deformed theory. From the numerical point of view we have to make a remark on how we computed the coefficient b_2. This is a rather noisy observable because it involves the computation of the fourth order momentum of the topological charge distribution.
If we want to measure b_2 at theta = 0, we need approximately 10^6 independent gauge configurations to get a value with an error of approx 20% and this is very expensive from the computational point of view. In order to have a good result with less numerical effort we used the imaginary theta method. We added to the Lagrangian an imaginary theta term and we performed simulation using different values of this imaginary theta term. The idea is to determine the cumulants of the topological charge as a function of the imaginary theta and then perform a combined fit of them to extract b_2. Topological susceptibility and b_2 are interesting observables for two reasons: first because they are related to the topological properties of the theory and topology is intrinsically non perturbative, second because both chi and b_2 show a different behaviour in the high temperature and in the low temperature regime. In the low temperature regime topological susceptibility is almost constant at the value obtained at zero temperature, then, after the phase transition, it falls down and it is largely suppressed. Instead the coefficient b_2 is negative and small at low-T, while, for high-T, the instanton gas approximation is valid and it becomes the predicted value of -1/12.

The question we want to answer is: are chi and b_2 computed in the deformed theory equivalent to the values obtained in the zero temperature, undeformed SU(3) YM theory?

We explored different values of the deformation and we performed simulations on lattices with different length of the compactified direction and different bare couplings. We started our simulations with the system deep in the high temperature regime and then we switched on the deformation. We noticed that when the parameter coupled to the deformation was large enough to restore centre symmetry both chi and b_2 reach a plateau and what we observed is that the values at the plateau are compatible with the zero temperature results up to values of the inverse length of the compactified direction of approximately 500 MeV (L approx 0.4 fm). This is quite a non trivial result because it means that the topological properties of the deformed theory are in agreement with the topological properties of the zero temperature SU(3) YM theory without the deformation. In the future it would be interesting to perform also simulations using lattices with a smaller compactification length in order to check if this agreement still holds. However, performing simulations on lattices with a short temporal direction is highly non trivial from the numerical point of view, since we need huge volumes in order to have a < Q^2> not too small (Q is the topological charge). In the deformed theory the instanton gas approximation (valid in the high temperature regime) is no more valid so we may ask what are the true topological objects now. In order to answer we should study the zero and near-zero modes of the spectrum of the Dirac operator.