• Gauge
• Chromodynamics
• Multiflavour

Lattice gauge theories provide an effective description of many fundamental phenomena in every branch of physics, ranging from emerging systems in condensed matter to fundamental mechanism in the Standard Model as confinement or Higgs mechanism. The role of symmetries is decisive to characterize the main features of a theory, such as the phase diagram, the spectrum etc..
In this work we particularly focus on the interplay between global and local symmetries in the zero-temperature limit of unconventional non linear sigma models.
Indeed, by means of Monte-Carlo simulations, we investigate numerically the critical behaviour of two-dimensional statistical theories associated to real valued Multiflavour Scalar Chromodynamics (MSC).

These theories are built up starting from a matricial formulation of the $O(N_cN_f)$ model, where the rows of the $\phi^{ia}_x$ field $i=1,..,N_c$ are the coloured degrees of freedom and the columns $a=1,..,N_f$ are the flavoured ones. We then implement a $SO(N_c)$ gauge theory with the introduction of the link matrix $V^{ij}_{x,\mu}$ between nearest neighbor sites and a pure gauge kinetical operator, according to Wilsonian formulation of gauge theories on the lattice. Throughout this gauging process the starting $O(N_cN_f)$ global symmetry is reduced to a residual $O(N_f)$ global one. As a result the model's degrees of freedom are also reduced and the system belongs to the coset space $S^{M-1}/SO(N_c)$, where $S^{M-1}$ is the surface of the $M=N_cN_f$ dimensional sphere.

H = -N_f\sum_{x,\mu} \Tr \phi^t_x V_{x,\mu} \phi_{x+\mu} - \frac{\gamma}{N_c} \sum_{x, \mu>\nu} \Tr V_{x,\mu} V_{x+\mu,\nu} V^t_{x+\nu,\mu}V^t_{x,\nu}

\Tr \phi^t_x \phi_x=1

As we are dealing with two-dimensional models, the Mermin-Wagner theorem forbids the breaking of a continuous symmetry at finite temperature. For this reason we expect these theories to develop a divergent correlation length only in the limit of zero-temperature, the same limit where standard 2D non linear sigma models exhibit asymptotic freedom. This property, together with non-abelian gauge symmetry, is shared with four-dimensional QCD, the theory of strong interactions: these features invite us even more to deepen the knowledge of these lattice field theories.

Motivated also by the results obtained in recent works where the complex MSC and the Abelian-Higgs models were investigated in two-dimensions, we put forward the following general conjecture: the Renormalization-Group flow describing the asymptotic low-temperature behaviour of these theories may be controlled by the 2D statistical field theory associated with the symmetric space that has the same global symmetry of the multi-flavoured model under analysis.
According to our conjecture, the critical behaviour should be independent of the number of colours of the gauge group associated to the starting hamiltonian, whose only role is to determine the residual global symmetry that will be broken at the critical point. For the real MSC under investigation, the target universality class would be the one associated to the $RP^{N_f-1}$ models, which effectively describe a theory of projectors onto the $N_f-1$ dimensional sphere surface.

To pursue and check the mentioned hypothesis we investigate two-dimensional MSC on the lattice with the same numerical Finite-Size Scaling (FSS) strategy of the mentioned works. We associate the breaking of the residual $O(N_f)$ symmetry to the condensation of a spin-2 gauge-invariant order parameter and we define a proper Binder cumulant to classify the universality classes in the critical domain. Indeed in the FSS limit, different models exhibit the same asymptotic scaling curve at a continuous transition, if they belong to the same universality class.

As for $N_c=2$ the model is exactly mapped into the scalar Abelian-Higgs lattice model already analyzed and the cases with $N_f=2$ may present a peculiar BKT transition-like, we began our analysis from $N_c=3,4$ and $N_f=3$ (and $\gamma=0$). This will be the core of our numerical work.\\
For the case $N_c=3$ and $N_f=3$ we also discuss the relevance (in the Renormalization Group meaning of the word) of the plaquette operator for small finite values of the inverse gauge coupling $\gamma=\pm 1$. In addition, to strengthen our conjecture, we analyze the minimum energy configurations which dominate the partition function in the low-temperature limit. The analysis of the energy density associated to the plaquette operator, together with the expectation value of $\expval{\Tr P^2_x}$ (being $P^{ab}_x=\phi^{ia}_x\phi^{ib}_x$) may confirm our ansatz, making the MSC an effective field theory of projectors in the zero-temperature limit.\\
Numerical data fully support our conjecture, as will be discussed at length in the thesis.
We then draw our conclusions and discuss a few questions that remain unanswered and will be relevant for future works.