Riassunto Tesi di Laurea Magistrale
Candidato: Claudio Bonanno
Relatore: Prof. Massimo D’Elia
Investigation of the topological properties of the CP
N −1model using Monte Carlo simulations
Quantum Chromodynamics (QCD) is the physical theory that describes strong inter-actions. The theory is part of the Standard Model (SM) of particles and describes strong forces inside the general framework of Quantum Field Theories in terms of fundamental interactions between quarks and gluons. Unlike Quantum Electrodynamics, QCD is an asymptotic-free theory, id est, quarks and gluons weakly couple at high energies. This allows, in this regime, to use the perturbative expansion to study the behaviour of the the-ory. However, many other peculiar properties enjoyed by QCD emerge in the low-energy regime, where non-perturbative effects are dominant, for example quarks confinement. Another important non-perturbative property of QCD is the existence of a non-trivial topological structure. In fact, one can introduce a non-perturbative contribution in the QCD Lagrangian, called topological term (depending on the parameter θ and on the topo-logical charge Q), which is strictly connected with many interesting physical aspects such as the explanation of the U (1)a anomaly or the physics of the η0 meson and whose
con-tribution does not emerge in perturbation theory. Besides, there are some open problems concerning beyond-SM physics which are connected with the θ-dependence of QCD, such as the strong-CP violation or the dynamics of the axion. Therefore, non-perturbative QCD has been the core of numerous researches in the latest years.
It is in this context that the CPN −1 model has emerged and has gained a great impor-tance. In fact, this two-dimensional quantum field theory is an interesting toy model that shares many fundamental properties with QCD such as confinement, asymptotic freedom, θ-term and topological properties. Therefore, it has been used as a theoretical laboratory to check non-trivial non-perturbative scenarios by means of analytical calculations. In fact, unlike in QCD, in the CPN −1 model it is possible to make a systematic 1/N
ex-pansion keeping the coupling constant fixed (which is analogous to the large number of colours limit in QCD) which allows an analytical investigation of the theory even in the non-perturbative regime. Thus, the study of the CPN −1 model has led to a better
com-prehension of the mechanisms that underlie non-perturbative QCD, such as confinement or the θ-dependence of the theory (a consequence of topology).
The CPN −1 model is also interesting for numerical studies. In fact, thanks to the
de-velopment of supercomputers, numerical Monte Carlo (MC) simulations of lattice QCD have become one of the main tools used to study non-perturbative QCD. Since they present many non-trivial computational challenges, which are suffered by MC simulations of CPN −1 model too, the latter has been the object of many numerical studies. Their goal is to test new algorithms that could solve such computational problems and to verify large-N predictions obtained by analytical studies.
One of the main computational problems that is still unsolved is the critical slowing down (CSD), namely, the local algorithms usually employed in MC simulations of lattice QCD experience an exponential growth of the machine time needed to generate two decorrelated measures of topological observables (called autocorrelation time) when approaching the
continuum limit (lattice spacing a → 0). This has the practical consequence of preventing to get closer to the continuum limit beyond a certain lattice spacing threshold. The CSD is a consequence of the existence of a non-trivial topological structure in the theory; for this reason this problem affects MC simulations of lattice CPN −1 too.
The main goal of our studies is to investigate the topological properties of the CPN −1 model using numerical Monte Carlo simulations; in particular, we aim to measure the first three terms of the θ-expansion of the vacuum free energy density f (θ), parametrized by the coefficients χ, b2 and b4, and to compare them with the large-N analytical results. To
do so, it is of utmost importance to study a new algorithm to deal with the CSD. In fact, the slowing down worsens exponentially as N grows and this fact has prevented precise numerical verifications of the large-N limit in past studies.
At first, we have implemented a Monte Carlo simulation which employs the usual local over-heat-bath algorithm in order to check its characteristics and limitations. In particu-lar, we studied the behaviour of the autocorrelation time of topological observables with N and with the lattice spacing. This study was pursued including larger values of N com-pared to past studies and exploring a different ansatz, obtaining a better mathematical modelling of the CSD.
Then, we have studied and implemented an improvement of the local algorithm in order to dampen the effects of the CSD: the simulated tempering. The main idea this algorithm is based on is to make simulations with a dynamical coupling constant g instead of a fixed one, as it is usually done. Changing the lattice coupling constant results in a change of the lattice spacing thanks to the asymptotic freedom of the theory (g → 0 =⇒ a → 0). When the system is far from the continuum limit, the algorithm correctly decorrelates the configurations and the change of g also allows finer lattice spacings to exploit it. The simulated tempering algorithm is based on rather loose hypothesis and can be easily for-mulated for a generic quantum field theory; thus, it is easily extendible to the case of lattice QCD or to other theories with a topological structure.
We made a systematic study of the characteristics and of the performances of the simu-lated tempering and compared its efficiency with the one of the local algorithm, finding a significant reduction by an order of magnitude in the autocorrelation time and a signif-icant improvement in measurement accuracy of topological observables.
Using this algorithm, along with the imaginary-θ method originally proposed in lattice QCD and first applied to the CPN −1 model in this work, we achieved precise
measure-ments of χ, b2 and b4 for various values of N . This has allowed a more precise comparison
between lattice data and large-N analytical predictions compared to past studies.
Since the use of the simulated tempering method resulted in more efficient simulations of the model and in more precise measures of topological related quantities, we plan, in the next future, to apply this algorithm to the physical case of lattice QCD. In particular, we aim to study the θ-dependence of the theory, which is related to axion physics and to the strong-CP violation problem, and the sector of hadron physics whose dynamics is influenced by topology, such as the η0 meson physics.
