Uno studio sull’insorgenza di grandi fluttuazioni in Teoria delle Perturbazioni Numerico Stocastica

Perturbation theory is universally recognized as a fundamental tool in modern theoretical physics. In the functional integral formalism, perturbation theory provides a method for studying field theories, offering both mathematical rigor and substantial physical insights. It is challenging to name an area of theoretical physics where perturbation theory does not play a fundamental role: even in theories specifically designed for a non-perturbative approach like lattice gauge theories, perturbation theory remains relevant and interesting. Lattice gauge theories offer a powerful framework for understanding non-perturbative aspects of quantum field theories. By discretizing space-time on a lattice, these theories enable detailed Monte Carlo simulations that are crucial for probing phenomena beyond the reach of perturbation theory, shedding light on subtle features such as quark confinement in QCD and many others. In the mid-1990s, a new method was developed that in a sense integrates traditional perturbation theory with Monte Carlo simulations of lattice field theories (in particular lattice gauge theories, for which traditional diagrammatic perturbation theory is cumbersome). This approach is known as Numerical Stochastic Perturbation Theory (NSPT). NSPT offers a fully automated stochastic method for calculating loop corrections in lattice field theories, using the power of Monte Carlo simulations. Its numerical implementation requires minimal changes with respect to traditional Monte Carlo simulations; (also due to this) NSPT enables the calculation of loop corrections at very high perturbative orders. The ease of implementation and advanced capability explain why NSPT has captured the attention of lattice practitioners. A not-well explored feature of NSPT is the freedom to choose any vacuum for calculating perturbation theory, in principle without encountering the complexities of the diagrammatic perturbation theory. If one had to make a natural choice, low-dimensional models are the best candidates for exploratory analysis of the feasibility of perturbative expansions on top of non-trivial vacua. This way, one immediately encounters problems: it is known that NSPT simulations exhibit large fluctuations in low-dimensional models. As the perturbative order increases, huge fluctuations show up, completely obscuring the signal at even not-so-high perturbative orders. In this thesis, we discuss NSPT simulations for a class of highly interesting low-dimensional models, the two-dimensional O(N) Non-Linear Sigma Models (NLSMs). O(N) non-linear sigma models can be regarded as a valuable theoretical laboratory in quantum field theory, as they display in a relatively simple framework interesting features like asymptotic freedom. From a more phenomenology oriented point of view, NLSM proved to be effective in modeling different features in different contexts. As we will see, in this work our interest for O(N) models is motivated by the possibility to tune N. On general grounds we expect that huge fluctuations in simulations of low-dimensional models are somehow connected to the limited number of degrees of freedom. From this perspective, O(N) NLSMs are an ideal laboratory: in fact we can modify the number of degrees of freedom by tuning the parameter N. Our numerical results show that in the large N limit NSPT simulations are not affected by the large fluctuations issue at high orders, in contrast to what occurs in the small N regime. Our conclusions are supported by extensive numerical studies of the properties of NSPT distributions as function of the perturbative order n and the parameter N. While a fundamental comprehension is admittedly lacking, we will consider different indicators for assessing if (and to what extent) large enough N computations are to be regarded as safe at a given perturbative order n. In particular, the study of relative errors has been particularly fruitful: in this context, the onset of fluctuations has been probed through violations of very generally expected scaling behaviors. Our numerical simulations strongly suggest that indeed for each perturbative order n, an NSPT computation in O(N) can always be found safe with respect to fluctuations if we take a large enough N. As a result, the larger the value of N, the more perturbative corrections we could compute, significantly extending the previously known results from lattice diagrammatic perturbation theory. Once for large enough N high perturbative orders can be safely computed, we expect we can explore the asymptotic behavior of perturbative expansions. In the context of lattice gauge theories, NSPT has proven to be effective in probing infrared renormalons. In the final part of the thesis, we discuss O(N) renormalon effects in the large N limit. We will perform computations on a pretty small lattice size, but we will provide new insights on the role of finite-volume effects. In particular, by explicitly taking into account the infrared cutoff, we obtained an analytic (first-principles) estimate of finite-volume effects, assessing how they modify the factorial scaling of coefficients. Once we have such a modeling, we can compare analytical predictions and numerical results, finding agreement in the asymptotic perturbative region. We stress that this will be a parameter-free comparison (there is no space for any parameter to adjsut). Large N NSPT simulations for O(N) models can also be regarded as a preliminary step towards going back to perturbative expansions around non-trivial vacua. Quite interestingly, such computations in the (quite close) CP(N-1) models are connected to resurgence scenarios.