PhD thesis
Topology and θ-dependence in QCD and QCD-like theories
PhD candidate: Claudio Bonanno Supervisor: Prof. Massimo D’Elia
Abstract
The study of the topological properties of gauge theories is a very active field of research because of their connection with interesting theoretical aspects and for their phenomenological implications for the Standard Model and beyond. In Quantum Chromo-Dynamics (QCD), for instance, topology is related to the open issue known as the strong CP problem and to one of its proposed solution, the Peccei–Quinn axion, and also to aspects of hadron physics such as the properties of the η0 meson. Moreover, QCD topology is also tightly connected with intriguing theoretical features of the model such as the dependence on the topological parameter θ or the 1/Nc expansion. For these reasons, these topics have also been investigated in QCD-related theories such as the SU (Nc) pure-gauge theories or the 2d CPN −1models.
Being topological properties purely non-perturbative, the lattice approach is a natural first- principle tool to investigate them, however, several non-trivial computational problems have to be properly faced to make substantial progresses in the present state of the art, such as: the rare fluctuations of the topological charge when studying hot QCD (due to the suppression of the susceptibility at high temperatures), the presence of large lattice corrections affecting the topo- logical susceptibility when adopting a non-chiral fermion discretization (related to the presence of the fermion determinant in the path-integral weight), the severe topological slowing down occurring when approaching the continuum limit or the large-Nc limit (due to the divergence of the free-energy barriers among different topological sectors in these limits) or the degrad- ing of the signal-to-noise ratio of higher-order cumulants of the topological charge probability distribution in the thermodynamic limit (due to the central limit theorem).
The goal of this thesis is to improve the state of the art of the study of topological properties in full QCD, in SU (Nc) pure Yang–Mills theories and in 2d CPN −1 models by means of lattice Monte Carlo numerical simulations employing new algorithms, methods and approaches in order to overcome the computational problems listed before.
In the first part of this thesis, we have discussed the extension of the spectral projectors method to the case of staggered fermions. This method allows to define a singularity-free fermionic definition of the topological susceptibility which tends to the same continuum limit as the usual gluonic one but that, unlike the latter, allows to control the magnitude of lattice artifacts through the choice of the threshold mass, i.e., the maximum eigenvalue included in the spectral sums defining the projectors. First, we have tested the method for the pure SU (3) gauge theory, and we have shown that the spectral definition yields consistent results with the gluonic one, both at zero and at finite temperature. Then, we have presented preliminary results for high- temperature full QCD simulations with staggered fermions. In particular, we have shown that the combination of the multicanonical method to overcome the problem of rare topological fluctuations and of the spectral projectors method to reduce lattice artifacts allows to better control the impact of the systematics affecting the continuum extrapolation of the susceptibility.
In the second part of this thesis we have presented a systematic study of the θ-dependence 1
of the vacuum energy of the 2d CPN −1 models both in the large-N and in the small-N limit up to O(θ4) terms. The large-N regime has been investigated using the Hasenbusch parallel tempering algorithm. Putting together simulations with periodic and open boundaries in a parallel tempering framework to strongly mitigate topological freezing occurring in the large-N limit, this algorithm has allowed to reach values of N which are practically unfeasible with standard methods. Combining this algorithm with the imaginary-θ method, we have studied the behavior of the susceptibility and of the quartic coefficient b2 beyond the leading order in 1/N , finding agreement with large-N analytic predictions once the presence of large higher-order corrections is taken into account. Concerning the small-N regime instead, we have performed extensive simulations for N = 2 and N = 3 with standard local algorithms, as well as a small- N extrapolation of the N < 10 determinations, to investigate the critical behavior of the θ- dependence of the vacuum energy when N → 2, which could be related to the slow convergence of the 1/N series observed at large N .
Finally, in the last part of this dissertation we have presented results for the θ-dependence of the vacuum energy of 4d SU (Nc) gauge theories in the large-Nc limit up to O(θ4) terms, obtained implementing the Hasenbusch algorithm in this case and again combining it with the imaginary-θ method. First, we have tested and compared our implementation of the Hasenbusch algorithm with standard local ones, showing that it allows to obtain a dramatic gain in efficiency without much tuning of the free parameters of the algorithm, strongly mitigating the topological slowing down occurring in the large-Nc limit. This has allowed to improve the study of the large-Nc behavior of b2; in particular, we have been able to obtain for the first time a solid and controlled continuum extrapolation for b2for Nc= 6, as well as to refine previous results for this quantity for Nc= 4. Thanks to these improvements, we have been able to confirm with percent precision that the expected leading-order large-Ncscaling for b2 holds already for Nc> 2, while higher-order corrections in 1/Ncappear to be negligible in this range within our precision.
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