Abstract
Thimble regularisation is a possible solution to the sign problem, which is evaded by formulating quantum field theories on manifolds where the imaginary part of the action stays constant (Lefschetz thimbles). A major obstacle is due to the fact that one in general needs to collect contributions coming from more than one thimble. Here we explore the idea of performing Taylor expansions on Lefschetz thimbles. We show that in some cases we can compute expansions in regions where only the dominant thimble contributes to the result in such a way that these (different, disjoint) regions can be bridged. This can most effectively be done via Pad\'e approximants. In this way multi-thimble simulations can be circumvented. The approach can be trusted provided we can show that the analytic continuation we are performing is a legitimate one, which thing we can indeed show. We briefly discuss two prototypal computations, for which we obtained a very good control on the analytical structure (and singularities) of the results. All in all, the main strategy that we adopt is supposed to be valuable not only in the thimble approach, which thing we finally discuss.
Highlights
THIMBLE REGULARIZATION AND SINGLETHIMBLE DOMINANCELattice regularization provides an effective framework for a nonperturbative definition of quantum field theories
We argue that computations on the dominant thimble at those points provide us with the complete result by checking that the results obtained by Taylor expansions smoothly join the result we get at μ m
The idea is to explore the space of the parameters describing the theory and find two points at which the dominant thimble accounts for the full result: as we saw, there could be different strategies to attain this
Summary
Lattice regularization provides an effective framework for a nonperturbative definition of quantum field theories It enables numerical computations: in the Euclidean formulation, a lattice-regularized QFT resembles a statistical physics problem, the functional integral defines a decent probability measure and Monte Carlo simulations are viable. The main idea is to compute Taylor expansions in different regions of the parameter space of a given theory, namely around points where only the dominant thimble contributes to the result one is interested in. This could seem somehow a lucky scenario, but we argue that this can quite often be the case. THIMBLE DECOMPOSITION AND STOKES PHENOMENA: A STORY OF CONTINUITY AND DISCONTINUITIES
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