Abstract
We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the Källén-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the Källén-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularise this problem we implement an appropriate version of Tikhonov regularisation supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two—mathematically equivalent—versions of the Källén-Lehmann spectral integral.
Highlights
A plethora of practical Quantum Field Theory calculational tools, both analytical and numerical, have been developed in a Euclidean setting, despite living in a Minkowski spacetime
Whereas in perturbation theory the analytical continuation of the Euclidean correlation functions into the entire complex momenta Argand plane relies on the usual Wick rotation, it is not clear that the same rule can be applied for the non-perturbative regime
This paper reports on applying the procedure outlined above to gluon and ghost two-point functions obtained from lattice QCD
Summary
A plethora of practical Quantum Field Theory calculational tools, both analytical and numerical, have been developed in a Euclidean setting, despite living in a Minkowski spacetime. Because the physical bound state spectrum is gauge invariant despite being constructed from gauge variant gluon, ghost and quark propagators [6], the spectral function of a bound state propagator must be nontrivially influenced by the analytic structure of the underlying constituents, see for instance [6,7,8,9,10,11] for discussions and examples This explains the relevance of various studies devoted to the spectral properties of a priori unphysical degrees of freedom. It is worth noting that Tikhonov regularisation using the Morozov criterion can alternatively be understood from a Bayesian approach as “historic MEM”, with a default model m = 0 This default model choice is well motivated, as the UV asymptotics of the ghost and gluon spectral functions predict that they will tend to zero, corresponding with a default model m = 0 in the UV.
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