Abstract
We discuss the relation of a variety of different methods to determine energy levels in lattice QCD simulations: the generalised eigenvalue, the Prony, the generalised pencil of function and the Gardner methods. All three former methods can be understood as special cases of a generalised eigenvalue problem. We show analytically that the leading corrections to an energy E_l in all three methods due to unresolved states decay asymptotically exponentially like exp (-(E_{n}-E_l)t). Using synthetic data we show that these corrections behave as expected also in practice. We propose a novel combination of the generalised eigenvalue and the Prony method, denoted as GEVM/PGEVM, which helps to increase the energy gap E_{n}-E_l. We illustrate its usage and performance using lattice QCD examples. The Gardner method on the other hand is found less applicable to realistic noisy data.
Highlights
In lattice field theories one is often confronted with the task to extract energy levels from noisy Monte Carlo data for Euclidean correlation functions, which have the theoretical form ∞ C(t) = ck e−Ek t (1)k=0 with real and distinct energy levels Ek+1 > Ek and real coefficients ck
While a systematic comparison is beyond the scope of this article, we show in Fig. 11 a comparison of generalised pencil of function (GPOF) versus generalised eigenvalue method (GEVM)/Prony GEVM (PGEVM) for the case of the η-meson
In this paper we have first discussed the relation among the generalised eigenvalue, the Prony and the generalised pencil of function methods: they are all special cases of a generalised eigenvalue method
Summary
In lattice field theories one is often confronted with the task to extract energy levels from noisy Monte Carlo data for Euclidean correlation functions, which have the theoretical form ∞ C(t) = ck e−Ek t (1). It is well known that this task represents an ill-posed problem because the exponential functions do not form an orthogonal system of functions. With corrections exponentially suppressed with increasing t due to ground state dominance. In lattice quantum chromodynamics, the non-perturbative approach to quantum chromodynamics (QCD), the signal to noise ratio for C(t) deteriorates exponentially with increasing t [1]. [2]) to the correlation functions, which, if not accounted for, render the data at large t useless. Once one is interested in excited energy levels Ek , k > 0, alternatives to the ground state dominance principle need to be found
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