Abstract
We propose the sparse modeling method to estimate the spectral function from the smeared correlation functions. We give a description of how to obtain the shear viscosity from the correlation function of the renormalized energy-momentum tensor (EMT) measured by the gradient flow method (C (t, τ )) for the quenched QCD at finite temperature. The measurement of the renormalized EMT in the gradient flow method reduces a statistical uncertainty thanks to its property of the smearing. However, the smearing breaks the sum rule of the spectral function and the over-smeared data in the correlation function may have to be eliminated from the analyzing process of physical observables. In this work, we demonstrate the sparse modeling analysis in the intermediate-representation basis (IR basis), which connects between the Matsubara frequency data and real frequency data. It works well even using very limited data of C (t, τ ) only in the fiducial window of the gradient flow. We utilize the ADMM algorithm which is useful to solve the LASSO problem under some constraints. We show that the obtained spectral function reproduces the input smeared correlation function at finite flow-time. Several systematic and statistical errors and the flow-time dependence are also discussed.
Highlights
Ratio between the shear viscosity (η) to the thermal entropy (s) is very small, η/s ≤ 0.4 [11]
We have proposed the sparse modeling analysis to estimate the spectral function from the smeared correlation functions
We have described how to obtain the shear viscosity from the correlation function of the renormalized energy-momentum tensor (EMT) measured by using the gradient flow method for the quenched QCD at finite temperature
Summary
We give a brief review of the sparse modeling method in the IR basis following refs. [49, 50] (see a review paper [48]). By introducing some threshold scut, we can drop such a component l > lcut, where sl < scut This truncation leads us to obtain a stable solution which is robust against the noise of C(τ ). We would like to find the solution ρl in eq (2.8), where many components of ρ takes zero to be consistent with the truncation To search for such a solution ρl, we consider the cost function with an L1 regularization term,. [48, 50], the spectral function becomes featureless for λ > λopt, while artificial spikes appear for λ < λopt In other words, the former case corresponds to under-fitting, where the L1 regularization term is too strong and the number of components ρl is too reduced.
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