We propose a method to reconstruct smeared spectral functions from two-point correlation functions measured on the Euclidean lattice. Arbitrary smearing function can be considered as far as it is smooth enough to allow an approximation using Chebyshev polynomials. We test the method with numerical lattice data of Charmonium correlators. The method provides a framework to compare lattice calculation with experimental data including excited state contributions without assuming quark-hadron duality.
The ratio $R(E)$ of the cross-sections for $e^+e^-\to$ hadrons and $e^+e^-\to μ^+μ^-$ is a valuable energy-dependent probe of the hadronic sector of the Standard Model. Moreover, the experimental measurements of $R(E)$ are the inputs of the dispersive calculations of the leading hadronic vacuum polarization contribution to the muon $g-2$ and these are in significant tension with direct lattice calculations and with the muon $g-2$ experiment. In this talk we discuss the results of our first-principles lattice study of $R(E)$. By using a recently proposed method for extracting smeared spectral d…
A novel proposal is outlined to determine scattering amplitudes from finite-volume spectral functions. The method requires extracting smeared spectral functions from finite-volume Euclidean correlation functions, with a particular complex smearing kernel of width $ε$ which implements the standard $iε$-prescription. In the $L \to \infty$ limit these smeared spectral functions are therefore equivalent to Minkowskian correlators with a specific time ordering to which a modified LSZ reduction formalism can be applied. The approach is presented for general $m \to n$ scattering amplitudes (above arb…
In Euclidean four-dimensional SU(N) pure gauge theory, eigenvalue distributions of Wilson loop parallel transport matrices around closed spacetime curves show non-analytic behavior (a 'large-N phase transition') at a critical size of the curve. We focus mainly on an observable composed of traces of the Wilson loop operator in all totally antisymmetric representations, which is regularized with the help of smearing. By studying sequences of square Wilson loops on a hypercubic lattice with standard Wilson action, it is shown that this observable has a nontrivial continuum limit as a function of …
The smeared $R$-ratio in isoQCD from first-principles lattice simulations
The Cyprus Institute（塞浦路斯研究所）；Johannes Gutenberg-Universität Mainz（美因茨约翰内斯·古腾堡大学）；GSI Helmholtz Centre for Heavy Ion Research（GSI重离子研究中心）；University of Cyprus（塞浦路斯大学）
A recently re-discovered variant of the Backus-Gilbert algorithm for spectral reconstruction enables the controlled determination of smeared spectral densities from lattice field theory correlation functions. A particular advantage of this approach is the \emph{a priori} specification of the kernel with which the underlying spectral density is smeared, allowing for variation of its peak position, smearing width, and functional form. If the unsmeared spectral density is sufficiently smooth in the neighborhood of a particular energy, it can be obtained from an extrapolation to zero smearing-kern…
This work employs the spectral reconstruction approach of Ref. [1] to determine an inclusive rate in the $1+1$ dimensional O(3) non-linear $σ$-model, analogous to the QCD part of ${e}^+{e}^- \rightarrow \rm {hadrons}$. The Euclidean two-point correlation function of the conserved current $j$ is computed using Monte Carlo lattice field theory simulations for a variety of spacetime volumes and lattice spacings. The spectral density of this correlator is related to the inclusive rate for $j \rightarrow {\rm X}$ in which all final states produced by the external current are summed. The ill-posed i…
We study charmonium correlators in pseudoscalar and vector channels at finite temperature using lattice QCD simulation in the quenched approximation. Anisotropic lattices are used in order to have sufficient numbers of degrees of freedom in the Euclidean temporal direction. We focus on the low energy structure of the spectral function, corresponding to the ground state in the hadron phase, by applying the smearing technique to enhance the contribution to the correlator from this region. We employ two analysis procedures: the maximum entropy method (MEM) for the extraction of the spectral funct…
We present the lattice QCD calculation of inclusive semileptonic $B_s$-meson decays. We follow a recently proposed method, which is based on the extraction of smeared spectral densities from Euclidean correlation functions and on the numerical reconstruction of the integration kernel relevant for the inclusive decay rate calculation. We compute four-point Euclidean correlation functions using JLQCD and ETM gauge ensembles with unphysically light $b$-quark masses, and apply two different methods for the integration kernel reconstruction. Finally, we show that the lattice results obtained in thi…
We present a non-perturbative computation of inclusive rates of semileptonic decays of heavy mesons from lattice QCD simulations. The calculation is based on the extraction of smeared spectral functions obtained from four-point Euclidean correlation functions computed on configuration ensembles of the JLQCD and ETM collaborations. We compare our results for the inclusive decay rates with analytical predictions from the operator-product expansion, finding a good agreement for the calculation of the inclusive decay rate. This opens the path to the theoretical determination of the magnitude of th…
We show how the sphaleron rate (the Minkowski rate for topological charge diffusion) can be determined by analytical continuation of the Euclidean topological-charge-density two-point function, which we investigate on the lattice, using gradient flow to reduce noise and provide improved operators which more accurately measure topology. We measure the correlators on large, fine lattices in the quenched approximation at $1.5\,T_c$ with high precision. Based on these data we first perform a continuum extrapolation at fixed physical flow time and then extrapolate the continuum estimates to zero fl…
A kernel-derived orthogonal basis for spectral functions from Euclidean correlators
High Energy Accelerator Research Organization (KEK)（高能加速器研究机构）；Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK)（高能加速器研究机构粒子核研究所在）；Graduate University for Advanced Studies (SOKENDAI)（高级研究大学院大学）
Extraction of spectral densities from lattice correlators: decoupling signal from noise
Aix Marseille Univ, Université de Toulon, CNRS, CPT（艾克斯马赛大学、土伦大学、法国国家科学研究中心、CPT）；Dipartimento di Fisica, Università di Roma Tor Vergata and INFN, Sezione di Roma Tor Vergata（罗马托尔维加塔大学物理系和意大利国家核物理研究所罗马托尔维加塔分部）
The prospects of extracting transport coefficients from euclidean lattice simulations are discussed. Some general comments on the reconstruction of spectral functions using the Maximal Entropy Method are given as well.
We present a novel approach for the reconstruction of spectra from Euclidean correlator data that makes close contact to modern Bayesian concepts. It is based upon an axiomatically justified dimensionless prior distribution, which in the case of constant prior function $m(ω)$ only imprints smoothness on the reconstructed spectrum. In addition we are able to analytically integrate out the only relevant overall hyper-parameter $α$ in the prior, removing the necessity for Gaussian approximations found e.g. in the Maximum Entropy Method. Using a quasi-Newton minimizer and high-precision arithmetic…
We present a novel approach to the inference of spectral functions from Euclidean time correlator data that makes close contact with modern Bayesian concepts. Our method differs significantly from the maximum entropy method (MEM). A new set of axioms is postulated for the prior probability, leading to an improved expression, which is devoid of the asymptotically flat directions present in the Shanon-Jaynes entropy. Hyperparameters are integrated out explicitly, liberating us from the Gaussian approximations underlying the evidence approach of the MEM. We present a realistic test of our method …
We present a Stochastic Optimization Method (SOM) for the reconstruction of the spectral functions (SPFs) from Euclidean correlation functions. In this approach the SPF is parameterized as a sum of randomly distributed boxes. By varying the width, location and height of the boxes stochastically an optimal SPF can be obtained. Using this approach we reproduce mock SPFs fairly well, which contain sharp resonance peaks, transport peaks and continuum spectra. We also analyzed the charmonium correlators obtained from $N_τ$=96, 48, 32 lattices using SOM and found similar conclusion on the dissociati…
Hadronic spectral densities are important quantities whose non-perturbative knowledge allows for calculating phenomenologically relevant observables, such as inclusive hadronic cross-sections and non-leptonic decay-rates. The extraction of spectral densities from lattice correlators is a notoriously difficult problem because lattice simulations are performed in Euclidean time and lattice data are unavoidably affected by statistical and systematic uncertainties. In this paper we present a new method for extracting hadronic spectral densities from lattice correlators. The method allows for choos…
The extraction of the finite temperature heavy quark potential from lattice QCD relies on a spectral analysis of the real-time Wilson loop. Through its position and shape, the lowest lying spectral peak encodes the real and imaginary part of this complex potential. We benchmark this extraction strategy using leading order hard-thermal loop (HTL) calculations. I.e. we analytically calculate the Wilson loop and determine the corresponding spectrum. By fitting its lowest lying peak we obtain the real- and imaginary part and confirm that the knowledge of the lowest peak alone is sufficient for obt…
