Stochastic reconstructions of spectral functions: Application to lattice QCD

We present a detailed study of the applications of two stochastic approaches, stochastic optimization method (SOM) and stochastic analytical inference (SAI), to extract spectral functions from Euclidean correlation functions. SOM has the advantage that it does not require prior information. On the other hand, SAI is a more generalized method based on Bayesian inference. Under mean field approximation SAI reduces to the often-used maximum entropy method (MEM) and for a specific choice of the prior SAI becomes equivalent to SOM. To test the applicability of these two stochastic methods to lattice QCD, firstly, we apply these methods to various reasonably chosen model correlation functions and present detailed comparisons of the reconstructed spectral functions obtained from SOM, SAI and MEM. Next, we present similar studies for charmonia correlation functions obtained from lattice QCD computations using clover-improved Wilson fermions on large, fine, isotropic lattices at 0.75 and $1.5T_c$, $T_c$ being the deconfinement transition temperature of a pure gluon plasma. We find that SAI and SOM give consistent results to MEM at these two temperatures.

Figure 1

A typical structure of ${\chi }^2-\alpha$ curve and its kink point.

Figure 2

Left: Shift a $\delta$-function. Right: Residue sharing between two $\delta$-functions.

Figure 3

Spectral function constructed from overlapping boxes in SOM.

Figure 4

Left: Shift a box. Middle: Change a box. Right: Share height between two boxes.

Figure 5

A comparison of spectral functions obtained from SOM and SAI using DM $K^{-1}(\widehat{\omega })$ (left) and a rescaled free Wilson spectral function (right) as the default model. The black solid curve is the input model spectral function, and the red dashed curve denotes the default model. The red and green solid curves represent the output spectral functions obtained using the SAI and SOM, respectively.

Figure 6

Dependence on $N_{\tau }$ and noise level $\epsilon$ of output spectral functions obtained from SOM, SAI and MEM. The black solid curve is the input spectral function, and the black dashed curve denotes the DM. The other colored curves are output spectral functions. From top to bottom $N_{\tau }=48$, 64 and 96. From left to right $\epsilon =10^{-2}$, $5\times 10^{-3}$ and $10^{-5}$.

Figure 7

The ratio of the correlators reconstructed from the output spectral functions obtained by three methods to the input ones at $N_{\tau }=96$, $\epsilon =5\times 10^{-3}$. The error bars shown in the figure are only from the input correlators.

Figure 8

Default model dependencies of $\widehat{\rho }(T<T_c)$ obtained from SAI (bottom panel) and MEM (top panel). SOM results are also shown with green solid curves. The left panels show the spectral functions in the small energy region while the right ones show the spectral function in the whole energy region. The model correlators are produced using ${\widehat{\rho }}_{\text{below}}(\widehat{\omega })$ with $N_{\tau }=96$ and a noise level $\epsilon =2.5\times 10^{-3}$.

Figure 9

Dependences on default models at $T>T_c$ with $N_{\tau }=48$ and $\epsilon =5\times 10^{-3}$. Top two figures: Results obtained using $DM1$. Bottom two figures: Results obtained using $DM2$. Both $DM1$ and $DM2$ are rescaled free Wilson spectral functions. The value of the quark mass $m$ is set to be 0.06 in $DM1$, and it is 0.02 in $DM2$.

Figure 10

Dependences on default models at $T>T_c$ ($N_{\tau }=48$) with a noise level $\epsilon =5\times 10^{-3}$. Top two figures: Results obtained using $DM3$. Bottom two figures: Results obtained using $DM4$. $DM3$ and $DM4$ have almost the same transport peak and large $\widehat{\omega }$ part but have different resonance peak locations. In $DM3$ the resonance peak locates at $M=0.155$ while in $DM4$ it locates at $M=0.300$.

Figure 11

Dependence on default models tests at $T>T_c$ ($N_{\tau }=48$) in noise level $\epsilon =5\times 10^{-3}$. Top two figures: Results of MEM. Bottom two figures: Results of SAI. $DM4–DM6$ have the same resonance peak and a large $\widehat{\omega }$ part. For a transport peak they have the same width but a different height. $DM5$ has $c_{\text{trans}}=5\times 10^{-5}/8$. $DM6$ has $c_{\text{trans}}=5\times 10^{-5}\times 2$. $DM7$ has $c_{\text{trans}}=5\times 10^{-5}\times 16$. Here $5\times 10^{-5}$ is the $c_{\text{trans}}$ of the input spectral function.

Figure 12

Dependence on default models tests at $T>T_c$ ($N_{\tau }=48$) in noise level $\epsilon =5\times 10^{-3}$. Top two figures: Results of MEM. Bottom two figures: Results of SAI. $DM6$ and $DM8$ have the same resonance peak and a large $\widehat{\omega }$ part. For the transport peak they have the same height but a different width. $DM6$ has $\eta =0.006$. $DM8$ has $\eta =0.003$.

Figure 13

Spectral functions obtained by SOM, SAI and MEM at $0.75T_c$ for the pseudoscalar channel (${\eta }_c$). Top two figures: The results of MEM and SOM. Bottom two figures: The results of SAI and SOM.

Figure 14

Dependence of the location of the possible ${\eta }_c$ resonance peak on various DMs at $1.5T_c$.

Figure 15

Dependence of the transport peak on the DMs at $1.50T_c$ in the pseudoscalar (${\eta }_c$) channel.

Figure 16

Spectral functions obtained by SOM, SAI and MEM at $0.75T_c$ for the vector channel ($J/\psi$). Top two figures: The results of MEM and SOM. Bottom two figures: The results of SAI and SOM.

Figure 17

Dependence of the location of the possible $J/\psi$ resonance peak on various DMs at $1.50T_c$.

Figure 18

The dependence on the transport peak in the DMs at $1.50T_c$ for the vector channel.

Figure 19

Left: Significance of the strength of the resonancelike peak structure in ${\eta }_c$ spectral function obtained using SOM, SAI and MEM at $0.75T_c$. Right: Same as left, but for the case at $1.5T_c$.

Figure 20

Same as Fig. 19, but for the vector channel.

Figure 21

Spectral functions in the pseudoscalar channel (left) and vector channel (right) obtained by three different methods from the reconstructed correlators $G_{\text{rec}}(\tilde{\tau },T=1.5T_c;T^{\prime }=0.75T_c)$. The default models used here are the same as the first ones of those used in the analysis of ${\eta }_c$ and $J/\psi$ correlators at $0.75T_c$. The solid curves are results obtained by corresponding methods from correlators at $0.75T_c$ with $DM1$ for comparison.

Figure 22

Significance of the strength of the resonancelike peak structure in the ${\eta }_c$ (left) and $J/\psi$ (right) spectral functions obtained using SOM, SAI and MEM from the reconstructed correlators.

Figure 23

Left: DoS calculated by replica exchange Wang-Landau sampling. Right: A typical distribution of $P[\alpha |G,D]$. Both are from model data tests.