Novel analysis method for excited states in lattice QCD: The nucleon case

We employ a novel method to analyze Euclidean correlation functions entering the calculation of hadron energies in lattice QCD. The method is based on the sampling of all possible solutions allowed by the spectral decomposition of the hadron correlators. We demonstrate the applicability of the method by studying the nucleon excited states in the positive- and negative-parity channels over a pion mass range of about 400 to 150 MeV. The results are compared to the standard variational approach routinely used to study excited states within lattice QCD. The main advantage of our new approach is its ability to unambiguously determine all excited states to which the Euclidean time correlation function is sensitive.

Figure 1

The correlator at a given $t_j$ is computed using 5000 measurements, having an average value of $C(t_j)=3.2\times 10^{-8}$. The standard jackknife error is $1.8\times 10^{-10}$, and the weighted standard deviation is ${\sigma }_j/\sqrt{N}=1.7\times 10^{-10}$, thus demonstrating that the measurements are uncorrelated.

Figure 2

The PDFs of $E_n$ for different values of the truncation parameter $n_{\max }$ (Eq. (6). A well-defined distribution is only present for the parameters that contribute to the PDF of Eq. (8). Parameters without any contribution lead to a uniform distribution.

Figure 3

The PDFs of the energies for an eight parameter case (four amplitudes and four energies). The horizontal axis is the energy in lattice units, and the vertical axis is the sampled probability of Eq. (7). The fitting range is varied by changing $t_0$. Exponentials with larger exponents (excited states) are suppressed, but the one with smaller exponents (low-lying spectrum) remains unaffected.

Figure 4

The PDFs of the amplitudes for an eight parameter case (four amplitudes and four masses). The horizontal axis is the amplitudes, and the vertical axis is the sampled probability of Eq. (7). The fitting range is varied by changing $t_0$. The suppressed excited-state amplitudes appear with a uniform distribution.

Figure 5

Contour plots showing correlations among the parameters. The contours are color coded according to the ${\chi }^2$ value. The actual data used correspond to the nucleon correlator $C_{11}^{S,S}(t)$ of the B55.32 ensemble described in Sec. 3.

Figure 6

The PDFs for the ground and excited energies (top) and amplitudes (bottom) for smeared-smeared and local-smeared correlators using the B55.32 ensemble. As expected the amplitudes depend on the smearing level while the energies are statistically equivalent (a reduction in the level of smearing leads to more well-defined distributions for the excited states).

Figure 7

The spectrum from AMIAS when using the diagonal $C_{11}^{(1,1)}$ and $C_{11}^{(5,5)}$ correlators as well as the $3\times 3$ correlation matrices $C_{1,1}^{(i,j)}$, $i,j=1,2,3$ and $C_{1,1}^{(i,j)}$, $i,j=1,2,4$. Results are also shown for the $5\times 5$ matrix $C_{1,1}^{(i,j)}$, $i,j=1,\dots ,5$. The inclusion of a variational basis that includes small and large smearings leads to an improved excited-state spectrum.

Figure 8

The points show the effective masses determined from the GEVP analysis, while the bands show the values obtained by AMIAS for the same correlation matrix. The left panel shows results extracted the $3\times 3$ correlation matrix $C_{11}^{(1,2,4)}$, while the left panel shows results extracted from the $3\times 3$ correlation matrix, $C_{11}^{(1,2,3)}$.

Figure 9

The nucleon ground state and first excited state in the positive-parity channel. The values obtained using AMIAS (filled back circles and filled green diamond) are compared to the results extracted from GEVP [8]. The results are obtained from the analysis of a $4\times 4$ correlation matrix $C_{ab}^{(i,j)}$, $a,b=1,2$, $i,j=3,4$, containing both $J_1$ and $J_2$, of the nucleon interpolator, each with two smearing levels.

Figure 10

The effective mass for the negative-parity ground state for all of our ensembles. Shaded error bands obtained by AMIAS for various values of $t_0$ are shown. The dotted line indicates the $N\pi$ ground state energy which is obtained from the sum of the nucleon and pion masses for the particular ensemble. The results are obtained from the analysis of a $4\times 4$ correlation matrix $C_{ab}^{(i,j)}$, $a,b=1,2$, $i,j=3,4$, containing both $J_1$ and $J_2$, of the nucleon interpolator, each with two smearing levels.

Figure 11

The same as in Fig. 9 but for the negative-parity channel. The results are obtained from the analysis of a $4\times 4$ correlation matrix $C_{ab}^{(i,j)}$, $a,b=1,2$, $i,j=3,4$, containing both $J_1$ and $J_2$, of the nucleon interpolator, each with two smearing levels.