# Analysis of spectra using matrices of correlation functions based on irreducible baryon operators

###### Abstract

We present results for ground and excited-state nucleon masses in quenched lattice QCD using anisotropic lattices. Group theoretical constructions of local and nonlocal straight-link irreducible operators are used to obtain suitable sources and sinks. Matrices of correlation functions are diagonalized to determine the eigenvectors. Both chi-square fitting and Bayesian inference with an entropic prior are used to extract masses from the correlation functions in a given channel. We observe clear separation of the excited state masses from the ground state mass. States of spin have been isolated by use of operators.

## 1 Introduction

Reproducing the spectrum of baryon resonances with spin-1/2 and spin-3/2 and both parities is an important test of lattice QCD. For that we require three-quark operators that transform irreducibly under the spinorial rotation group of the lattice [1]. Local and nonlocal straight-link operators corresponding to irreducible representations (IRs) , and are constructed to obtain suitable sources and sinks [2]. Here we analyze spectra using these operators.

To determine the excited states, a matrix of correlation functions is computed in the quenched approximation to QCD using irreducible baryon interpolating operators of definite quantum numbers,

(1) |

The matrices for and states for each parity are constructed using local operators, smeared local operators and smeared straight-link operators. For , we have one operator which is of smeared straight-link type (Tables 1, 3 in [2]).

The computation of masses of the lowest-lying resonances is based on the variational method applied to the matrix of correlation functions. In this paper we solve the generalized eigenvalue equation,

(2) |

and determine eigenvectors for each , with close to the source time. Then the masses of states correspond to the eigenvalues of Eq. (2): 3]. The effective masses are determined from [

(3) |

Another way to extract spectrum information is to calculate the spectral mass density from lattice correlation functions using the Maximum Entropy Method (MEM) [4],

(4) |

One of the advantages of MEM is that it utilizes data on a wide range of available time slices and has been shown to yield results even for noisy data [4]. This feature may be helpful in extracting masses of excited states.

## 2 Results

We use an ensemble of 287 quenched, anisotropic lattices with renormalized anisotropy and , corresponding to GeV [5]. We use the anisotropic Wilson action. The parameters of the Wilson fermion action are tuned nonperturbatively so as to satisfy the continuum dispersion relation at a pion mass MeV. To improve the coupling of operators to the lower mass states we employ gauge-covariant smearing of the quark fields on both source and sink: , where is the three dimensional Laplacian and denotes APE-smeared link variables. The parameters used to smear the quark fields are and .

The effective masses are calculated from matrices with . In Fig. 1 we choose to show a few low-lying states that are clearly separated. However, the details of the states above the ground state are under study. The plot shows a good plateau for the ground state and statistically significant splittings for a couple of excited states.

In Figs. 2 and 3 we have collected the effective mass plots of the lowest states of both parities for , and . The ratio of lowest masses for and is roughly in accordance with experiment, for spin-1/2 states, the mass being higher. The effective masses for are obtained using a matrices of correlation functions. The lowest negative parity state has smaller mass than the lowest positive parity state. This is compatible with the pattern found in nature for spin-3/2. However, the masses of and overlap within errors.

IRs | Fit range | Mass (MeV) | |
---|---|---|---|

9 – 20 | 0.208 (4) | 1250 | |

8 – 12 | 0.321 (4) | 1930 | |

9 – 12 | 0.410 (2) | 2460 | |

6 – 15 | 0.315 (4) | 1890 | |

9 – 14 | 0.409 (7) | 2450 | |

8 – 15 | 0.475 (7) | 2850 |

Our result for masses also reveals reasonable separations of the and masses, being lower. From Fig. 2, it is evident that the effective mass for (allowed spin , , ) is very similar to that for (allowed spin , , ). These states are orthogonal. One possibility for this is that the lowest state has spin- and its mass is accidentally close to that of the lowest state. Another possibility is that the lowest state is spin-, in which case the same state must be present in and , but not in . Study over different values of lattice spacing is required to decide.

Finally, we present the MEM spectral function in Figure 4. We find that the peak of the spectral density roughly corresponds to the effective mass value.

In Table 1 we summarize our preliminary estimates of the lowest masses for the different representations extracted from single-exponential fits to the of Eqn. 1. The effective masses for the lowest states of , and for both parities, whether obtained from the variational method or preliminary MEM analysis, show a spectrum of distinct masses. However, the behavior of the spectrum with and the sensitivity of the spectrum to variations in the lattice volume has yet to be studied.

This work is supported by US National Science Foundation under Awards PHY-0099450 and PHY-0300065, and by US Department of Energy under contract DE-AC05-84ER40150 and DE-FG02-93ER-40762.

## References

- [1] C. Morningstar et al., Nucl. Phys. B (Proc. Suppl.) 129 (2004) 236 and these proceedings
- [2] I. Sato et al. Nucl. Phys. B (Proc. Suppl.) 129 (2004) 209 and these proceedings
- [3] C. Michael, Nucl. Phys. B259 (1985) 58; M. Lüscher et al., Nucl. Phys. B339 (1990) 222; C.R. Alton et al., Phys. Rev. D47 (1993) 5128
- [4] H.R. Fiebig, Phys. Rev. D65 (2002) 094512
- [5] R.G. Edwards et al., Nucl. Phys. B (Proc. Suppl.) 119 (2003) 305