5-2018

Honors Thesis

#### Degree Name

Bachelors of Science (BS)

Physics

Jozef Dudek

#### Committee Members

Irina Novikova

Andreas Stathopoulos

#### Abstract

We extracted a discrete energy spectrum corresponding to scattering of two pions in a finite volume in Quantum Chromodynamics by analyzing matrices of two-point correlation functions computed within lattice QCD. We solved the generalized eigenvalue problem $C(t)v=\lambda C(t_0)v$, where the eigenvectors correspond to the optimal linear combination of basis operators to interpolate each state in the spectrum and the time dependence of the eigenvalues is controlled by the state energy. In order to solve the generalized eigenvalue problem, we used the properties of positive definite matrices to decompose $C(t_0)$ into its eigensystem and rewrite the GEVP as an ordinary eigenvalue problem with a Hermitian matrix. In order to fit the time dependence of the eigenvalues, it was necessary to identify corresponding eigenvalues across times and across configurations. This was accomplished by comparing the inner products of the corresponding eigenvectors.

The energies were extracted from the eigenvalues by applying nonlinear fitting of the form $\lambda(t)=(1-A)e^{-E(t-t_0)}+Ae^{-E'(t-t_0)}$ to the to the time-dependent generalized eigenvalues of the correlation matrix, where the second exponential with the $E'$ parameter accounts for excited states not captured by the limited basis of operators used.