Characterization of resonances using finite size effects

Abstract

We develop methods to extract resonance widths from finite volume spectra of 1+1 dimensional quantum field theories. Our two methods are based on Luscher's description of finite size corrections, and are dubbed the Breit-Wigner and the improved "mini-Hamiltonian" method, respectively. We establish a consistent framework for the finite volume description of sufficiently narrow resonances that takes into account the finite size corrections and mass shifts properly. Using predictions from form factor perturbation theory, we test the two methods against finite size data from truncated conformal space approach, and find excellent agreement which confirms both the theoretical framework and the numerical validity of the methods. Although our investigation is carried out in 1+1 dimensions, the extension to physical (3+1) space-time dimensions appears straightforward, given sufficiently accurate finite volume spectra.

Figures (9)

• Figure 3.1: Behaviour of levels at $\lambda=0$ and $\lambda\neq0$, illustrated using actual numerical data for the Ising case. (Energy and distance are measured in units given by the value that the mass of the lightest particle ($m_{1}$) takes at $\lambda=0$.)
• Figure 3.2: Illustrating the behaviour of two-particle states in the integrable case $\lambda=0$ and with a resonance present ($\lambda\neq0$).
• Figure 3.3: Phase shift functions $\delta_{n}(E)$ extracted from the two-particle levels illustrated in figure \ref{['cap:two_part_state_illustration']} around the resonant energy $E_{c}$ .
• Figure 3.4: Illustration of the residual finite size effects in the Ising model, for the levels pertaining to $A_{4}$ and two-particle states $A_{1}A_{1}$. The volume dependence of the two-particle levels follows very precisely the description given in eq. (\ref{['eq:Luscher_quant']}), but the variation of $A_{4}$ is entirely due to the residual finite size effects that we neglected. The plot shows the integrable case $t=0$ when the line crossings are exact, but the only result of switching on a small value for $t$ is a slight shift in the lines and resolution of the degeneracy at the level crossings.
• Figure 4.1: Corrections to the masses of $A_{1}$ and $A_{2}$. TCSA data are indicated with crosses (the numerical uncertainties are too small to be displayed), while the lines give the FFPT predictions.