The scattering of spinning hadrons from lattice QCD

Abstract

Hadron spectroscopy is predominantly the study of resonances that decay via the strong interaction into a multitude of stable hadrons, such as the pion. The vast majority of resonances decay via an intermediate hadron with non-zero intrinsic spin. In this thesis, I will present the results of scattering calculations featuring mesons with non-zero intrinsic spin. Before doing so, I will first give a brief introduction to QCD and review the framework necessary to perform lattice QCD calculations in Chapters 1 and 2.

In Chapter 3, I present the first lattice calculation of $\rho \pi$ scattering in isospin-2. Here, $\rho\pi$ features in dynamically-coupled $^3{S}_1$ and $^3{D}_1$ partial-waves with $J^P=1^+$. No resonance enhancement is anticipated in the flavour exotic isospin-2 channel and as such it provides an ideal testing ground for this first calculation. I work at heavier than physical quark masses at the $\text{SU}(3)_{\text{F}}$ point where the up, down and strange quarks are mass degenerate. Finite-volume spectra are calculated and, utilising the relationship between the discrete energy spectrum and the infinite-volume scattering amplitudes, partial-wave amplitudes with $J \le 3$ and the degree of dynamical mixing between the coupled $^3{S}_1$ and $^3{D}_1$ channels are determined.

In Chapter 4, I investigate $\rho\pi$ in isospin-1 where the $a_1$ axial-vector resonance is expected to feature. Here, I present a discussion on $G$-parity and Bose-symmetry at the $\text{SU}(3)_{\text{F}}$ point. Working at heavier than physical quark masses, the resulting finite volume spectrum suggests that the $a_1$ is a bound-state and that the $^3{S}_1$- and $^3{D}_1$-wave, $\rho\pi$ scattering amplitudes are similar to those in isospin-2.

I present the first calculation of coupled $\pi\omega$ and $\pi\phi$ scattering in Chapter 5 where resonant enhancement is seen experimentally in the $J^P=1^+$ channel. Working at a somewhat lighter pion mass than in previous chapters, the finite-volume spectra are determined and the scattering amplitudes are calculated. Analytically continuing the amplitudes into the complex energy plane, a resonance pole is found, interpreted as the analogue of the $b_1$ axial-vector, which couples dominantly to $^3{S}_1$-wave $\pi\omega$, with a much-suppressed coupling to $^3{D}_1$-wave $\pi\omega$, and a negligible coupling to $\pi\phi$.

In Chapter 6, the exotic $J^{PC}=1^{-+}$ channel is studied. These quantum numbers are not allowed in the quark model but can be obtained, for example, through a gluonic excitation coupled to a quark-antiquark pair. In this exploratory calculation, performed at the $\text{SU}(3)_\text{F}$ point, the finite-volume spectra and coupled-channel scattering amplitudes are presented. A single resonance pole is found, interpreted as the exotic $\pi_1$, and couplings to meson-meson channels, including for example $\pi\eta\{^1{P}_1\}$, $\pi\eta'\{^1{P}_1\}$ and $\rho\pi\{^3{P}_1\}$, are calculated for the first time in lattice QCD.

In order to minimally present the contents of a unitary $n$-channel scattering matrix, I introduce, in Chapter 7, an $n$-channel generalisation of the traditional two-channel Stapp parameterisation.