A formalism for a consistent treatment of two-pion interactions in heavy meson decays

Abstract

The best-established theory of particle physics up to now is the Standard Model. The theory of strong interactions called quantum chromodynamics (QCD) is a part of it. QCD leads to a wide range of phenomena, which still need to be understood. For example, many experiments showed that the spectrum of QCD bound states contains many states that are inconsistent with the most simple quark model. To gain a better insight into their nature, theoretically sound studies are required, which preserve fundamental physical properties such as Lorentz invariance, causality and probability conservation. The problem is complicated because the standard perturbation theory approach in the strong coupling constant is not applicable at the hadronic scale. A proper alternative method is dispersion theory, which includes analyticity, unitarity and crossing by construction. However, its application is restricted to low energies due to the opening of inelastic channels. Therefore, this thesis aims to construct an effective high-energy extension of the dispersive framework in two subprojects. <br />The first project introduces a scattering matrix and form factor parametrization for ππ-interactions, which is applicable from threshold to about 2 GeV. We focus primarily on the rich spectrum provided by the isoscalar, scalar final state rescattering of the channels ππ and K<span style="text-decoration: overline">K</span>. In our method additional channels to those two are coupled by s-channel resonance exchange. An application to the decays <span style="text-decoration: overline">B</span>_s⁰ → J/ψ ππ and <span style="text-decoration: overline">B</span>_s⁰ → J/ψ K<span style="text-decoration: overline">K</span> allows for the extraction of the scalar, isoscalar form factor up to about 2.2 GeV. In this approach the J/ψ acts as a spectator and hence does not interact with the other final states. An analytic continuation of the form factor allows us to extract resonance poles for the f₀(1500) and f₀(2020). <br />The second project expands on providing a fully analytic and unitary solution to the three-particle decay amplitudes. While the previous project assumes one particle to be a spectator, we now allow for pairwise rescattering of all three particles. For this we consider the decay Υ(5S) → Υ(nS) ππ with n=1,2,3. While the ππ scalar spectrum is fixed by the previous analysis we include the exotic resonances Z_b^±(10610) and Z_b^±(10650) in the Υ(nS)π spectrum. Due to their closeness to the B<span style="text-decoration: overline">B</span>^* and B^*<span style="text-decoration: overline">B</span>^* thresholds, respectively they are modeled as hadronic molecules by dynamic rescattering of these channels and further inelastic effects. Two approaches in order to determine the ππ S-wave amplitude for the Υ(1S) ππ final state are employed. The first method is the standard integration along the Khuri-Treiman path. On the other hand, the second approach employs a spectral density integral for the crossed-channel amplitude to determine the partial-wave projection analytically. Both are consistent with each other, allowing us to use the second approach for the Υ(nS) ππ final states with n=2,3. An application to the Dalitz plot data is in progress.