Jansen, Maximilian Bernhard (2016)
Effective field theory approaches for tensor potentials.
Technische Universität Darmstadt
Dissertation, Erstveröffentlichung

Kurzbeschreibung (Abstract)

Effective field theories are a widely used tool to study physical systems at low energies. We apply them to systematically analyze two and three particles interacting via tensor potentials. Two examples are addressed: pion interactions for DD* scattering to dynamically generate the X(3872) and dipole interactions for two and three bosons at low energies. For the former, the one-pion exchange and for the latter, the long-range dipole force induce a tensor-like structure of the potential. We apply perturbative as well as non-perturbative methods to determine low-energy observables.

The X(3872) is of major interest in modern high-energy physics. Its exotic characteristics require approaches outside the range of the quark model for baryons and mesons. Effective field theories represent such methods and provide access to its peculiar nature. We interpret the X(3872) as a hadronic molecule consisting of neutral D and D* mesons. It is possible to apply an effective field theory with perturbative pions. Within this framwork, we address chiral as well as finite volume extrapolations for low-energy observables, such as the binding energy and the scattering length. We show that the two-point correlation function for the D* meson has to be resummed to cure infrared divergences. Moreover, next-to-leading order coupling constants, which were introduced by power counting arguments, appear to be essential to renormalize the scattering amplitude. The binding energy as well as the scattering length display a moderate dependence on the light quark masses. The X(3872) is most likely deeper bound for large light quark masses. In a finite volume on the other hand, the binding energy significantly increases. The dependence on the light quark masses and the volume size can be simultaneously obtained.

For bosonic dipoles we apply a non-perturbative, numerical approach. We solve the Lippmann-Schwinger equation for the two-dipole system and the Faddeev equation for three bosonic dipoles. Scattering amplitudes are ultraviolet divergent and have to be regularized. A single, isotropic S-wave operator is insufficient to properly renormalize and a linear combination of, in general anisotropic, short-range interactions has to be introduced. Bound states can be classified by parity and projection of the orbital angular momentum on the dipole moment. Both quantum numbers are conserved. Moreover, binding energies can be divided into different sets characterized by multiple angular momentum quantum numbers. However, different sets couple among each other and the spectrum displays avoided level crossings whenever two of them come close. We further determine the bound-state spectrum for three bosonic dipoles. The Faddeev equation decouples if the two-body threshold is dominated by a particular projection of orbital angular momentum. We solve it for the case of vanishing projection quantum number. It appears that bound states are universally determined by two-body low-energy observables and no explicit three-body forces have to be included to assure regulator independence. Furthermore, we derive a reformulation of the Faddeev equation, which is numerically beneficial. For a proof of concept we implement it for the S-wave projected 1/r^3 potential.

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