B.Sc. Thesis:

Properties of light quarks and π-mesons in a moving frame of reference described in Bethe-Salpeter formalism

In order to obtain my Bachelor of Science degree in physics, I worked on a project in the group of Prof. Fischer with the above topic as thesis. It mainly served as a means of getting into the field of theoretical hadron physics and the Dyson-Schwinger and Bethe-Salpeter framework. Therefore, the first part of the project consisted of reading about the relevant topics and reproducing already well-known results. These include the dynamical generation of effective quark masses, spontaneous chiral symmetry breaking and the pion as a Goldstone boson.

The part about the project, that was actually new, was to generalize the results from the numerical calculations to a moving frame, giving the pion a non-zero three-momentum. One would expect, that physical properties, such as the pions mass and decay constant, are Lorentz-invariant and therefore should not be affected from a change to a moving frame. Indeed, this is the result that I got out in the end, but in order to derive there, one has to be careful with some subtleties. In particular, the UV cutoff used to regularize divergent integrals has to be chosen not to violate Poincaré-invariance of the theory.

The equations, which have been used to calculate said properties are the Dyson-Schwinger equation for the quark propagator and the homogeneous Bethe-Salpeter equation for the pion Bethe-Salpeter amplitude. The infinite tower of DSEs has been truncated using the Rainbow truncation, together with the Ladder truncation for the quark-antiquark scattering kernel. As a model for the gluon propagator, a contact interaction has been used, which is equivalent to the Nambu-Jona-Lasinio model and therefore only well defined with a fixed UV cutoff scale. In the thesis I show, that using a hard UV cutoff violates Lorentz-invariance of the pions mass and decay constant. I also show, that using a proper-time regulator fixes this issue and results in Lorentz-invariant physical properties.

The thesis itself is available here, if you have any questions about it or the source code used, feel free to contact me. Below is the abstract of the thesis:

Abstract

In this thesis we will investigate the properties of light quarks and the π-meson (pion) based on the corresponding Dyson-Schwinger equation for quarks and the Bethe-Salpeter equation for bound states of quarks and antiquarks. We investigate, how mass is generated dynamically due to spontaneously broken chiral symmetry in the underlying quantum field theory, quantum chromodynamics (QCD), and we will calculate the effects on the mass and the leptonic decay constant of the pion. Furthermore we will investigate, how these properties behave under Lorentz transformation into a moving frame of reference in our model and what effect different approaches to regularization of the ultraviolet divergence of the theory have.