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The colour-singlet axial-vector vertex plays a pivotal role in understanding dynamical chiral symmetry breaking and numerous hadronic weak interactions, yet scant model-independent information is available. We therefore use longitudinal and transverse Ward–Green–Takahashi (WGT) identities, together with kinematic constraints, in order to ameliorate this situation and expose novel features of the axial vertex: amongst them, Ward-like identities for elements in the transverse piece of the vertex, which complement and shed new light on identities determined previously for components in its longitudinal part. Such algebraic results are verified via solutions of the Bethe–Salpeter equation for the axial vertex obtained using two materially different kernels for the relevant Dyson–Schwinger equations. The solutions also provide insights that suggest a practical Ansatz for the axial-vector vertex.
This thesis explores the phase diagrams of the Nambu--Jona-Lasinio (NJL) and quark-meson (QM) model in the mean-field approximation and beyond. The focus lies in the investigation of the interplay between inhomogeneous chiral condensates and two-flavor color superconductivity.
In the first part of this thesis, we study the NJL model with 2SC diquarks in the mean-field approximation and determine the dispersion relations for quasiparticle excitations for generic spatial modulations of the chiral condensate in the presence of a homogeneous 2SC-diquark condensate, provided that the dispersion relations in the absence of color superconductivity are known. We then compare two different Ansätze for the chiral order parameter, the chiral density wave (CDW) and the real-kink crystal (RKC). For both Ansätze we find for specific diquark couplings a so-called coexistence phase where both the inhomogeneous chiral condensate and the diquark condensate coexist. Increasing the diquark coupling disfavors the coexistence phase in favor of a pure diquark phase.
On the other hand, decreasing the diquark coupling favors the inhomogeneous phase over the coexistence phase.
In the second part of this thesis the functional renormalization group is employed to study the phase diagram of the quark-meson-diquark model. We observe that the region of the phase diagram found in previous studies, where the entropy density takes on unphysical negative values, vanishes when including diquark degrees of freedom. Furthermore, we perform a stability analysis of the homogeneous phase and compare the results with those of previous studies. We find that an increasing diquark coupling leads to a smaller region of instability as the 2SC phase extends to a smaller chemical potential. We also find a region where simultaneously an instability occurs and a non-vanishing diquark condensate forms, which is an indication of the existence of a coexistence phase in accordance with the results of the first part of this work.
In this work we study compact stars, i.e. neutron stars, as cosmic laboratories for the nuclear matter. With a mass of around 1 - 3 solar masses and a radius of around 10km, compact stars are very dense and, besides nucleons, can contain exotic matter such as hyperons or quark matter. The KaoS collaboration studied nuclear matter for densities up to 2-3 times saturation density by analysing kaon multiplicities from Au+Au and C+C collisions. The results show that nuclear matter in the corresponding density region is very compressible, with a compressibility of <200MeV. For such soft nuclear equations of state the maximum masses of neutron stars are ca. 1.8 - 1.9 solar masses, whereas the central densities are higher than 5 times nuclear saturation density and therefore point towards a possible phase transition to quark matter. If quark matter would be present in the interior of neutron stars, so-called hybrid stars, it could be produced already during their birth in supernova explosions. To study this we implement a quark matter phase transition in a hadronic equation of state which is used in supernova simulations. Supernova simulations of low and intermediate mass progenitors and two different bag constants show a collapse of the proto neutron star due to the softening of the equations of state in the quark-hadron mixed phase. The stiffening of the equation of state for pure quark matter halts the collapse and leads to the production of a second shock wave. The second shock wave is energetic enough to lead to an explosion of the star and produces a neutrino burst when passing the neutrinospheres. Furthermore, first studies of the longtime cooling of hybrid stars show, that colour superconductivity can significantly influence the cooling behaviour of hybrid stars, if all quarks form Cooper Pairs. For the so-called CSL phase (colour-spin locking) with pairing energies of several MeV, the cooling of the quark phase is suppressed and the hybrid star appears as a pure hadronic star.