Baryons as Relativistic Bound States of Quark and Diquark by M. Ottel

By M. Ottel

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Invariance is not guaranteed for an arbitrary truncation in the Chebyshev expansion. In the determination of observables the explicit solutions for wave or vertex function enter. We will therefore have a look at their convergence properties as well. The Chebyshev moments of the scalar functions S1 and S2 building the vertex function show a very similar convergence pattern for the different choices of η. Each subsequent moment drops in its magnitude by a factor of 7 . . 10 as compared to the magnitude of the previous one.

138) The new propagator matrix (g0′ )ij and the modified kernel (H ′ )ij are the result of taking Dirac traces after multiplication with the respective adjoint Dirac components. 5 Numerical solutions 49 ˆ Pˆ , z = pˆ·Pˆ between the vectors k, p, P which are can be expressed as k 2 , p2 , z ′ = k· ˆ p ˆ. 140) 1 − z2 dk (H ′ )ij (p2 , k 2 , z, z ′ , y ′) × (2π)4 2 π in−m Yˆjn (k 2 ) n=0 Tn (z ′ )Tm (z) √ . 141) Of course, in the practical calculation the Chebyshev expansion can be taken into m th account only up to a mth max moment for the Yi and up to a nmax moment for the Yˆin .

3 The diquark-quark Bethe-Salpeter equation 33 This incorporates from the beginning an effective quark mass in the order of magnitude of several hundred MeV which is believed to be dynamically generated in QCD. The property of confinement is neither manifest in eq. 64) nor in the form of the diquark propagators, eqs. 46). Effective parametrizations of confinement in the model propagators and its influence on results will be investigated in chapter 5. 1 Nucleon The Faddeev amplitude of eq. 35) takes the following form, Γαβγ = χ5βγ D Φ5 u α + χµβγ D µν (Φν u)α .

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