Master's Thesis in Theoretical Physics - Universiteit Utrecht

This term is precisely the derivative of θ times the zeroth component of the axial vectorcurrent J 5µ ,J 5µ = χγ µ γ 5 χ = χ L γ µ χ L − χ R γ µ χ R . (6.57)This term explicitly violates C and CP (it is odd under CP). If we now derive the field equationsfrom the fermion action with the additional term (6.56), we find the field equationswhere the mass m = f |φ| and whereiχ ′ L,h + h ˜k h χ L,h − amχ R,h = 0iχ ′ R,h − h ˜k h χ R,h − amχ L,h = 0, (6.58)˜k h = k − h ˙θ2 . (6.59)If ˙θ is constant, we find that we have obtained the field equations (6.48) with the differencethat the mass is now real and the momentum k is shifted by ˙θ. These are precisely the fieldequations that are solved by Koksma and Prokopec in [38]. We can therefore simply copytheir results and replace all the momenta k by ˜k. In calculating the propagator one has toinsert the solutions in the definition of the propagator Eq. (6.39). Roughly this means onehas to integrate the mode functions χ(k,η) over the momenta k, analogous to the calculationof the scalar propagator in Eq. (5.92). We can write this as an integration over k, butremember that the Hankel functions are functions of ˜k. One can then try to solve this byshifting the integration variable k to ˜k, which means we have to do an extra integrationover the mode functions from −h ˙θ˙θ2to 0. Another way to solve this is by considering the h2as a small parameter, such that we can write ˜k = k + δk and do a Taylor expansion for δksmall.The study on the calculation of the massive fermion propagator with a shifted momentum˜k was only started towards the end of this thesis. Therefore we cannot present any rigorousresults here, but only give an outline for further study. We hope that the shifted momentumwill give an extra term in the propagator that is similar to the axial vector current and isCP violating. This axial vector current can then be converted by sphaleron processes intoa baryon asymmetry and could source baryogenesis. If this is the case, then the two-Higgsdoublet model would not only be a good model for inflation, but could also be used to explainthe baryon asymmetry of the universe. We hope to do a rigorous study on the fermionpropagator in future work.This concludes our study on the massive fermion propagator derived from a fermion actionwhere the mass is generated by a complex scalar field. The calculations in this sectionwhere all focused on solving the Dirac equation for the fermion fields at tree-level. In otherwords, we tried to calculate the fermion propagator, but did not yet include any loop effects.In the next section we will in fact calculate the one-loop fermion self-energy.6.3 One-loop fermion self-energyIn quantum field theory, an interaction is seen as a small perturbation to the quadratic partof the action. By expanding the path integral in terms of the small perturbation, one canderive the Feynman rules and Feynman diagrams for the interactions. Instead of takingonly the "classical" tree-level propagator, the quantum propagator is constructed by takinginto account all loop corrections of arbitrary order. Since each loop is proportional to afactor ħ, the biggest loop contribution to the propagator will be the one-loop effect. In Fig.