Dynamical generation of fermion mass in a scalar-fermion theory with ${\lambda}{\phi}^4$ interaction

Abstract

The effective potential for a scalar theory with $\lambda\phi^4$ interaction, coupled to a massless fermion through Yukawa interaction is calculated by summing over infinite number of two particle irreducible (2PI) diagrams of two different types and a 2PI diagram of a third type using Cornwall, Jackiw and Tomboulis (CJT) method. There is an inversion symmetry present in the effective potential under $\phi\rightarrow -\phi$. When the value of coupling constant falls beyond an open set of positive real numbers, the effective potential exhibits both maxima and minima above and below the zero potential line respectively on either side of its minimum at $\phi=0$. The fermion acquires a mass in this region of coupling constant when the system settles into the minimum at positive, non-zero $\phi$ breaking the inversion symmetry of the vacuum. However, the effective potential exhibits a minimum only at $\phi=0$ and also the fermion remains massless when the coupling constant assumes any value from this open set.