Dirac Equations With All Possible Combinations of Scalar-Scalar, Vector-Vector and Pseudoscalar-Pseudoscalar Interactions

Abstract

We obtain exact solutions of the nonlinear Dirac (NLD) equation in $1+1$ dimensions of the form $ψ(x,t) = e^{-iωt} ψ(x)$ for all possible combinations of scalar-scalar (SS), vector-vector (VV) and pseudoscalar-pseudoscalar (PP) interactions at nonlinearity parameter $κ= 1$, which corresponds to the most general 4-Fermion interactions. We look for solutions where $0 < ω< m$ where $ω, m$ are frequency and mass respectively. Remarkably, we show that at $κ= 1$, all the NLD models with SS, VV and PP interactions can be reduced to suitable combinations of SS and VV interactions and hence all solitary waves in these models must also be bound states. We show that two of the ten possible combinations are equivalent to $SS -(1/p) VV$ interactions while two other combinations are equivalent to $VV -(1/q) SS$ interactions for which we already know the results. For the remaining six combinations of SS, VV and PP, we show that depending on the parameters, whereas in some cases entire range of $0 < ω< m$ is allowed, in some other cases $ω$ is restricted to a smaller range $ω_{min} < ω< m$. Further, depending on the parameters, while the solitary waves in some cases have only one hump, in other cases the solitary waves have both single and double hump depending on the value of $ω$. We compute the charge $Q$ and the energy $E$ for each of the solitary wave solutions and show that while $E$ and $Q$ depend on coupling constant $g$, their ratio $E/Q$ is independent of $g$. When $κ= 1$, all the NLD models with SS, VV and PP interactions can be reduced to suitable combinations of SS and VV interactions. As a result all solitary waves in these models are bound states with $E/Q<m$.