Dynamical symmetry breaking in a non-polynomial four-fermion model

Core claim

The paper demonstrates that the one-loop effective potential for two auxiliary vector fields $A_\mu$ and $B_\mu$ (and an auxiliary scalar $\sigma$) derived from the non-polynomial four-fermion Lagrangian (1) takes the closed form $V_{\rm eff} = -M\upsilon - 2^{-D/2}\pi^{1-D/2}\mu^{4-D}\csc(\pi D/2) M^D (1-4g^2B^2)^{-(D+1)/2} / (D\Gamma(D/2))$ after dimensional regularization. Upon renormalization and with $\upsilon=0$ at the minimum, the potential reduces to $V_{\rm eff} = (1+2eg A\cdot B)^4 m_R^4 (1-4g^2B^2)^{-5/2}$. For constant $A_\mu = a_\mu$, the potential has a global minimum at $a\cdot B = -1/(2eg)$ with $V_{\rm eff}=0$, provided $1-4g^2B^2>0$ (automatically satisfied for spacelike $B_\mu$, and requiring $B^2<1/(4g^2)$ for timelike $B_\mu$). To fourth order in the fields, the potential is bounded from below with nontrivial minima in both spacelike and timelike cases. Additionally, the low-energy effective action is computed using the derivative expansion, yielding Maxwell kinetic terms for $A_\mu$ and $B_\mu$ plus a mixed term, as shown in Eq. (42). This establishes a concrete mechanism for the dynamical generation of two interacting vector fields with spontaneous Lorentz symmetry breaking.