Schwinger-Dyson approach for a Lifshitz-type Yukawa model

We consider a $3+1$ dimensional field theory at a Lifshitz point for a dynamical critical exponent $z=3$, with a scalar and a fermion field coupled via a Yukawa interaction. Using the nonperturbative Schwinger-Dyson approach we calculate quantum corrections to the effective action. We demonstrate that a first order derivative kinetic term as well as a mass term for the fermion arise dynamically. This signals the restoration of Lorentz symmetry in the IR regime of the single fermion model, although for theories with more than one fermionic species such a conclusion will require fine-tuning of couplings. The limitations of the model and our approach are discussed.

Figure 1

The fermion self-energy given by the Schwinger-Dyson equation in the rainbow approximation. A solid thick line represents the dressed fermion propagator, a solid thin line the bare fermion propagator, and a dashed line represents the dressed scalar propagator (which, in our approximation, is like the bare propagator, but with the renormalized mass instead of the bare one). As one can see, the fermion self-energy is obtained as a resummation of an infinite number of graphs, which is at the origin of the nonperturbative feature of the results.

Figure 2

The parameters $\mu =m_f/m_b$, $l^{1/2}={\lambda }^{1/2}/m_b$ as a function of $\epsilon =g/(2\pi m_b^3)$. The system of Eqs. (31) has a unique solution for $\epsilon >{\epsilon }_c$, which for $\epsilon \to {\epsilon }_c$ tends asymptotically to $\mu =0$ and $l^{1/2}\simeq 0.529$. For $\epsilon \le {\epsilon }_c$ we have checked numerically that the system (31) has no solution.