Field theoretic treatment of an effective action for a model of catalyzed autoamplification

Reaction-diffusion models can exhibit continuous phase transitions in behaviors, and their dynamics at criticality often exhibit scalings with key parameters that can be characterized by exponents. While models with only a single field that transitions between absorbing and nonabsorbing states are well characterized and typically fall in the directed percolation universality class, the effects of coupling multiple fields remain poorly understood. We recently introduced a model which has three fields: one of which relaxes exponentially, one of which displays critical behavior, and one of which has purely diffusive dynamics but exerts an influence on the critical field [Tchernookov et al., J. Chem. Phys. 130, 134906 (2009)]. Simulations suggested that this model is in a universality class distinct from other reaction-diffusion systems studied previously. Although the three fields give rise to interesting physics, they complicate analysis of the model with renormalization-group methods. Here, we show how to systematically simplify the action for this model such that analytical expressions for the exponents of this universality class can be obtained by standard means. We expect the approach taken here to be of general applicability in reaction-diffusion systems with coupled order parameters that display qualitatively different behaviors close to criticality.

Figure 1

Illustration of a Lagrangian which is not complete. (a) Inclusion of the term ${\overline{\varphi }}^2\tau$ in the full Lagrangian [Eq. (17)] would enable the construction of this diagram. Solid, dashed, and wiggly lines represent the propagators for the $\varphi$, $\tau$, and $\psi$ fields, respectively. In these diagrams, time advances from left to right. (b) The diagram in (a) is a correction to this vertex which would have to be introduced by hand into the Lagrangian when performing the large mass reduction.

Figure 2

Vertices used in the renormalization-group calculation. Lines correspond to the same fields as in Fig. 1.

Figure 3

Diagrammatic expansions for the two-point vertex functions to one-loop order. Note that formally the diagrammatic expressions here are actually contributions to the correlation function $G_{\overline{\varphi }\varphi }={\Gamma }_{\overline{\varphi }\varphi }^{-1}$. To obtain the expression in Eq. (26), we Taylor expand $G_{\overline{\varphi }\varphi }^{-1}$ to first order in $ϵ$, which simply changes the sign of the two rightmost diagrams in this figure.

Figure 4

Diagrammatic expansions for the three-point vertex functions to one-loop order.