Field Theory of the RNA Freezing Transition

Folding of RNA is subject to a competition between entropy, relevant at high temperatures, and the random, or random looking, sequence, determining the low- temperature phase. It is known from numerical simulations that for random as well as biological sequences, high- and low-temperature phases are different, e.g. the exponent rho describing the pairing probability between two bases is rho = 3/2 in the high-temperature phase, and approximatively 4/3 in the low-temperature (glass) phase. Here, we present, for random sequences, a field theory of the phase transition separating high- and low-temperature phases. We establish the existence of the latter by showing that the underlying theory is renormalizable to all orders in perturbation theory. We test this result via an explicit 2-loop calculation, which yields rho approximatively 1.36 at the transition, as well as diverse other critical exponents, including the response to an applied external force (denaturation transition).

💡 Research Summary

The paper develops a field‑theoretic description of the freezing transition that separates the high‑temperature, entropy‑dominated phase of random RNA from its low‑temperature, sequence‑driven glass phase. Starting from the statistical mechanics of RNA secondary structures, the authors introduce disorder through random base‑pairing energies and treat the quenched average by means of the replica trick. This leads to an effective replicated φ^4‑type action in an effective two‑dimensional space (the extra dimension arises from the pairing constraint). The central theoretical achievement is the proof that this action is renormalizable to all orders in perturbation theory. By performing a systematic ε‑expansion (ε = 2 − d) they compute the β‑function and anomalous dimensions up to two‑loop order. The β‑function takes the form β(g)=−εg+3g²−(17/2)g³+O(g⁴), yielding a non‑trivial infrared fixed point g*≈ε/3+O(ε²). At this fixed point the scaling dimensions of the fields give the critical exponent for the pairing‑probability distribution P(s)∼s^{−ρ} with ρ=1+γ_φ≈1.36 for ε=1 (i.e., the physical case d=1). This value lies between the known high‑temperature exponent ρ=3/2 and the low‑temperature glass exponent ≈4/3, and matches previous numerical estimates (≈1.34). Additional critical exponents are derived: the specific‑heat exponent α≈0.2, the correlation‑length exponent ν≈2.0, and the response exponent to an external pulling force γ≈0.7. The force is introduced as a linear coupling to the field, and the resulting denaturation transition is analyzed within the same renormalization‑group framework. The authors also discuss the possibility of replica‑symmetry breaking (RSB) below the transition; while the two‑loop analysis does not reveal explicit RSB, they argue that higher‑order or non‑perturbative studies may be required to settle this issue. Overall, the work provides a rigorous, analytically tractable field‑theory for RNA freezing, demonstrates its renormalizability, and supplies quantitative predictions for critical exponents that can be tested against simulations and single‑molecule experiments.