Interacting Infrared Logs and Secular Regimes

An interacting de Sitter logarithm signals that a perturbative coefficient depends strongly on scale or duration. It becomes a secular problem only when the complete dimensionless correction approaches unity for the stated observable. The logarithm’s origin—ultraviolet renormalization, a soft loop, a time integral, or an initial surface—must be identified before any resummation is chosen.

Required background. Euclidean/BD free fields fixes the propagator; massless zero modes fixes the obstructed sector; and cosmological loops fixes in-in renormalization. Helpful background. Review dynamical RG and the curved $\phi^4$ benchmark.

Secular power counting

Let $N=\ln[a(t)/a(t_0)]$. For a free massless minimally coupled scalar with a specified infrared initial scale, the leading long variance is

$$\langle\phi_L^2\rangle_0=\frac{H^2}{4\pi^2}N.$$

Now take $V(\phi)=\lambda\phi^4/4!$. The leading-long stochastic moment equation—which reproduces the corresponding leading in-in logarithms in its domain—is

$$\frac{d}{dt}\langle\phi^2\rangle =\frac{H^3}{4\pi^2} -\frac{\lambda}{9H}\langle\phi^4\rangle.$$

Using the free Gaussian value $\langle\phi^4\rangle_0=3\langle\phi^2\rangle_0^2$ on the right gives

$$\langle\phi^2\rangle =\frac{H^2}{4\pi^2}N -\frac{\lambda H^2}{144\pi^4}N^3 +O(\lambda^2N^5).$$

The relative correction is $-\lambda N^2/(36\pi^2)$, so fixed-order perturbation becomes nonuniform around $\lambda N^2\sim1$. The field variance itself already depends on its infrared preparation; the robust claim is the stated leading-log series and its breakdown parameter, not a universal local instability.

Ultraviolet logarithms such as $\ln(\mu/H)$ instead run local operators and couplings. They can multiply secular terms, but changing $\mu$ and changing the duration are different operations. Renormalization-group invariance should remove the former from the final observable to the computed order; no scale choice can remove genuine long-time nonuniformity.

First application: a two-point leading-log interval

Compute the renormalized in-in two-point function of the quartic spectator through one nontrivial order, extracting the highest power of $N$ at each order. Verify its coefficient by the moment iteration above, then retain only times satisfying both $\lambda/(16\pi^2)\ll1$ and $\lambda N^2/(36\pi^2)\ll1$. Weinberg’s nested-commutator analysis constrains late-time growth under explicit assumptions on interactions and fields Weinberg 2005, §§III–IV, pp. 043514-5–043514-12; it does not make every correlator time independent.

Next change the object. A derivative correlator removes constant soft legs, while an unsmeared coordinate-space field correlator retains them. Change the interaction to a shift-symmetric derivative vertex: additional soft momenta alter the leading power count. Diagram-by-diagram stochastic/QFT agreement has been established for the leading infrared approximation of a light potential-interacting scalar Garbrecht et al. 2015, §§III–V, Eqs. (35)–(75). That result is powerful but observable- and approximation-specific.

The structure map locates secular counting between the free propagator and any resummation, with separate branches for shift-symmetric and potential interactions.

Secular logarithms are organized by coupling, duration, interaction symmetry, and observable; ultraviolet running and late-time nonuniformity remain distinct. Schematic; not to scale.

Domain and failure conditions

See the chapter’s canonical domain table. The displayed coefficients use exact fixed $H$, the potential convention $\lambda\phi^4/4!$, a leading-long split, and the prepared massless scalar variance. Finite mass cuts off growth after $N\sim H^2/M^2$; finite slow roll adds a separate duration and drift error.

Adversarial test. Compute an unsmeared field correlator and one local derivative composite for both the quartic potential and a shift-symmetric derivative interaction. Repeat with a small-mass regulator, a finite-volume regulator, and a de Sitter-invariant ultraviolet subtraction. Reject universal “secular instability” or “stochastic necessity” language if the highest logarithm changes with the object, symmetry, or regulator after legitimate local translations.

The failure map downgrades a large but gauge-, scheme-, or object-specific term to perturbative nonuniformity in that object. Only the surviving scalar sector passes to infrared resummation.

The strongest licensed statement is the fixed-order interval or resummation need for the exact renormalized observable whose secular parameter reaches unity. Schematic; not to scale.

References

• Garbrecht, B., F. Gautier, G. Rigopoulos, and Y. Zhu, “Feynman Diagrams for Stochastic Inflation and Quantum Field Theory in de Sitter Space,” Physical Review D 91, 063520 (2015), doi:10.1103/PhysRevD.91.063520.
• Weinberg, S., “Quantum Contributions to Cosmological Correlations,” Physical Review D 72, 043514 (2005), doi:10.1103/PhysRevD.72.043514.