Infrared Resummation and Dynamical Mass

Infrared resummation reorganizes a nonuniform expansion; it does not create one method-independent particle mass. A “dynamical mass” may denote an equal-time variance scale, a gap-equation parameter, a late-time decay exponent, or a pole of a retarded correlator. Agreement is meaningful only after those observables and coupling conventions are translated.

Required background. Interacting infrared logs supplies the breakdown parameter; dynamical RG supplies secular reorganization; and large-N saddles supplies the controlled limit. Helpful background. Review massless zero modes and nonuniform large-N corrections.

A declared large-N convention

Consider the $O(N)$ spectator potential

$$V(\boldsymbol\phi)=\frac{m^2}{2}\boldsymbol\phi^2 +\frac{\lambda}{4!N}(\boldsymbol\phi^2)^2.$$

At leading large $N$, define the per-component coincidence function

$$F=\frac1N\langle\boldsymbol\phi^2\rangle.$$

The self-consistent mass in this normalization is

$$M^2=m^2+\frac{\lambda}{6}F.$$

For $0<M^2\ll H^2$, the leading de Sitter infrared variance is $F\simeq3H^4/(8\pi^2M^2)$. Hence the massless gap equation gives

$$M^4=\frac{\lambda H^4}{16\pi^2}, \qquad M^2=\frac{\sqrt\lambda}{4\pi}H^2.$$

The coefficient changes if the quartic coupling or $F$ is normalized differently; the scaling $M^2\sim\sqrt\lambda H^2$ is the more portable statement. The mass screens the equal-time infrared enhancement after a relaxation interval of order $H^2/M^2$ e-folds.

The same result appears from the leading stochastic equilibrium. Write $r^2=\boldsymbol\phi^2=N\rho$. The radial probability includes both

$$\exp\!\left[-\frac{8\pi^2V(r)}{3H^4}\right]$$

and the Jacobian $r^{N-1}$. At large $N$, the radial saddle gives $\rho=3H^2/(2\pi\sqrt\lambda)$, and substituting $F=\rho$ in the gap equation reproduces $M^2=\sqrt\lambda H^2/(4\pi)$. Omitting the radial measure gives the wrong saddle.

First application: translate three resummations

Compute the late equal-time two-point function in three descriptions: the large-N gap equation, the Euclidean zero-mode integral, and the stochastic stationary distribution. Translate all of them to the potential above and to the same $F$. At leading large $N$, their infrared variance and $\sqrt\lambda$ mass scale agree. Beneke and Moch show how nonperturbative Euclidean zero-mode treatment generates this scaling and how its diagrammatic recovery requires infinitely many graphs Beneke and Moch 2013, §§2–5, Eqs. (10)–(46).

Now compare an unequal-time correlator. Its slowest decay eigenvalue in the Fokker–Planck problem need not equal $M^2/(3H)$ beyond the Gaussian or leading-large-N limit. A self-energy extracted from a retarded function can also have frequency dependence. Therefore an equal-time match licenses an infrared variance scale, not automatically a common spectral or relaxation mass.

Large-N resummation also organizes vertices and anomalous infrared powers. Serreau and Parentani obtain a nonperturbative large-N resummation of bubble diagrams with explicit ultraviolet renormalization Serreau and Parentani 2013, §§III–VI, pp. 085012-5–085012-16. Its domain is the controlled large-N deep infrared, not arbitrary finite $N$ or subhorizon momentum.

The structure map places each method beside the observable it resums. Inspect the convention translation before comparing reported masses.

For the declared $O(N)$ normalization, controlled methods reproduce $M^2=\sqrt\lambda H^2/(4\pi)$ at leading large $N$; unequal-time and finite-N claims require additional tests. Schematic; not to scale.

Domain and failure conditions

Use the chapter’s canonical domain table. The worked coefficient assumes exact de Sitter, the Euclidean/BD state, minimal coupling, massless bare theory, the displayed $O(N)$ convention, leading $1/N$, and the light infrared approximation. Finite duration may end before the gap relaxes.

Adversarial test. Move to finite $N$, include the first subleading correction, and compare equal-time variance, unequal-time decay, and a retarded response in the same renormalization scheme. Reject a universal mass claim if only $F$ agrees or if translating the coupling removes the numerical equality. Also vary the ultraviolet matching scale: surviving dependence beyond the computed order signals incomplete renormalization rather than infrared physics.

The failure map narrows the result to its controlled observable. The matched equal-time sector can pass to stochastic coarse-graining; spectral and response statements require their own real-time calculation.

Resummations agree only on translated observables within their expansion parameters; one equal-time variance does not define a universal de Sitter quasiparticle mass. Schematic; not to scale.

References

• Beneke, M., and P. Moch, “On ‘Dynamical Mass’ Generation in Euclidean de Sitter Space,” Physical Review D 87, 064018 (2013), doi:10.1103/PhysRevD.87.064018.
• Serreau, J., and R. Parentani, “Nonperturbative Resummation of de Sitter Infrared Logarithms in the Large-N Limit,” Physical Review D 87, 085012 (2013), doi:10.1103/PhysRevD.87.085012.