The IR stability of de Sitter: Loop corrections to scalar propagators

TL;DR

The paper addresses infrared stability in de Sitter space for interacting massive scalar fields by constructing Lorentz-invariant correlators through analytic continuation from the Euclidean vacuum. It develops two computational tools—embedding-distance in position space and Watson–Sommerfeld transformations in momentum space—to obtain absolutely convergent integral representations of 1-loop corrections to propagators, valid for masses in the complementary and principal series. The main result is that these 1-loop propagators are finite and decay at large timelike separations, often faster than any free propagator with $M^2>0$, indicating well-defined IR behavior and stability of the de Sitter-invariant vacua. The work provides a practical framework for higher-loop extensions and numerical computations, and it aligns with stochastic inflation results in the $M\ell\ll1$ limit, with potential extensions to gravitational fluctuations and higher $n$-point functions in future work.

Abstract

We compute 1-loop corrections to Lorentz-signature de Sitter-invariant 2-point functions defined by the interacting Euclidean vacuum for massive scalar quantum fields with cubic and quartic interactions. Our results apply to all masses for which the free Euclidean de Sitter vacuum is well-defined, including values in both the complimentary and the principal series of SO(D,1). In dimensions where the interactions are renormalizeable we provide absolutely convergent integral representations of the corrections. These representations suffice to analytically extract the leading behavior of the 2-point functions at large separations and may also be used for numerical computations. The interacting propagators decay at long distances at least as fast as one would naively expect, suggesting that such interacting de Sitter invariant vacuua are well-defined and are well-behaved in the IR. In fact, in some cases the interacting propagators decay faster than any free propagator with any value of $M^2> 0$.

Figures (7)

• Figure 1: On-shell values of $\sigma$ in the complex plane for massive scalar fields. The solid line denotes the path of $\sigma$ for increasing $M^2$ starting from at $\sigma=0$ for $M^2 = 0$. The dotted line shows the path of $-(\sigma+d)$ for increasing $M^2$ starting from $-(\sigma+d) = -d$ for $M^2 = 0$. Relatively light fields with $0 < M^2\ell^2 < d^2/4$ correspond to values of $\sigma$ and $-(\sigma+d)$ on the negative real axis and belong to the complementary series. Heavier fields with $M^2\ell^2 \ge d^2/4$ correspond to complex values of $\sigma$ and $-(\sigma+d)$ on the line $\Gamma_P$ and belong to the principal series.
• Figure 2: Even the above tree-level diagrams diverge for complimentary series fields with $\sigma$ close enough to zero.
• Figure 3: An example of the contour prescription for computing $\Delta^\sigma(Z)$. The contour $C_2$ is an arbitrary straight line through the reflection point $L = -d/2$. Sample $\sigma$-poles are drawn for the principal series (boxes) and complementary series (circles).
• Figure 4: Tree-level contributions to (a) $\left\langle \phi_1(x_1)\phi_2(x_2) \right\rangle$ and (b) $\left\langle \phi_1(x_1)\phi_1(x_2) \right\rangle$.
• Figure 5: $O(g)$ contributions to $\left\langle \phi_1(x_1)\phi_1(x_2) \right\rangle$. Diagram (a) is the $1$-loop contribution while (b) is a possible counterterm.