Perturbative S-matrix for massive scalar fields in global de Sitter space

TL;DR

This work constructs a perturbative S-matrix for interacting massive scalar fields in global de Sitter space, defined through Hartle–Hawking asymptotic states and two complementary formulations: a Lorentz-signature LSZ-like extraction and a pole-residue representation from Euclidean correlators. It proves key properties—unitarity, de Sitter invariance, CPT covariance, and perturbative field-redefinition invariance—and recovers the Minkowski S-matrix in the flat-space limit. The authors develop a Schwinger–Keldysh-based perturbation theory on the global dS contour and verify the optical theorem to $O(g^2)$ in a cubic heavy-field model, including both tree- and loop-level contributions. A central technical advance is the $R_\sigma$ projector, which resolves IR issues for light fields and yields well-defined asymptotic states, enabling a consistent treatment of both heavy and light fields. The results imply that, unlike certain claims in the literature, complementary-series particles generically decay at tree level, and they lay groundwork for extensions to gauge fields and perturbative string theories on de Sitter backgrounds.

Abstract

We construct a perturbative S-matrix for interacting massive scalar fields in global de Sitter space. Our S-matrix is formulated in terms of asymptotic particle states in the far past and future, taking appropriate care for light fields whose wavefunctions decay only very slowly near the de Sitter conformal boundaries. An alternative formulation expresses this S-matrix in terms of residues of poles in analytically-continued Euclidean correlators (computed in perturbation theory), making it clear that the standard Minkowski-space result is obtained in the flat-space limit. Our S-matrix transforms properly under CPT, is invariant under the de Sitter isometries and perturbative field redefinitions, and is unitary. This unitarity implies a de Sitter version of the optical theorem. We explicitly verify these properties to second order in the coupling for a general cubic interaction, including both tree- and loop-level contributions. Contrary to other statements in the literature, we find that a particle of any positive mass may decay at tree level to any number of particles, each of arbitrary positive masses. In particular, even very light fields (in the complementary series of de Sitter representations) are not protected from tree-level decays.

Figures (7)

• Figure 1: Left: A finite piece of global $D$-dimensional de Sitter space represented as a timelike hyperboloid in $D+1$ Minkowski space. The diagonal line is a cosmological horizon. Center: A conformal (Carter-Penrose) diagram for global de Sitter marked with horizontal cross-sections, each representing an $S^{D-1}$. The left and right edges are the poles. Right: The dS conformal diagram showing a cosmological horizon ${\cal H}$. Each point on the diagram represents an $S^{D-2}$ which contracts to zero size at the left and right edges.
• Figure 2: The time contour used to construct our global de Sitter S-matrix. The surface $\Sigma_0$ represents a cosmological horizon. See section \ref{['sec:Lorentz']} for details.
• Figure 3: Scalar de Sitter UIRs are depicted by the solid gray line in the complex $\sigma$ plane. For complementary series representations the weight $\sigma$ takes values along the negative real axis $\sigma \in (-(D-1)/2, 0)$ while for the principal series $\sigma$ takes complex values $\sigma = -(D-1)/2 + i \rho$, $\rho \ge 0$. For each series the 'conjugate weights' $-(\sigma+D-1)$ are depicted with a dashed gray line. We denote by $\Gamma_p$ the $\mathop{\rm Re}\sigma = -(D-1)/2$ contour. Representations with $\sigma$ values to the left of $\Gamma_p$ are reducible; they may be represented as an integral over the principal series UIRs. Representations with $\mathop{\rm Re} \sigma > -(D-1)/2$ and $\mathop{\rm Im} \sigma \neq 0$ are not unitary.
• Figure 4: An example contour of integration for the Lehmann-Kallën 2-point function. The contour is traversed from $-i\infty$ to $+i\infty$ in the strip to the left of the imaginary axis and to the right of any singularities (x's for our example) in the Källén-Lehmann weight with $\mathop{\rm Re} \sigma < 0$. The particular example involves the theory discussed in section \ref{['sec:Sheavy']}, which involves three species of scalar field. We have shown the poles in $\langle \phi_1(x)\phi_1(y)\rangle$ at $\mathcal{O}(g^2)$ for a case where $\phi_1,\phi_2$ lie in the principle series and $\phi_3$ lies in the complementary series.
• Figure 5: Feynman graphs contributing to connected corrections at $\mathcal{O}(g)$ and $\mathcal{O}(g^2)$ in $g\phi_3\phi_2\phi_1(x)$ theory. We use solid lines for $\phi_1(x)$, dashed lines for $\phi_2(x)$, and curly lines for $\phi_3(x)$. In the top row, from left to right: the corrections $\left\langle \phi_3(x_3)\phi_2(x_2)\phi_1(x_1) \right\rangle^{(1)}$, $\left\langle \phi_2({\overline{x}}_2)\phi_1({\overline{x}}_1)\phi_2(x_2)\phi_1(x_1) \right\rangle^{(1)}$, and $\left\langle \phi_1({\overline{x}})\phi_1(x) \right\rangle^{(2)}$. On the bottom row are the mass and field renormalization counterterms present in $\left\langle \phi_1({\overline{x}})\phi_1(x) \right\rangle^{(2)}$. There are also graphs of the same topology but with different permutations of the fields.