Infrared stability of de Sitter space: Loop corrections to scalar propagators

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 $M^2>0$.

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) $⟨{\varphi }_1(x_1){\varphi }_2(x_2)⟩$ and (b) $⟨{\varphi }_1(x_1){\varphi }_1(x_2)⟩$.

Figure 5

$O(g)$ contributions to $⟨{\varphi }_1(x_1){\varphi }_1(x_2)⟩$. Diagram (a) is the 1-loop contribution while (b) is a possible counterterm.

Figure 6

$O(g^2)$ corrections to $⟨{\varphi }_1(x_1){\varphi }_1(x_2)⟩$. Diagram (a) is the 1-loop contribution, (b) the counterterm due to field renormalization, and (c) the counterterm due to mass renormalization. The slash in diagram (b) denotes the action of ${\nabla }^2$.

Figure 7

Two examples of the complex $L$-plane. Poles at ${\sigma }_1$ and $-({\sigma }_1+2\alpha )$ are marked by $\square$’s, while the pole at (${\sigma }_2+{\sigma }_3$) is marked by an $\circ$. In example (a) ${\varphi }_1(x)$ and ${\varphi }_2(x)$ are in the complementary series while ${\varphi }_3(x)$ is in the principal series. Here $\Gamma ={\Gamma }_P$. In example (b) ${\varphi }_2(x)$ and ${\varphi }_3(x)$ are in the complementary series while ${\varphi }_1(x)$ is in the principal series. Here $\Gamma$ differs from ${\Gamma }_P$ since we take $\Gamma$ to pass to the left of poles with $\mathrm{Re}L=-\alpha$.