The Fate of the Alpha-Vacuum

TL;DR

The paper addresses whether the family of de Sitter-invariant $\alpha$-vacua yield a consistent interacting QFT. Using the Schwinger-Keldysh formalism to track finite-time evolution from an $\alpha$-vacuum, it shows that one-loop corrections introduce a linearly divergent UV term proportional to $e^{\alpha+\alpha^*}$ that cannot be absorbed by any de Sitter-invariant counterterm, while logarithmic divergences remain renormalizable via a mass counterterm; the Euclidean vacuum avoids both issues. This establishes a fundamental pathology for true $\alpha$-vacua in interacting theories, highlighting that only the Euclidean (Bunch-Davies) vacuum yields a well-defined renormalizable framework in de Sitter space. The results also suggest that truncated $\alpha$-vacua, which cut off high-energy modes, can be finite and may be used to study how non-standard initial states could affect inflationary observables, though they remove the linear divergence by construction. Overall, the work reinforces the robustness of the Euclidean vacuum for inflation and provides a precise, non-equilibrium approach to initial-state questions in curved spacetime QFT.

Abstract

de Sitter space-time has a one complex parameter family of invariant vacua for the theory of a free, massive scalar field. For most of these vacua, in an interacting scalar theory the one loop corrections diverge linearly for large values of the loop momentum. These divergences are not of a form that can be removed by a de Sitter invariant counterterm, except in the case of the Euclidean, or Bunch-Davies, vacuum.

Figures (6)

• Figure 1: The contour used to evaluate the evolution of operators over a finite time interval. The initial state is an eigenstate of the Hamiltonian until $\eta_0$ at which time the interactions are turned on. We double the field content so that separate copies of the fields are used for the upper and lower parts of the contour.
• Figure 2: The coefficient $J$ of the linear counterterm is chosen to cancel insertions of tadpoles. The dashed line represents a line in a general diagram.
• Figure 3: After choosing the linear term to cancel graphs containing a tadpole, only these terms contribute at order $\lambda^2$ to the expectation value of $[H_I,\tilde{a}_{\vec{k}}^\dagger \tilde{a}_{\vec{k}}]$ in the $\alpha$-vacuum. The first represents a self-energy graph while the second is from the mass counterterm.
• Figure 4: The order $\lambda^2$ corrections to the expectation value of $[H_0,a_{\vec{k}}^{E\dagger} a_{\vec{k}}^E]$ in the $\alpha$-vacuum. Again, the first represents a self-energy graph while the second is from the mass counterterm.
• Figure 5: Examples of divergent diagrams in theories with quartic (left) or quintic (right) interactions. Any loop that contains only two lines will exhibit a UV divergence similar to that in Eq. (\ref{['loopdivg']}). This result follows from the structure of the propagator in the $\alpha$-vacuum and not the form of the interactions.