[Paper Review] Anomaly breaking of de Sitter symmetry

Myron Bander|arXiv (Cornell University)|Nov 2, 2010

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper demonstrates that interacting scalar fields on de Sitter space exhibit an infrared anomaly at one-loop order, breaking de Sitter symmetry for all vacua except the Euclidean (Bunch-Davies) vacuum. The anomaly arises because the propagator's antipodal singularity leads to non-conservation of de Sitter current operators, with the divergence at point $x$ receiving a non-zero contribution from its antipodal point $\bar{x}$, explicitly breaking symmetry via a $\sim \partial_\mu S^\mu \sim \left(\tau\partial_\tau + x_i\partial_i\right) \partial^2_\phi V(\phi(\bar{x}))$ term.

ABSTRACT

To one loop order, interacting boson fields on de Sitter space have an "infrared" anomaly that breaks the de Sitter symmetry for all vacua save the Euclidian one. The divergence of a symmetry current at point $x$ has a non-zero contribution at the antipodal point ${\bar x}$.

Motivation & Objective

• To investigate the quantum anomalies in de Sitter symmetry for interacting scalar fields in $D=(1+d)$ dimensions.
• To determine whether de Sitter symmetry is preserved in interacting quantum field theories at one-loop order.
• To analyze the role of the propagator proposed by Polyakov in breaking de Sitter symmetry via infrared divergences.
• To examine the dependence of the anomaly on vacuum choice, particularly contrasting the Euclidean vacuum with other $α$-vacua.
• To assess the implications of this anomaly for cosmological models, especially during inflation.

Proposed method

• Use of the flat slicing metric in de Sitter space with conformal time $\tau$, parameterizing the geometry via embedding in a $(D+1)$-dimensional Minkowski space.
• Construction of de Sitter isometry generators $S^{(a;i)}_\nu$, $S^{(b)}_\nu$, $S^{(c;i,j)}_\nu$, and $S^{(d;i)}_\nu$ from the canonical energy-momentum tensor $\Theta_{\mu\nu}$.
• Adoption of Polyakov's propagator $D_m(x_1,x_2) = C(1-z_{12}^2)^{-(D-2)/4} \mathcal{Q}^{-1/2+i\nu(m)}_{(D-2)/2}(z_{12})$, which exhibits an antipodal singularity at $z_{12} = -1$.
• Regularization via subtraction of a reference propagator $D_M$ with $M \to \infty$, ensuring equal residues at the antipodal point to isolate the anomaly.
• Asymptotic evaluation of loop integrals in the large $M$ limit, showing peak localization at the antipodal point $\bar{x}$, allowing replacement of $D_m(z,y_i)$ by $D_m(\bar{x},y_i)$.
• Use of analyticity and contour integration in the embedding space to evaluate the $z_0$-integral, leading to a non-vanishing anomaly proportional to $D_m(\bar{x},y_1)D_m(\bar{x},y_2)$.

Experimental results

Research questions

• RQ1 Does the one-loop effective action for interacting scalar fields on de Sitter space break de Sitter symmetry due to infrared anomalies?
• RQ2 How does the choice of vacuum, particularly the Polyakov propagator, affect the conservation of de Sitter current operators?
• RQ3 Is the anomaly dependent on the regularization scheme used for the antipodal singularity in the propagator?
• RQ4 Do higher-order quantum corrections preserve or further enhance the anomaly observed at one-loop order?
• RQ5 Can the anomaly be generalized to fermionic or other interacting fields beyond scalar theories?

Key findings

• The de Sitter current $S^{(\cdot,i)}_\nu$ is not conserved at one-loop order due to a non-vanishing divergence $\eta^{\mu\nu}\partial_\mu S^{(\cdot,i)}_\nu \sim \left(\tau\partial_\tau + x_i\partial_i\right) g\phi(\bar{x})^2$, indicating explicit breaking of de Sitter symmetry.
• The anomaly arises from the antipodal singularity in the propagator, which is present in the Polyakov (non-Euclidean) vacuum but absent in the Euclidean vacuum.
• The regularization procedure, involving subtraction of a high-mass reference propagator $D_M$ with $M \to \infty$, isolates the anomaly and confirms its non-vanishing nature in the large-$M$ limit.
• The one-loop correction to the current divergence scales as $\sim D_m(\bar{x},y_1)D_m(\bar{x},y_2)$, confirming that the anomaly is localized at the antipodal point.
• The anomaly is absent in the Euclidean vacuum due to the absence of the antipodal singularity in its propagator, making it the only vacuum preserving de Sitter symmetry in this framework.
• The result suggests that the Polyakov propagator may not be suitable for perturbative quantum field theory on de Sitter space, as it leads to a fundamental violation of de Sitter symmetry.