Summing inflationary logarithms in nonlinear sigma models

Abstract

We consider two nonlinear sigma models on de Sitter background which involve the same derivative interactions as quantum gravity but without the gauge issue. The first model contains only a single field, which can be reduced to a free theory by a local field redefinition; the second contains two fields and cannot be so reduced. Loop corrections in both models produce large temporal and spatial logarithms which cause perturbation theory to break down at late times and large distances. Many of these logarithms derive from the “tail” part of the propagator and can be summed using a variant of Starobinsky’s stochastic formalism involving a curvature-dependent effective potential. The remaining logarithms derive from the ultraviolet and can be summed using a variant of the renormalization group based on a special class of curvature-dependent renormalizations. Explicit results are derived at 1-loop and 2-loop orders.

Perturbative Quantum Gravity Induced Scalar Coupling to Electromagnetism

“…That cannot happen in perturbative quantum gravity because the graviton remains massless to all orders. However, constant scalars can also contribute by changing a field strength [20]. In our case, a constant scalar background changes the cosmological constant, and the graviton propagator in de Sitter background depends upon this cosmological constant [12,13,14,15,16].…”

How Inflationary Gravitons Affect the Force of Gravity

“…This led to the speculation that resummation might be accomplished by combining a variant of Starobinsky's stochastic formalism [37,38] with a variant of the renormalization group. This speculation was recently confirmed in the context of nonlinear sigma models on a nondynamical de Sitter background [39], which possess the same kinds of derivative interactions as quantum gravity and exhibit the same mixture of "tail" and ultraviolet logarithms. The technique has been applied to explain graviton loop corrections to the exchange potential of a massless, minimally coupled scalar [35], and strenuous efforts are underway to devise similar explanations for the collection of large graviton logarithms that have been patiently accumulated by direct computation over the course of two decades.…”

How Inflationary Gravitons Affect the Force of Gravity

“…This led to the speculation that resummation might be accomplished by combining a variant of Starobinsky's stochastic formalism [38,39] with a variant of the renormalization group. This speculation was recently confirmed in the context of nonlinear sigma models on a nondynamical de Sitter background [40], which possess the same kinds of derivative interactions as quantum gravity and exhibit the same mixture of "tail" and ultraviolet logarithms. The technique has been applied to explain graviton loop corrections to the exchange potential of a massless, minimally coupled scalar [36], and strenuous efforts are underway to devise similar explanations for the collection of large graviton logarithms that have been patiently accumulated by direct computation over the course of two decades.…”

“…Even though the graviton field is Hermitian, the factors nonlocal factors (48) and (38)(39)(40) are neither real nor causal because the Feynman diagrams from which they derive are in-out matrix elements rather than expectation values. We can derive true expectation values using the Schwinger-Keldysh formalism [14][15][16][17][18] which is a diagrammatic technique that is almost as simple as the Feynman rules.…”

“…Implementing these rules is straightforward. First, recall that the only dependence on the coordinate interval ∆x 2 in the nonlocal results of Tables 5, 6 and 7 comes through the integrable functions A 1−3 , B 12 and C 1−2 , which were defined in expressions (38)(39)(40) and (48). We can eliminate the factors of 1/∆x 2 using identities (128-136) of the Appendix.…”