Abstract

In the context of Horava gravity, the most promising known scenarios to recover Lorentz invariance at low energy are the possibilities that (1) the renormalization group flow of the system leads to emergent infrared Lorentz invariance, and (2) that supersymmetry protects infrared Lorentz invariance. A third scenario proposes that a classically Lorentz invariant matter sector with controlled quantum corrections may simply co-exist with Horava gravity under certain conditions. However, for non-projectable Horava gravity in 3+1 dimensions it is known that, in the absence of additional structures, this mechanism is spoiled by unexpected power-law divergences. We confirm this same result in the projectable version of the theory by employing the recently found gauge-fixing term that renders the shift and graviton propagators regular. We show that the problem persists for all dimensions $D\geq 3$, and that the degree of fine tuning in squared sound speeds between a U(1) gauge field and a scalar field increases with $D$. In particular, this difference in the zero external momentum limit is proportional to $\Lambda^{D-1}$ for $D\geq 3$, where $\Lambda$ is the ultraviolet momentum cutoff for loop integrals, while the power-law divergences are absent for $D=1$ and $D=2$. These results suggest that not only the gravity sector but also the matter sector should exhibit a transition to Lifshitz scaling above some scale, and that there should not be a large separation between the transition scales in the gravity and matter sectors. We close with a discussion of other more promising scenarios, including emergent Lorentz invariance from supersymmetry/strong dynamics, and pointing out challenges where they exist.

Highlights

  • Despite the empirical successes of General Relativity (GR), there are reasons to believe that it is incomplete

  • These results suggest that the gravity sector and the matter sector should exhibit a transition to Lifshitz scaling above some scale and that there should not be a large separation between the transition scales in the gravity and matter sectors

  • Renormalization group (RG) flow is organized around a fixed point with “Lifshitz scaling,” which acts anisotropically on space and time, so that higher order spatial derivatives scale the same way as second order time derivatives in the ultraviolet [1,2]

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Summary

INTRODUCTION

Despite the empirical successes of General Relativity (GR), there are reasons to believe that it is incomplete. Renormalization group (RG) flow is organized around a fixed point with “Lifshitz scaling,” which acts anisotropically on space and time, so that higher order spatial derivatives scale the same way as second order time derivatives in the ultraviolet [1,2].1 We refer to such theories as Horava gravity. A technical but crucial difference between the two theories is that only in the projectable case does the gravity sector possess propagators that are regular, explicitly respect the Lifshitz scaling in the ultraviolet, and have no instantaneous modes, provided we adopt the gauge fixing of [18]. Our second goal is to discuss possible resolutions of these problems These fall into three classes: mixed derivative terms in the projectable Horava gravity, supersymmetry, and strong dynamics. V summarizes our conclusions and discusses interesting directions for future work

SETUP AND NOTATION
Gravity action
Matter action
Propagators
ONE-LOOP CORRECTIONS IN D SPATIAL DIMENSIONS
Shift field loop corrections to the scalar action
Graviton loop corrections to scalar action
Shift field loop corrections to gauge field action
Graviton loop corrections to gauge field action
Lorentz-violating quantum corrections
Mixed derivatives
Supersymmetry
Strong dynamics
SUMMARY
Full Text
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