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arxiv: 2512.14864 · v1 · pith:JKCSYPPAnew · submitted 2025-12-16 · ✦ hep-th · gr-qc

Quantizing non-projectable Hov{r}ava gravity with Lagrangian path integral

D. Blas , F. Del Porro , M. Herrero-Valea , J. Radkovski , S. Sibiryakov This is my paper

Pith reviewed 2026-05-22 11:28 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Horava gravitynon-projectablepath integralone-loop divergencesbeta functionsNewton constantrenormalizability
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The pith

Non-projectable Hořava gravity yields local beta functions after shift divergences cancel at one loop.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets up non-projectable Hořava gravity as a quantum theory via a Lagrangian path integral whose measure is ultra-local in time but non-local in space. Auxiliary fields are added to rewrite the measure locally while preserving bosonic and fermionic symmetries. A one-loop computation is performed in 2+1 dimensions on a background with flat spatial metric, unit lapse and non-trivial shift, without truncating perturbations, using diagrams and heat-kernel techniques. Linear-in-frequency divergences in the shift two-point function are isolated and shown to cancel, leaving a local divergent quadratic effective action. Beta functions for the Newton constant and the essential coupling λ are extracted from this action.

Core claim

The central claim is that the potentially dangerous non-localities from linear-in-frequency terms in the shift two-point function cancel explicitly, leaving a fully local expression for the divergent part of the quadratic effective action in non-projectable Hořava gravity, from which the one-loop beta functions of the Newton constant and the essential coupling λ can be extracted.

What carries the argument

The central mechanism is the localization of the ultra-local-in-time but non-local-in-space field-dependent measure via auxiliary fields that obey bosonic and fermionic symmetries, combined with a diagrammatic technique and heat-kernel method to compute one-loop divergences without perturbation truncations on a non-trivial shift background.

Load-bearing premise

The non-local-in-space measure can be rewritten locally using auxiliary fields that preserve the necessary symmetries, and the chosen background with flat spatial slices and non-trivial shift captures all divergences relevant for the beta functions.

What would settle it

Non-cancellation of the linear-in-frequency divergences in the two-point function of the shift on the given background would produce spatial non-localities and block extraction of local beta functions.

Figures

Figures reproduced from arXiv: 2512.14864 by D. Blas, F. Del Porro, J. Radkovski, M. Herrero-Valea, S. Sibiryakov.

Figure 1
Figure 1. Figure 1: Graphic notation for the fields. • The ghosts: ⟨c¯icj ⟩ = −⟨cic¯j ⟩ = −iG P1(δij − qˆiqˆj ) + P2qˆiqˆj  . (4.14) For completeness, we also give in Appendix B the propagators of the auxiliary fields ⟨AA⟩ and ⟨ηη¯ ⟩. All other propagators vanish identically. In the above expressions, we have suppressed the arguments of the fields, except in the case of the mixed propagator ⟨niπj ⟩, where one has to be care… view at source ↗
Figure 3
Figure 3. Figure 3: The momentum-space expressions associated with these vertices are cumbersome and we do [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: Interaction vertices with one background shift vector [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Interaction vertices with two background shift vectors [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bubble diagrams contributing into the two-point function of [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematically, a bubble diagram will lead to a loop integral of the form [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Irregular diagrams with the lapse n. Γ 1 AA = Γ 1 ηη¯ = Γ 4 AA = Γ 4 ηη¯ = [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Irregular diagrams with auxiliary fields [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
read the original abstract

We formulate the quantum version of non-projectable Ho\v{r}ava gravity as a Lagrangian theory with a path integral in the configuration space with an ultra-local in time, but non-local in space, field-dependent measure. Using auxiliary fields, we cast the measure into a local form satisfying several bosonic and fermionic symmetries. We perform an explicit one-loop computation in the theory in $(2+1)$ dimensions, using for the case study the divergent part of the action on a background with non-trivial shift vector; the background spatial metric is taken to be flat and the background lapse function is set to 1. No truncations are assumed at the level of perturbations, for which we develop a diagrammatic technique and a version of the heat-kernel method. We isolate dangerous linear-in-frequency divergences in the two-point function of the shift, which can lead to spatial non-localities, and explicitly verify their cancellation. This leaves a fully local expression for the divergent part of the quadratic effective action, from which we extract the beta functions for the Newton constant and the essential coupling $\lambda$ in the kinetic term of the metric. We formulate the questions that need to be addressed to prove perturbative renormalizability of the non-projectable Ho\v{r}ava gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims to provide a path integral formulation for quantizing non-projectable Hořava gravity using a configuration space measure that is ultra-local in time but non-local in space. Auxiliary fields are introduced to localize the measure while maintaining symmetries. An explicit one-loop calculation is carried out in 2+1 dimensions on a background with non-trivial shift, flat spatial metric, and lapse set to 1, without any truncations on perturbations. Dangerous linear divergences in the shift two-point function are shown to cancel, leading to a local divergent quadratic effective action from which beta functions for the Newton constant and λ are extracted. Open questions for perturbative renormalizability are formulated.

Significance. If the cancellation is verified and the beta functions are correctly extracted, this work advances the quantization program for Hořava gravity by demonstrating how to handle the measure and avoid non-localities at one loop. The development of a diagrammatic technique and heat-kernel method for this theory is a technical strength. However, the significance is tempered by the restricted background used, which may not fully determine the renormalization group flow for all relevant operators.

major comments (1)
  1. [One-loop computation] The computation is performed on a background with flat spatial metric and lapse fixed to 1 (as stated in the abstract and the description of the case study). This choice allows verification of the cancellation in the two-point function of the shift but does not excite operators involving spatial curvature or lapse fluctuations. Since the divergent part is used to extract beta functions for G and λ, the paper needs to address whether these beta functions receive contributions from such operators or provide a justification for the sufficiency of this background.
minor comments (1)
  1. The abstract mentions formulating questions for perturbative renormalizability; these should be listed explicitly in the conclusion for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the scope of the one-loop computation. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [One-loop computation] The computation is performed on a background with flat spatial metric and lapse fixed to 1 (as stated in the abstract and the description of the case study). This choice allows verification of the cancellation in the two-point function of the shift but does not excite operators involving spatial curvature or lapse fluctuations. Since the divergent part is used to extract beta functions for G and λ, the paper needs to address whether these beta functions receive contributions from such operators or provide a justification for the sufficiency of this background.

    Authors: The background with flat spatial metric, unit lapse, and non-trivial shift was deliberately chosen to isolate and verify the cancellation of linear-in-frequency divergences in the shift two-point function while working with the full set of perturbations and no truncations. This setup generates the quadratic divergent terms from which the beta functions for the Newton constant and λ are extracted. We agree that this background does not excite operators involving spatial curvature or lapse fluctuations, so it does not capture possible additional contributions to these beta functions from those sectors. In the revised manuscript we will add an explicit discussion (in the conclusions or a dedicated subsection) clarifying the scope of the calculation, stating that the extracted beta functions are those appearing on this background, and noting that the potential contributions from curvature and lapse operators constitute one of the open questions for perturbative renormalizability already formulated in the paper. This provides the requested justification without changing the reported results. revision: yes

Circularity Check

0 steps flagged

One-loop beta function extraction via explicit computation is self-contained

full rationale

The paper formulates the path integral with an ultra-local-in-time non-local-in-space measure, localizes it via auxiliary fields obeying bosonic and fermionic symmetries, then computes the one-loop divergent part of the quadratic effective action on a flat-spatial-metric/unit-lapse/non-trivial-shift background using diagrammatic techniques and a heat-kernel method with no truncations. Linear-in-frequency divergences in the shift two-point function are isolated and shown to cancel, leaving a local expression from which beta functions for G and λ are read off. This chain is a direct perturbative QFT calculation from the action and propagators; it contains no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations that reduce the central result to prior unverified claims by the same authors. The background restriction is a methodological choice for the case study rather than a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The formulation rests on standard quantum-field-theory path-integral assumptions plus the specific choice of a field-dependent measure that is made local by auxiliary fields; no free parameters are fitted in the reported one-loop result, and no new particles or forces are postulated.

axioms (2)
  • domain assumption Existence of a well-defined path-integral measure for the non-projectable Hořava theory that is ultra-local in time and can be localized in space via auxiliary fields while preserving bosonic and fermionic symmetries.
    Invoked when casting the measure into local form and when performing the one-loop computation.
  • standard math Standard assumptions of perturbative quantum field theory on a curved background with a non-trivial shift vector, including the validity of the heat-kernel expansion and diagrammatic expansion without truncations.
    Underlying the explicit one-loop calculation and divergence analysis.

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Forward citations

Cited by 1 Pith paper

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    Higher-derivative corrections in projectable Hořava gravity do not yield static planar-symmetric solutions that can serve as endpoints for the Minkowski infrared instability.

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