# Towards the graviton from spinfoams: Higher order corrections in the 3D toy model

## Abstract

We consider the recent calculation by Rovelli of the graviton propagator in the spinfoam formalism. Within the 3D toy model introduced by Speziale, we test how the spinfoam formalism can be used to construct the perturbative expansion of graviton amplitudes. Although the 3D graviton is a pure gauge, one can choose to work in a gauge where it is not zero and thus reproduce the structure of the 4D perturbative calculations. We compute explicitly the next-to-leading and next-to-next-to-leading orders, corresponding to one-loop and two-loop corrections. We show that while the first arises entirely from the expansion of the Regge action around the flat background, the latter receives contributions from the microscopic, non-Regge-like, quantum geometry. Surprisingly, this new contribution reduces the magnitude of the next-to-next-to-leading order. It thus appears that the spinfoam formalism is likely to substantially modify the conventional perturbative expansion at higher orders. This result supports the interest in this approach. We then address a number of open issues in the rest of the paper. First, we discuss the boundary state ansatz, which is a key ingredient in the whole construction. We propose a way to enhance the ansatz in order to make the edge lengths and dihedral anglesmore »

- Authors:

- Laboratoire de Physique, ENS Lyon, CNRS UMR 5672, 46 Allee d'Italie, 69364 Lyon (France) and Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5 (Canada)
- (Canada)

- Publication Date:

- OSTI Identifier:
- 21010908

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.75.024038; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ACTION INTEGRAL; AMPLITUDES; CORRECTIONS; COSMOLOGY; FOUR-DIMENSIONAL CALCULATIONS; GAUGE INVARIANCE; GEOMETRY; GRAVITONS; PROPAGATOR; QUANTUM FIELD THEORY; QUANTUM GRAVITY; REGGE POLES; THREE-DIMENSIONAL CALCULATIONS; TOPOLOGY

### Citation Formats

```
Livine, Etera R., Speziale, Simone, Willis, Joshua L., Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, and Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7.
```*Towards the graviton from spinfoams: Higher order corrections in the 3D toy model*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.024038.

```
Livine, Etera R., Speziale, Simone, Willis, Joshua L., Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, & Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7.
```*Towards the graviton from spinfoams: Higher order corrections in the 3D toy model*. United States. doi:10.1103/PHYSREVD.75.024038.

```
Livine, Etera R., Speziale, Simone, Willis, Joshua L., Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, and Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7. Mon .
"Towards the graviton from spinfoams: Higher order corrections in the 3D toy model". United States.
doi:10.1103/PHYSREVD.75.024038.
```

```
@article{osti_21010908,
```

title = {Towards the graviton from spinfoams: Higher order corrections in the 3D toy model},

author = {Livine, Etera R. and Speziale, Simone and Willis, Joshua L. and Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5 and Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7},

abstractNote = {We consider the recent calculation by Rovelli of the graviton propagator in the spinfoam formalism. Within the 3D toy model introduced by Speziale, we test how the spinfoam formalism can be used to construct the perturbative expansion of graviton amplitudes. Although the 3D graviton is a pure gauge, one can choose to work in a gauge where it is not zero and thus reproduce the structure of the 4D perturbative calculations. We compute explicitly the next-to-leading and next-to-next-to-leading orders, corresponding to one-loop and two-loop corrections. We show that while the first arises entirely from the expansion of the Regge action around the flat background, the latter receives contributions from the microscopic, non-Regge-like, quantum geometry. Surprisingly, this new contribution reduces the magnitude of the next-to-next-to-leading order. It thus appears that the spinfoam formalism is likely to substantially modify the conventional perturbative expansion at higher orders. This result supports the interest in this approach. We then address a number of open issues in the rest of the paper. First, we discuss the boundary state ansatz, which is a key ingredient in the whole construction. We propose a way to enhance the ansatz in order to make the edge lengths and dihedral angles conjugate variables in a mathematically well-defined way. Second, we show that the leading order is stable against different choices of the face weights of the spinfoam model; the next-to-leading order, on the other hand, is changed in a simple way, and we show that the topological face weight minimizes it. Finally, we extend the leading-order result to the case of a regular, but not equilateral, tetrahedron.},

doi = {10.1103/PHYSREVD.75.024038},

journal = {Physical Review. D, Particles Fields},

number = 2,

volume = 75,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}