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

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.

Figure 1

The dynamical tetrahedron as evolution between two hyperplanes, which is useful to study the correlation between edge lengths. The labels give the physical lengths as ${\ell }_1=C(j_1)$, ${\ell }_2=C(j_2)$, ${\ell }_t=C(j_t)$, and $T=t_f-t_i={\ell }_t/\sqrt{2}$.

Figure 2

The prism interpolating between two hyperplanes, and its decomposition into three tetrahedra. This setting can be used to study the correlation between triangular areas.

Figure 3

The plots showing the numerical evaluation of the 2-point function. Left plot: the leading order. The (red) crosses are the numerical evaluations of (4); the dashed (green) line is the analytic result (13). Right plot: the leading order is plotted again, this time together with the next-to-leading order. The numerical evaluation of the latter shown as (blue) squares, whereas the analytic result (23) is the (purple) dashed line.

Figure 4

The plots showing the numerical evaluation of the 2-point function with $\alpha =0$. Left plot: the leading order. The (red) crosses are the numerical evaluations of (4); the dashed (green) line is the analytic result in (32). Right plot: the numerical evaluations of the next-to-leading order, shown as (blue) squares, together with the first two terms of the analytic result (23). The dashed (green) line is the leading order, whereas the dashed (purple) line shows the next-to-leading order.

Figure 5

The plots for the real boundary state (35). Left plot: the leading order. The (red) crosses are the numerical evaluations; the dashed (green) line is the analytic result in (37). Right plot: the numerical evaluations of the next-to-leading order, shown as (blue) squares, together with the analytic result of the next-to-leading order in (37).

Figure 6

We plot the three considered Gaussian distributions on SU(2) peaked around $\pm \mathbb{1}$: $f_1(\varphi )=e^{-(2/\alpha ){sin}^2(\varphi /2)}$, $f_2(\varphi )=\sum _j{\chi }_j(\varphi )e^{-(\alpha /2)j^2}$, and $f_3(\varphi )=e^{-(1/2\alpha ){\varphi }^2}$ (for $\varphi \in [0,\pi ]$ and its mirror image $e^{-(1/2\alpha )(2\pi -\varphi {)}^2}$ for $\varphi \in [\pi ,2\pi ]$). The two plots, respectively, correspond to the values $\alpha =2$ and $\alpha =0.2$. On the first plot, it is clear that the three Gaussians are distinct and that $f_3$ is not smooth. On the second, we see that $f_2$ converges toward the distribution $f_3$ as $\alpha$ goes to 0 (high $j_0$), thus justifying (39), while the distribution $f_1$ dual to the Bessel functions remains slightly different. This deviation of $f_1$ from the standard $f_2$ will be responsible for the different NLO of the 2-point function when using it as the boundary state (see below).

Figure 7

Leading and next-to-leading orders using (44). Left plot: the leading order. In red is the numerical evaluation; the dashed (green) line is the analytic result in (32). Right plot: the numerical evaluations of the next-to-leading order, shown as (blue) squares, together with the leading-order plot.

Figure 8

The relative correction $W^{\mathrm{NLO}}/W^{\mathrm{LO}}$, as a function of the measure weight $k$. On the left, the case with boundary state (5); the quadratic function is the real part, while the linear function is the imaginary part. We see that $k=1$ is the integer value closest to the minimum of the real part. On the right is the real case with (35) as the phase of the boundary state. We see that $k=0$ is the integer value closest to the minimum.

Figure 9

Exact numerical evaluation of $|W_{1122}|$ as a function of $j_t$, for $r=2.5$, 25, and 50. On the left is the leading order, computed in (59). On the right is the next-to-leading order, not computed analytically here.