Self-Energy of the Lorentzian EPRL-FK Spin Foam Model of Quantum Gravity

TL;DR

This paper analyzes the leading divergent contribution of the self-energy (melon) graph in the Lorentzian EPRL-FK spin foam model. By applying a large-spin stationary-phase analysis to the Lorentzian amplitude, it identifies two non-degenerate sectors (Euclidean and Lorentzian) and a degenerate sector, and shows that the dominant divergence scales as Λ^{6(μ-1)} with μ(j) describing the spin-face weight; for the face-splitting invariant choice μ(j)=2j+1 this reduces to a logarithmic divergence. Importantly, the leading boundary-data dependence is governed by the EPRL-FK Y_γ map, and a distinct Lorentzian contribution involving non-commuting SL(2,ℂ) elements appears, indicating a richer radiative structure than in BF theory. The results connect geometric interpretations (two parity-related tetrahedra) to radiative corrections and suggest nontrivial implications for the renormalization group and the semiclassical limit of the theory, with potential relevance to the role of a cosmological constant via q-deformation. Overall, the work provides a concrete, controllable step toward understanding divergences in spin-foam gravity and their physical consequences for quantum gravity renormalization and effective dynamics.

Abstract

We calculate the most divergent contribution to the non-degenerate sector of the self-energy (or "melonic") graph in the context of the Lorentzian EPRL-FK Spin Foam model of Quantum Gravity. We find that such a contribution is logarithmically divergent in the cut-off over the SU(2)-representation spins when one chooses the face amplitude guaranteeing the face-splitting invariance of the foam.We also find that the dependence on the boundary data is different from that of the bare propagator. This fact has its origin in the non-commutativity of the EPRL-FK Y-map with the projector onto SL(2,C)-invariant states. In the course of the paper, we discuss in detail the approximations used during the calculations, its geometrical interpretation as well as the physical consequences of our result.

Figures (9)

• Figure 1: The vertex and edge structure of the self-energy cellular complex considered in this paper (see \ref{['melon_simple']} for its face structure).
• Figure 2: The melon graph. On the right, its faces and extra structure entering the LS representation are put into evidence. Triangles point in the direction of the action of the ${SU(2)}$ elements $\{h, h_a,\tilde{h}, \tilde{h}_a\}$. Dots represent insertions of the resolution of the identity. External faces are drawn in a dashed line.
• Figure 3: The figure shows the combinatoric structure of the internal faces of the melon graph, and how it corresponds to two tetrahedra with faces identified.
• Figure 4: The EPRL-FK melon graph. With respect to \ref{['melon_simple']} lines are added close to the triangles indicating group averaging: these lines represent the EPRL-FK ${\mathrm{Y}_\gamma}$-map. Integrations over the $\mathbb{CP}^1$ spinors $\{z_{ab},\tilde{z}_{ab}\}$ are not shown in the figure.
• Figure 5: A graphical representation of the vectors $\{h_a\rhd\vec{\ell}_{ab}\}$ in the Euclidean sector. Remark how they close into triangular faces ( \ref{['closure_h']}) and into a tetrahedron ( \ref{['transp_h']}).