Infrared divergences in the EPRL-FK Spin Foam model

Abstract

We provide an algorithm to estimate the divergence degree of the Lorentzian EPRL-FK spin foam amplitudes for arbitrary 2-complexes. We focus on the "self-energy" and "vertex renormalization" diagrams and find an upper bound estimate. We argue that our upper bound must be close to the actual value, and explain what numerical improvements are needed to verify this numerically. For the self-energy, this turns out to be significantly more divergent than the lower bound estimate present in the literature. We support the validity of our algorithm using 3-stranded versions of the amplitudes (corresponding to a toy 3d model) for which our estimates are confirmed numerically. We also apply our methods to the simplified EPRLs model, finding a completely convergent behavior, and to BF theory, independently recovering the divergent estimates present in the literature.

Figures (8)

• Figure 1: We represent here the two-complex of the four diagrams we will study in the paper. The two diagrams on the top have three stranded edges. On the contrary, the diagrams on the bottom have four stranded edges and we will call them four dimensional, each edge is dual to a tetrahedron. We will refer to the diagrams on the left as bubble diagrams and to the diagrams on the right as ball diagrams. In each picture, we highlight in red an internal face and in green an external one.
• Figure 2: Numerical scaling of booster functions. Left panel: Non-isotropic scaling of the booster function $B_3\left(j_ 1,j_ 2+\lambda,j_ 3+\lambda \right)$ compared with the best fit $f(\lambda) = 2.6 \lambda^{-1}$. We rescaled the booster function by its $\lambda=0$ value. Right panel: Non-isotropic scaling of the booster function $B_4\left(j_1,j_2+\lambda,j_3+\lambda , j_4 + \lambda ; i + \lambda , k + \lambda \right)$ compared with the best fit $f(\lambda) = 5.012\ \lambda^{-2.38}$. We rescaled the booster function by its $\lambda=0$ value. The difference in the range is due to additional resources needed to compute the $B_4$ respect to the $B_3$. We also expect, comparing with the behavior of the $B_3$s, that the proper asymptotic region for the $B_4$ boosters functions is reached for larger spins of the one plotted. To give an idea to the reader while we were able to compute all the points in the left panel on a normal laptop, the plot on the right required 64 cores in a cluster working for approximately 80 hours of walltime.
• Figure 3: Numerical evaluation of the transition amplitude \ref{['amplitudeSE3DEPRLs']} as a function of the cutoff in logarithmic scale. We choose the external spins to be $k_1 = 1$, $k_2 = 2$, $k_3 = 3$, Immirzi parameter $\gamma = 1.2$. Left panel: for face weight $\mu = 1$ the amplitude is convergent to the best fit $\mathcal{W}=9.513 \cdot 10^{-8}$ in red. The plot is rescaled to allow a clearer reading. Right panel: for face weight $\mu = 3$, the amplitude diverge cubically. We plot for comparison the best fit function $3.385 \cdot 10^{-7}\, \Lambda^3$ in red.
• Figure 4: Numerical evaluation of the transition amplitude \ref{['amplitudevert3DEPRLs']} as a function of the cutoff in logarithmic scale. We choose the external spins to be $k_1 = 1$, $k_2 = 2$, $k_3 = 1$, $k_4 = 2$, $k_5 = 1$, $k_6 = 1$ and the Immirzi parameter is set to $\gamma = 1.2$. Left panel: for face weight $\mu = 1$ the amplitude is convergent to the best fit $\mathcal{W}=1.032 \cdot 10^{-17}$ in red. The plot is rescaled to allow a clearer reading. Right panel: for face weight $\mu = 4$ the amplitude diverge cubically. We plot for comparison the best fit function $6.811 \cdot 10^{-15}\, \Lambda^3$ in red.
• Figure 5: Numerical scaling of booster as a function of the magnetic spins $l$s. Left panel: Non-isotropic scaling of the booster function $B_3\left(j_ 1,j_ 2+\Delta l,j_ 3+\Delta l\right)$ in the auxiliary spins $\Delta l$ compared with the curve $f(\Delta l) = 4.2 \Delta l^{-1/2}$. We rescaled the booster function by its $\Delta l=0$ value. Right panel: Non-isotropic scaling of the booster function $B_4\left(j_1,j_2+\Delta l,j_3+\Delta l, j_4 +\Delta l; i , k +\Delta l\right)$ compared with the curve $f(\Delta l) = 9.5 \Delta l^{-2}$. We rescaled the booster function by its $\lambda=0$ value. We would prefer to accumulate more point to have a more definite estimate since by comparing with the plot on the left the asymptotic region is reached at larger spins, unfortunately our software needs to be improved to treat boosters with spins larger than 100 with sufficient precision. Luckily the analysis we are going to perform is not very sensitive to the value of this coefficient.