Infrared divergences in the EPRL-FK Spin Foam model
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 an utterly convergent behavior, and to BF theory, independently recovering the divergent estimates present in the literature.
- 1 Introduction
- 2 The EPRL model and its connection with BF theory
- 3 The diagrams
- 4 Divergences estimation in SU(2) BF spin foam model
- 5 Divergences estimation in the simplified EPRL model
- 6 Divergences estimation in the full EPRL model
- 7 Conclusions
- A SU(2) Symbols and Boosters
- B Divergences of SU(2) BF
- C Vertex amplitudes of the 4D Ball diagrams
- D Details of the numeric analysis
The spin foam formalism is an attempt to define the dynamics of loop quantum gravity in a background independent and Lorentz covariant way [1, 2]. It defines transition amplitudes for spin network states of the canonical theory in a form of a sum (or equivalently a refinement ) over all the possible two-complexes having the chosen (projected) spin networks as boundary. This is equivalent to a sum over histories of quantum geometries providing in this way a regularised version of the quantum gravity path integral.
The state of the art is the model proposed by Engle, Pereira, Rovelli and Livine (EPRL) [4, 5, 6] and independently by Freidel and Krasnov (-FK)  and its extension to arbitrary spin network states [8, 9]. The model admits a quantum group deformation conjectured to describe the case of non-vanishing cosmological constant [10, 11] and notably, the large spin asymptotics of the 4-simplex vertex amplitude contains exponentials of the Regge action [12, 13]. The model is free of ultraviolet divergences because there are no trans-Planckian degrees of freedom, however, there are potential large-volume infrared divergences.
The presence of divergences may require some sort of renormalization procedure, and in general, their study and understanding is important in the definition of the continuum limit. This has been the subject of many studies and can be achieved in many ways: via refining of the 2-complex as proposed in [14, 15, 16], or via a resummation, defined for instance using group field theory/random tensor models as proposed in [17, 18, 19, 20]. The properties of these divergences have been studied in the context of the Ponzano-Regge model of 3d quantum gravity and discrete BF theory , group field theory  and EPRL model: with both Euclidean [23, 24] and Lorentzian signature .
In particular  is, to our knowledge, the only analytic estimate of divergences in the Lorentzian EPRL model. It considers the “self
energy” (see Figure 0(c)), finding a logarithmic divergence as a lower bound. The computation is rather involved and relies on the techniques developed for the asymptotic analysis of the vertex amplitude of the model . This approach requires an independent study of each geometrical sector: crucially, the logarithmic divergence is obtained by looking at the non-degenerate geometries, resulting in a lower bound estimate only. Our results suggest that this lower bound is close to 9 powers short. Moreover, even if in principle the same technique of  applies to any spin foam diagram, doing it is a very challenging task.
On the other hand, the various estimates provided in  for the Euclidean model of both the “self-energy” diagram and the “vertex renormalization” diagram, (see Figures 0(c) and 0(d)) just rely on the scaling for large spins of SU(2) invariants, and they are easily applicable to any spin foam diagram. Nevertheless, the extension of this technique to the Lorentzian model is not at all straightforward, due to the non-compactness of the Lorentz group.
In this work, we develop a simple algorithm to systematically determine the potential divergence of all spin foam diagrams within the EPRL model. Instead of approaching it directly in its generality we proceed by increasing complexity a bit at a time: we will introduce our algorithm first for SU(2) BF theory, moving to a simplified version of the EPRL model and concluding with the full quantum gravity model. We review the three transition amplitudes and their relation in Section 2. In Section 3 we introduce the four diagrams in analysis. Again, we opted to increase complexity gradually: before approaching the four stranded diagrams corresponding to a four dimensional triangulation (each four stranded edge is dual to a tetrahedron) we warm up with the analog three stranded diagrams corresponding to a three dimensional triangulation (each edge is dual to a triangle). Three dimensional spin foam diagrams are simpler than their four dimensional counterpart for the absence of edge intertwiners and the overall smaller number of internal faces. We will consider both three and four dimensional bubble and ball diagrams. In Sections 4, 5, 6 we proceed with the study of the divergence of the diagrams one by one in order of complexity. We then conclude summarizing the algorithm and the results obtained. Let us for the impatient reader comment here the results. We estimate both the bubble and the ball amplitudes in the four dimensional EPRL model to be divergent with the same power of the cutoff of the analog diagrams for SU(2) BF model. Furthermore, we also find convergence for all the diagrams in the simplified EPRL model and the three dimensional ones for the EPRL model.
2 The EPRL model and its connection with BF theory
We assume that the reader is familiar with the EPRL-FK222from now on we will call it just EPRL for notation convenience. model, and refer to the original literature [4, 5, 6, 7] and existing reviews (e.g. [1, 2]) for motivations, details and its relation to Loop Quantum Gravity. In the following, we will use an unconventional notation for the partition function which was recently developed in .
Given a closed 2-complex the partition function is a state sum over spins and intertwiners , associated respectively with faces and edges :
We denoted with the face weights: the requirement that the path integral at fixed boundary graph compose correctly under convolution fixes the face weight to be  but to compare to various other models present in the literature we will use a generalized face weight (i.e. correspond to the choice made in the BF SU(2) model and the EPRL model, correspond to the BF model). To have more symmetric expressions we will also take the dimensions of the intertwiners on the edges to be . The main goal of this paper is to find a systematic way to study the convergence of the multidimensional infinite sum .
To each vertex of the two-complex a vertex amplitude is associated:
it is defined as a superposition of invariants 333the specific invariant depend on the details of the vertex, if the vertex is dual to a 4-simplex the invariant is the symbol. weighted by one booster functions per edge touching the vertex , with the valency of the edge . The sums run over a set of auxiliary spins 444that are effectively magnetic indices respect the group associated to each face containing the vertex , with , and a set of auxiliary intertwiners for each edge connected to the vertex . Notice that the formulas for the partition function (1) and (2) are extendable to generalized spin foams with 2-complexes dual to arbitrary tesselations done with polyhedra being careful of using the appropriate dimension of the intertwiner space instead of and (i.e. for three valent edges the intertwiner space associated to each edge is trivial and on those edges; for five valent edges the intertwiner space associated to each edge is determined by two spins and the proper dimension to use is ).
The booster functions encode all the details of the EPRL model, they are defined in the following way:
where the boost matrix elements for -simple irreducible representation of in the principal series, is the Immirzi parameter and the symbols are reported in Appendix A. We are using the notations used in . On one hand, the introduction of booster functions simplifies a lot the computation of spin foam transition amplitudes because it trades the problem of dealing with many high oscillatory integrals with the study a family of one dimensional integrals, which are easier to handle and manipulate. Analytical and numerical properties of these functions are work in progress [26, 28, 29, 30]. On the other hand, the explicit evaluation of booster functions in spite of their rather simple form is still a very involved task: For we employ an expression for (3) in terms of finite sums of functions, for details see [32, 26]; for a similar formula exists but features an integration over virtual labels555See Equation (41) of ., and in the end we found it less time consuming to numerically integrate directly the boost integrals. A C numerical code for the virtual irreps formula has been recently developed in . The asymptotic behavior for large spins is still unknown: the properties we will need for our analysis will be inferred from numerical analysis.
As suggested in , we introduce here a simplified version of the EPRL model, we will denote it EPRLs where s stays for simplified. The reformulation of the EPRL amplitude as in (2) traded the major complexity of multiple integrals over the non-compact group with multiple infinite sums over the auxiliary spins . We can for the moment put aside the proliferation of spin labels and fix all the new spins to their minimal values :
We can also try to give a geometrical interpretation to this model. By removing the sums we fix the areas of the polyhedra on the edges the be fixed to the minimal ones, on the other hand, the shapes (associated to the intertwiners) are still allowed to be boosted from a vertex to the other. This is a dramatic simplification and it is not clear if this model can capture any feature of the full one, nevertheless it is a useful playground to study some properties in a simplified environment. There are some indications that the vertex amplitude of this model is dominated by Euclidean four dimensional geometries .
Furthermore, notice that with the additional simplification the vertex amplitude reduces to the one of the BF spin foam model:
In the following, we will study the divergences of these three models starting from the simpler one, BF model, for which the computation of the divergence of any diagram is also possible analytically, moving to the more complex EPRLs and finishing with the physically relevant EPRL.
3 The diagrams
In this Section, we will describe the four diagrams we will focus on in the rest of the paper. In spin foam models divergences turn out to be associated with bubbles in the triangulation. A bubble is a collection of faces in the cellular complex forming a closed 2-surface. Here we study the most elementary of such bubbles, and the potential divergences they give rise to, leaving the detailed characterization of all divergences of the whole theory to future works.
We will focus on two classes of those diagrams represented in Figure 1: the bubble diagram (or to use the Feynman diagrams’ language the self-energy), and the ball diagram (or vertex renormalization). The divergence of these two classes of diagrams can be viewed as the divergence on particularly simple triangulations with boundaries or more in general as the divergence arising from a sub-triangulations of a larger triangulation.
Even if the physical implication of the three stranded diagrams on the top of (1) is not clear, we will look at them as a simpler prototype of the four stranded ones where is easier to test our algorithm and some of the assumptions we will make. We will refer to them as three dimensional because we can imagine the dual to the three stranded edge to be a triangle.
3D bubble diagram.
The two-complex associated to the 3D bubble (Figure 0(a)) is composed by two vertices, three edges, three internal faces (one per couple of edges) and three external faces (one per edge). The dual triangulation is formed by two tetrahedra joined by three triangles and its boundary is formed by two triangles joined by all their sides. Therefore, the boundary graph consists of two three valent nodes joined by all their links.
We will in the following use a general convention denoting with s the boundary spins, s the face spins, s the boundary intertwiners and s the edge intertwiners. In this specific case, the boundary graph is completely determined by the three spins of the boundary links , . One spin is also associated to each internal face , .
3D ball diagram.
The two-complex associated to the 3D ball (Figure 0(b)) is composed by four vertices, six edges, four internal faces (one per triple of vertices) and six external faces (one per internal edge). It can be interpreted as a tetrahedron expanded with a 1-4 Pachner move. The boundary of the dual triangulation is formed by four triangles joined to form a tetrahedron. Therefore, the boundary graph consists of four three-valent nodes joined in a complete graph. We associate a spin , where , to each link of the boundary graph and a spin with to each internal face.
4D bubble diagram.
The two-complex associated to the 4D bubble (Figure 0(c)) is composed by two vertices, four edges, six internal faces (one per couple of edges) and four external faces (one per edge). The dual triangulation is formed by two 4-simplices joined by four tetrahedra. The boundary of the dual triangulation is formed by two tetrahedra joined by all their four faces, therefore the boundary graph is formed by two four valent nodes joined by all the links. Therefore, the boundary graph consists of two four valent node joined by all their links. We denote with , where the spins of the boundary graph links and and the intertwiners at the two nodes in the recoupling base . We attach a spin with to each face and an intertwiner with to each edge.
4D ball diagram.
Finally, the two-complex associated to the 4D ball (Figure 0(d)) is composed by five vertices, ten edges, ten internal faces (one per triple of vertices) and ten external faces (one per internal edge). It can be interpreted as a 4-simplex expanded with a 1-5 Pachner move into five 4-simplices. Such graph corresponds to a triangulation of a 3-ball with five 4-simplices and its divergence can be associated to the vertex renormalization of a simplicial spinfoam model. The boundary of the dual triangulation is formed by five tetrahedra joined in a 4-simplex. Therefore, the boundary graph consists of five four-valent nodes connected in a complete graph. We denote with , where the spins of the boundary graph links and with the intertwiners of the five nodes, we will not specify the base choice for the moment. We attach a spin with to each face and an intertwiner with to each edge.
4 Divergences estimation in SU(2) BF spin foam model
We warm up by testing our techniques with the simplest of the three models we are going to look at: the SU(2) BF spin foam model. For this model is possible to compute any diagram analytically, we refer to Appendix B for the analytic evaluation of the diagrams considered in this section. The vertex amplitude (5) for three stranded edges spin foams is a symbol while for four stranded edges spin foams is a symbol.
4.1 3D bubble diagram - self-energy
The transition amplitude for the 3D bubble diagram (Figure 0(a)) is:
Not all the sums are unbounded, to isolate them is useful to make a change of variable: , , . Triangular inequalities implies that the sums over and are bounded. We can rewrite (6) in terms of these new variables and obtain
Our final goal is to study the convergence of the infinite sum over the face spins. With that scope in mind we can assume that is arbitrarily large and drop any contribution small respect to . At this stage the summand does not depend anymore on the bounded variables and , so we can perform the sum explicitly and then omit the multiplicative factor that is irrelevant for our purposes and cumbersome to keep track of. We use the symbol to indicate this equivalence. The asymptotic behavior of the symbol with small spins and large spins is well known :
where we are ignoring an irrelevant multiplicative factor. If we introduce a cutoff to the sum over and use the asymptotic expression (9) we obtain an estimate for the divergence of the amplitude:
For a trivial face amplitude we reproduce the divergence we can compute analytically (see Appendix B for more details).
4.2 3D ball diagram - vertex renormalization
By carefully placing the internal and external spins, the transition amplitude for the 3D ball diagram (Figure 0(b)) is:
We follow closely the discussion of Section 4.1, the first step is to identify and isolate the unbounded sums performing the following change of variables , , and . Triangular inequalities implies that the sums over the new variables , and are bounded, tighter bounds are possible but they are not relevant for our analysis. In terms of this new variables we can rewrite the amplitude as:
Neglecting all the small contributions respect to , the variable of the only unbounded sum, and neglecting irrelevant multiplicative factors we obtain:
We put a cutoff on the sum over and we approximate the symbol with its large spin expression (9) to get the estimate:
Setting a trivial face amplitude () our estimate agrees with the analytical computation (see Appendix B for more details).
4.3 4D bubble diagram - self-energy
The transition amplitude for the 4D bubble diagram (Figure 0(c)) is:
where the specification of the symbol depends on the choice of intertwiner base of each spin foam edge. Even if the full amplitude is independent of this choice, it is convenient to choose the intertwiner bases that lead to a reducible symbols, as already noted in , to easily derive the scaling for large spins of the symbol:
Each edge carries a boundary spin, three face spins and an intertwiner. Triangular inequalities constrain the intertwiner to assume values in an interval centered on a face spin, implying that the sums over these intertwiners are bounded. In analogy to the three dimensional case it is useful to perform a change of variables to make it manifest. We define new variables for the spin faces for and for the intertwiners , , , . The sums over the intertwiners are in fact bounded: , , and . The sums over the are all unbounded contrary to the three dimensional case. The symbol (18) can be rewritten in terms of this new variables and the large spins asymptotic can be found in the literature [33, 35, 36, 37, 38]:
where is the volume of a Euclidean tetrahedron having for sides with . We are ignoring the oscillatory behavior of the symbol: since the summand is proportional to the square of this oscillation, disruptive interference between terms is not possible and we expect the leading order of the divergence to be unaffected.
Notice that this formula is not valid for values of the spins such that . In these cases, the semiclassical approximation used to derive the asymptotic formula for the symbol in (19) needs to be modified . The set of spins for which this happens form a measure zero set in the bigger set of face spins, so we expect they will not affect the divergence. For this reason, we ignore those points completely in the following analysis.
We can rewrite the whole amplitude in the new variables and expand at the leading order in :
To proceed with the estimate we will assume that the only kind of relevant divergence, if any, comes from the radial direction of the sum and will neglect any angular contribution. The divergence of this diagram can be computed analytically and has been extensively studied in the literature , we will use these results to test our assumption. We wanted to stress that this hypothesis is not new: all the other computation of divergences within the EPRL model in the literature also assume it [25, 23].
Calling this radial coordinate and introducing a factor as measure volume element and a cutoff :
For trivial face amplitude () we can compare our estimate with the analytical evaluation (see Appendix B for more details). We find perfect agreement, this corroborates our hypothesis that the divergence gets contribution mainly from the radial direction of the sum. This assumption will be also used in the estimates of the divergences of amplitudes in the EPRL model where, unfortunately, any alternative computation or checks are not possible.
4.4 4D ball diagram - vertex renormalization
The transition amplitude for the 4D ball diagram (Figure 0(d)) is:
We choose the intertwiner basis of the ten edges in order to get the following symbols:
Analougusly to the analysis performed in the previous section we define new variables for the spin faces for and for the edge intertwiners:
In terms of these new variables all sums over are manifestly bounded,while the sums over are all unbounded. Even if the invariants in (27) do not have the small spins in the same places their large spin scaling, omitting again the oscillations, are similar:
where are all the face spins entering in the -th vertex, i.e. for the 4-th vertex .
We assume also in this case that there is no angular contribution to the divergence and we change to radial coordinates. Imposing a cutoff to the radial summation
If we set a trivial face amplitude we do not reproduce the divergence obtained with analytical methods (see Appendix B for more details). We stress that all our estimate are upper bounds since we are neglecting any oscillations. Even if we are overestimating the divergence, neglecting the interference between the terms of the sum, we still get a result very close to the analytic evaluation.
5 Divergences estimation in the simplified EPRL model
Before trying to estimate divergences in the full EPRL model, it is useful to test our technique on the simpler EPRLs model we introduced at the end of Section 2. The vertex amplitude of the EPRLs model (4) differs from the SU(2) BF one in the introduction of the booster functions and in the extra summations over a set of auxiliary “boosted” intertwiners per vertex. While the latter requires minimal modification in the logic described in the previous Sections, how to deal with the booster functions will be the main novelty of this Section.
The main ingredient of the recipe we will describe in the following is the large spins scaling of both the and booster functions, where a spin is kept small and the others become large uniformly. The analytic study of the booster functions is very difficult and it is still work in progress . This forces us to employ numerical methods to extract the scaling we are looking for. A similar property is already been investigated in  and we independently confirm it here. We infer from our numerics the following scaling for the booster functions (refer to Figure 2 and Appendix D for more details):
with and or . To keep the expressions compact, we employed, and we will employ in the rest of the paper, a short-hand notation for the booster functions:
5.1 3D bubble diagram - self-energy
The transition amplitude associated to this diagram in the EPRLs model is the following:
We can estimate the divergence of this diagram following the strategy used in Section 4.1. We proceed by performing the same change of variable to isolate the unbounded summations and we drop all the irrelevant multiplicative terms:
We introduce a cutoff in the unbounded sum over and we approximate the summand with its asymptotic behavior obtained combining the large spin scaling of the symbol (9) and of the booster functions (34):
Notice that for the standard choice of face weight the amplitude is convergent, where for the SU(2) BF model it was cubically divergent.
We do not have in this case an analytical computation to compare to, but the system is simple enough to allow us to evaluate numerically the amplitude (38) as a function of the cutoff . We show the numerical result in Figure 3, we see a remarkable agreement with our estimate. To have a better comparison we artificially make the amplitude divergent by setting .
One can wonder where and if there is any dependence in the Immirzi parameter. Our analysis is not sensitive to it since it mainly focuses on the power of the cutoff. It will for sure play a role in the multiplicative factor that we ignored.
5.2 3D ball diagram - vertex renormalization
The transition amplitude associated to the 3D ball diagram in the EPRLs model is the following:
We proceed by performing the same change of variable of Section 4.2 to isolate the unbounded summations and we drop all the irrelevant multiplicative terms:
As we did in the previous section we introduce a cutoff in the unbounded sum over and we approximate the summand with its asymptotic behavior obtained combining the large spin scaling of the symbol (9) and of the booster functions (34):
The amplitude is convergent for the standard choice of face weight while is cubically divergent for . The amplitude (41) is also simple enough to allow us to evaluate it exactly as a function of the cutoff . The results are shown in Figure 4 and we see an excellent agreement with our estimate for both and .
5.3 4D bubble diagram - self-energy
The transition amplitude associated to the 4D bubble diagram (Figure 0(d)) in the EPRLs model is: