Universal opening of four-loop scattering amplitudes to trees

The perturbative approach to quantum field theories has made it possible to obtain incredibly accurate theoretical predictions in high-energy physics. Although various techniques have been developed to boost the efficiency of these calculations, some ingredients remain specially challenging. This is the case of multiloop scattering amplitudes that constitute a hard bottleneck to solve. In this Letter, we delve into the application of a disruptive technique based on the loop-tree duality theorem, which is aimed at an efficient computation of such objects by opening the loops to nondisjoint trees. We study the multiloop topologies that first appear at four loops and assemble them in a clever and general expression, the N$^4$MLT universal topology. This general expression enables to open any scattering amplitude of up to four loops, and also describes a subset of higher order configurations to all orders. These results confirm the conjecture of a factorized opening in terms of simpler known subtopologies, which also determines how the causal structure of the entire loop amplitude is characterized by the causal structure of its subtopologies. In addition, we confirm that the loop-tree duality representation of the N$^4$MLT universal topology is manifestly free of non-causal threshold singularities, thus pointing towards a remarkably more stable numerical implementation of multi-loop scattering amplitudes.


INTRODUCTION
The impressive progress in the understanding of the fundamental building blocks of Nature was due to the ability to extract theoretical predictions from Quantum Field Theories (QFT). The perturbative framework has proven to be extremely efficient for that purpose, nevertheless, the continuous effort to reach better predictions has revealed some challenges. The main bottleneck to automate higher perturbative orders is the study of vacuum quantum fluctuations associated to Feynman loop diagrams. These mathematical objects, defined in Minkowski space, exhibit a complex behaviour of physical and unphysical singularities, which prevents straightforward numerical calculations. Likewise, the high luminosity achieved by collider machines such as the CERN's Large Hadron Collider (LHC) [1] and future colliders [2][3][4][5][6][7][8] is pushing the precision frontier towards even more accurate theoretical predictions and better understanding of the behavior of such quantum objects.
Nowadays, accurate observables have been computed using several techniques based on the mathematical properties of scattering amplitudes. In particular, predictions ranging from next-to-leading to even next-to-next-to-next-to leading order have been calculated for several processes of interest at high energy colliders [9][10][11][12][13][14][15][16]. Since the numerical evaluation of integrals at multi-loop level requires, in general, a careful treatment of singularities, new methods need to be proposed to achieve better theoretical predictions.
It was recently conjectured in Ref. [23] that LTD straightforwardly leads to extremely compact and causal representations of scattering amplitudes to all orders. This pattern was explicitly proven for a series of multiloop topologies, the maximal loop topology (MLT), next-to-maximal (NMLT) and next-to-next-to-maximal (N 2 MLT) that are respectively characterized by L+1, L+2 and L+3 sets of propagators. Each set of propagators is categorized by its dependence on a specific loop momentum or a linear combination of the L independent loop momenta. These three topologies are sufficient to open any scattering amplitude of up to three loops and, in fact, the N 2 MLT opening identity embraces the other two topologies. Remarkably, their analytic dual representations are manifestly free of unphysical singularities, and the causal structure can be interpreted in terms of entangled causal thresholds [49].
In this Letter, we extend the application of LTD to a collection of multiloop topologies that first appear at four loops. This includes nonplanar diagrams for the first time. All these topologies are unified into a single one whose LTD representation describes at once the opening of any four-loop scattering amplitude to nondisjoint trees. Its multiloop version is part of the topologies that are necessary at higher loop orders.
The document is organized as follows. We begin by considering the main features and setting the LTD notation. Then, we present the N 4 MLT universal topology together with its dual decomposition in terms of simpler topologies. We continue with the interpretation of the dual representation in terms of causal singularities and, finally we present our conclusions and discuss future research directions.

LOOP-TREE DUALITY
A generic L-loop scattering amplitude with N external legs, {p j } N , is encoded in the Feynman representation as an integral in the Minkowski space on the L loop momenta, { s } L , over the product of Feynman propagators, G F (q i ) = (q 2 i −m 2 i +ı0) −1 , and numerators given by the Feynman rules of the specific theory, with (2) The integration measure in dimensional regularization [50,51] with d the number of space-time dimensions. In Eq. (2), we have introduced a shorthand notation to denote the product of Feynman propagators of one set of propagators that depend on a specific loop momentum or the union of several sets that depend on different linear combinations of the loop momenta, with a i arbitrary powers. It is important to remark that from now on the powers a i will appear only implicitly. Also, the LTD representations that will be presented do not require to detail the internal configuration of each set. The LTD representation is obtained by integrating out one degree of freedom per loop through the Cauchy residue theorem. This results in a modification of the infinitesimal complex prescription of the Feynman propagators [17], that needs to be considered carefully to preserve the causal structure of the amplitude. In the context of multiloop scattering amplitudes, the LTD representation is written in terms of nested residues [23]   where A (L) is the integrand in the Feynman representation, Eq. (2). The Cauchy countours are always closed on the lower half plane such that the poles with negative imaginary components are selected. This is implemented through the futurelike vector η that selects which component of the loop momenta are integrated. The usual choice is η µ = (1, 0), which is equivalent to integrate out the loop energies and has some advantages because the remaining integration domain is Euclidean. The LTD representations presented in the following are, however, independent of the coordinate system.
All the sets in Eq. (4) before the semicolon contain one propagator that has been set on shell, while all the propagators belonging the sets that appear after the semicolon remain off shell. The sum over all possible on shell configurations is implicit. For example, the LTD representation of the MLT topology has the very compact and symmetric form [23] A (L) The bars in Eq. (6) indicate a reversal of momentum flow, q is = −q is , which is necessary to preserve causality. More details can be found in Ref. [52] THE N 4 MLT UNIVERSAL TOPOLOGY In this work, we study the multiloop topologies that appear for the first time at four loops. They are characterized by multiloop diagrams with L + 4 and L + 5 sets of propagators. According to the classification scheme introduced in Ref. [23], they correspond to the next-to-next-to-next-to maximal loop topology (N 3 MLT) and next-to-next-to-next-to-next-to maximal loop topology (N 4 MLT). Actually, N 4 MLT embraces in a natural way all N k−1 MLT configurations, with k ≤ 4.
This arrangement allows to restrict the overall assessment to the N 4 MLT family that consists of three main topologies. These topologies were checked with QGRAF [53] and are shown in Fig. 1. Two of them are planar and one is nonplanar. Nicely, we observe the similarity of these topologies with the insertion of a four-point subamplitude with trivalent vertices into a larger topology. Therefore, in order to achieve a unified description we can interpret each of the three N 4 MLT topologies as the t-, sand u-kinematic channels, respectively, of a universal topology.
The three topologies contain L + 4 common sets of propagators, and one extra set which is different for each of them. Each of the first L sets depends on one characteristic loop momentum s , with s ∈ {1, . . . , L}, and the momenta of their propagators have the form q is = s + k is . The remaining four common sets are established as linear combinations of all the loop momenta, explicitly The momenta k is , k i (L+1) , k i12 , k i123 and k i234 are linear combinations of external momenta. The extra sets are the distinctive key to each of the channels in the universal topology. We identify the momenta of their propagators as different linear combinations of 2 , 3 and 4 , writing them as For the sake of simplicity, we will denote each set by s, with s ∈ {1 . . . , L + 1, 12, 123, 234, 23, 34, 24}. To assemble the three N 4 MLT channels in a single topology we define the current J that includes the three different type of sets, Notice that due to momentum conservation, the three subsets cannot contribute to the same individual Feynman diagram but they all contribute at amplitude level. Based on the development of this framework, the Feynman representation of the N 4 MLT universal topology can be expressed as, The dual opening of this topology fulfills a factorization identity in terms of simpler topologies which is very similar to the factorization identities presented in Ref. [23] for NMLT and N 2 MLT,  This factorization identity has a clear graphical interpretation as shown in Fig. 2. The convolution symbol indicates that each of the two convoluted components is open independently, but the on-shell conditions from both components act together on the propagators that remain off-shell. In order to make the notation lighter, A N k−1 MLT will refer in the following to the integrand of the corresponding topology in the LTD representation; integration over the L loop momenta will be implicitly understood.
An essential restriction that the selected on-shell propagators must meet concerns the feasibility of generating disjoint trees due to the dual opening. The term A The factorization identity in Eq. (11) is the main result of this Letter, and is the universal identity that opens any multiloop N 4 MLT topology to nondisjoint trees. It also accounts properly for all the N k−1 MLT configurations with k ≤ 4, and therefore it is the only master expression required to open any four-loop scattering amplitude to nondisjoint trees, independently of its internal configuration. Beyond four loops, new topologies arise that, for consistency, should include this universal topology as a particular case.
We have to mention that there is a certain arbitrariness in the expression Eq. (11) due to the freedom in the order of integration in applying the Cauchy residue theorem in succession to all loops, and in the reversal of momentum flows. Although there are at least L! possibilities, all potential LTD representations are equivalent and lead to the same expression once they are explicitly written in terms of dual propagators, and in particular if the corresponding causal representation is achieved [49].
The four-loop subtopology in Eq. (11) is opened as well through a factorization identity which is written in terms of simpler topologies, The diagrammatic representation of Eq. (12) is depicted in which has a similar structure to Eq. (12). The diagrammatic representation of Eq. (13) is depicted in Fig. 4. Similarly to Fig. 3, the first diagram on the rhs of Fig. 4, which represents the first term on the rhs of Eq. (13), is a threeloop NMLT subtopology and all the propagators in J are off shell, while the remaining three diagrams are specific to each of the three channels. The NMLT subtopology is made up of 7 subsets of momenta grouped into 5 sets as follows {1 ∪ 234, 2, 3, 4 ∪ 123, 12}. This construction prevents, for example, that propagators in the sets 1 and 234 are set on shell simultaneously.
Turning back into Eqs. (12) and (13) in a more detailed way, the first terms on the rhs of both equations are composed of dual contributions where all the propagators in J remain off shell. These propagators in J actually act as spectators in relation to the opening of the accompanying subtopology, and can eventually be replaced by a contact interaction to deduce the opening rule of these contributions to trees.
In the case of the four-loop N 2 MLT subamplitude in Eq. (12), its LTD representation is given by the factorized expression  All the MLT subamplitudes that involve a number of loops equal to the number of sets require to set on shell propagators in both sets. The rest of NMLT and MLT subtopologies are open according to know expressions (see Ref. [23] and Eq. (6)). To simplify the presentation, we have omitted in Eq. (14) the explicit reference to the sets with all their propagators off shell; for instance, the element A This notation will be used in the following; the omitted sets are understood to be off shell.
The three-loop NMLT subtopology in Eq. (13) is generated from 7 subsets clustered as {1 ∪ 234, 2, 3, 4 ∪ 123, 12} and its LTD representation is where the first term is a convolution of two MLT subtopologies, and in the second term all the propagators in the set 12 are off shell.

The t channel
The second terms in Eqs. (12) and (13), distinguish the dual configurations arising for each of the three channels because propagators in J are set on shell. We begin analyzing the terms related exclusively to the topology known as t channel and shown in Fig. 1 (left). There is one four-loop subtopology and one three-loop subtopology that contribute to Eqs. (12) and (13), respectively. The bold in 23 indicates that the two momentum flows of 23 should be considered, the original one 23 and the reversed 23; this fact will also be present in the s and u channels.

The s channel
In order to obtain the terms that characterize the s channel, shown in Fig. 1 (center), we set on shell a propagator in the set 34. The four-loop subtopology is given by    This expression is more involved than the corresponding expression in the t channel, because the loop momentum 4 is now present in three sets, while in the t channel 4 is was found in two sets only.
On the contrary, for the three-loop subamplitude we observe a very symmetric structure which allows to avoid any momentum flow reversion. In this case, we end up with an expression that only depends on the original momentum flow of the set 34, Given the structure of this subtopology, it is straightforward to realize that propagators in 12 and 34 cannot be on shell simultaneously without generating a disjoint tree.

The u channel
Moving on to the last terms associated to the nonplanar topology known as u channel, the LTD representation of the four-loop subamplitude with on-shell propagators in the set 24 is given by    This subtopology is also not as compact as the expression for the t channel because 4 is also present in three different sets. For the three-loop subamplitude, we find, = A MLT (1 ∪ 234 ∪ 3, 2, 12) + A MLT (3, 4 ∪ 123) MLT (4 ∪ 123, 12) + A All these results are consistent with the fact that no disjoint trees are generated. We have to clarify that selfenergy insertions can generate disjoint trees because the same propagator appears in different places of the corresponding Feynman diagram. From our point of view, propagators that are duplicated are treated as one single propagator raised to some power, and are not considered as leading to disjoint trees. Let us mention that the number of trees in the LTD forest can also be computed through the combinatorial exercise of selecting, from the full list of sets, all possible subsets of L elements that cannot generate disjoint trees when their propagators are set on shell. For the individual t, s and u channels the number of terms calculated in this way are 5(8L − 17), 15(3L − 7) and 9(5L − 11), respectively, and 82L − 187 for the N 4 MLT universal topology, in agreement with the number of dual contributions generated by Eq. (11). The momentum flows of the on-shell propagators, however, can only be determined through the nested residues.
In addition, following Ref. [49], we have explicitly considered the analytic reconstruction of N 4 MLT in terms of exclusively causal propagators. For all the internal configurations considered, we have achieved analytic expressions which are free of noncausal singularities. For example, the LTD representation of the simplest t channel configuration with only one propagator in each set can be written in terms of a sum of products of five entangled causal thresholds. The explicit expressions, which are too lengthy to be presented here, confirm the conjecture of Ref. [23] regarding the manifest absence of noncausal singularities in the LTD representation.

CONCLUSIONS
We have analized the multiloop topologies that appear for the first time at four loops and have found a general representation, the N 4 MLT universal topology, which describes their opening to nondisjoint trees through the loop-tree duality. The opening to trees admits a very structured and compact cascade interpretation in terms of convolutions of known subtopologies, that finally determine the internal causal structure of the entire amplitude. The LTD representation presented in this Letter is valid in arbitrary coordinate systems and space-time dimensions.
The N 4 MLT topology is called universal because it unifies in a single expression all the necessary ingredients to open any scattering amplitude of up to four loops. Beyond four loops, it is expected that the multiloop version of this topology will be embedded in more complex topologies, so that the methodology presented here can be used as a guide to achieve higher orders.
We have verified that the LTD representation of N 4 MLT is causal, namely, that the explicit LTD analytic expression is manifestly free of noncausal singularities. On the one hand, this supports the applicability and generalization of four-dimensional unsubtraction to higher orders. On the other hand, it allows a more efficient numerical evaluation of multiloop scattering amplitudes than other integrand representations. These results extend by one perturbative order the causal analysis of Ref. [49], and the interpretation of LTD in terms of entangled causal thresholds. In addition, they confirm the all-order conjectures of Ref. [23]. We expect that similar conclusions can be established at higher orders, thus leading to a noticeable improvement in the available toolkit for computing highly-precise theoretical predictions.