Open loop amplitudes and causality to all orders and powers from the loop-tree duality
J. Jesus Aguilera-Verdugo, Felix Driencourt-Mangin, Roger J. Hernandez-Pinto, Judith Plenter, Selomit Ramirez-Uribe, Andres E. Renteria-Olivo, German Rodrigo, German F. R. Sborlini, William J. Torres Bobadilla, Szymon Tracz
IIFIC/20-02
Open loop amplitudes and causality to all orders and powers from the loop-tree duality
J. Jes´us Aguilera-Verdugo ( a ) , F´elix Driencourt-Mangin ( a ) , Roger J. Hern´andez-Pinto ( b ) ,Judith Plenter ( a ) , Selomit Ram´ırez-Uribe ( a,b,c ) , Andr´es E. Renter´ıa-Olivo ( a ) , Germ´anRodrigo ( a ) , Germ´an F. R. Sborlini ( a ) , William J. Torres Bobadilla ( a ) , and Szymon Tracz ( a ) a Instituto de F´ısica Corpuscular, Universitat de Val`encia – Consejo Superior deInvestigaciones Cient´ıficas, Parc Cient´ıfic, E-46980 Paterna, Valencia, Spain. b Facultad de Ciencias F´ısico-Matem´aticas, Universidad Aut´onoma de Sinaloa, Ciudad Universitaria, CP 80000 Culiac´an, Mexico. c Facultad de Ciencias de la Tierra y el Espacio, Universidad Aut´onoma de Sinaloa, Ciudad Universitaria, CP 80000 Culiac´an, Mexico. (Dated: May 1, 2020)Multiloop scattering amplitudes describing the quantum fluctuations at high-energy scattering processes arethe main bottleneck in perturbative quantum field theory. The loop-tree duality is a novel method aimed at over-coming this bottleneck by opening the loop amplitudes into trees and combining them at integrand level withthe real-emission matrix elements. In this Letter, we generalize the loop-tree duality to all orders in the per-turbative expansion by using the complex Lorentz-covariant prescription of the original one-loop formulation.We introduce a series of mutiloop topologies with arbitrary internal configurations and derive very compact andfactorizable expressions of their open-to-trees representation in the loop-tree duality formalism. Furthermore,these expressions are entirely independent at integrand level of the initial assignments of momentum flows inthe Feynman representation and remarkably free of noncausal singularities. These properties, that we conjectureto hold to other topologies at all orders, provide integrand representations of scattering amplitudes that exhibitmanifest causal singular structures and better numerical stability than in other representations.
INTRODUCTION
Precision modeling of fundamental interactions reliesmostly on perturbative quantum field theory. Quantum fluc-tuations in perturbative quantum field theory are encoded byFeynman diagrams with closed loop circuits. These loopdiagrams are the main bottleneck to achieve higher pertur-bative orders and therefore more precise theoretical predic-tions for high-energy colliders [1, 2]. Whereas loop inte-grals are defined in the Minkowski space of the loop four-momenta, the loop-tree duality (LTD) [3–22] exploits theCauchy residue theorem to reduce the dimensions of the in-tegration domain by one unit in each loop. In the most gen-eral version of LTD the loop momentum component that isintegrated out is arbitrary [3, 4]. In numerical implementa-tions [7, 8, 10, 11, 13, 14, 16, 21, 22] and asymptotic ex-pansions [12, 17], it is convenient to select the energy com-ponent because the remaining integration domain, the loopthree-momenta, is Euclidean.LTD opens any loop diagram to a forest (a sum) of nondis-joint trees by introducing as many on-shell conditions on theinternal loop propagators as the number of loops, and is re-alized by modifying the usual infinitesimal complex prescrip-tion of the Feynman propagators. The new propagators withmodified prescription are called dual propagators. LTD athigher orders proceeds iteratively, or in the words of Feyn-man [23, 24], by opening the loops in succession . Whilethe position of the poles of Feynman propagators in the com-plex plane is well defined, i.e., the positive (negative) energymodes feature a negative (positive) imaginary component dueto the momentum independent + ı imaginary prescription,the dual prescription of dual propagators is momentum de-pendent. Therefore, after applying LTD to the first loop, theposition of the poles in the complex plane of the subsequent loop momenta is modified. The solution found in Refs. [4, 5]was to reshuffle the imaginary components of the dual propa-gators by using a general identity that relates dual with Feyn-man propagators in such a way that propagators entering thesecond and successive applications of LTD are Feynman prop-agators only. This procedure requires to reverse the momen-tum flow of a few subsets of propagators in order to keep acoherent momentum flow in each LTD round. n = P ni =1 i − ii + 1 n
12 123 n n m FIG. 1. Maximal loop topology (left) and the corresponding opendual representation (right). An arbitrary number of external legs isattached to each loop line. All the propagators in the set i on therhs are off shell, while the dashed line represents the on-shell cutover the other n − sets: one on-shell propagator in each set and animplicit sum over all possible on-shell configurations. Bars indicatea reversal of the momentum flow. Recent papers have proposed alternative dual representa-tions [19–22]. In Refs. [19, 20], an average of all the possi-ble momentum flows is proposed, which requires a detailedcalculation of symmetry factors. We show in this Letter thatthis average is unnecessary. In Refs. [21, 22], the Cauchyresidue theorem is applied iteratively by keeping track of theactual position of the poles in the complex plane. The proce-dure requires to close the Cauchy contours at infinity fromeither below or above the real axis, in order to cancel the a r X i v : . [ h e p - ph ] M a y dependence on the position of the poles. In this Letter, wefollow a new strategy to generalize LTD to all orders, andwith the original Lorentz-covariant prescription [3, 4]. Asin Refs. [4, 5, 13, 14], we reverse sets of internal momentawhenever it is necessary to keep a coherent momentum flow,and we close the Cauchy contours always in the lower com-plex half-plane. Causality [6, 15, 25–30] is also used as apowerful guide to select which kind of dual contributions areendorsed, and then construct suitable Ans¨atze that are provenby induction. This procedure allows us to obtain explicit andvery compact analytic expressions of the LTD representationfor a series of loop topologies to all orders and arbitrary inter-nal configurations.
LOOP-TREE DUALITY TO ALL ORDERS AND POWERS
The internal propagators of any multiloop integral or scat-tering amplitude can be classified into different sets or looplines, each set collecting all the propagators that depend onthe same single loop momentum or a linear combination ofthem. To simplify the notation, s labels the set of all theinternal propagators i s ∈ s carrying momenta of the form q i s = (cid:96) s + k i s , where (cid:96) s is the loop momentum identifyingthis set, and where k i s is a linear combination of external mo-menta { p j } N . Note that (cid:96) s may be a linear combination ofloop momenta, so long as it is the same fixed combination inall the elements in the set s . The usual Feynman propagatorof one single internal particle is G F ( q i s ) = 1 q i s , − (cid:16) q (+) i s , (cid:17) , (1)where q (+) i s , = (cid:113) q i s + m i s − ı , (2)with q i s , and q i s the time and spatial components of the mo-mentum q i s , respectively, m i s its mass, and ı the usual Feyn-man’s infinitesimal imaginary prescription. We extend thisdefinition to encode in a compact way the product of the Feyn-man propagators of one set or the union of several sets: G F (1 , . . . , n ) = (cid:89) i ∈ ∪···∪ n ( G F ( q i )) a i . (3)Here, we contemplate the general case where the Feynmanpropagators are raised to arbitrary powers. Still, the powers a i will appear only implicitly in the following. A typical L -loopscattering amplitude is expressed as A ( L ) N (1 , . . . , n ) = (cid:90) (cid:96) ,...,(cid:96) L N ( { (cid:96) i } L , { p j } N ) G F (1 , . . . , n ) (4)in the Feynman representation, i.e. as an integral in theMinkowski space of the L -loop momenta over the product ofFeynman propagators and the numerator N ( { (cid:96) i } L , { p j } N ) , which is given by the Feynman rules of the theory. The inte-gration measure reads (cid:82) (cid:96) i = − ı µ − d (cid:82) d d (cid:96) i / (2 π ) d in dimen-sional regularization [31, 32], with d the number of space-timedimensions.Beyond one loop, any loop subtopology involves at leasttwo loop lines that depend on the same loop momentum. Wedefine the dual function that accounts for the sum of residuesin the complex plane of the common loop momentum as G D ( s ; t ) = − πı (cid:88) i s ∈ s Res ( G F ( s, t ) , Im( η · q i s ) < , (5)where G F ( s, t ) represents the product of the Feynman prop-agators that belong to the two sets s and t . Each of the Feyn-man propagators can be raised to an arbitrary power. Noticethat in Eq. (5) only the propagators that belong to the set s are set consecutively on shell. The Cauchy contour is alwaysclosed from below the real axis, Im( η · q i s ) < . The vector η is futurelike and was introduced in the original formulationof LTD [3] to regularize the dual propagators in a Lorentz-covariant form. For single power propagators, s = t and η = (1 , ) , Eq. (5) provides the customary dual function atone loop with the energy component integrated out G D ( s ) = − (cid:88) i s ∈ s ˜ δ ( q i s ) (cid:89) js (cid:54) = is j s ∈ s q (+) i s , + k j s i s , ) − ( q (+) j s , ) , (6)with k j s i s = q j s − q i s , and ˜ δ ( q i s ) = 2 πı θ ( q i s , ) δ ( q i s − m i s ) selecting the on-shell positive energy mode, q i s , > . Ifsome of the Feynman propagators are raised to multiple pow-ers, then Eq. (5) leads to heavier expressions [5] but the lo-cation of the poles in the complex plane is the same as in thesingle power case.Then, we construct the dual function of nested residues in-volving several sets of momenta G D (1 , . . . , r ; n ) = − πı × (cid:88) i r ∈ r Res ( G D (1 , . . . , r − r, n ) , Im( η · q i r ) < . (7)In the rhs of Eqs. (5) and (7), we can introduce numerators orreplace the Feynman propagators by the integrand of Eq. (4)to define the corresponding unintegrated open dual amplitudes A ( L ) D (1 , . . . , r ; n ) . An example of dual amplitudes at twoloops was presented in Ref. [13].In the next sections, we will derive the LTD representationof the multiloop scattering amplitude in Eq. (4) and we willpresent explicit expressions for several benchmark topologiesto all orders. The notation introduced above allows us to ex-press the LTD representations in a very compact way, since itonly requires to label and specify the overall structure of theloop sets, regardless of their internal and specific configura-tion. MAXIMAL LOOP TOPOLOGY
The maximal loop topology (MLT), see Fig. 1, is definedby L -loop topologies with n = L + 1 sets of propagators,where the momenta of the propagators belonging to the first L sets depend on one single loop momentum, q i s = (cid:96) s + k i s with s ∈ { , . . . , L } , and the momenta of the extra set, de-noted by n , are a linear combination of all the loop momenta, q i n = − (cid:80) Ls =1 (cid:96) s + k i n . The minus sign in front of the sumis imposed by momentum conservation. The momenta k i s and k i n are linear combinations of external momenta. At twoloops ( n = 3 ), this is the only possible topology, and thereforesufficient to describe any two-loop scattering amplitude.The LTD representation of the multiloop MLT amplitude,starting at two loops, is extremely simple and symmetric A ( L )MLT (1 , . . . , n )= (cid:90) (cid:96) ,...,(cid:96) L n (cid:88) i =1 A ( L ) D (1 , . . . , i − , i + 1 , . . . , n ; i ) , (8)with A ( L ) D (2 , . . . , n ; 1) and A ( L ) D (1 , . . . , n − n ) as the firstand the last elements of the sum, respectively. The bar in s indicates that the momentum flow of the set s is reversed( q i s → − q i s ), which is equivalent to selecting the on-shellmodes with negative energy of the original momentum flow.The compact expression in Eq. (8) was obtained by first eval-uating the nested residues, Eq. (7), of several representativemultiloop integrals. The derived expressions were then usedto formulate an Ansatz to all orders that was proven by in-duction. It is noteworthy that there is no dependence in thisexpression on the position of the poles in the complex plane.In each term of the sum in the integrand of Eq. (8) thereis one set i with all its propagators off shell, and there is oneon-shell propagator in each of the other n − sets. This isthe necessary condition to open the multiloop amplitude intonondisjoint trees. Note also that there is an implicit sum overall possible on-shell configurations of the n − sets. The LTDrepresentation in Eq. (8) is displayed graphically in Fig. 1, andrepresents the basic building block entering other topologies.The causal behavior of Eq. (8) is also clear and manifest.The dual representation in Eq. (8) becomes singular when oneor more off-shell propagators eventually become on shell andgenerate a disjoint tree dual subamplitude. If these propaga-tors belong to a set where there is already one on-shell prop-agator then the discussion reduces to the one-loop case [6].We do not comment further on this case. The interesting caseoccurs when the propagator becoming singular belongs to theset with all the propagators off shell [15]. For example, thefirst element of the sum in Eq. (8) features all the propaga-tors in the set off shell. One of those propagators might be-come on shell, and there are two potential singular solutions,one with positive energy and another with negative energy,depending on the magnitude and direction of the external mo-menta [6, 15]. The solution with negative energy representsa singular configuration where there is at least one on-shell propagator in each set. Therefore, the amplitude splits intotwo disjoint trees, with all the momenta over the causal on-shell cut pointing to the same direction. Abusing notation: A ( L ) D (2 , . . . , n ; 1) − shell → A ( L ) D (1 , , . . . , n ) . (9)The on-shell singular solution with positive energy, however,is locally entangled with the next term in Eq. (8) such that thefull LTD representation remains nonsingular in this configu-ration: A ( L ) D (2 , , . . . , n ; 1) + A ( L ) D (1 , , . . . , n ; 2) (10) (1 ,
2) on − shell → A ( L ) D (1 , , , . . . , n ) − A ( L ) D (1 , , , . . . , n ) . These local cancellations also occur with multiple powerpropagators. They are the known dual cancellations of un-physical or noncausal singularities [6, 13–15] and their can-cellation is essential to support that the remaining causal andanomalous thresholds as well as infrared singularities arerestricted to a compact region of the loop three-momenta.Causality determines that the only surviving singularities fallon ellipsoid surfaces in the loop three-momenta space [7, 8,22], that collapse to finite segments for massless particlesleading to infrared singularities. These causal singularitiesare bounded by the magnitude of the external momenta, thusenabling the simultaneous generation with the tree contribu-tions describing emission of extra radiation through suitablemomentum mappings, as defined in four-dimensional unsub-straction (FDU) [9–11]. Another potential causal singularityoccurs from the last term in Eq. (8) when all the on-shell mo-menta are aligned in the opposite direction over the causalon-shell cut, A ( L ) D (1 , . . . , n − n ) n on − shell → A ( L ) D (1 , . . . , n ) .It is also interesting to note the remarkable structure that theLTD representation exhibits when expressed in terms of dualpropagators. For example, the scalar MLT integral with onlyone single propagator in each set, e.g. the sunrise diagram attwo loops, reduces to the extremely compact expression A ( L )MLT (1 , . . . , n ) = − (cid:90) (cid:126)(cid:96) ,...,(cid:126)(cid:96) L q (+) n, (cid:32) λ − ,n + 1 λ +1 ,n (cid:33) , (11)where λ ± ,n = (cid:80) ni =1 q (+) i, ± k ,n , with k n = (cid:80) ni =1 q i , and (cid:82) (cid:126)(cid:96) s = − µ − d (cid:82) d d − (cid:96) s / (2 π ) d − / (2 q (+) s, ) . The most notableproperty of this expression is that it is explicitly free of un-physical singularities, and the causal singularities occur, asexpected, when either λ +1 ,n or λ − ,n vanishes, depending onthe sign of the energy component of k n , in the loop three-momenta region where the on-shell energies are bounded, q (+) i, < | k ,n | . This property also holds for powered propa-gators, nonscalar integrals, and more than one propagator ineach set. Furthermore, Eq. (11) is independent of the initialmomentum flows in the Feynman representation.
12 123 n = ⊗ n + 12 12 ⊗ n n = ⊗ n + 23 12 123 ⊗ n FIG. 2. Next-to-maximal loop topology (left) and its convoluted dualrepresentation (right). Each MLT subtopology opens according toEq. (8). Only the on-shell cut of the last MLT-like subtopology withreversed momentum flow is shown.
NEXT-TO-MAXIMAL LOOP TOPOLOGY
The next multiloop topology in complexity, see Fig. 2, con-tains one extra set of momenta, denoted by , that dependson the sum of two loop momenta, q i = − (cid:96) − (cid:96) + k i . Wecall it next-to-maximal loop topology (NMLT). This topologyappears for the first time at three loops, i.e. n + 1 sets with L = n − and n ≥ , and its LTD representation is given bythe compact and factorized expression A ( L )NMLT (1 , . . . , n,
12) = A (2)MLT (1 , , ⊗ A ( L − (3 , . . . , n )+ A (1)MLT (1 , ⊗ A (0) (12) ⊗ A ( L − (3 , . . . , n ) . (12)The first term on the rhs of Eq. (12) represents a convolutionof the two-loop MLT subtopology involving the sets (1 , , with the rest of the amplitude, which is also MLT. Each MLTcomponent of the convolution opens according to Eq. (8). Inthe second term on the rhs of Eq. (12), the set remains offshell while there are on-shell propagators in either or , andall the inverted sets from to n contain on-shell propagators.For example, at three loops ( n = 4 ), these convolutions areinterpreted as A (2)MLT (1 , , ⊗ A (1)MLT (3 , (cid:90) (cid:96) ,(cid:96) ,(cid:96) (cid:16) A (3) D (2 , ,
4; 1 ,
3) + A (3) D (1 , ,
4; 2 , A (3) D (1 , ,
4; 12 ,
3) + (4 ↔ (cid:17) , (13)and A (1)MLT (1 , ⊗ A (0) (12) ⊗ A (2)MLT (3 , (14) = (cid:90) (cid:96) ,(cid:96) ,(cid:96) (cid:16) A (3) D (2 , ,
4; 1 ,
12) + A (3) D (1 , ,
4; 2 , (cid:17) . The two sets after the semicolon remain off shell. In total, thenumber of terms generated by Eq. (12) is L − .Causal thresholds and infrared singularities are then de-termined by the singular structure of the A (2)MLT (1 , ,
12 123 n = ⊗ n + 12 12 ⊗ n n = ⊗ n + 23 12 123 ⊗ n FIG. 3. Next-to-next-to-maximal loop topology (left) and its con-voluted dual representation (right). Opening according to Eq. (15).Only the on-shell cut of the last MLT-like subtopology with reversedmomentum flow is shown. subtopology, and by the singular configurations that split theNMLT topology into two disjoint trees with all the on-shellmomenta aligned over the causal cut. Again, the singular sur-faces in the loop three-momenta space are limited by the ex-ternal momenta, and all the noncausal singular configurationsthat arise in individual contributions undergo dual cancella-tions.
NEXT-TO-NEXT-TO-MAXIMAL LOOP TOPOLOGY
The last multiloop topology that we consider explicitly isthe next-to-next-to-maximal loop topology (N MLT) shownin Fig. 3. At three loops, it corresponds to the so-calledMercedes-Benz topology. Besides the -set, there is anotherset denoted by with q i = − (cid:96) − (cid:96) + k i . Its LTD repre-sentation is given by the following convolution of factorizedsubtopologies A ( L )N MLT (1 , . . . , n, , (15) = A (3)NMLT (1 , , , , ⊗ A ( L − (4 , . . . , n )+ A (2)MLT (1 ∪ , , ∪ ⊗ A ( L − (4 , . . . , n ) . The sets (1 , , , , form a NMLT subtopology. There-fore, the first component of the first term on the rhs of Eq. (15)opens iteratively as A (3)NMLT (1 , , , ,
23) = A (2)MLT (1 , , ⊗ A (1)MLT (3 , (cid:90) (cid:96) ,(cid:96) ,(cid:96) (cid:16) A (3) D (1 , ,
23; 2 ,
12) + A (3) D (12 , ,
23; 1 , (cid:17) . (16)The last two terms on the rhs of Eq. (16) are fixed by the con-dition that the sets (2 , , cannot generate a disjoint sub-tree. The second term on the rhs of Eq. (15) contains a two-loop subtopology made of five sets of momenta, A (2)MLT (1 ∪ , , ∪ , which are grouped into three sets and dualizedthrough Eq. (8). For example, propagators in the sets and are not set simultaneously on shell. The number of termsgenerated by Eq. (15) is L − . As for the NMLT, thecausal singularities of the N MLT topology are determinedby its subtopologies and by the singular configurations thatsplit the open amplitude into disjoint trees with all the on-shellmomenta aligned over the causal cut. Any other singular con-figuration is entangled among dual amplitudes and cancels.We would like to emphasize that Eq. (15) accounts properlyfor the NMLT and MLT topologies as well, if either or both and are taken as empty sets. At three loops, therefore,Eq. (15) emerges as the LTD master topology for opening anyscattering amplitude from its Feynman representation.Finally, let us comment on more complex topologies athigher orders. For example, let’s consider the multiloop topol-ogy made of one MLT and two two-loop NMLT subtopolo-gies that appears for the first time at four loops. This topologyopens into nondisjoint trees by leaving three loop sets off shelland by introducing on-shell conditions in the others under cer-tain conditions: either one off-shell set in each subtopology ortwo in one NMLT subtopology and one in the other with on-shell propagators in all the sets of the MLT subtopology. Oncethe loop amplitude is open into trees, the singular causal struc-ture is determined by the causal singularities of its subtopolo-gies, and all entangled noncausal singularities of the forestcancel. CONCLUSIONS
We have reformulated the loop-tree duality at higher or-ders and have obtained very compact open-into-tree analyti-cal representations of selected loop topologies to all orders.These loop-tree dual representations exhibit a factorized cas-cade form in terms of simpler subtopologies. Since this fac-torized structure is imposed by the opening into nondisjointtrees and by causality, we conjecture that it holds to all looporders and topologies. Remarkably, specific multiloop config-urations are described by extremely compact dual representa-tions which are, moreover, free of unphysical singularities andindependent of the initial momentum flow. This property hasbeen tested with all the topologies and several internal config-urations. We also conjecture that analytic dual representationsin terms of only causal denominators are always plausible.The explicit expressions presented in this Letter are suffi-cient to describe any scattering amplitude up to three loops.Other topologies that appear for the first time at four loopsand beyond have been anticipated, and will be presented in aforthcoming publication. This reformulation allows for a di-rect and efficient application to physical scattering processes,and is also advantageous to unveil formal aspects of multiloopscattering amplitudes.
Acknowledgements:
We thank Stefano Catani for very stim-ulating discussions. This work is supported by the SpanishGovernment (Agencia Estatal de Investigaci´on) and ERDFfunds from European Commission (Grants No. FPA2017-84445-P and No. SEV-2014-0398), Generalitat Valenciana (Grant No. PROMETEO/2017/053), Consejo Superior de In-vestigaciones Cient´ıficas (Grant No. PIE-201750E021) andthe COST Action CA16201 PARTICLEFACE. JP acknowl-edges support from ”la Caixa” Foundation (No. 100010434,LCF/BQ/IN17/11620037), and the European Union’s H2020-MSCA Grant Agreement No. 713673; SRU from CONACyTand Universidad Aut´onoma de Sinaloa; JJAV from Gener-alitat Valenciana (GRISOLIAP/2018/101); WJT and AEROfrom the Spanish Government (FJCI-2017-32128, PRE2018-085925); and RJHP from Departament de F´ısica Te`orica,Universitat de Val`encia, CONACyT through the project A1-S-33202 (Ciencia B´asica) and Sistema Nacional de Investi-gadores. [1] A. Dainese, M. Mangano, A. B. Meyer, A. Nisati, G. Salam andM. A. Vesterinen, doi:10.23731/CYRM-2019-007[2] A. Abada et al. [FCC Collaboration], Eur. Phys. J. C (2019)no.6, 474.[3] S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J. C. Winter,JHEP (2008) 065 [arXiv:0804.3170 [hep-ph]].[4] I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, JHEP (2010) 073 [arXiv:1007.0194 [hep-ph]].[5] I. Bierenbaum, S. Buchta, P. Draggiotis, I. Malamos and G. Ro-drigo, JHEP (2013) 025 [arXiv:1211.5048 [hep-ph]].[6] S. Buchta, G. Chachamis, P. Draggiotis, I. Malamos and G. Ro-drigo, JHEP (2014) 014 [arXiv:1405.7850 [hep-ph]].[7] S. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, Eur.Phys. J. C (2017) no.5, 274 [arXiv:1510.00187 [hep-ph]].[8] S. Buchta, PhD thesis, Universitat de Val`encia, 2015,arXiv:1509.07167 [hep-ph].[9] R. J. Hern´andez-Pinto, G. F. R. Sborlini and G. Rodrigo, JHEP (2016) 044 [arXiv:1506.04617 [hep-ph]].[10] G. F. R. Sborlini, F. Driencourt-Mangin, R. Hern´andez-Pintoand G. Rodrigo, JHEP (2016) 160 [arXiv:1604.06699[hep-ph]].[11] G. F. R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, JHEP (2016) 162 [arXiv:1608.01584 [hep-ph]].[12] F. Driencourt-Mangin, G. Rodrigo and G. F. R. Sborlini, Eur.Phys. J. C (2018) no.3, 231 [arXiv:1702.07581 [hep-ph]].[13] F. Driencourt-Mangin, G. Rodrigo, G. F. R. Sborliniand W. J. Torres Bobadilla, JHEP (2019) 143[arXiv:1901.09853 [hep-ph]].[14] F. Driencourt-Mangin, PhD thesis, Universitat de Val`encia,2019, arXiv:1907.12450 [hep-ph].[15] J. J. Aguilera-Verdugo, F. Driencourt-Mangin, J. Plen-ter, S. Ram´ırez-Uribe, G. Rodrigo, G. F. R. Sborlini,W. J. Torres Bobadilla and S. Tracz, JHEP (2019) 163arXiv:1904.08389 [hep-ph].[16] F. Driencourt-Mangin, G. Rodrigo, G. F. R. Sborlini andW. J. Torres Bobadilla, arXiv:1911.11125 [hep-ph].[17] J. Plenter, Acta Phys. Polon. B (2019) 1983.doi:10.5506/APhysPolB.50.1983[18] E. T. Tomboulis, JHEP (2017) 148 [arXiv:1701.07052[hep-th]].[19] R. Runkel, Z. Szor, J. P. Vesga and S. Weinzierl, Phys. Rev.Lett. (2019) no.11, 111603 Erratum: [Phys. Rev. Lett. (2019) no.5, 059902] [arXiv:1902.02135 [hep-ph]].[20] R. Runkel, Z. Szor, J. P. Vesga and S. Weinzierl,arXiv:1906.02218 [hep-ph]. [21] Z. Capatti, V. Hirschi, D. Kermanschah and B. Ruijl, Phys. Rev.Lett. (2019) no.15, 151602 [arXiv:1906.06138 [hep-ph]].[22] Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl,arXiv:1912.09291 [hep-ph].[23] R. P. Feynman, Acta Phys. Polon. (1963) 697.[24] R. P. Feynman, In *Brown, L.M. (ed.): Selected papers ofRichard Feynman* 867-887. World Scientific Publishing Co.Pte. Ltd., Singapore, 2000.[25] R. E. Cutkosky, J. Math. Phys. (1960) 429. [26] S. Mandelstam, Phys. Rev. Lett. (1960) 84.[27] L. D. Landau, Nucl. Phys. (1959) 181.[28] R. E. Cutkosky, Rev. Mod. Phys. 33 (1961) 448.[29] S. Coleman and R. E. Norton, Nuovo Cim. (1965) 438.[30] D. Kershaw, Phys. Rev. D (1972) 1976.[31] C. G. Bollini and J. J. Giambiagi, Nuovo Cim. B (1972) 20.[32] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B44