Interplay between the loop-tree duality and helicity amplitudes

Interplay between the loop-tree duality and helicity amplitudes F. Driencourt-Mangin,1,* G. Rodrigo ,1,†G. F. R. Sborlini ,1,2,‡and W. J. Torres Bobadilla 1,3,§ 1Instituto de Física Corpuscular, Universitat de Val`encia—Consejo Superior de Investigaciones Científicas, Parc Científic, E-46980 Paterna, Valencia, Spain 2Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany 3Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany (Received 30 March 2021; revised 2 December 2021; accepted 2 January 2022; published 12 January 2022) The spinor-helicity formalism has proven to be very efficient in the calculation of scattering amplitudes in quantum field theory, while the loop-tree duality (LTD) representation of multiloop integrals exhibits appealing and interesting advantages with respect to other approaches. In view of the most recent developments in LTD, we exploit the synergies with the spinor-helicity formalism to analyze illustrative oneand two-loop scattering processes. We focus our discussion on the local UV renormalization of IR and UV finite helicity amplitudes and present a fully automated numerical implementation that provides efficient expressions, which are integrable directly in four space-time dimensions. DOI: 10.1103/PhysRevD.105.016012 I. INTRODUCTION In order to unveil the fundamental components of matter and their interactions, it is necessary to analyze highlyprecise experimental data obtained from colliders using accurate theoretical predictions. However, the established theoretical frameworks, i.e., the Standard Model, involves very complicated mathematical equations, whose exact solutions are unknown in many physically relevant processes. Thus, most of the computations performed nowadays rely on the perturbative approach, which naturally leads to Feynman loop amplitudes and loop integrals. With the purpose of achieving a higher accuracy in the theoretical predictions, it is mandatory to explore higher perturbative orders and compute multiloop amplitudes with high multiplicity. To tackle these calculations, several methods have been developed in the last years. On the one hand, there has been enormous progress in the algebraic handling of scattering amplitudes in gauge theories by using alternative kinematic variables as the ones provided by the spinor-helicity formalism [1]. Also, there was an important improvement due to the study of the mathematical properties of the scattering amplitudes, for instance, in the color sector [2–4] and the development of new regularization strategies [5,6]. These techniques lead to a much more efficient treatment of the scattering amplitudes, exploiting several symmetries to simplify the underlying expressions. On the other hand, there were also great advances in the calculation of multiloop Feynman integrals, both analytically and numerically [7]. In particular, pointing towards a more efficient numerical implementation, we have been developing a novel strategy based on the loop-tree duality (LTD) theorem [9–18]. This theorem allows to decompose any loop amplitude (or loop integral) as the sum of treelevel-like objects integrated over a proper phase-space region. From the physical point of view, loop particles are converted into real-radiation ones. From the mathematical side, the integration domain is transformed from a Minkowski to an Euclidean space. In fact, the numerical evaluation of multiloop integrals through LTD is, with respect to the approaches that pass by Feynman parametrization or Mellin-Barnes transformations, more efficient as the number of integrations to be performed does not scale with the number of external particles. Very recently, a novel LTD-inspired representation of multiloop multileg scattering amplitudes was presented in Refs. [14–17,19,20]. This strategy, based on the nested residue strategy, leads to very compact integrand-level representations, which are free of unphysical or noncausal singularities. Likewise, alternative studies of LTD have been presented in Refs. [21–26]. In this paper, we apply the LTD formalism to the calculation of multiloop helicity amplitudes. We exploit the fact that LTD works at the level of denominators, and the structure of the numerator does not generate any additional difficulties. To this end, we start considering *[email protected].es †[email protected] ‡[email protected] §[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 105, 016012 (2022) 2470-0010=2022=105(1)=016012(13) 016012-1 Published by the American Physical Society illustrative examples in which the simplicity of the latter is displayed. We make use of the spinor-helicity formalism, where we write definite helicity states for the external particles. On top of it, we also use the momentum twistors’ variables [27] to implement simplifications in the integrand of the helicity amplitudes. These variables, due to their mathematical properties, allow us to express any kinematic process in terms of the minimal set of variables. In other words, for a process with Nexternal massless particles we have 3N−10 invariants to deal with [28,29]. Likewise, the extension to massive particles is straightforward. Besides the clearness LTD offers us to compute any multiloop amplitude, in this paper, we also insist on the local UV renormalization. Hence, for the sake of simplicity, we consider scattering amplitudes that are IR and UV safe but might still exhibit a local singular UV behavior. For the latter, it is known that UV finite integrals might be locally divergent in the high-energy region [30]. Therefore, a careful treatment in the UV has to be performed. For instance, at one-loop level, we refer the reader to Refs. [31–35] (and references therein) and to Ref. [36] beyond one loop. We remark that the idea of performing a local UV renormalization is to obtain welldefined integrands in four space-time dimensions that allow a straightforward numerical evaluation. The paper is organized as follows. In Sec. II, we recall the basis of the LTD formalism, with special emphasis on the formulae applied in this work. We briefly present its extension to the multiloop case. In Sec. III, we provide a description of the generation of kinematical variables by using the spinor-helicity formalism. In Sec. III A,wefocus on the parametrization of the loop three-momenta to integrate the dual contributions when external momenta are complex. The introduction of local UV renormalization counterterms is reviewed in Sec. IV. The main part of this manuscript is presented in Sec. V, where we show numerical results for explicit examples at one-loop level. Special emphasis is put on the local cancellation of UV singularities, to render the expressions integrable in four space-time dimensions. Then, we present a careful study of the local UV counterterms for H→gg at two loops in Sec. VI.For this purpose, we make use of the computational tools developed throughout this paper. This allows us to support the feasibility of the LTD-based numerical strategy with realistic scattering processes, as well as its efficiency. Conclusions and future research directions are analyzed in Sec. VII. II. LOOP-TREE DUALITY IN A NUTSHELL The LTD theorem [9–17] rewrites any loop integral in terms of tree-level-like expressions that correspond to cutting, i.e., setting on shell, a number of internal particles equal to the number of loops. It relies on a suitable application of the Cauchy’s residue theorem to reduce one degree of freedom for each loop. We usually apply it on the energy component of the loop momenta, which translates into reducing the original Minkowski integration domain into the Euclidean space of the loop threemomenta, although it could be used to remove any other component of the loop momenta. In order to explain the formalism, let us consider a generic L-loop N-particle scattering amplitude, where the external momenta are labeled as piwith i∈f1;…;Ng. We have Lindependent primitive integration variables, fljgj¼1;…;L, and the momenta associated to the different internal lines can be written as qis¼lsþkis, where lsis a linear combination of the primitive loop momenta, and kis is a linear combination of external momenta. All the propagators depending on the same linear combination of primitive loop momenta lsare enclosed together inside the set s. With this notation, a generic amplitude is given by AðLÞ Nð1;…;nÞ¼Zl1;…;lLXN×GFð1;…;nÞ;ð1Þ where Nrepresents an arbitrary numerator, and GFð1;…;nÞ¼ Y j∈1∪…∪nðGFðqjÞÞαjð2Þ is a product of Feynman propagators spanned over all the possible momenta sets. Each scalar Feynman propagator can be written as GFðqiÞ¼ 1 q2 i−m2 iþ{0¼1 q2 i;0−ðqðþÞ i;0Þ2;ð3Þ where qðþÞ i;0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 iþm2 i−{0 pcorresponds to the positive on-shell energy of the associated internal particle. In Eq. (2), we also take into account the possibility of arbitrary powers of the propagators, through the parameters fαjg. Regarding the integration measure, we have Zl ≡−{μ4−dZddl ð2πÞd;ð4Þ with μan arbitrary energy scale to restore the proper units after the extension to a d-dimensional space-time. The LTD representation is obtained by defining the nested residues and closing the integration contour in the lower part of the complex plane. Explicitly, if AðLÞ Frepresents the integrand of Eq. (1) in the Feynman representation, then the first iteration of Cauchy’s theorem leads to AðLÞ Dð1; 2;…;nÞ¼X i1∈1 ResðAðLÞ F;Imðη·qi1Þ<0Þ;ð5Þ where we sum over all the possible configurations containing one on-shell propagator of the first set 1, whilst propagators in the remaining sets are left off shell. In this expression, ηis F. DRIENCOURT-MANGIN et al. PHYS. REV. D 105, 016012 (2022) 016012-2 a futurelike vector that determines which degree of freedom of the loop momenta is integrated out. For the sake of simplicity, we choose ημ¼ð1;  0Þ, which corresponds to apply the nested residues on the energy component of the propagator’smomentaqi;0. After the rth iteration, we end up with AðLÞ Dð1;…;r;rþ1;…;nÞ ¼X ir∈r ResðAðLÞ Dð1;…;r−1; r; …;nÞ;Imðη·qirÞ<0Þ; ð6Þ where all the propagators in the sets to the right of the semicolon on the lhs of Eq. (6) are left off shell, and we set exactly one propagator on shell in each one of the first rsets. Once we apply the Cauchy’s theorem in the energy component of the primitive momenta, the integration measure turns into Zl →Z l ≡−μd−4Zdd−1l ð2πÞd−1;ð7Þ i.e., transforming the d-dimensional Minkowski space into a (d−1)-dimensional Euclidean one. The dual LTD representation is obtained after the Lth iteration of the residue computation, i.e., AðLÞ Nð1;…;nÞ¼Z l1…  lLX σ AðLÞ Dðσ1;…;σL;σLþ1;…;σnÞ; ð8Þ where we sum over all the possible combinations of simultaneous Lcuts in different momenta sets. This is equivalent to perform as many cuts as loops, in order to open the loop amplitude into a set of nondisjoint trees [9]. At one loop, the aforementioned formulae reduce to the usual LTD representation given in Refs. [9,10]. It can be easily obtained by summing over all the possible single cuts and replacing the Feynman propagators by the so-called dual propagators, namely GDðqi;qjÞ¼ 1 q2 j−m2 j−{0η·kji ;ð9Þ where qjis the momenta flowing through the line, qi corresponds to the one that is set on shell, and kji ¼qj−qi. As in Eq. (5),ηis a generic futurelike vector, which is usually chosen as ημ¼ð1;  0Þ. It is important to notice that the dual prescription accounts for the information contained in the multiple cuts defined within the Feynman tree theorem [37]. Also, that within the representation introduced in Refs. [14–17], there is no need to explicitly deal with the complex prescription of the dual propagators since this operation is encoded within the definition of the nested residues. In this paper, we present practical applications up to the two-loop level, although the underlying algorithms can be extended to any loop order. In the particular two-loop case, we have lj¼fl1;l2g, and there are three sets of propagators, s∈f1;2;12g, with qi12 ¼l1þl2þki12 , as shown in Fig. 1. More details about the underlying subtleties of the two-loop case are available in Refs. [10,36]. A. Multiple poles and IBPs Multiloop integrals and local UV counterterms can contribute with multiple powers of the propagators. Using integration-by-parts identities (IBPs) [38–40], they can be reduced to linear combinations of other integrals containing only single powers of the propagators, as done in [11]. However, this modifies the local behavior of the integrands and might spoil the point-by-point cancellation of IR singularities present in the real-emission contribution. Thus, we will stick to the local approach and avoid using IBPs. The master formula given in Eq. (8) handles also amplitudes with multiple powers of the propagators since its definition relies directly on the nested application of the Cauchy’s theorem. A careful discussion about the computation of the residue is presented in Refs. [9,11].Itis important to take care of the dual prescription for the contributions associated to the original amplitude since it may contain thresholds in the low energy region. In that FIG. 1. Diagrammatic representation of a generic two-loop diagram with Nexternal particles. INTERPLAY BETWEEN THE LOOP-TREE DUALITY AND …PHYS. REV. D 105, 016012 (2022) 016012-3 case, the propagators associated to off-shell propagators must be promoted to dual propagators. In contrast, when applying the LTD formalism to the UV counterterms, we can neglect the complex prescriptions and straightforwardly use the Cauchy’s formula for computing the residue. It is worth noticing that compact formulae to obtain the LTD representation with higher powers are presented in Ref. [15]. These expressions are obtained considering the derivatives, with respect to ðqðþÞ i;0Þ2, and taking advantage of Eq. (3) to express propagators in terms of on-shell energies. III. GENERATION OF THE KINEMATICS In order to provide helicity amplitudes, we take advantage of the momentum twistor parametrization proposed in Ref. [27], where the standard spinor products, h••i;½••, are replaced by a minimal set of independent variables zi. The number of variables in the latter depends on the kinematic process. In particular, any n-point massless amplitude can be expressed in terms of 3n−10 independent variables. Hence, the extension to amplitudes with massive particles is straightforward. In Appendix A, we briefly recall the main features of these variables. Within the LTD approach, the evaluation of integrals is performed in the momentum space instead of using Feynman parameters or, equivalently, Mellin transformations. Then, the most suitable way of computing helicity amplitudes is through the form factors’decomposition, which has been applied within the LTD framework in Refs. [30,36]. Very recently, some alternative methods to bypass this decomposition were proposed [41,42]. Since the representation of the polarization vectors may be an obstacle depending on the regularization scheme being applied, we use the one in which external wave functions are kept in four dimensions, i.e., ’t Hooft-Veltman (HV) [43] and four-dimensional helicity (FDH) [44,45]. The use of HV and FDH allows us to project objects in ddimensions into a four-dimensional space by making use of the following properties [49]: qi;½d·pj;½4¼qi;½4·pj;½4;ð10aÞ qi;½d·εj;½4¼qi;½4·εj;½4;ð10bÞ where we contracted the loop momentum with external momenta or polarization vectors. Therefore, we only need to keep track of squared loop momenta, qi;½d·qj;½d. It turns out that due to the cuts performed within the LTD formalism, we can easily remove this dependence and work with objects in four dimensions. Let us also remark that, within this approach, we do not need to include extradimensional products, i.e., qi;½d−4·qj;½d−4. Then, working in four space-time dimensions, we can parametrize the loop momenta in terms of a fourdimensional basis, i.e., E¼feig. Therefore, to reduce as much as possible the number of scalar products to be evaluated, we choose E¼fp1;p 2;ε12;ε21g. With this choice, the loop momenta is expressed as qα i¼xi;1pα 1þxi;2pα 2þxi;3εα 12 þxi;4εα 21;ð11Þ where p1and p2are the massless momenta built from the parametrization obtained from the momentum twistors of an n-point kinematics and εα ij ¼1 2hijγαjj. We remark that for the elements of the basis, we explicitly work with the components of the four-vectors. Hence, with this decomposition, all the scalar products involving external momenta or polarization vectors contracted with the loop momenta can always be expressed in terms of scalar products among the elements of the basis and the loop momenta, i.e., qi·ej. The aim of this refinement is twofold, firstly, to reduce the number of scalar products required for the computation, and in the second place, to cancel redundant expressions that appear at integrand level. This prevents some noncontributing terms that pop up in intermediate steps of the computation before performing an explicit evaluation. A. Parametrization of the loop momentum As discussed in Sec. II, once LTD is applied to any loop integral or virtual amplitude, the integration over the loop energy component is removed, and the remaining one is performed over an Euclidean space. Thus, the loop threemomentum needs to be properly parametrized to improve the computational efficiency. We remark that we are considering a complex-valued parametrization of the external momenta. Explicitly, the second component of the three-momentum is purely imaginary; this is due to the method applied to build their representation starting from scalar invariants [50]. It is worth noticing, however, that the scalar products among themselves do not contain any complex phase (i.e., they are purely real), as expected in any physical kinematic configuration. Hence, to overcome any possible issue when using a real parametrization of the loop three-momentum, we express it in cylindrical coordinates, li¼ðξicos ϕi;ρi;ξisin ϕiÞ;ð12Þ for the ith loop three-momentum. Then, the resulting integral is given by Ii¼Z∞ 0 ξidξiZ2π 0 dϕiZ∞ −∞ dρiIiðξi;ϕi;ρiÞ;ð13Þ where Iiis the integrand after plugging the explicit parametrization of the loop three-momentum (12).We note that carefully integrating Iiover ρibrings large cancellations, in particular, when considering kinematical configurations below threshold. This is because the imaginary part introduced by the prescriptions must cancel in F. DRIENCOURT-MANGIN et al. PHYS. REV. D 105, 016012 (2022) 016012-4 these configurations, and the ρivariable captures all the imaginary contributions due to the explicit functional form of the parametrization of the external momenta. Of course, we are excluding from this claim the presence of imaginary terms introduced by the numerators (for instance, originated by the polarization vectors). Hence, to account for the simplifications that occur in the ρiintegration, we rewrite Eq. (13) as Ii¼Z∞ 0 ξidξiZ2π 0 dϕiZ∞ 0 dρi ×½Iiðξi;ϕi;ρiÞþIiðξi;ϕi;−ρiÞ;ð14Þ which turns out to be equivalent to consider the real-part of the integrand in the previously mentioned conditions. Furthermore, we notice that the ðξi;ρiÞ−plane can be compactified by changing variables and using polar coordinates. Explicitly, we define ðξi;ρiÞ→ xi 1−xiðcos θi;sin θiÞ;ð15Þ with 0≤xi<1and 0≤θi<π=2. In the last part, we restricted the angular integration to the first quadrant because both ξiand ρiare positive. IV. LOCAL UV RENORMALIZATION Since we are aiming for a complete numerical implementation, it is necessary to build integrand-level counterterms, in order to cancel the local singular behavior at very high energies of the amplitudes under consideration. In the following, we recall how to generate these counterterms very easily from the original amplitudes [30,36]. For a given loop momentum lj, we consider the integrand-level replacement Sj;UV∶fl2 jjlj·kig→fλ2q2 j;UV þð1−λ2Þμ2 UVjλqj;UV ·kig; ð16Þ where μUV is an arbitrary scale that can be identified with the renormalization scale and qj;UV ¼ljþkj;UV. The vector kj;UV is arbitrary, and we can set kj;UV ≡0without any loss of generality. By applying Sj;UV to an unintegrated and uncut one-loop amplitude Að1Þ Nwith loop momentum lj, and then expanding in λaround infinity up to logarithmic degree (this operation will be represented by the operator Lλin the following), we directly obtain an integrand-level expression that cancels the local UV singularities exhibited by Að1Þ N. This procedure is equivalent to expanding around the UV propagator [31,32,34,35], i.e., GFðqj;UVÞ¼ 1 q2 j;UV −μ2 UV þi0;ð17Þ and then keeping only the most divergent terms: the resulting object mimics locally the UV behavior of the original expression. It is important to note, though, that this counterterm may generate a finite part after integration, which must be fixed through a scheme fixing parameter dj;UV. Therefore, the counterterm reads, Að1Þ j;UV ¼LλðAð1Þ NjSj;UV Þ−dj;UVμ2 UV ZljðGFðqj;UVÞÞ3;ð18Þ where the integral multiplying dj;UV integrates to the same finite quantity in both four and dspace-time dimensions. By construction, the first term in the rhs of Eq. (18) exactly contains the UV divergent part of the original amplitude, extracted through the operator Lλat integrand level. Thus, the quantity Að1Þ j;UV locally cancels the UV behavior of Að1Þ, while giving the required finite part (which is 0, for instance, in the MS scheme). For a two-loop amplitude Að2Þ N, the general local renormalization procedure has been extended in Ref. [36].In the two-loop case, it is necessary to consider three UV divergent configurations involving the two internal momenta, l1and l2. For instance, we can consider the regimes jl1j→∞ jl2jfixed ;jl1jfixed jl2j→∞;jl1j→∞ jl2j→∞:ð19Þ The counterterms relative to the singular behavior of the first two regimes can be generated using the replacement in Eq. (16). These contributions are known as the single UV counterterms because only one integration loop momenta goes to infinity, whilst the other remains subdominant. To obtain the local counterterm needed to cancel the third one, we need the additional replacement, SUV2∶ l2 j→λ2q2 j;UV þð1−λ2Þμ2 UV; lj·lk→λ2qj;UV ·qk;UV þð1−λ2Þμ2 UV=2; lj·ki→λqj;UV ·ki;ð20Þ to build the counterterm Að2Þ UV2¼LλAð2Þ N−Xj¼1;2Að2Þ j;UVSUV2 −dUV2μ4 UV Zl1l2ðGFðq1;UVÞÞ3ðGFðq2;UVÞÞ3;ð21Þ where once again, the term proportional to the schemefixing coefficient dUV2integrates to a finite quantity. INTERPLAY BETWEEN THE LOOP-TREE DUALITY AND …PHYS. REV. D 105, 016012 (2022) 016012-5 This last terms accounts for the so-called double UV divergence, i.e., when both integration loop momenta go to infinity. The oneand two-loop versions of this algorithm were explicitly implemented in a MATHEMATICA code [36].Itis fully process independent and can be directly applied to any scattering amplitude, producing the appropriate local counterterm to regularize the divergent behavior in the high-energy region. V. APPLICATIONS AT ONE LOOP In this section, we give explicit examples in which the techniques described in Secs. III and IV are applied. We explore the computational advantages of the LTD-based representation combined with the spinor-helicity formalism, which constitutes the central part of this paper. We focus on processes that contain two to four kinematic invariants, and we consider the nonvanishing helicity configurations. We summarize the description of our examples in Table I. In the following, the kinematic invariants are implicitly given in GeV2. Since we are aiming at a calculation performed purely in four space-time dimensions, we restrict the analysis presented in this paper to helicity amplitudes that are simultaneously IR and UV finite. Although the processes under consideration exhibit these features, they might still posses a local UV-divergent behavior that prevents to perform the calculation directly in four space-time dimensions, without introducing any additional regularization. This is because, in the most general case, the associated integrands turn out to be nonintegrable functions in the high-energy limit (or UV limit). Eventually, in the context of dimensional regularization, setting d¼4from the beginning of the calculation can generate wrong results. This situation was exhaustively discussed in Ref. [30] for the computation of the decay width of H→γγ at leading order. Therefore, we need to build local UV counterterms that take care of the singularities that appear at integrand level in the UV limit. In other words, we need to locally renormalize our amplitude, as explained in Sec. IV, to render the expressions integrable in four space-time dimensions. The calculation of the amplitude H→γγ performed in Ref. [30], through the form factor decomposition, exploited several analytical properties in order to simplify the results. In particular, due to gauge invariance, it was possible to remove vanishing terms at integrand level. In contrast, in the present calculation, we directly generate the proper UV counterterm to render the amplitude integrable in four space-time dimensions, without taking into account any kind of analytical property to achieve further simplifications. The numerical integration performed by LTD was compared with the analytic expression of the amplitude. For the latter, we rely on two MATHEMATICA packages, the integral reduction provided by FEYNCALC [51–53] and the analytic expressions for the one-loop scalar integrals collected in PACKAGE - X [54]. Our results are shown in Fig. 2, where we plot the value of the amplitude as a function of the fermion internal mass m2 ffor different values of s12. An excellent agreement is found, as expected from our previous studies of this process [30,36]. Regarding the scale of the vertical axis of Fig. 2, we used the default normalization provided by the aforementioned packages, and all the invariants are expressed in GeV2. The same choice is applied for the remaining plots shown in this article. For the processes including more kinematic scales, namely γγ →γγ and H→ggg, we do not rely on FEYNCALC because it becomes inefficient when the rank of the loop momentum in the numerator starts increasing. Therefore, instead of decomposing the integrals, we work at the integrand level by reducing the amplitudes to scalar one-loop integrals. In order to do so, we follow the Ossola-Papadopoulos-Pittau method [55] together with the integrand reduction algorithm [56–63]. For the evaluation of the scalar one-loop integrals, we keep using PACKAGE - X . Our results are shown in Figs. 3and 4, where we plot the amplitudes as a function of the fermion TABLE I. Processes considered at one-loop level with their kinematic scales. We indicate the nonvanishing helicity configurations. Process Kinematic scales Helicity configuration H→γγ s12;m 2 fþþ γγ →γγ s; t; m2 fþþþþ−þþþ−−þþ H→ggg s12;s 13;s 23;m 2 fþþþ−þþ s12=8 s12=10 s12=12 Helicity: ++ 100 120 140 160 180 200 0.0030 0.0025 0.0020 0.0015 0.0010 mf 2 A3 (1) FIG. 2. H→γγ at one loop as a function of the internal mass m2 f. We plot the predictions for s12 ∈f8;10;12g. The solid blue lines correspond to the analytical results, while the red points are computed through the LTD-based numerical approach. F. DRIENCOURT-MANGIN et al. PHYS. REV. D 105, 016012 (2022) 016012-6 internal mass m2 f.Forγγ →γγ, we fixed s¼−5and considered t¼f−8;−10;−12g. In the case of H→ggg, s12 ¼−1=3and s23 ¼−1=7remained fixed, while we varied s13 ∈½8;12. The agreement is very good for both processes, in all the kinematical and helicity configurations that we explored. Small numerical instabilities arise for m2 f>180 in H→gggg, although they can be fixed by slightly increasing the numerical precision of the integration. Let us stress that in the processes we consider within the LTD approach, we do not perform any integral or integrand reduction. We directly evaluate them with the proper inclusion of the UV local counterterms, as explained in Sec. IV. Regarding the evaluation of the required integrals, t12 t10 t8 Helicity: s5 100 120 140 160 180 200 0.2 0.4 0.6 0.8 1.0 1.2 1.4 mf 2 A4 (1) t8 t10 t12 Helicity: s5 100 120 140 160 180 200 0.04 0.03 0.02 0.01 0.00 mf 2 A4 (1) t8 t10 t12 Helicity: s5 100 120 140 160 180 200 0.12 0.10 0.08 0.06 0.04 mf 2 A4 (1) FIG. 3. One-loop contributions to the process γγ →γγ,asa function of the internal mass m2 f. We consider all the possible helicity configurations for a fixed ordering of the external legs: þþþþ,−þþþand −−þþ:In each case, we fix s¼−5 and plot the predictions for t∈f−8;−10;−12g. The solid blue lines correspond to the analytical results, while the red points were computed through the LTD-based numerical approach. s13 12 s13 10 s13 8 Helicity: s12 1/3, s23 1/7 100 120 140 160 180 200 0.004 0.006 0.008 0.010 0.012 0.014 mf 2 A4 (1) s13 8 s13 10 s13 12 Helicity: s12 1/3, s23 1/7 100 120 140 160 180 200 6. 10 9 8. 10 9 1. 10 8 1.2 10 8 1.4 10 8 mf 2 A4 (1) FIG. 4. One-loop contributions to the process H→ggg,asa function of the internal mass m2 f. We consider all the possible configurations for a given helicity amplitude, þþþand −þþ: In each case, we show the predictions for s13 ¼f8;10;12g. The solid blue lines corresponds to the analytical results, while the red points were computed through the LTD-based numerical approach. INTERPLAY BETWEEN THE LOOP-TREE DUALITY AND …PHYS. REV. D 105, 016012 (2022) 016012-7 we use the built-in MATHEMATICA function NIntegrate on a desktop machine with an Intel i7 (3.4 GHz) processor with eight cores and 16 GB of RAM. The computing time for each phase-space point was Oð3000Þ. It is worth appreciating that we did not implement any further optimization for performing the numerical integration. In future developments, we plan to improve this point, by using parallel integration strategies or optimized integrators to speed up the computation. In any case, we would like to highlight that the main advantage of our framework relies on the smooth behavior of the integrand and the fully local cancellation of singularities. VI. A TWO-LOOP EXAMPLE: H→gg In the previous section, we have demonstrated the viability of the LTD approach to tackle one-loop locally UV-divergent helicity amplitudes in four space-time dimensions. Here, we will show that it is also a reliable strategy for two-loop processes. So, we focus on the computation of Oðe3g2 SÞcorrections to the decay process H→gg, where the internal particles are massive top quarks and Zbosons. In this case we are dealing with a finite amplitude, whose contributions are given by the diagrams shown in Fig. 5. There are neither IR nor UV singularities, but it is mandatory to perform a local renormalization to smoothly pass from dto four space-time dimensions when evaluating this amplitude. In the same spirit of Ref. [36], we start considering a minimal set of independent denominators. This is done in order to match the structures of the planar (P) and nonplanar (NP) contributions at integrand level. Hence, we express all Feynman diagrams in terms of the following families of integrals, IP=NP ¼Zl1Zl2 1 Dν1 1Dν2 2Dν3 3Dν4 4Dν5 5Dν6 6Dν7 7 ;ð22Þ with D1¼ðl1þl2Þ2−m2 Zþ{0;ð23aÞ D2¼ðl1þl2þp1þp2Þ2−m2 Zþ{0;ð23bÞ D3¼l2 1−m2 tþ{0;ð23cÞ D4¼ðl1þp1Þ2−m2 tþ{0;ð23dÞ D5¼ðl1þp1þp2Þ2−m2 tþ{0;ð23eÞ D6¼l2 2−m2 tþ{0;ð23fÞ D7¼ðl2−p1Þ2−m2 tþ{0;ð23gÞ where the auxiliary propagators are D7and D3in the planar and nonplanar topologies, respectively. In the previous expressions, we also consider the presence of nontrivial numerators by allowing νi<0. In the following, we discuss the extraction and the structure of the single and double UV local counterterms. In particular, we have (i) Single UV counter-terms, Að2Þ 1;UV ¼Að2Þ 2;UV ¼0:ð24Þ Let us remark that single UV counterterms vanish at integrand level, as a consequence of the way in which the propagators have been labeled. A different choice might lead to nonvanishing integrand-level expressions. In order to profit from this property, we labeled the propagators in such a way that the most UV-divergent contributions depend on l1 and l2, whilst those contributions depending on l1þl2≡l12 exhibit a less UV-divergent behavior. (ii) Double UV counterterm, Að2Þ UV2¼2{ðd−2Þgfδa1a2Zl1l2−s12 D2 1UVD2UVD12UV þs12 D2 1UVD2 12UV þðε1·l12Þðε2·l2Þ D1UVD2 2UVD2 12UV −ðε1·l12Þðε2·l1Þ D2 1UVD2UVD2 12UV þ2ðε1·l1Þðε2·l1Þ D3 1UVD2UVD12UV −2ðε1·l1Þðε2·l1Þ D3 1UVD2 12UV −ðε1·l1Þðε2·l2Þ D2 1UVD2 2UVD12UV¼0;ð25Þ where δa1a2accounts for the color factor originated from the two external gluons, and gf¼ egg0g2 SmZðg2 Lþg2 RÞcorresponds to the overall coupling constant. In order to refer to the UV limit of the Feynman propagators, we introduced the shorthand notation FIG. 5. Two-loop diagrams for the process H→gg with internal massive top quarks and Zbosons. F. DRIENCOURT-MANGIN et al. PHYS. REV. D 105, 016012 (2022) 016012-8 DjUV ¼ðGFðqj;UVÞÞ−1;ð26Þ with qj;UV ¼ljwith j¼f1;2g, and q12;UV ¼l12. It is easy to check that after applying integrationby-parts identities on Að2Þ UV2, it vanishes in ddimensions. In fact, as expected from the finiteness of the amplitude, the UV singularities that emerge from the planar and nonplanar diagrams cancel exactly at this point. The latter is indeed in agreement with the above discussion since we are including an additional term that does not alter the behavior in ddimensions, but it collects several local features in d¼4. In other words, we end up with an integrable function in four space-time dimensions. Subtracting the counterterms in Eqs. (24) and (25) from the original H→gg amplitude, we tested the UV limit [64]. We parametrized the loop momenta and studied the large energy limit by performing a series expansion. Analytically, we found that all the nonintegrable powers of the loop-energy cancel, supporting the validity of the local renormalization proposed here. To test the numerical behavior of the locally renormalized amplitude, we started by reparametrizing the energy component of the loop momenta. Following the notation of Sec. III, we used polar coordinates to implement the transformation, ðξ1;ξ2Þ→ðrcos α;rsin αÞ;ð27Þ with r∈½0;∞Þand α∈½0;π=2. In this way, the highenergy limit corresponds to r→∞, whilst the angle α allows to control the overlapping singularities associated to l1and l2. Thus, the amplitude and the double UV counterterm can be generally written as Að2Þ H→gg ¼Z∞ 0 drr Zπ=2 0 dαZdΩfð2Þðr; α;ΩÞ; Að2Þ UV2¼Z∞ 0 drr Zπ=2 0 dαZdΩfð2Þ 12;UV2ðr; α;ΩÞ;ð28Þ where Ωdenotes all the angular variables describing the loop momenta. In Fig. 6, we present a graphical representation of the integrand functions fð2Þand fð2Þ 12;UV2for arbitrary fixed values of the angular variables and the helicity configuration þþ:In these plots, δAcorresponds to the integrands of Eq. (28), whilst δAREN ¼rðfð2Þ−fð2Þ 12;UV2Þ;ð29Þ i.e., the integrand after subtracting the UV counterterm. Also, we set mH¼125 GeV, mt¼175 GeV, and mZ¼91 GeV, together with μUV ¼mHas the renormalization scale. In Fig. 6(a), we appreciate that the amplitude and the counterterm exhibit the same behavior in the high-energy limit. Additionally, in Fig. 6(b), we notice that the asymptotic limit satisfies jrðfð2Þ−fð2Þ 12;UV2Þj ≤1=rνfor r→∞;ð30Þ with ν>1since the renormalized integrand (represented by the solid line) decreases faster than 1=r2(dashed line). This constitutes a numerical proof of the convergence of the integral in the UV region. Furthermore, the corresponding analytic expressions for the renormalized amplitude, before integration, are available from the authors upon request. VII. CONCLUSIONS AND FUTURE DIRECTIONS In this paper, we have presented an efficient numerical implementation of helicity amplitudes in the LTD representation. To achieve this, we exploited the prominent fact that LTD changes the loop integration domain from a Minkowski to an Euclidean space. This leads to important numerical simplifications in the integrand-level representation of scattering amplitudes and loop integrals due to the manifest cancellation of unphysical or noncausal (a) rf(2) rf12,UV2 (2) 10 100 1000 104105106107 8. 10 8 6. 10 8 4. 10 8 2. 10 8 0 2. 10 8 r A (b) r(f (2)-f 12,UV2 (2) ) 1/r1 1/r2 1/r3 10 100 1000 104105106107 10 19 10 17 10 15 10 13 10 11 10 9 10 7 r AREN FIG. 6. Integrand-level study of the local cancellation of UV singularities. (a) The UV counterterm (dotted) exactly reproduces the high-energy behavior of the amplitude (dashed). (b) The dotted line scales like 1=r (nonintegrable function); the dashed one corresponds to 1=r2(integrable function). The locally renormalized amplitude (solid line) decreases faster than 1=r2 for high energies. INTERPLAY BETWEEN THE LOOP-TREE DUALITY AND …PHYS. REV. D 105, 016012 (2022) 016012-9