We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion. arXiv:1912.09291v1 [hep-ph] 19 Dec 2019 7 Results 53 7.1 Multi-loop finite integrals 54 7.2 Divergent one-loop four-and five-point scalar integrals 69 7.3 One-loop amplitude for qq → γ 1 γ 2 and qq → γ 1 γ 2 γ 3 69 -i -8 Conclusion 74 9 Acknowledgements 75 A Loop-Tree Duality example at two loops 76 B Expression for the qq → γ 1 γ 2 γ 3 amplitude and its counterterms 79 C Loop-Tree Duality with raised propagators 80 1 As discussed in ref.[83], our final expression in eq. (2.5) is also correct in the case of complex-valued external momenta, due to the fact that the right-most column of the matrix appearing in eq. (2.4) does not include the imaginary part Im[p 0 i ] of the external momenta. We note however, that the correct interpretation of the absence of this term in eq. (2.4) for complex-valued external kinematics is that the energy integrals are no longer performed along the real line but instead along a path including only one out of the two complex energy solutions of each propagator.3)The quantity ∇ k η( k), henceforth denoted as ∇η, is the outward pointing normal vector to the surface η( k) = 0. The contour deformation is defined in the (3n)-dimensional complex space and we parametrise it as k − i κ( k). It must satisfy constraints affecting two of its key characteristics, the direction and magnitude of the vector field κ( k):Direction: The deformation vector κ( k) must induce a sign of the imaginary part of the E-surface equation that matches the sign enforced by the causal prescription whenever k lies on a singular E-surfaces. This imposes conditions on the direction of the vector field κ( k). We derive these conditions by comparing the sign of the LTD prescription

Loop Tree Duality (LTD) offers a promising avenue to numerically integrate multi-loop integrals directly in momentum space. It is well-established at one loop, but there have been only sparse numerical results at two loops. We provide a formal derivation for a novel multi-loop LTD expression and study its threshold singularity structure. We apply our findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed. We also lay down the ground work for constructing such a deformation. Our results serve as an important stepping stone towards a generalised and efficient numerical implementation of LTD, applicable to the computation of virtual corrections.

Loop-Tree Duality (LTD) is a framework in which the energy components of all loop momenta of a Feynman integral are integrated out using residue theorem, resulting in a sum over tree-like structures. Originally, the LTD expression exhibits cancellations of non-causal thresholds between summands, also known as dual cancellations. As a result, the expression exhibits numerical instabilities in the vicinity of non-causal thresholds and for large loop momenta. In this work we derive a novel, generically applicable, Manifestly Causal LTD (cLTD) representation whose only thresholds are causal thresholds, i.e. it manifestly realizes dual cancellations. Consequently, this result also serves as a general proof for dual cancellations. We show that LTD, cLTD, and the expression stemming from Time Ordered Perturbation Theory (TOPT) are locally equivalent. TOPT and cLTD both feature only causal threshold singularities, however cLTD features better scaling with the number of propagators. On top of the new theoretical perspectives offered by our representation, it has the useful property that the ultraviolet (UV) behaviour of the original 4D integrand is maintained for every summand. We show that the resulting cLTD integrand expression is completely stable in the UV region which is key for practical applications of LTD to the computation of amplitudes and cross sections. We present explicit examples of the cLTD expression for a variety of up to four-loop integrals and show that its increased computational complexity can be efficiently mitigated by optimising its numerical implementation. Finally, we provide a computer code that automatically generates the cLTD expression for an arbitrary topology.

We propose a new approach that allows for the separate numerical calculation of the real and imaginary parts of finite loop integrals. We find that at one-loop the real part is given by the Loop-Tree Duality integral supplemented with suitable counterterms and the imaginary part is a sum of two-body phase space integrals, constituting a locally finite representation of the generalised optical theorem. These expressions are integrals in momentum space, whose integrands were specially designed to feature local cancellations of threshold singularities. Such a representation is well suited for Monte Carlo integration and avoids the drawbacks of a numerical contour deformation around remaining singularities. Our method is directly applicable to a range integrals with certain geometric properties but not yet fully generalised for arbitrary one-loop integrals. We demonstrate the computational performance with examples of one-loop integrals with various kinematic configurations, which gives promising prospects for an extension to multi-loop integrals.