FIRE6: Feynman Integral REduction with modular arithmetic

107

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

Section: Continuity Constraintmentioning

confidence: 99%

Numerical Loop-Tree Duality: contour deformation and subtraction

Capatti

Hirschi

Kermanschah

et al. 2020

107

“…The two-point integrals will generically have numerators with tensor structures, but we can follow the strategy of [21] to reduce them to scalar integrals. Once that is accomplished we use LiteRed [57] and FIRE [58] to implement IBP identities and reduce to a basis of simpler master integrals, which were obtained in [59].…”

Section: Ope Analysismentioning

confidence: 99%

Non-planar data of $$ \mathcal{N} $$ = 4 SYM

Fleury

Pereira

2020

The four-point function of length-two half-BPS operators in N = 4 SYM receives nonplanar corrections starting at four loops. Previous work relied on the analysis of symmetries and logarithmic divergences to fix the integrand up to four constants. In this work, we compute those undetermined coefficients and fix the integrand completely by using the reformulation of N = 4 SYM in twistor space. The final integrand can be written as a combination of finite conformal integrals and we have used the method of asymptotic expansions to extract non-planar anomalous dimensions and structure constants for twist-two operators up to spin eight. Some of the results were already know in the literature and we have found agreement with them.

“…Combining the master integrals into complete amplitudes requires the solution of increasingly complicated linear systems of IBP equations. Considerable effort has led to a variety of efficient solutions [3,[35][36][37][38] and public implementations [39][40][41][42][43]. Applications to five-particle problems have been possible though yielded large IBP reduction tables [44,45].…”

Section: Introductionmentioning

confidence: 99%

A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons

Hartanto

Badger

Brønnum-Hansen

et al. 2019

We present the first numerical results for the two-loop helicity amplitudes for the scattering of four partons and a W -boson in QCD. We use a finite field sampling method to reduce directly from Feynman diagrams to the coefficients of a set of master integrals after applying integration-by-parts identities. Since the basis of master integrals is not yet fully known analytically, we identify a set of master integrals with a simple divergence structure using local numerator insertions. This allows for accurate numerical evaluation of the amplitude using sector decomposition methods.