Related papers: Differential equations on unitarity cut surfaces
In this paper we describe a method of calculation of master integrals based on the solution of systems of difference equations in one variable. Various explicit examples are given, as well as the generalization to arbitrary diagrams.
By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved…
The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method builds upon the formulation introduced in Bertalmio et al., J. Comput. Phys., 174 (2001),…
We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then,…
In this talk we discuss how ideas from the theory of mixed Hodge structures can be used to find differential equations for Feynman integrals. In particular we discuss the two-loop sunrise graph in two dimensions and show that these methods…
Scattering amplitudes at loop level can be reduced to a basis of linearly independent Feynman integrals. The integral coefficients are extracted from generalized unitarity cuts which define algebraic varieties. The topology of an algebraic…
We propose a novel method, called the dimension-changing transformation (DCT), to compute one-loop Feynman integrals and recently introduced fixed-branch integrals to arbitrary orders in $\epsilon$. The DCT relates one-loop Feynman…
Integration By Parts (IBP) is an important method for computing Feynman integrals. This work describes a formulation of the theory involving a set of differential equations in parameter space, and especially the definition and study of an…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
We propose a framework for calculating two-loop Feynman diagrams which appear within a renormalizable theory in the general mass case and at finite external momenta. Our approach is a combination of analytical results and of high accuracy…
We investigate the analytic structure of functions defined by integrals with integrands singular on a finite union of quadrics. The main motivation comes from Feynman integrals which belong to this class. Using isotopy techniques we derive…
We study a set of two-loop non-planar master integrals needed for the NNLO QCD corrections to diphoton and dijet production at hadron colliders. The top-sector topology contains an internal massive fermion loop and is known to contain…
Partial Differential Equation (PDE)-constrained optimization problems often take the form of an optimization of an objective function given as a sum of loss terms. Each function or gradient evaluation requires one or more PDE solves, which…
Singularities hidden in the collinear region around an external massless leg may lead to conformal symmetry breaking in otherwise conformally invariant finite loop momentum integrals. For an $\ell$-loop integral, this mechanism leads to a…
Evaluation of a wide variety of Feynman diagrams with multi-loop integrals and physical parameters and its comparison with high energy experiments are expected to investigate new physics beyond the Standard Model. We have been developing a…
Recently the Cachazo-He-Yuan (CHY) approach has been extended to loop level, but the resulting loop integrand has propagators that are linear in the loop momentum unlike Feynman's. In this note we present a new technique that directly…
In this paper, the reduction of Feynman integrals in the parametric representation is considered. This method proves to be more efficient than the integration-by-part (IBP) method in the momentum space. Tensor integrals can directly be…
Generalized-unitarity calculations of two-loop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of generalized cuts. In this paper, we present a…
We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…
We study the symbology of planar Feynman integrals in dimensional regularization by considering geometric configurations in momentum twistor space corresponding to their leading singularities (LS). Cutting propagators in momentum twistor…