arXiv subjects

Alessandro Georgoudis

Publications and source records attributed to Alessandro Georgoudis.

26 records · Page 2Linked to original sources

arXiv2019

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficiently automatized and automatically parallelized by modeling the algorithm in GPI-Space using the language of Petri-nets. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.

hep-th↗

arXiv2018

Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

hep-th↗

arXiv2018

Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections

We present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of $73$ master integrals.

hep-th↗

arXiv2018

Five-loop massless propagator integrals

We develop a method to obtain $ε$-expansions of massless two-point integrals in position space, based on the constraints implied by symmetries of the asymptotic expansion of conformal four-point integrals. Together with parametric integration, we are able to fix the expansions of 170 genuine five-loop master integrals. In particular, we computed the expansions of all planar master integrals up to transcendental weight 9.

hep-th↗

arXiv2017

Cristal and Azurite: new tools for integration-by-parts reductions

Scattering amplitudes computed at a fixed loop order, along with any other object computed in perturbative quantum field theory, can be expressed as a linear combination of a finite basis of loop integrals. To compute loop amplitudes in practice, such a basis of integrals must be determined. We discuss Azurite (A ZURich-bred method for finding master InTEgrals), a publicly available package for finding bases of loop integrals. We also discuss Cristal (Complete Reduction of IntegralS Through All Loops), a future package that produces the complete integration-by-parts reductions.

hep-ph↗

arXiv2017

Konishi OPE coefficient at the five loop order

We use the method of asymptotic expansions to study the OPE limit of a four-point function of protected operators in N=4 SYM. We use a new method for evaluating the resulting propagator-type integrals and then extract the OPE coefficient with Konishi at the five loop order.

hep-th↗

arXiv2017

Azurite: An algebraic geometry based package for finding bases of loop integrals

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package {\sc Azurite} ({\bf A ZUR}ich-bred method for finding master {\bf I}n{\bf TE}grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems {\sc Singular} and {\sc Mathematica}. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts.

hep-th↗

arXiv2015

Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.

hep-th↗