This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Collier: A fortran-based complex one-loop library in extended regularizations

ZZ production at hadron colliders in NNLO QCD

Kira—A Feynman integral reduction program

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space–time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE.

https://iopscience.iop.org/article/10.1088/1751-8113/48/15/153001/pdf

Lectures on differential equations for Feynman integrals

We compute the full set of two-loop Feynman integrals appearing in massless two-loop four-point functions with two off-shell legs with the same invariant mass. These integrals allow to determine the two-loop corrections to the amplitudes for vector boson pair production at hadron colliders, $ q\overline{q} $
 → VV , and thus to compute this process to next-to- next-to-leading order accuracy in QCD. The master integrals are derived using the method of differential equations, employing a canonical basis for the integrals. We obtain analytical results for all integrals, expressed in terms of multiple polylogarithms. We optimize our results for numerical evaluation by employing functions which are real valued for physical scattering kinematics and allow for an immediate power series expansion.

/pdf/the-two-loop-master-integrals-for-q-overline-q-vv-4684xppv61.pdf

The two-loop master integrals for $ q\overline{q} $ → VV

We compute the full set of two-loop Feynman integrals appearing in massless two-loop four-point functions with two off-shell legs with the same invariant mass These integrals allow to determine the two-loop corrections to the amplitudes for vector boson pair production at hadron colliders, $q \bar{q} \to V V$, and thus to compute this process to next-to-next-to-leading order accuracy in QCD The master integrals are derived using the method of differential equations, employing a canonical basis for the integrals We obtain analytical results for all integrals, expressed in terms of multiple polylogarithms We optimize our results for numerical evaluation by employing functions which are real valued for physical scattering kinematics and allow for an immediate power series expansion

/pdf/the-two-loop-master-integrals-for-q-bar-q-to-v-v-kmmqzl5f1i.pdf

The Two-Loop Master Integrals for $q \bar{q} \to V V$

The two-loop QCD corrections to vector boson pair production at hadron colliders involve a new class of Feynman integrals: two-loop four-point functions with two off-shell external legs. We describe their reduction to a small set of master integrals by solving linear relations among them. We then use differential equations in the external invariants to compute all master integrals that are relevant to planar Feynman amplitudes. Our results are expressed analytically in terms of generalised harmonic polylogarithms. The calculation relies heavily on techniques that exploit the algebraic structure of these functions, which we describe in detail.

/pdf/two-loop-master-integrals-for-q-bar-q-to-v-v-the-planar-4o8s74gz3e.pdf

Two-Loop Master Integrals for $q \bar{q} \to V V$: the Planar Topologies

The two-loop QCD corrections to vector boson pair production at hadron colliders involve a new class of Feynman integrals: two-loop four-point functions with two off-shell external legs. We describe their reduction to a small set of master integrals by solving linear relations among them. We then use differential equations in the external invariants to compute all master integrals that are relevant to planar Feynman amplitudes. Our results are expressed analytically in terms of generalized harmonic polylogarithms. The calculation relies heavily on techniques that exploit the algebraic structure of these functions, which we describe in detail.

/pdf/two-loop-master-integrals-for-q-overline-q-to-vv-the-planar-1r2icgjvvp.pdf

Erich Weihs

Papers

ZZ production at hadron colliders in NNLO QCD

The two-loop master integrals for \( q\overline{q} \) → VV

The Two-Loop Master Integrals for $q \bar{q} \to V V$

Two-Loop Master Integrals for $q \bar{q} \to V V$: the Planar Topologies

Two-loop master integrals for \( q\overline{q}\to VV \): the planar topologies