# Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case

@article{Abreu2017DiagrammaticHA, title={Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case}, author={Samuel Abreu and Ruth Britto and Claude Duhr and Einan Gardi}, journal={Journal of High Energy Physics}, year={2017}, volume={2017}, pages={1-74} }

A bstractWe construct a diagrammatic coaction acting on one-loop Feynman graphs and their cuts. The graphs are naturally identified with the corresponding (cut) Feynman integrals in dimensional regularization, whose coefficients of the Laurent expansion in the dimensional regulator are multiple polylogarithms (MPLs). Our main result is the conjecture that this diagrammatic coaction reproduces the combinatorics of the coaction on MPLs order by order in the Laurent expansion. We show that our…

## 48 Citations

### Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction.

- MathematicsPhysical review letters
- 2017

The algebraic and analytic structure of Feynman integrals is studied by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour, and it is demonstrated that it can be given a diagrammatic representation purely in terms of operations on graphs.

### Hopf algebra structure of the two loop three mass nonplanar Feynman diagram

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The method of using Hopf algebras for calculating Feynman integrals developed by Abreu et al. is applied to the two-loop non-planar on-shell diagram with massless propagators and three external mass…

### The diagrammatic coaction and cuts of the double box

- MathematicsSciPost Physics Proceedings
- 2022

The diagrammatic coaction encodes the analytic structure of Feynman integrals by mapping any given Feynman diagram into a tensor product of diagrams defined by contractions and cuts of the original…

### The diagrammatic coaction and the algebraic structure of cut Feynman integrals

- Mathematics
- 2018

We present a new formula for the coaction of a large class of integrals. When applied to one-loop (cut) Feynman integrals, it can be given a diagrammatic representation purely in terms of pinches and…

### Decomposition of Feynman integrals on the maximal cut by intersection numbers

- MathematicsProceedings of 14th International Symposium on Radiative Corrections — PoS(RADCOR2019)
- 2019

A bstractWe elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application…

### The diagrammatic coaction

- MathematicsProceedings of Loops and Legs in Quantum Field Theory — PoS(LL2022)
- 2022

The diagrammatic coaction underpins the analytic structure of Feynman integrals, their cuts and the diﬀerential equations they admit. The coaction maps any diagram into a tensor product of its…

### Coaction for Feynman integrals and diagrams

- MathematicsProceedings of Loops and Legs in Quantum Field Theory — PoS(LL2018)
- 2018

We propose a general coaction for families of integrals appearing in the evaluation of Feynman diagrams, such as multiple polylogarithms and generalized hypergeometric functions. We further…

### Motivic Galois coaction and one-loop Feynman graphs

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- 2019

Following the work of Brown, we can canonically associate a family of motivic periods -- called the motivic Feynman amplitude -- to any convergent Feynman integral, viewed as a function of the…

### Cluster algebras and the subalgebra constructibility of the seven-particle remainder function

- MathematicsJournal of High Energy Physics
- 2019

A bstractWe review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar N=4$$ \mathcal{N}=4 $$ supersymmetric Yang-Mills theory. In…

### Diagrammatic Coaction of Two-Loop Feynman Integrals

- MathematicsProceedings of 14th International Symposium on Radiative Corrections — PoS(RADCOR2019)
- 2019

It is known that one-loop Feynman integrals possess an algebraic structure encoding some of their analytic properties called the coaction, which can be written in terms of Feynman integrals and their…

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