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… 

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References

SHOWING 1-10 OF 110 REFERENCES

Algebraic Structure of Cut Feynman Integrals and the Diagrammatic Coaction.

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.

Cuts from residues: the one-loop case

A bstractUsing the multivariate residue calculus of Leray, we give a precise definition of the notion of a cut Feynman integral in dimensional regularization, as a residue evaluated on the variety

A Feynman integral via higher normal functions

We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two

Iterative structure of finite loop integrals

In this paper we develop further and refine the method of differential equations for computing Feynman integrals. In particular, we show that an additional iterative structure emerges for finite loop

From trees to loops and back

We argue that generic one-loop scattering amplitudes in supersymmetric Yang-Mills theories can be computed equivalently with MHV diagrams or with Feynman diagrams. We first present a general proof of

Local mirror symmetry and the sunset Feynman integral

We study the sunset Feynman integral defined as the scalar two-point self-energy at two-loop order in a two dimensional space-time. We firstly compute the Feynman integral, for arbitrary internal

Leading singularities and off-shell conformal integrals

A bstractThe three-loop four-point function of stress-tensor multiplets in $ \mathcal{N}=4 $ super Yang-Mills theory contains two so far unknown, off-shell, conformal integrals, in addition to the

The Massless Higher-Loop Two-Point Function

We introduce a new method for computing massless Feynman integrals analytically in parametric form. An analysis of the method yields a criterion for a primitive Feynman graph G to evaluate to
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