# Building bases of loop integrands

@article{Bourjaily2020BuildingBO, title={Building bases of loop integrands}, author={Jacob L. Bourjaily and Enrico Herrmann and Cameron Langer and Jaroslav Trnka}, journal={Journal of High Energy Physics}, year={2020} }

Abstract
We describe a systematic approach to the construction of loop-integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of ‘power-counting’ for multi-loop integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In…

## 17 Citations

### Gravity loop integrands from the ultraviolet

- PhysicsSciPost Physics
- 2021

We demonstrate that loop integrands of
(super-)gravity scattering amplitudes possess surprising properties in the
ultraviolet (UV) region. In particular, we study the scaling of multi-particle…

### Prescriptive unitarity for non-planar six-particle amplitudes at two loops

- MathematicsJournal of High Energy Physics
- 2019

Abstract
We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, integrand-level representations of six-particle MHV scattering amplitudes…

### Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space

- MathematicsJournal of High Energy Physics
- 2020

It has recently been demonstrated that Feynman integrals relevant to a wide range of perturbative quantum field theories involve periods of Calabi-Yau manifolds of arbitrarily large dimension. While…

### Locally-finite quantities in sYM

- MathematicsJournal of High Energy Physics
- 2021

A locally-finite quantity is one for which there is no region of divergence anywhere in the space of real loop momenta; it can therefore be computed (in principle) without regularization. In this…

### All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops

- Physics, Education
- 2019

We give a closed-form, prescriptive representation of all-multiplicity two-loop MHV amplitude integrands in fully-color-dressed (non-planar) maximally supersymmetric Yang-Mills theory.

### Elliptic, Yangian-Invariant "Leading Singularity".

- MathematicsPhysical review letters
- 2021

We derive closed formulas for the first examples of nonalgebraic, elliptic "leading singularities" in planar, maximally supersymmetric Yang-Mills theory and show that they are Yangian invariant.

### Prescriptive unitarity with elliptic leading singularities

- PhysicsPhysical Review D
- 2021

Jacob L. Bourjaily, Nikhil Kalyanapuram, Cameron Langer, Kokkimidis Patatoukos aInstitute for Gravitation and the Cosmos, Department of Physics, Pennsylvania State University, University Park, PA…

### The Stratification of Rigidity

- Mathematics
- 2022

We show that a master integrand basis exists for all planar, two-loop amplitudes in massless four-dimensional theories which is fully stratiﬁed by rigidity—with each integrand being either pure and…

### Illustrations of integrand-basis building at two loops

- PhysicsJournal of High Energy Physics
- 2022

Abstract
We outline the concrete steps involved in building prescriptive master integrand bases for scattering amplitudes beyond the planar limit. We highlight the role of contour choices in such…

### Integrands of less-supersymmetric Yang-Mills at one loop

- MathematicsJournal of High Energy Physics
- 2022

Abstract
We construct a prescriptive, bubble power-counting basis of one-loop integrands suitable for representing amplitude integrands in less-supersymmetric (1 ≤$$ \mathcal{N} $$
N
≤ 4)…

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