# Local integrals for planar scattering amplitudes

@article{ArkaniHamed2012LocalIF, title={Local integrals for planar scattering amplitudes}, author={Nima Arkani-Hamed and Jacob L. Bourjaily and Freddy Cachazo and Jaroslav Trnka}, journal={Journal of High Energy Physics}, year={2012}, volume={2012}, pages={1-70} }

A bstractRecently, an explicit, recursive formula for the all-loop integrand of planar scattering amplitudes in $ \mathcal{N} = {4} $ SYM has been described, generalizing the BCFW formula for tree amplitudes, and making manifest the Yangian symmetry of the theory. This has made it possible to easily study the structure of multi-loop amplitudes in the theory. In this paper we describe a remarkable fact revealed by these investigations: the integrand can be expressed in an amazingly simple and…

## 252 Citations

### Iterative structure of finite loop integrals

- Mathematics
- 2014

A bstractIn 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…

### Scattering Amplitudes and the Positive Grassmannian

- Mathematics
- 2012

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical…

### Iterative structure of finite loop integrals

- Mathematics
- 2014

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…

### Leading singularities and off-shell conformal integrals

- Mathematics
- 2013

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…

### Uniqueness of two-loop master contours

- Mathematics
- 2012

A bstractGeneralized-unitarity calculations of two-loop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of…

### Scattering amplitudes — Wilson loops duality for the first non-planar correction

- MathematicsJournal of High Energy Physics
- 2018

A bstractWe study the first non-planar correction to gluon scattering amplitudes in N=4$$ \mathcal{N}=4 $$ SYM theory. The correction takes the form of a double trace partial amplitude and is…

### Two-loop integrals for planar five-point one-mass processes

- Mathematics
- 2020

We present the computation of a full set of planar five-point two-loop master integrals with one external mass. These integrals are an important ingredient for two-loop scattering amplitudes for…

### The double pentaladder integral to all orders

- MathematicsJournal of High Energy Physics
- 2018

A bstractWe compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N$$ \mathcal{N} $$…

### Comments on all-loop constraints for scattering amplitudes and Feynman integrals

- MathematicsJournal of High Energy Physics
- 2022

We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar N $$ \mathcal{N} $$ =…

### Generalised Unitarity, Integrand Decomposition, and Hidden properties of QCD Scattering Amplitudes in Dimensional Regularisation

- Physics
- 2017

In this thesis, we present new developments for the analytic calculation of tree- and multi-loop level amplitudes. Similarly, we study and extend their analytic properties.
We propose a…

## References

SHOWING 1-10 OF 88 REFERENCES

### The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM

- Mathematics
- 2010

We give an explicit recursive formula for the all ℓ-loop integrand for scattering amplitudes in $ \mathcal{N} = 4 $ SYM in the planar limit, manifesting the full Yangian symmetry of the theory. This…

### Notes on the scattering amplitude — Wilson loop duality

- Mathematics
- 2011

We consider the duality between the four-dimensional S-matrix of planar maximally supersymmetric Yang-Mills theory and the expectation value of polygonal shaped Wilson loops in the same theory. We…

### The complete planar S-matrix of $ \mathcal{N} = 4 $ SYM as a Wilson loop in twistor space

- Physics
- 2010

We show that the complete planar S-matrix of $ \mathcal{N} = 4 $ super Yang-Mills — including all NkMHV partial amplitudes to all loops — is equivalent to the correlation function of a supersymmetric…

### The Grassmannian origin of dual superconformal invariance

- Mathematics, Physics
- 2010

A dual formulation of the S Matrix for $$ \mathcal {N} $$ = 4 SYM has recently been presented, where all leading singularities of n-particle Nk−2MHV amplitudes are given as an integral over the…

### Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

- Physics
- 2005

We compute the leading-color (planar) three-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4 - 2{epsilon} dimensions, as a Laurent expansion about {epsilon} = 0 including the…

### Scattering into the fifth dimension of $$ \mathcal{N} $$ = 4 super Yang-Mills

- Physics
- 2009

We study an alternative to dimensional regularisation of planar scattering amplitudes in $$ \mathcal{N} $$ = 4 super Yang-Mills theory by going to the Coulomb phase of the theory. The infrared…

### What is the simplest quantum field theory?

- Mathematics
- 2010

Conventional wisdom says that the simpler the Lagrangian of a theory the simpler its perturbation theory. An ever-increasing understanding of the structure of scattering amplitudes has however been…

### An operator product expansion for polygonal null Wilson loops

- Mathematics
- 2010

We consider polygonal Wilson loops with null edges in conformal gauge theories. We derive an OPE-like expansion when several successive lines of the polygon are becoming aligned. The limit…

### Amplitudes at weak coupling as polytopes in AdS5

- Mathematics
- 2011

We show that one-loop scalar box functions can be interpreted as volumes of geodesic tetrahedra embedded in a copy of AdS5 that has dual conformal spacetime as boundary. When the tetrahedron is…

### Eliminating spurious poles from gauge-theoretic amplitudes

- Physics
- 2009

A bstractThis note addresses the problem of spurious poles in gauge-theoretic scattering amplitudes. New twistor coordinates for the momenta are introduced, based on the concept of dual conformal…