Built on

G. ’t Hooft and M. Veltman, Scalar one-loop integrals
1979
Earlier work this paper cites.
J. Milnor, Hyperbolic geometry: the first 150 years
1982
Earlier work this paper cites.
A. Denner, U. Nierste, and R. Scharf, A compact expression for the scalar one loop four point function
1991
Earlier work this paper cites.
A. Davydychev and N. Usyukina, An approach to the evaluation of three and four point ladder diagrams
1993
Earlier work this paper cites.
D. Broadhurst, Summation of an infinite series of ladder diagrams
1993
Earlier work this paper cites.
Z. Bern, L. Dixon, D. Dunbar, and D. Kosower, One-loop n n -point gauge theory amplitudes, unitarity and collinear limits
1994
Earlier work this paper cites.
A. Davydychev and R. Delbourgo, A geometrical angle on Feynman integrals
1998
Earlier work this paper cites.
Y. Cho and H. Kim, On the volume formula for hyperbolic tetrahedra
1999
Earlier work this paper cites.
G. Duplancic and B. Nizic, Ir finite one-loop box scalar integral with massless internal lines
2002
Earlier work this paper cites.

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A. Isaev, Multi-loop Feynman integrals and conformal quantum mechanics
2003
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R. Britto, F. Cachazo, and B. Feng, Generalized unitarity and one loop amplitudes in 𝒩 = 4 \mathcal{N}=4 super Yang-Mills
2005
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J. Murakami and M. Yano, On the volume of a hyperbolic and spherical tetrahedron
Original
2005
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J. Drummond, J. Henn, V. Smirnov, and E. Sokatchev, Magic identities for conformal four-point integrals
2007
Cited alongside, same era.
Z. Bern, J. J. Carrasco, H. Johansson, and D. Kosower, Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
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2007
Cited alongside, same era.

Then

N. Berkovits and J. Maldacena, Fermionic T-duality, dual superconformal symmetry and the amplitude/Wilson loop connection
Original
2008
Later among the works it cites.
J. M. Drummond and J. M. Henn, All Tree-Level Amplitudes in 𝒩 = 4 \mathcal{N}=4 SYM
Original
2009
Later among the works it cites.
H. Elvang, D. Z. Freedman, and M. Kiermaier, Dual conformal symmetry of 1-loop NMHV amplitudes in 𝒩 = 4 \mathcal{N}=4 SYM theory
Original
2009
Later among the works it cites.
A. Brandhuber, P. Heslop, and G. Travaglini, Proof of the dual conformal anomaly of one-loop amplitudes in 𝒩 = 4 \mathcal{N}=4 SYM
Original
2009
Later among the works it cites.
J. Drummond, J. Henn, and J. Plefka, Yangian symmetry of scattering amplitudes in 𝒩 = 4 \mathcal{N}=4 super Yang-Mills

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2010

Amplitudes at Weak Coupling as Polytopes in AdS_5

Mason, Lionel, Skinner, David

Understand

We show that one-loop scalar box functions can be interpreted as volumes of geodesic tetrahedra embedded in a copy of AdS_5 that has dual conformal space-time as boundary.

When the tetrahedron is space-like, it lies in a totally geodesic hyperbolic three-space inside AdS_5, with its four vertices on the boundary.
It is a classical result that the volume of such a tetrahedron is given by the Bloch-Wigner dilogarithm and this agrees with the standard physics formulae for such box functions.
The combinations of box functions that arise in the n-particle one-loop MHV amplitude in N=4 super Yang-Mills correspond to the volume of a three-dimensional polytope without boundary, all of whose vertices are attached to a null polygon (which in other formulations is interpreted as a Wilson loop) at infinity.

Built on

G. ’t Hooft and M. Veltman, Scalar one-loop integrals
1979
Earlier work this paper cites.
J. Milnor, Hyperbolic geometry: the first 150 years
1982
Earlier work this paper cites.
A. Denner, U. Nierste, and R. Scharf, A compact expression for the scalar one loop four point function
1991
Earlier work this paper cites.
A. Davydychev and N. Usyukina, An approach to the evaluation of three and four point ladder diagrams
1993
Earlier work this paper cites.
D. Broadhurst, Summation of an infinite series of ladder diagrams
1993
Earlier work this paper cites.
Z. Bern, L. Dixon, D. Dunbar, and D. Kosower, One-loop n n -point gauge theory amplitudes, unitarity and collinear limits
1994
Earlier work this paper cites.
A. Davydychev and R. Delbourgo, A geometrical angle on Feynman integrals
1998
Earlier work this paper cites.
Y. Cho and H. Kim, On the volume formula for hyperbolic tetrahedra
1999
Earlier work this paper cites.
G. Duplancic and B. Nizic, Ir finite one-loop box scalar integral with massless internal lines
2002
Earlier work this paper cites.

Similar

A. Isaev, Multi-loop Feynman integrals and conformal quantum mechanics
2003
Cited alongside, same era.
R. Britto, F. Cachazo, and B. Feng, Generalized unitarity and one loop amplitudes in 𝒩 = 4 \mathcal{N}=4 super Yang-Mills
2005
Cited alongside, same era.
J. Murakami and M. Yano, On the volume of a hyperbolic and spherical tetrahedron
Original
2005
Cited alongside, same era.
J. Drummond, J. Henn, V. Smirnov, and E. Sokatchev, Magic identities for conformal four-point integrals
2007
Cited alongside, same era.
Z. Bern, J. J. Carrasco, H. Johansson, and D. Kosower, Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
Original
2007
Cited alongside, same era.