Built on

J. L. Bourjaily, F. Dulat, and E. Panzer, “Manifestly Dual-Conformal Loop Integration,” arXiv:1901.02887 [hep-th]
Original
1901
Earlier work this paper cites.
J. Golden, A. J. McLeod, M. Spradlin, and A. Volovich, “The Sklyanin Bracket and Cluster Adjacency at All Multiplicity,” JHEP
Original
1902
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. von Hippel, A. J. McLeod, and G. Papathanasiou, “Six-Gluon Amplitudes in Planar 𝒩 = 4 \mathcal{N}\!=\!4 super-Yang-Mills Theory at Six and Seven Loops,” JHEP
Original
1903
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. Von Hippel, A. J. McLeod, and G. Papathanasiou, “The Cosmic Galois Group and Extended Steinmann Relations for Planar 𝒩 = 4 \mathcal{N}=4 SYM Amplitudes,” arXiv:1906.07116 [hep-th]
Original
1906
Earlier work this paper cites.
J. Mago, A. Schreiber, M. Spradlin, and A. Volovich, “Yangian Invariants and Cluster Adjacency in 𝒩 = 4 \mathcal{N}\!=\!4 Yang-Mills,” JHEP
Original
1906
Earlier work this paper cites.
J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante, and L. Tancredi, “An Analytic Solution for the Equal-Mass Banana Graph,” arXiv:1907.03787 [hep-th]
Original
1907
Earlier work this paper cites.
D. Rudenko, “Rational Elliptic Surfaces and the Trigonometry of Tetrahedra,” arXiv:1908.01141 [math.AG]
Original
1908
Earlier work this paper cites.
M. Besier, D. Festi, M. Harrison, and B. Naskrecki, “Arithmetic and Geometry of a K3 Surface Emerging from Virtual Corrections to Drell-Yan Scattering,” arXiv:1908.01079 [math.AG]
Original
1908
Earlier work this paper cites.
E. Herrmann and J. Parra-Martinez, “Logarithmic Forms and Differential Equations for Feynman Integrals,” arXiv:1909.04777 [hep-th]
Original
1909
Earlier work this paper cites.
J. L. Bourjaily, A. J. McLeod, C. Vergu, M. Volk, M. Von Hippel, and M. Wilhelm, “Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory,” arXiv:1910.14224 [hep-th]
Original
1910
Earlier work this paper cites.
J. L. Bourjaily, A. J. McLeod, C. Vergu, M. Volk, M. Von Hippel, and M. Wilhelm, “Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space,” arXiv:1910.01534 [hep-th]
Original
1910
Earlier work this paper cites.
J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod, and J. Trnka, “All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops,” arXiv:1911.09106 [hep-th]
Original
1911
Earlier work this paper cites.
F. Loebbert, D. Müller, and H. Münkler, “Yangian Bootstrap for Conformal Feynman Integrals,” arXiv:1912.05561 [hep-th]
Original
1912
Earlier work this paper cites.
N. Arkani-Hamed, T. Lam, and M. Spradlin, “Non-Perturbative Geometries for Planar 𝒩 = 4 \mathcal{N}\!=\!4 SYM Amplitudes,” arXiv:1912.08222 [hep-th]
Original
1912
Earlier work this paper cites.
P. A. M. Dirac, “Wave Equations in Conformal Space,” Annals Math
1936
Earlier work this paper cites.
O. Steinmann, “Über den Zusammenhang Zwischen den Wightmanfunktionen und der Retardierten Kommutatoren,” Helv. Physica Acta
1960
Earlier work this paper cites.
O. Steinmann, “Wightman-Funktionen und Retardierte Kommutatoren. II,” Helv. Physica Acta
1960
Earlier work this paper cites.
A. C.-T. Wu, “On the Analytic Properties of the 4-Point Function in Perturbation Theory,” Matematisk-fysiske Meddelelser udgivet af Det Kongelige Dnaske Videnskabernes Selskab
1961
Earlier work this paper cites.
G. Mack and A. Salam, “Finite Component Field Representations of the Conformal Group,” Annals Phys
1969
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, and R. Gatto, “Tensor Representations of Conformal Algebra and Conformally Covariant Operator Product Expansion,” Annals Phys
1973
Earlier work this paper cites.
S. Ferrara, “Supergauge Transformations on the Six-Dimensional Hypercone,” Nucl. Phys
1974
Earlier work this paper cites.
K. E. Cahill and H. P. Stapp, “Optical Theorems and Steinmann Relations,” Annals Phys
1975
Earlier work this paper cites.
K. Aomoto, “Analytic Structure of Schläfli Function,” Nagoya Math J
1977
Earlier work this paper cites.
B. G. Nickel, “Evaluation of Simple Feynman Graphs,” Journal of Mathematical Physics
1978
Earlier work this paper cites.
G. ’t Hooft and M. J. G. Veltman, “Scalar One Loop Integrals,” Nucl. Phys
1979
Earlier work this paper cites.
A. Denner, U. Nierste, and R. Scharf, “A Compact Expression for the Scalar One-Loop Four-Point Function,” Nucl. Phys
1991
Earlier work this paper cites.
R. Kellerhals, “On Schläfli’s Reduction Formula,” Math. Z
1991
Earlier work this paper cites.
K. Aomoto, “Hyperlogarithmic Expansion and the Volume of a Hyperbolic Simplex,” Banach Center Publications
1992
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and D. A. Kosower, “Dimensionally Regulated One Loop Integrals,” Phys. Lett
1993
Earlier work this paper cites.

Similar

E. B. Vinberg, “Volumes of non-Euclidean Polyhedra,” Uspekhi Mat. Nauk
1993
Cited alongside, same era.
G. J. Heckman, “The Volume of Hyperbolic Coxeter Polytopes of Even Dimension,” Indag. Math. (N.S.)
1995
Cited alongside, same era.
O. V. Tarasov, “Connection between Feynman Integrals having Different Values of the Space-Time Dimension,” Phys. Rev
1996
Cited alongside, same era.
A. I. Davydychev and R. Delbourgo, “A Geometrical Angle on Feynman Integrals,” J. Math. Phys
1998
Cited alongside, same era.
A. Goncharov, “Volumes of Hyperbolic Manifolds and Mixed Tate Motives,” J. Amer. Math. Soc
1999
Cited alongside, same era.
J. Murakami and M. Yano, “On the Volume of a Hyperbolic and Spherical Tetrahedron,” Comm. Anal. Geom

Then

J. Murakami, “Volume Formulas for a Spherical Tetrahedron,” Proc. Amer. Math. Soc
2012
Later among the works it cites.
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, and J. Trnka, “Local Integrals for Planar Scattering Amplitudes,” JHEP
Original
2012
Later among the works it cites.
A. Hodges, “The Box Integrals in Momentum-Twistor Geometry,” JHEP
Original
2013
Later among the works it cites.
D. Nandan, M. F. Paulos, M. Spradlin, and A. Volovich, “Star Integrals, Convolutions and Simplices,” JHEP
Original
2013
Later among the works it cites.
L. J. Dixon, J. M. Drummond, M. von Hippel, and J. Pennington, “Hexagon Functions and the Three-Loop Remainder Function,” JHEP
Original
2013

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Reading the bibliography…

2019

All-Mass $n$-gon Integrals in $n$ Dimensions

Bourjaily, Jacob L., Gardi, Einan, McLeod, Andrew J. et al.

Understand

We explore the correspondence between one-loop Feynman integrals and (hyperbolic) simplicial geometry to describe the "all-mass" case: integrals with generic external and internal masses.

Specifically, we focus on $n$-particle integrals in exactly $n$ space-time dimensions, as these integrals have particularly nice geometric properties and respect a dual conformal symmetry.
In four dimensions, we leverage this geometric connection to give a concise dilogarithmic expression for the all-mass box in terms of the Murakami-Yano formula.
In five dimensions, we use a generalized Gauss-Bonnet theorem to derive a similar dilogarithmic expression for the all-mass pentagon.

Built on

J. L. Bourjaily, F. Dulat, and E. Panzer, “Manifestly Dual-Conformal Loop Integration,” arXiv:1901.02887 [hep-th]
Original
1901
Earlier work this paper cites.
J. Golden, A. J. McLeod, M. Spradlin, and A. Volovich, “The Sklyanin Bracket and Cluster Adjacency at All Multiplicity,” JHEP
Original
1902
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. von Hippel, A. J. McLeod, and G. Papathanasiou, “Six-Gluon Amplitudes in Planar 𝒩 = 4 \mathcal{N}\!=\!4 super-Yang-Mills Theory at Six and Seven Loops,” JHEP
Original
1903
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. Von Hippel, A. J. McLeod, and G. Papathanasiou, “The Cosmic Galois Group and Extended Steinmann Relations for Planar 𝒩 = 4 \mathcal{N}=4 SYM Amplitudes,” arXiv:1906.07116 [hep-th]
Original
1906
Earlier work this paper cites.
J. Mago, A. Schreiber, M. Spradlin, and A. Volovich, “Yangian Invariants and Cluster Adjacency in 𝒩 = 4 \mathcal{N}\!=\!4 Yang-Mills,” JHEP
Original
1906
Earlier work this paper cites.
J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante, and L. Tancredi, “An Analytic Solution for the Equal-Mass Banana Graph,” arXiv:1907.03787 [hep-th]
Original
1907
Earlier work this paper cites.
D. Rudenko, “Rational Elliptic Surfaces and the Trigonometry of Tetrahedra,” arXiv:1908.01141 [math.AG]
Original
1908
Earlier work this paper cites.
M. Besier, D. Festi, M. Harrison, and B. Naskrecki, “Arithmetic and Geometry of a K3 Surface Emerging from Virtual Corrections to Drell-Yan Scattering,” arXiv:1908.01079 [math.AG]
Original
1908
Earlier work this paper cites.
E. Herrmann and J. Parra-Martinez, “Logarithmic Forms and Differential Equations for Feynman Integrals,” arXiv:1909.04777 [hep-th]
Original
1909
Earlier work this paper cites.
J. L. Bourjaily, A. J. McLeod, C. Vergu, M. Volk, M. Von Hippel, and M. Wilhelm, “Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory,” arXiv:1910.14224 [hep-th]
Original
1910
Earlier work this paper cites.
J. L. Bourjaily, A. J. McLeod, C. Vergu, M. Volk, M. Von Hippel, and M. Wilhelm, “Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space,” arXiv:1910.01534 [hep-th]
Original
1910
Earlier work this paper cites.
J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod, and J. Trnka, “All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops,” arXiv:1911.09106 [hep-th]
Original
1911
Earlier work this paper cites.
F. Loebbert, D. Müller, and H. Münkler, “Yangian Bootstrap for Conformal Feynman Integrals,” arXiv:1912.05561 [hep-th]
Original
1912
Earlier work this paper cites.
N. Arkani-Hamed, T. Lam, and M. Spradlin, “Non-Perturbative Geometries for Planar 𝒩 = 4 \mathcal{N}\!=\!4 SYM Amplitudes,” arXiv:1912.08222 [hep-th]
Original
1912
Earlier work this paper cites.
P. A. M. Dirac, “Wave Equations in Conformal Space,” Annals Math
1936
Earlier work this paper cites.
O. Steinmann, “Über den Zusammenhang Zwischen den Wightmanfunktionen und der Retardierten Kommutatoren,” Helv. Physica Acta
1960
Earlier work this paper cites.
O. Steinmann, “Wightman-Funktionen und Retardierte Kommutatoren. II,” Helv. Physica Acta
1960
Earlier work this paper cites.
A. C.-T. Wu, “On the Analytic Properties of the 4-Point Function in Perturbation Theory,” Matematisk-fysiske Meddelelser udgivet af Det Kongelige Dnaske Videnskabernes Selskab
1961
Earlier work this paper cites.
G. Mack and A. Salam, “Finite Component Field Representations of the Conformal Group,” Annals Phys
1969
Earlier work this paper cites.
S. Ferrara, A. F. Grillo, and R. Gatto, “Tensor Representations of Conformal Algebra and Conformally Covariant Operator Product Expansion,” Annals Phys
1973
Earlier work this paper cites.
S. Ferrara, “Supergauge Transformations on the Six-Dimensional Hypercone,” Nucl. Phys
1974
Earlier work this paper cites.
K. E. Cahill and H. P. Stapp, “Optical Theorems and Steinmann Relations,” Annals Phys
1975
Earlier work this paper cites.
K. Aomoto, “Analytic Structure of Schläfli Function,” Nagoya Math J
1977
Earlier work this paper cites.
B. G. Nickel, “Evaluation of Simple Feynman Graphs,” Journal of Mathematical Physics
1978
Earlier work this paper cites.
G. ’t Hooft and M. J. G. Veltman, “Scalar One Loop Integrals,” Nucl. Phys
1979
Earlier work this paper cites.
A. Denner, U. Nierste, and R. Scharf, “A Compact Expression for the Scalar One-Loop Four-Point Function,” Nucl. Phys
1991
Earlier work this paper cites.
R. Kellerhals, “On Schläfli’s Reduction Formula,” Math. Z
1991
Earlier work this paper cites.
K. Aomoto, “Hyperlogarithmic Expansion and the Volume of a Hyperbolic Simplex,” Banach Center Publications
1992
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and D. A. Kosower, “Dimensionally Regulated One Loop Integrals,” Phys. Lett
1993
Earlier work this paper cites.

Similar

E. B. Vinberg, “Volumes of non-Euclidean Polyhedra,” Uspekhi Mat. Nauk
1993
Cited alongside, same era.
G. J. Heckman, “The Volume of Hyperbolic Coxeter Polytopes of Even Dimension,” Indag. Math. (N.S.)
1995
Cited alongside, same era.
O. V. Tarasov, “Connection between Feynman Integrals having Different Values of the Space-Time Dimension,” Phys. Rev
1996
Cited alongside, same era.
A. I. Davydychev and R. Delbourgo, “A Geometrical Angle on Feynman Integrals,” J. Math. Phys
1998
Cited alongside, same era.
A. Goncharov, “Volumes of Hyperbolic Manifolds and Mixed Tate Motives,” J. Amer. Math. Soc
1999
Cited alongside, same era.
J. Murakami and M. Yano, “On the Volume of a Hyperbolic and Spherical Tetrahedron,” Comm. Anal. Geom

Then

J. Murakami, “Volume Formulas for a Spherical Tetrahedron,” Proc. Amer. Math. Soc
2012
Later among the works it cites.
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, and J. Trnka, “Local Integrals for Planar Scattering Amplitudes,” JHEP
Original
2012
Later among the works it cites.
A. Hodges, “The Box Integrals in Momentum-Twistor Geometry,” JHEP
Original
2013
Later among the works it cites.
D. Nandan, M. F. Paulos, M. Spradlin, and A. Volovich, “Star Integrals, Convolutions and Simplices,” JHEP
Original
2013
Later among the works it cites.
L. J. Dixon, J. M. Drummond, M. von Hippel, and J. Pennington, “Hexagon Functions and the Three-Loop Remainder Function,” JHEP
Original
2013

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

2005

Cited alongside, same era.

J. Drummond, J. Henn, V. Smirnov, and E. Sokatchev, “Magic Identities for Conformal Four-Point Integrals,” JHEP

2007

Cited alongside, same era.

Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, “The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory,” Phys. Rev

2007

Cited alongside, same era.

Z. Bern, J. Carrasco, H. Johansson, and D. Kosower, “Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops,” Phys. Rev