2013

Star Integrals, Convolutions and Simplices

Nandan, Dhritiman, Paulos, Miguel F., Spradlin, Marcus et al.

Understand

We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space.

Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop $n$-gon integrals in $n$ dimensions.
These are known to be given by volumes of hyperbolic simplices.
We explicitly compute the five-dimensional pentagon integral in full generality using Schl\"afli's formula.

Built on

P. A. M. Dirac, “Wave equations in conformal space,” Annals Math. 37
1936
Earlier work this paper cites.
L. Brink, J. H. Schwarz and J. Scherk, “Supersymmetric Yang-Mills Theories,” Nucl. Phys. B 121
1977
Earlier work this paper cites.
F. Gliozzi, J. Scherk and D. I. Olive, “Supersymmetry, Supergravity Theories and the Dual Spinor Model,” Nucl. Phys. B 122
1977
Earlier work this paper cites.
A. I. Davydychev, “Some exact results for N point massive Feynman integrals,” J. Math. Phys. 32
1991
Earlier work this paper cites.
Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “One loop n point gauge theory amplitudes, unitarity and collinear limits,” Nucl. Phys. B 425
1994
Earlier work this paper cites.
A. B. Goncharov, “Polylogarithms and Motivic Galois groups,” Proc. Symp. Pure Math. 55
1994
Earlier work this paper cites.
Z. Bern, L. J. Dixon and D. A. Kosower, “Dimensionally regulated pentagon integrals,” Nucl. Phys. B 412
1994
Earlier work this paper cites.
Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, “Fusing gauge theory tree amplitudes into loop amplitudes,” Nucl. Phys. B 435
1995
Earlier work this paper cites.
A. B. Goncharov, “Multiple polylogarithms, cyclotomy and modular complexes,” Math. Res. Lett. 5
1998
Earlier work this paper cites.
A. I. Davydychev and R. Delbourgo, “A Geometrical angle on Feynman integrals,” J. Math. Phys. 39
1998
Earlier work this paper cites.
R. Britto, F. Cachazo and B. Feng, “New recursion relations for tree amplitudes of gluons,” Nucl. Phys. B 715
2005
Earlier work this paper cites.
R. Britto, F. Cachazo, B. Feng and E. Witten, “Direct proof of tree-level recursion relation in Yang-Mills theory,” Phys. Rev. Lett. 94
2005
Earlier work this paper cites.
A. B. Goncharov, “Galois symmetries of fundamental groupoids and noncommutative geometry,” Duke Math J. 128
2005
Earlier work this paper cites.
S. Laporta and E. Remiddi, “Analytic treatment of the two loop equal mass sunrise graph,” Nucl. Phys. B 704
2005
Earlier work this paper cites.
J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev, “Magic identities for conformal four-point integrals,” JHEP 0701
2007
Earlier work this paper cites.
Z. Bern, J. J. M. Carrasco, H. Johansson and D. A. Kosower, “Maximally supersymmetric planar Yang-Mills amplitudes at five loops,” Phys. Rev. D 76
Original
2007
Earlier work this paper cites.
L. F. Alday and J. M. Maldacena, “Gluon scattering amplitudes at strong coupling,” JHEP 0706
Original
2007
Earlier work this paper cites.
A. Brandhuber, P. Heslop and G. Travaglini, “MHV amplitudes in N=4 super Yang-Mills and Wilson loops,” Nucl. Phys. B 794
Original
2008
Earlier work this paper cites.
Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, M. Spradlin, C. Vergu and A. Volovich, “The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory,” Phys. Rev. D 78
Original
2008
Earlier work this paper cites.
J. M. Drummond, J. M. Henn and J. Plefka, “Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory,” JHEP 0905
Original
2009
Earlier work this paper cites.
J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, “Hexagon Wilson loop = six-gluon MHV amplitude,” Nucl. Phys. B 815
Original
2009
Earlier work this paper cites.
G. Mack, “D-dimensional Conformal Field Theories with anomalous dimensions as Dual Resonance Models,” Bulg. J. Phys. 36
Original
2009
Earlier work this paper cites.
J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, “Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory,” Nucl. Phys. B 828
Original
2010
Earlier work this paper cites.

Similar

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Original
2010
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Original
2010
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Then

J. M. Drummond, J. M. Henn and J. Trnka, “New differential equations for on-shell loop integrals,” JHEP 1104
Original
2011
Later among the works it cites.
L. J. Dixon, J. M. Drummond and J. M. Henn, “The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM,” JHEP 1106
Original
2011
Later among the works it cites.
L. Mason and D. Skinner, “Amplitudes at Weak Coupling as Polytopes in A d S 5 AdS_{5} ,” J. Phys. A 44
Original
2011
Later among the works it cites.
V. Del Duca, C. Duhr and V. A. Smirnov, “The massless hexagon integral in D = 6 dimensions,” Phys. Lett. B 703
Original
2011
Later among the works it cites.
V. Del Duca, C. Duhr and V. A. Smirnov, “The One-Loop One-Mass Hexagon Integral in D=6 Dimensions,” JHEP 1107
Original
2011
Later among the works it cites.
V. Del Duca, L. J. Dixon, J. M. Drummond, C. Duhr, J. M. Henn and V. A. Smirnov, “The one-loop six-dimensional hexagon integral with three massive corners,” Phys. Rev. D 84
Original
2011
Later among the works it cites.
L. F. Alday, “Some analytic results for two-loop scattering amplitudes,” JHEP 1107
Original
2011
Later among the works it cites.
P. Heslop and V. V. Khoze, “Wilson Loops @ 3-Loops in Special Kinematics,” JHEP 1111
Original
2011
Later among the works it cites.
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, “Local Integrals for Planar Scattering Amplitudes,” JHEP 1206
Original
2012
Later among the works it cites.
C. Duhr, H. Gangl and J. R. Rhodes, “From polygons and symbols to polylogarithmic functions,” JHEP 1210
Original
2012
Later among the works it cites.
A. Brandhuber, G. Travaglini and G. Yang, “Analytic two-loop form factors in N=4 SYM,” JHEP 1205
Original
2012
Later among the works it cites.
C. Duhr, “Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes,” JHEP 1208
Original
2012
Later among the works it cites.
L. J. Dixon, J. M. Drummond and J. M. Henn, “Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory,” JHEP 1201
Original
2012
Later among the works it cites.
L. J. Dixon, C. Duhr and J. Pennington, “Single-valued harmonic polylogarithms and the multi-Regge limit,” JHEP 1210
Original
2012
Later among the works it cites.
A. L. Fitzpatrick and J. Kaplan, “Analyticity and the Holographic S-Matrix,” JHEP 1210
Original
2012
Later among the works it cites.
D. Nandan, A. Volovich and C. Wen, “On Feynman Rules for Mellin Amplitudes in AdS/CFT,” JHEP 1205
Original
2012
Later among the works it cites.
A. L. Fitzpatrick and J. Kaplan, “Unitarity and the Holographic S-Matrix,” JHEP 1210
Original
2012
Later among the works it cites.
M. F. Paulos, M. Spradlin and A. Volovich, “Mellin Amplitudes for Dual Conformal Integrals,” JHEP 1208
Original
2012
Later among the works it cites.
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, “A Note on Polytopes for Scattering Amplitudes,” JHEP 1204
Original
2012
Later among the works it cites.
S. Caron-Huot and K. J. Larsen, “Uniqueness of two-loop master contours,” JHEP 1210
Original
2012
Later among the works it cites.
T. Goddard, P. Heslop and V. V. Khoze, “Uplifting Amplitudes in Special Kinematics,” JHEP 1210
Original
2012
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M. Spradlin and A. Volovich, “Symbols of One-Loop Integrals From Mixed Tate Motives,” JHEP 1111
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2024
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