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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.
D. J. Broadhurst, J. Fleischer, and O. V. Tarasov, “Two-Loop Two-Point Functions with Masses: Asymptotic Expansions and Taylor Series, in Any Dimension,” Z. Phys
1993
Earlier work this paper cites.
V. V. Batyrev, “Variations of the Mixed Hodge Structure of Affine Hypersurfaces in Algebraic Tori,” Duke Mathematical Journal
1993
Earlier work this paper cites.
N. I. Usyukina and A. I. Davydychev, “An Approach to the Evaluation of Three and Four Point Ladder Diagrams,” Phys. Lett
1993
Earlier work this paper cites.
N. I. Usyukina and A. I. Davydychev, “Exact Results for Three and Four Point Ladder Diagrams with an Arbitrary Number of Rungs,” Phys. Lett
1993
Earlier work this paper cites.
D. J. Broadhurst, “Summation of an Infinite Series of Ladder Diagrams,” Phys. Lett
1993
Earlier work this paper cites.
V. V. Batyrev and D. A. Cox, “On the Hodge Structure of Projective Hypersurfaces in Toric Varieties,” Duke Mathematical Journal
1994
Earlier work this paper cites.
World Scientific, Singapore, 1994
T. Hubsch, Calabi-Yau Manifolds: A Bestiary for Physicists · 1994
Earlier work this paper cites.
O. V. Tarasov, “Connection between Feynman integrals having different values of the space-time dimension,” Phys. Rev
1996
Earlier work this paper cites.
A. I. Davydychev and R. Delbourgo, “A Geometrical Angle on Feynman Integrals,” J. Math. Phys
1998
Earlier work this paper cites.
M. Caffo, H. Czyz, S. Laporta, and E. Remiddi, “The Master Differential Equations for the Two-Loop Sunrise Selfmass Amplitudes,” Nuovo Cim
1998
Earlier work this paper cites.
A. P. Isaev, “Multiloop Feynman Integrals and Conformal Quantum Mechanics,” Nucl. Phys
2003
Earlier work this paper cites.
V. A. Smirnov, “Evaluating Feynman Integrals,” Springer Tracts Mod. Phys
2004
Earlier work this paper cites.
S. Laporta and E. Remiddi, “Analytic Treatment of the Two-Loop Equal Mass Sunrise Graph,” Nucl. Phys
2005
Earlier work this paper cites.
B. A. Kniehl, A. V. Kotikov, A. Onishchenko, and O. Veretin, “Two-Loop Sunset Diagrams with Three Massive Lines,” Nucl. Phys
2006
Earlier work this paper cites.
N. Beisert, B. Eden, and M. Staudacher, “Transcendentality and Crossing,” J. Stat. Mech
2007
Earlier work this paper cites.
S. Groote, J. G. Korner, and A. A. Pivovarov, “On the Evaluation of a Certain Class of Feynman Diagrams in x x -Space: Sunrise-Type Topologies at Any Loop Order,” Annals Phys
2007
Earlier work this paper cites.
D. H. Bailey, J. M. Borwein, D. Broadhurst, and M. L. Glasser, “Elliptic Integral Evaluations of Bessel Moments,” J. Phys
Original
2008
Earlier work this paper cites.
D. J. Broadhurst and A. I. Davydychev, “Exponential Suppression with Four Legs and an Infinity of Loops,” Nucl. Phys. Proc. Suppl
Original
2010
Earlier work this paper cites.
S. Caron-Huot, “Superconformal Symmetry and Two-Loop Amplitudes in Planar 𝒩 = 4 \mathcal{N}\!=\!4 Super Yang-Mills,” JHEP
Original
2011
Earlier work this paper cites.
F. Brown and K. Yeats, “Spanning Forest Polynomials and the Transcendental Weight of Feynman Graphs,” Commun. Math. Phys
Original
2011
Earlier work this paper cites.

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2018

A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries

Bourjaily, Jacob L., McLeod, Andrew J., von Hippel, Matt et al.

Understand

We define the rigidity of a Feynman integral to be the smallest dimension over which it is non-polylogarithmic.

We argue that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show that this bound may be saturated for integrals that we call marginal: those with (L+1)D/2 propagators in (even) D dimensions.
We show that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries, and we give examples of finite four-dimensional Feynman integrals in massless $\phi^4$ theory that saturate our predicted bound in rigidity at all loop orders.

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.
D. J. Broadhurst, J. Fleischer, and O. V. Tarasov, “Two-Loop Two-Point Functions with Masses: Asymptotic Expansions and Taylor Series, in Any Dimension,” Z. Phys
1993
Earlier work this paper cites.
V. V. Batyrev, “Variations of the Mixed Hodge Structure of Affine Hypersurfaces in Algebraic Tori,” Duke Mathematical Journal
1993
Earlier work this paper cites.
N. I. Usyukina and A. I. Davydychev, “An Approach to the Evaluation of Three and Four Point Ladder Diagrams,” Phys. Lett
1993
Earlier work this paper cites.
N. I. Usyukina and A. I. Davydychev, “Exact Results for Three and Four Point Ladder Diagrams with an Arbitrary Number of Rungs,” Phys. Lett
1993
Earlier work this paper cites.
D. J. Broadhurst, “Summation of an Infinite Series of Ladder Diagrams,” Phys. Lett
1993
Earlier work this paper cites.
V. V. Batyrev and D. A. Cox, “On the Hodge Structure of Projective Hypersurfaces in Toric Varieties,” Duke Mathematical Journal
1994
Earlier work this paper cites.
World Scientific, Singapore, 1994
T. Hubsch, Calabi-Yau Manifolds: A Bestiary for Physicists · 1994
Earlier work this paper cites.
O. V. Tarasov, “Connection between Feynman integrals having different values of the space-time dimension,” Phys. Rev
1996
Earlier work this paper cites.
A. I. Davydychev and R. Delbourgo, “A Geometrical Angle on Feynman Integrals,” J. Math. Phys
1998
Earlier work this paper cites.
M. Caffo, H. Czyz, S. Laporta, and E. Remiddi, “The Master Differential Equations for the Two-Loop Sunrise Selfmass Amplitudes,” Nuovo Cim
1998
Earlier work this paper cites.
A. P. Isaev, “Multiloop Feynman Integrals and Conformal Quantum Mechanics,” Nucl. Phys
2003
Earlier work this paper cites.
V. A. Smirnov, “Evaluating Feynman Integrals,” Springer Tracts Mod. Phys
2004
Earlier work this paper cites.
S. Laporta and E. Remiddi, “Analytic Treatment of the Two-Loop Equal Mass Sunrise Graph,” Nucl. Phys
2005
Earlier work this paper cites.
B. A. Kniehl, A. V. Kotikov, A. Onishchenko, and O. Veretin, “Two-Loop Sunset Diagrams with Three Massive Lines,” Nucl. Phys
2006
Earlier work this paper cites.
N. Beisert, B. Eden, and M. Staudacher, “Transcendentality and Crossing,” J. Stat. Mech
2007
Earlier work this paper cites.
S. Groote, J. G. Korner, and A. A. Pivovarov, “On the Evaluation of a Certain Class of Feynman Diagrams in x x -Space: Sunrise-Type Topologies at Any Loop Order,” Annals Phys
2007
Earlier work this paper cites.
D. H. Bailey, J. M. Borwein, D. Broadhurst, and M. L. Glasser, “Elliptic Integral Evaluations of Bessel Moments,” J. Phys
Original
2008
Earlier work this paper cites.
D. J. Broadhurst and A. I. Davydychev, “Exponential Suppression with Four Legs and an Infinity of Loops,” Nucl. Phys. Proc. Suppl
Original
2010
Earlier work this paper cites.
S. Caron-Huot, “Superconformal Symmetry and Two-Loop Amplitudes in Planar 𝒩 = 4 \mathcal{N}\!=\!4 Super Yang-Mills,” JHEP
Original
2011
Earlier work this paper cites.
F. Brown and K. Yeats, “Spanning Forest Polynomials and the Transcendental Weight of Feynman Graphs,” Commun. Math. Phys
Original
2011
Earlier work this paper cites.