Built on

A. V. Smirnov and F. S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic , 1901.07808
Original
1901
Earlier work this paper cites.
H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi et al., Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers , 1901.11510
Original
1901
Earlier work this paper cites.
S. Badger, D. Chicherin, T. Gehrmann, G. Heinrich, J. M. Henn, T. Peraro et al., Analytic form of the full two-loop five-gluon all-plus helicity amplitude , Phys. Rev. Lett. 123
Original
1905
Earlier work this paper cites.
H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, Vector Space of Feynman Integrals and Multivariate Intersection Numbers , 1907.02000
Original
1907
Earlier work this paper cites.
C. Bogner, S. Müller-Stach and S. Weinzierl, The unequal mass sunrise integral expressed through iterated integrals on ℳ ¯ 1 , 3 \overline{\mathcal{M}}_{1,3} , Nucl. Phys. B 954
Original
1907
Earlier work this paper cites.
F. Brown and C. Dupont, Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions , 1907.06603
Original
1907
Earlier work this paper cites.
J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod and J. Trnka, Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops , JHEP 12
Original
1909
Earlier work this paper cites.
E. Herrmann and J. Parra-Martinez, Logarithmic forms and differential equations for Feynman integrals , 1909.04777
Original
1909
Earlier work this paper cites.
S. Mizera and A. Pokraka, From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals , JHEP 02
Original
1910
Earlier work this paper cites.
E. Casali, S. Mizera and P. Tourkine, Monodromy relations from twisted homology , 1910.08514
Original
1910
Earlier work this paper cites.
S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew, From positive geometries to a coaction on hypergeometric functions , JHEP 02
Original
1910
Earlier work this paper cites.
J. L. Bourjaily, E. Gardi, A. J. McLeod and C. Vergu, All-mass n n -gon integrals in n n dimensions , JHEP 08
Original
1912
Earlier work this paper cites.
S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew, Diagrammatic Coaction of Two-Loop Feynman Integrals , in 14th International Symposium on Radiative Corrections: Application of Quantum Field Theory to Phenomenology , 12, 2019, 1912.06561 , DOI
Original
1912
Earlier work this paper cites.
P. A. Dirac, Wave equations in conformal space , Annals Math. 37
1936
Earlier work this paper cites.
R. E. Cutkosky, Singularities and discontinuities of Feynman amplitudes , J. Math. Phys. 1
1960
Earlier work this paper cites.
D. Fotiadi, M. Froissart, J. Lascoux and F. Pham, Applications of an isotopy theorem , Topology 4
1965
Earlier work this paper cites.
Mathematical physics monograph series. Benjamin, New York, NY, 1966
R. C. Hwa and V. L. Teplitz, Homology and Feynman integrals · 1966
Earlier work this paper cites.
J. Lascoux, Perturbation Theory in Quantum Field Theory and Homology , in Battelle Rencontres , pp. 354–419, 1968
1968
Earlier work this paper cites.
G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields , Nuclear Physics B 44
1972
Earlier work this paper cites.
G. ’t Hooft, Dimensional regularization and the renormalization group , Nucl. Phys. B 61
1973
Earlier work this paper cites.
K. AOMOTO, On vanishing of cohomology attached to certain many valued meromorphic functions , J. Math. Soc. Japan 27
1975
Earlier work this paper cites.
K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops , Nucl. Phys. B192
1981
Earlier work this paper cites.
F. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions , Phys. Lett. B 100
1981
Earlier work this paper cites.
K. AOMOTO, Configurations and invariant gauss-manin connections of integrals i , Tokyo J. Math. 05
1982
Earlier work this paper cites.
K. AOMOTO, Configurations and invariant gauss-manin connections for integrals ii , Tokyo J. Math. 06
1983
Earlier work this paper cites.
K. Aomoto, Gauss-manin connection of integral of difference products , Journal of the Mathematical Society of Japan 39
1987
Earlier work this paper cites.
F. Cachazo, Sharpening The Leading Singularity , 0803.1988
Original
1988
Earlier work this paper cites.
R. Britto, B. Feng and P. Mastrolia, Closed-Form Decomposition of One-Loop Massive Amplitudes , Phys. Rev. D 78
Original
1989
Earlier work this paper cites.
A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation , Phys. Lett. B 254
1991
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.
Graduate Texts in Mathematics. 1995
R. Bott and L. Tu, Differential Forms in Algebraic Topology · 1995
Earlier work this paper cites.

Similar

K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted riemann’s period relations i , Nagoya Mathematical Journal 139
1995
Cited alongside, same era.
Z. Bern and A. G. Morgan, Massive loop amplitudes from unitarity , Nucl. Phys. B 467
1996
Cited alongside, same era.
P. A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application , Nucl. Instrum. Meth. A389
1997
Cited alongside, same era.
K. Aomoto, M. Kita, P. Orlik and H. Terao, Twisted de rham cohomology groups of logarithmic forms , Advances in Mathematics 128
1997
Cited alongside, same era.
E. Remiddi, Differential equations for Feynman graph amplitudes , Nuovo Cim. A 110
1997
Cited alongside, same era.

Then

Springer, 2011
K. Aomoto, M. Kita, T. Kohno and K. Iohara, Theory of hypergeometric functions · 2011
Later among the works it cites.
Springer Science & Business Media, 2011
F. Pham, Singularities of Integrals: Homology, hyperfunctions and microlocal Analysis · 2011
Later among the works it cites.
J. J. M. Carrasco and H. Johansson, Five-Point Amplitudes in N=4 Super-Yang-Mills Theory and N=8 Supergravity , Phys. Rev. D 85
Original
2012
Later among the works it cites.
K. Aomoto, Hypersphere arrangement and imaginary cycles for hypergeometric integrals , in Arrangements of Hyperplanes — Sapporo 2009 , (Tokyo, Japan), pp. 1–26, Mathematical Society of Japan, 2012, DOI
2012
Later among the works it cites.
S. Badger, H. Frellesvig and Y. Zhang, A Two-Loop Five-Gluon Helicity Amplitude in QCD , JHEP 12
Original
2013
Later among the works it cites.

Beyond the bibliography

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

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…

2021

Duals of Feynman Integrals, I: Differential Equations

Caron-Huot, Simon, Pokraka, Andrzej

Understand

We elucidate the vector space (twisted relative cohomology) that is Poincar\'e dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension.

The pairing between these spaces - an algebraic invariant called the intersection number - extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities.
Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries).
Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data.

Built on

A. V. Smirnov and F. S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic , 1901.07808
Original
1901
Earlier work this paper cites.
H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi et al., Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers , 1901.11510
Original
1901
Earlier work this paper cites.
S. Badger, D. Chicherin, T. Gehrmann, G. Heinrich, J. M. Henn, T. Peraro et al., Analytic form of the full two-loop five-gluon all-plus helicity amplitude , Phys. Rev. Lett. 123
Original
1905
Earlier work this paper cites.
H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, Vector Space of Feynman Integrals and Multivariate Intersection Numbers , 1907.02000
Original
1907
Earlier work this paper cites.
C. Bogner, S. Müller-Stach and S. Weinzierl, The unequal mass sunrise integral expressed through iterated integrals on ℳ ¯ 1 , 3 \overline{\mathcal{M}}_{1,3} , Nucl. Phys. B 954
Original
1907
Earlier work this paper cites.
F. Brown and C. Dupont, Lauricella hypergeometric functions, unipotent fundamental groups of the punctured Riemann sphere, and their motivic coactions , 1907.06603
Original
1907
Earlier work this paper cites.
J. L. Bourjaily, E. Herrmann, C. Langer, A. J. McLeod and J. Trnka, Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops , JHEP 12
Original
1909
Earlier work this paper cites.
E. Herrmann and J. Parra-Martinez, Logarithmic forms and differential equations for Feynman integrals , 1909.04777
Original
1909
Earlier work this paper cites.
S. Mizera and A. Pokraka, From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals , JHEP 02
Original
1910
Earlier work this paper cites.
E. Casali, S. Mizera and P. Tourkine, Monodromy relations from twisted homology , 1910.08514
Original
1910
Earlier work this paper cites.
S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew, From positive geometries to a coaction on hypergeometric functions , JHEP 02
Original
1910
Earlier work this paper cites.
J. L. Bourjaily, E. Gardi, A. J. McLeod and C. Vergu, All-mass n n -gon integrals in n n dimensions , JHEP 08
Original
1912
Earlier work this paper cites.
S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew, Diagrammatic Coaction of Two-Loop Feynman Integrals , in 14th International Symposium on Radiative Corrections: Application of Quantum Field Theory to Phenomenology , 12, 2019, 1912.06561 , DOI
Original
1912
Earlier work this paper cites.
P. A. Dirac, Wave equations in conformal space , Annals Math. 37
1936
Earlier work this paper cites.
R. E. Cutkosky, Singularities and discontinuities of Feynman amplitudes , J. Math. Phys. 1
1960
Earlier work this paper cites.
D. Fotiadi, M. Froissart, J. Lascoux and F. Pham, Applications of an isotopy theorem , Topology 4
1965
Earlier work this paper cites.
Mathematical physics monograph series. Benjamin, New York, NY, 1966
R. C. Hwa and V. L. Teplitz, Homology and Feynman integrals · 1966
Earlier work this paper cites.
J. Lascoux, Perturbation Theory in Quantum Field Theory and Homology , in Battelle Rencontres , pp. 354–419, 1968
1968
Earlier work this paper cites.
G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields , Nuclear Physics B 44
1972
Earlier work this paper cites.
G. ’t Hooft, Dimensional regularization and the renormalization group , Nucl. Phys. B 61
1973
Earlier work this paper cites.
K. AOMOTO, On vanishing of cohomology attached to certain many valued meromorphic functions , J. Math. Soc. Japan 27
1975
Earlier work this paper cites.
K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops , Nucl. Phys. B192
1981
Earlier work this paper cites.
F. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions , Phys. Lett. B 100
1981
Earlier work this paper cites.
K. AOMOTO, Configurations and invariant gauss-manin connections of integrals i , Tokyo J. Math. 05
1982
Earlier work this paper cites.
K. AOMOTO, Configurations and invariant gauss-manin connections for integrals ii , Tokyo J. Math. 06
1983
Earlier work this paper cites.
K. Aomoto, Gauss-manin connection of integral of difference products , Journal of the Mathematical Society of Japan 39
1987
Earlier work this paper cites.
F. Cachazo, Sharpening The Leading Singularity , 0803.1988
Original
1988
Earlier work this paper cites.
R. Britto, B. Feng and P. Mastrolia, Closed-Form Decomposition of One-Loop Massive Amplitudes , Phys. Rev. D 78
Original
1989
Earlier work this paper cites.
A. V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation , Phys. Lett. B 254
1991
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.
Graduate Texts in Mathematics. 1995
R. Bott and L. Tu, Differential Forms in Algebraic Topology · 1995
Earlier work this paper cites.

Similar

K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted riemann’s period relations i , Nagoya Mathematical Journal 139
1995
Cited alongside, same era.
Z. Bern and A. G. Morgan, Massive loop amplitudes from unitarity , Nucl. Phys. B 467
1996
Cited alongside, same era.
P. A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application , Nucl. Instrum. Meth. A389
1997
Cited alongside, same era.
K. Aomoto, M. Kita, P. Orlik and H. Terao, Twisted de rham cohomology groups of logarithmic forms , Advances in Mathematics 128
1997
Cited alongside, same era.
E. Remiddi, Differential equations for Feynman graph amplitudes , Nuovo Cim. A 110
1997
Cited alongside, same era.

Then

Springer, 2011
K. Aomoto, M. Kita, T. Kohno and K. Iohara, Theory of hypergeometric functions · 2011
Later among the works it cites.
Springer Science & Business Media, 2011
F. Pham, Singularities of Integrals: Homology, hyperfunctions and microlocal Analysis · 2011
Later among the works it cites.
J. J. M. Carrasco and H. Johansson, Five-Point Amplitudes in N=4 Super-Yang-Mills Theory and N=8 Supergravity , Phys. Rev. D 85
Original
2012
Later among the works it cites.
K. Aomoto, Hypersphere arrangement and imaginary cycles for hypergeometric integrals , in Arrangements of Hyperplanes — Sapporo 2009 , (Tokyo, Japan), pp. 1–26, Mathematical Society of Japan, 2012, DOI
2012
Later among the works it cites.
S. Badger, H. Frellesvig and Y. Zhang, A Two-Loop Five-Gluon Helicity Amplitude in QCD , JHEP 12
Original
2013
Later among the works it cites.

Beyond the bibliography

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

Open on alphaXiv

alphaXiv is searching for related work…

K. Matsumoto, Intersection numbers for logarithmic k k -forms , Osaka J. Math. 35

1998

Cited alongside, same era.

S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations , Int. J. Mod. Phys. A 15

2000

Cited alongside, same era.

T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions , Nucl. Phys. B 580

2000

Cited alongside, same era.

S. Mizera, Status of Intersection Theory and Feynman Integrals , 2002.10476

Original

2002

Cited alongside, same era.

C. Dlapa, J. Henn and K. Yan, Deriving canonical differential equations for Feynman integrals from a single uniform weight integral , JHEP 05

Original

2002

Cited alongside, same era.

K. Mimachi and M. Yoshida, Intersection numbers of twisted cycles associated with the selberg integral and an application to the conformal field theory , Communications in Mathematical Physics 250

2004

Cited alongside, same era.

Z. Bern, V. Del Duca, L. J. Dixon and D. A. Kosower, All non-maximally-helicity-violating one-loop seven-gluon amplitudes in N=4 super-yang-Mills theory , Phys. Rev. D 71

2005

Cited alongside, same era.

R. N. Lee and A. A. Pomeransky, Critical points and number of master integrals , JHEP 11

Original

2013

Later among the works it cites.

Springer Science & Business Media, 2013

M. Yoshida, Hypergeometric functions, my love: modular interpretations of configuration spaces , vol. 32 · 2013

Later among the works it cites.

J. M. Henn, Multiloop integrals in dimensional regularization made simple , Phys. Rev. Lett. 110

Original

2013