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.
S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization , 1906.02099
Original
1906
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.
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.
Q. Jin and H. Luo, Analytic Form of the Three-loop Four-gluon Scattering Amplitudes in Yang-Mills Theory , 1910.05889
Original
1910
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.
R. E. Cutkosky, Singularities and discontinuities of Feynman amplitudes , J. Math. Phys. 1
1960
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.
G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields , Nuclear Physics B 44
1972
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.
W. L. van Neerven and J. A. M. Vermaseren, LARGE LOOP INTEGRALS , Phys. Lett. B 137
1984
Earlier work this paper cites.
D. J. Broadhurst, Evaluation of a Class of Feynman Diagrams for All Numbers of Loops and Dimensions , Phys. Lett. B 164
1985
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.
Z. Bern and D. A. Kosower, The Computation of loop amplitudes in gauge theories , Nucl. Phys. B 379
1992
Earlier work this paper cites.
Z. Bern, L. J. Dixon and D. A. Kosower, One loop corrections to five gluon amplitudes , Phys. Rev. Lett. 70
1993
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.
K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted riemann’s period relations i , Nagoya Mathematical Journal 139
1995
Earlier work this paper cites.

Similar

Z. Bern and A. G. Morgan, Massive loop amplitudes from unitarity , Nucl. Phys. B 467
1996
Cited alongside, same era.
O. V. Tarasov, Connection between feynman integrals having different values of the space-time dimension , Physical Review D 54
1996
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.
D. J. Broadhurst and D. Kreimer, Feynman diagrams as a weight system: Four loop test of a four term relation , Phys. Lett. B 426
1998
Cited alongside, same era.
K. Matsumoto, Intersection numbers for logarithmic k k -forms , Osaka J. Math. 35
1998
Cited alongside, same era.
Springer, 1998

Then

F. Brown and K. Yeats, Spanning forest polynomials and the transcendental weight of Feynman graphs , Commun. Math. Phys. 301
Original
2011
Later among the works it cites.
Springer, 2011
K. Aomoto, M. Kita, T. Kohno and K. Iohara, Theory of hypergeometric functions · 2011
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.
H. Elvang, D. Z. Freedman and M. Kiermaier, Integrands for QCD rational terms and N=4 SYM from massive CSW rules , JHEP 06
Original
2012
Later among the works it cites.
R. N. Lee and A. A. Pomeransky, Critical points and number of master integrals , JHEP 11
Original

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2021

Duals of Feynman Integrals, 2: Generalized Unitarity

Caron-Huot, Simon, Pokraka, Andrzej

Understand

The first paper of this series introduced objects (elements of twisted relative cohomology) that are Poincar\'e dual to Feynman integrals.

We show how to use the pairing between these spaces -- an algebraic invariant called the intersection number -- to express a scattering amplitude over a minimal basis of integrals, bypassing the generation of integration-by-parts identities.
The initial information is the integrand on cuts of various topologies, computable as products of on-shell trees, providing a systematic approach to generalized unitarity.
We give two algorithms for computing the multi-variate intersection number.

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.
S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization , 1906.02099
Original
1906
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.
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.
Q. Jin and H. Luo, Analytic Form of the Three-loop Four-gluon Scattering Amplitudes in Yang-Mills Theory , 1910.05889
Original
1910
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.
R. E. Cutkosky, Singularities and discontinuities of Feynman amplitudes , J. Math. Phys. 1
1960
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.
G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields , Nuclear Physics B 44
1972
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.
W. L. van Neerven and J. A. M. Vermaseren, LARGE LOOP INTEGRALS , Phys. Lett. B 137
1984
Earlier work this paper cites.
D. J. Broadhurst, Evaluation of a Class of Feynman Diagrams for All Numbers of Loops and Dimensions , Phys. Lett. B 164
1985
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.
Z. Bern and D. A. Kosower, The Computation of loop amplitudes in gauge theories , Nucl. Phys. B 379
1992
Earlier work this paper cites.
Z. Bern, L. J. Dixon and D. A. Kosower, One loop corrections to five gluon amplitudes , Phys. Rev. Lett. 70
1993
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.
K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted riemann’s period relations i , Nagoya Mathematical Journal 139
1995
Earlier work this paper cites.

Similar

Z. Bern and A. G. Morgan, Massive loop amplitudes from unitarity , Nucl. Phys. B 467
1996
Cited alongside, same era.
O. V. Tarasov, Connection between feynman integrals having different values of the space-time dimension , Physical Review D 54
1996
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.
D. J. Broadhurst and D. Kreimer, Feynman diagrams as a weight system: Four loop test of a four term relation , Phys. Lett. B 426
1998
Cited alongside, same era.
K. Matsumoto, Intersection numbers for logarithmic k k -forms , Osaka J. Math. 35
1998
Cited alongside, same era.
Springer, 1998

Then

F. Brown and K. Yeats, Spanning forest polynomials and the transcendental weight of Feynman graphs , Commun. Math. Phys. 301
Original
2011
Later among the works it cites.
Springer, 2011
K. Aomoto, M. Kita, T. Kohno and K. Iohara, Theory of hypergeometric functions · 2011
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.
H. Elvang, D. Z. Freedman and M. Kiermaier, Integrands for QCD rational terms and N=4 SYM from massive CSW rules , JHEP 06
Original
2012
Later among the works it cites.
R. N. Lee and A. A. Pomeransky, Critical points and number of master integrals , JHEP 11
Original

Beyond the bibliography

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

Open on alphaXiv

alphaXiv is searching for related work…

A. Matsuo, Summary of the theory of primitive forms , in Topological Field Theory, Primitive Forms and Related Topics , pp. 337–363 · 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.

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

Original

2002

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. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills , Nucl. Phys. B 725

2005

Cited alongside, same era.

E. I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N=4 super Yang-Mills , JHEP 11

2005

Cited alongside, same era.

S. Bloch, H. Esnault and D. Kreimer, On Motives associated to graph polynomials , Commun. Math. Phys. 267

2006

Cited alongside, same era.

2013