Fetching the paper…

Reading the bibliography…

2022

Algorithms for minimal Picard-Fuchs operators of Feynman integrals

Lairez, Pierre, Vanhove, Pierre

Original
PDF
Post on X

Understand

In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space.

By using an algorithm that extends the Griffiths--Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations, the Picard--Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals.
With this approach we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders.
Using recent factorisation algorithms we give the minimal order differential operator in most of the cases studied in this paper.

Algorithms for minimal Picard-Fuchs operators of Feynman integrals · Around

Built on

L. de la Cruz, “Feynman Integrals as A-Hypergeometric Functions,” JHEP 12
Original
1907
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,” JHEP 01
Original
1910
Earlier work this paper cites.
R. P. Klausen, “Hypergeometric Series Representations of Feynman Integrals by Gkz Hypergeometric Systems,” JHEP 04
Original
1910
Earlier work this paper cites.
A. Klemm, C. Nega and R. Safari, “The l l -loop Banana Amplitude from Gkz Systems and Relative Calabi–Yau Periods,” JHEP 04
Original
1912
Earlier work this paper cites.
T. F. Feng, C. H. Chang, J. B. Chen and H. B. Zhang, “Gkz-Hypergeometric Systems for Feynman Integrals,” Nucl. Phys. B 953
Original
1912
Earlier work this paper cites.
S. Weinberg, “High-Energy Behavior in Quantum Field Theory,” Phys. Rev. 118
1960
Earlier work this paper cites.
V. E. Asribekov, “Choice of Invariant Variables for the ‘’Many-Point” Functions,” J. Exp. Theor. Phys. 15
1962
Earlier work this paper cites.
On the zeta function of a hypersurface
B. Dwork · 1962
Earlier work this paper cites.
On the zeta function of a hypersurface: II
B. Dwork · 1964
Earlier work this paper cites.
F. Pham, “Introduction à l’étude topologique des singularités de Landau”, Paris : Gauthier-Villars; 1967
1967
Earlier work this paper cites.
E. R. Speer, “Generalized Feynman Amplitudes,” vol. 62 of Annals of Mathematics Studies. Princeton University Press, New Jersey, Apr., 1969
1969
Earlier work this paper cites.
On the periods of certain rational integrals
P. A. Griffiths · 1969
Earlier work this paper cites.
Noboru Nakanishi, “Graph Theory and Feynman Integrals”, Gordon & Breach Science Publishers Ltd (1971)
1971
Earlier work this paper cites.
E. R. Speer, “Ultraviolet and Infrared Singularity Structure of Generic Feynman Amplitudes,” Ann. Inst. H. Poincare Phys. Theor. 23
1975
Earlier work this paper cites.
V. A. Golubeva, “Some Problems In The Analytic Theory Of Feynman Integrals” , Russ. Math. Surv. 31
1976
Earlier work this paper cites.
C. Itzykson and J. B. Zuber,“Quantum Field Theory,” McGraw-Hill, New York, 1980
1980
Earlier work this paper cites.
I. M. Gelfand, M. M. Kapranov, And A. V. Zelevinsky, “Generalized Euler Integrals and A-Hypergeometric Functions”, Advances In Mathematics 84
1990
Earlier work this paper cites.
D. J. Broadhurst, “The Master Two Loop Diagram With Masses,” Z. Phys. C 47
1990
Earlier work this paper cites.
D. J. Broadhurst and D. Kreimer, “Knots and Numbers in Ph i 4 i^{4} Theory to 7 Loops and Beyond,” Int. J. Mod. Phys. C 6
1995
Earlier work this paper cites.
O. V. Tarasov, “Connection Between Feynman Integrals Having Different Values of the Space-Time Dimension,” Phys. Rev. D 54
1996
Earlier work this paper cites.
H. Verrill, Root lattices and pencils of varieties , J. Math. Kyoto Univ. 36 (2) (1996), 423-446
1996
Earlier work this paper cites.

Similar

D. J. Broadhurst and D. Kreimer, “Association of Multiple Zeta Values with Positive Knots via Feynman Diagrams Up to 9 Loops,” Phys. Lett. B 393
1997
Cited alongside, same era.
W. Fakler, “On second order homogeneous linear differential equations with Liouvillian solutions”, Theoretical Computer Science 187 (1997) 27-48
1997
Cited alongside, same era.
V. V. Batyrev, I. Ciocan-Fontanine, B. Kim and D. van Straten, “Conifold transitions and mirror symmetry for Calabi–Yau complete intersections in Grassmannians,” Nucl. Phys. B 514
1998
Cited alongside, same era.
D. R. Morrison, “Picard-Fuchs Equations and Mirror Maps for Hypersurfaces,” AMS/IP Stud. Adv. Math. 9
1998
Cited alongside, same era.
M. Kontsevich and D. Zagier, “Periods”, in Engquist, Björn; Schmid, Wilfried, Mathematics unlimited – 2001 and beyond, Berlin, New York: Springer-Verlag, pp. 771-808

Then

S. Bloch, M. Kerr and P. Vanhove, “A Feynman Integral via Higher Normal Functions,” Compos. Math. 151
Original
2015
Later among the works it cites.
J. M. Henn, “Lectures on differential equations for Feynman integrals,” J. Phys. A 48
Original
2015
Later among the works it cites.
S. Bloch and P. Vanhove, “The Elliptic Dilogarithm for the Sunset Graph,” J. Number Theor. 148
Original
2015
Later among the works it cites.
P. Lairez, “Computing periods of rational integrals”, Math. Comp. 85
Original
2016
Later among the works it cites.
M. Mezzarobba, “Rigorous Multiple-Precision Evaluation of D-Finite Functions in SageMath”,5th International Congress on Mathematical Software (ICMS 2016), Jul 2016, Berlin, Germany, [arXiv:1607.01967]

Beyond the bibliography

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

Open on alphaXiv

alphaXiv is searching for related work…

2001

Cited alongside, same era.

S. Laporta, “Calculation of Master Integrals by Difference Equations,” Phys. Lett. B 504

2001

Cited alongside, same era.

S. Mizera, “Status of Intersection Theory and Feynman Integrals,” PoS MA2019

2002

Cited alongside, same era.

R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, “The analytic S-matrix,” Cambridge University Press, 2002

2002

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.

K. Bönisch, F. Fischbach, A. Klemm, C. Nega and R. Safari, “Analytic Structure of All Loop Banana Integrals,” JHEP 05

2008

Cited alongside, same era.

C. Bogner and S. Weinzierl, “Feynman Graph Polynomials,” Int. J. Mod. Phys. A 25

2010

Cited alongside, same era.

C. Koutschan. “HolonomicFunctions (user’s guide).” Technical Report 10-01, RISC Report Series, Johannes Kepler University, Linz, Austria, 2010. http://www.risc.jku.at/research/ combinat/software/HolonomicFunctions/

2010

Cited alongside, same era.

2016

Later among the works it cites.

E. Remiddi and L. Tancredi, “Differential Equations and Dispersion Relations for Feynman Amplitudes. the Two-Loop Massive Sunrise and the kite Integral,” Nucl. Phys. B 907

2016

Later among the works it cites.

L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, “The kite Integral to All Orders in Terms of Elliptic Polylogarithms,” J. Math. Phys. 57

2016

Later among the works it cites.

S. Bloch, M. Kerr and P. Vanhove, “Local Mirror Symmetry and the Sunset Feynman Integral,” Adv. Theor. Math. Phys. 21

2017

Later among the works it cites.

C. Bogner, A. Schweitzer and S. Weinzierl, “Analytic Continuation and Numerical Evaluation of the kite Integral and the Equal Mass Sunrise Integral,” Nucl. Phys. B 922

2017

Later among the works it cites.

J. L. Bourjaily, Y. H. He, A. J. Mcleod, M. Von Hippel and M. Wilhelm, “Traintracks Through Calabi–Yau Manifolds: Scattering Amplitudes Beyond Elliptic Polylogarithms,” Phys. Rev. Lett. 121

2018

Later among the works it cites.

J. L. Bourjaily, A. J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, “Elliptic Double-Box Integrals: Massless Scattering Amplitudes Beyond Polylogarithms,” Phys. Rev. Lett. 120

2018

Later among the works it cites.

C. Duhr, “Function Theory for Multiloop Feynman Integrals,” Ann. Rev. Nucl. Part. Sci. 69

2019

Later among the works it cites.

J. L. Bourjaily, A. J. McLeod, M. von Hippel and M. Wilhelm, “Bounded Collection of Feynman Integral Calabi–Yau Geometries,” Phys. Rev. Lett. 122

2019

Later among the works it cites.

T. Bitoun, C. Bogner, R. P. Klausen and E. Panzer, “Feynman Integral Relations from Parametric Annihilators,” Lett. Math. Phys. 109

2019

Later among the works it cites.

P. Vanhove,”Mirroring towers of Feynman integrals: Fibration and degeneration in Feynman integral Calabi–Yau geometries”, (String Math 2019)

2019

Later among the works it cites.

J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, “Elliptic Feynman Integrals and Pure Functions,” JHEP 01

2019

Later among the works it cites.

P. Vanhove “Differential Equations for Feynman Integrals.” Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation, 21-26. https://doi.org/10.1145/3452143.3465512

2021

Later among the works it cites.

S. Bloch, “Double Box Motive,” SIGMA 17

2021

Later among the works it cites.

R. P. Klausen, “Kinematic singularities of Feynman integrals and principal A-determinants,” JHEP 02

2022

Closest in time.

C. Duhr, A. Klemm, C. Nega and L. Tancredi, “The Ice Cone Family and Iterated Integrals for Calabi-Yau Varieties,” JHEP 02

2023

Closest in time.