Abstract
We consider the analytic properties of Feynman integrals from the perspective of general A-discriminants and \( \mathcal{A} \)-hypergeometric functions introduced by Gelfand, Kapranov and Zelevinsky (GKZ). This enables us, to give a clear and mathematically rigorous description of the singular locus, also known as Landau variety, via principal A-determinants. We also comprise a description of the various second type singularities. Moreover, by the Horn-Kapranov-parametrization we give a very efficient way to calculate a parametrization of Landau varieties. We furthermore present a new approach to study the sheet structure of multivalued Feynman integrals by the use of coamoebas.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Adolphson, Hypergeometric Functions and Rings Generated by Monomials, Duke Math. J. 73 (1994) 269.
C. Anastasiou, S. Beerli and A. Daleo, Evaluating multi-loop Feynman diagrams with infrared and threshold singularities numerically, JHEP 05 (2007) 071 [hep-ph/0703282] [INSPIRE].
C. Berkesch, J. Forsgård and M. Passare, Euler-Mellin Integrals and A-Hypergeometric Functions, arXiv:1103.6273.
C. Berkesch Zamaere, L.F. Matusevich and U. Walther, Singularities and Holonomicity of Binomial D-Modules, arXiv:1308.5898.
T. Binoth and G. Heinrich, Numerical evaluation of multiloop integrals by sector decomposition, Nucl. Phys. B 680 (2004) 375 [hep-ph/0305234] [INSPIRE].
T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer, Feynman integral relations from parametric annihilators, Lett. Math. Phys. 109 (2019) 497 [arXiv:1712.09215] [INSPIRE].
J.-E. Björk, Analytic D-Modules and Applications, Springer, Dordrecht Netherlands (1993).
J.D. Bjorken, Experimental Tests of Quantum Electrodynamics and Spectral Representations of Green’s Functions in Perturbation Theory, Ph.D. Thesis, Stanford University, Stanford CA (1959) [INSPIRE].
S. Bloch and D. Kreimer, Cutkosky Rules and Outer Space, arXiv:1512.01705 [INSPIRE].
C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys. A 25 (2010) 2585 [arXiv:1002.3458] [INSPIRE].
K. Bönisch, C. Duhr, F. Fischbach, A. Klemm and C. Nega, Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives, arXiv:2108.05310 [INSPIRE].
K. Bönisch, F. Fischbach, A. Klemm, C. Nega and R. Safari, Analytic structure of all loop banana integrals, JHEP 05 (2021) 066 [arXiv:2008.10574] [INSPIRE].
S. Borowka, J. Carter and G. Heinrich, Numerical Evaluation of Multi-Loop Integrals for Arbitrary Kinematics with SecDec 2.0, Comput. Phys. Commun. 184 (2013) 396 [arXiv:1204.4152] [INSPIRE].
F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].
K.E. Cahill and H.P. Stapp, Optical Theorems and Steinmann Relations, Ann. Phys. 90 (1975) 438.
E. Cattani, Three Lectures on Hypergeometric Functions, (2006).
S. Coleman and R.E. Norton, Singularities in the physical region, Nuovo Cim. 38 (1965) 438 [INSPIRE].
J. Collins, A new and complete proof of the Landau condition for pinch singularities of Feynman graphs and other integrals, arXiv:2007.04085 [INSPIRE].
D.A. Cox, J.B. Little and D. O’Shea, Using Algebraic Geometry, 2nd edition, in Graduate Texts in Mathematics 185, Springer, New York NY U.S.A. (2005).
M.A. Cueto and A. Dickenstein, Some Results on Inhomogeneous Discriminants, math/0610031.
R. Curran and E. Cattani, Restriction of A-Discriminants and Dual Defect Toric Varieties, math/0510615.
R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].
L. de la Cruz, Feynman integrals as A-hypergeometric functions, JHEP 12 (2019) 123 [arXiv:1907.00507] [INSPIRE].
J.A. De Loera, J. Rambau and F. Santos, Triangulations, in Algorithms and Computation in Mathematics 25, Springer (2010).
A. Dickenstein, E.M. Feichtner and B. Sturmfels, Tropical Discriminants, math/0510126.
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The Analytic S-Matrix, Cambridge University Press, Cambridge U.K. (1966).
A. Esterov, Newton Polyhedra of Discriminants of Projections, arXiv:0810.4996.
J. Forsgård, On Hypersurface Coamoebas and Integral Representations of A-Hypergeometric Functions, Filosofie licentiatavhandling, Stockholm University, Stockholm Sweden (2012).
J. Forsgård, Tropical Aspects of Real Polynomials and Hypergeometric Functions, Ph.D. Thesis, Stockholm University, Stockholm Sweden (2015).
J. Forsgård and P. Johansson, On the Order Map for Hypersurface Coamoebas, Ark. Mat. 53 (2015) 79.
J. Forsgård and L.F. Matusevich, A Zariski Theorem for Monodromy of A-Hypergeometric Systems, arXiv:2005.00275.
I.M. Gelfand, M.I. Graev and V.S. Retakh, General Hypergeometric Systems of Equations and Series of Hypergeometric Type, Russ. Math. Surv. 47 (1992) 1.
I.M. Gelfand, A.V. Zelevinsky and M.M. Kapranov, Discriminants of Polynomials in Several Variables and Triangulations of Newton Polyhedra, Leningrad Math. J. 2 (1991) 449.
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, A-Discriminants and Cayley-Koszul Complexes, Sov. Math. Dokl. 40 (1990) 239.
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Hypergeometric Functions, Toric Varieties and Newton Polyhedra, in ICM-90 Satellite Conference Proceedings: Special Functions, M. Kashiwara and T. Miwa eds., Springer, Tokyo Japan (1991), pp. 104–121.
I.M. Gelfand, A.V. Zelevinsky and M.M. Kapranov, Hypergeometric Functions and Toral Manifolds, Funct. Anal. Appl. 23 (1989) 94.
I.M. Gelfand, A.V. Zelevinsky and M.M. Kapranov, Discriminants of Polynomials in Many Variables, Funct. Anal. Appl. 24 (1990) 1.
I.M. Gelfand, A.V. Zelevinsky and M.M. Kapranov, Equations of Hypergeometric Type and Newton Polyhedra, Sov. Math. Dokl. 37 (1988) 678.
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Generalized Euler Integrals and A-Hypergeometric Functions, Adv. Math. 84 (1990) 255.
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Newton Polytopes of the Classical Resultant and Discriminant, Adv. Math. 84 (1990) 237.
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston MA U.S.A. (1994).
D.R. Grayson and M.E. Stillman, Macaulay2, a Software System for Research in Algebraic Geometry, (2022) http://www.math.uiuc.edu/Macaulay2/.
R. Hartshorne, Algebraic Geometry, in Graduate Texts in Mathematics 52, Springer, New York NY U.S.A. (1977).
A. Holme, A Royal Road to Algebraic Geometry, Springer (2012).
R.C. Hwa and V.L. Teplitz, Homology and Feynman Integrals, W.A. Benjamin (1966).
C. Itzykson and J.B. Zuber, Quantum Field Theory, in International Series in Pure and Applied Physics, McGraw-Hill International Book Co, New York NY U.S.A. (1980).
P. Johansson, Coamoebas, Research Reports in Mathematic Number 6, Stockholm University, Stockholm Sweden (2010).
P. Johansson, The Argument Cycle and the Coamoeba, Complex Var. Elliptic Equ. 58 (2013) 373.
M.M. Kapranov, A Characterization of A-Discriminantal Hypersurfaces in Terms of the Logarithmic Gauss Map, Math. Ann. 290 (1991) 277.
M.M. Kapranov, Thermodynamics and the Moment Map, arXiv:1108.3472.
M. Kashiwara and T. Kawai, On a Conjecture of Regge and Sato on Feynman Integrals, Proc. Jpn. Acad. 52 (1976) 161.
L. Kaup, B. Kaup and G. Barthel, Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory, in De Gruyter Studies in Mathematics 3, W. de Gruyter (1983).
R.P. Klausen, Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems, JHEP 04 (2020) 121 [arXiv:1910.08651] [INSPIRE].
D. Kreimer and K. Yeats, Properties of the corolla polynomial of a 3-regular graph, arXiv:1207.5460 [INSPIRE].
L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 [INSPIRE].
R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].
S.B. Libby and G.F. Sterman, Mass Divergences in Two Particle Inelastic Scattering, Phys. Rev. D 18 (1978) 4737 [INSPIRE].
S. Mizera, Crossing symmetry in the planar limit, Phys. Rev. D 104 (2021) 045003 [arXiv:2104.12776] [INSPIRE].
M. Mühlbauer, Momentum Space Landau Equations Via Isotopy Techniques, arXiv:2011.10368 [INSPIRE].
N. Nakanishi, Ordinary and Anomalous Thresholds in Perturbation Theory, Prog. Theor. Phys. 22 (1959) 128.
N. Nakanishi, Graph Theory and Feynman Integrals, in Mathematics and Its Applications 11, Gordon and Breach, New York NY U.S.A. (1971).
E. Nasrollahpoursamami, Periods of Feynman Diagrams and GKZ D-Modules, arXiv:1605.04970.
L. Nilsson and M. Passare, Mellin Transforms of Multivariate Rational Functions, arXiv:1010.5060.
M. Nisse, Geometric and Combinatorial Structure of Hypersurface Coamoebas, arXiv:0906.2729.
M. Nisse and M. Passare, Amoebas and Coamoebas of Linear Spaces, arXiv:1205.2808.
M. Nisse and F. Sottile, Higher Convexity of Coamoeba Complements, arXiv:1405.4900.
T. Oaku, Computation of the Characteristic Variety and the Singular Locus of a System of Differential Equations with Polynomial Coefficients, Jpn. J. Ind. Appl. Math. 11 (1994) 485.
E. Panzer, Feynman integrals and hyperlogarithms, arXiv:1506.07243 [INSPIRE].
M. Passare, D. Pochekutov and A. Tsikh, Amoebas of Complex Hypersurfaces in Statistical Thermodynamics, Math. Phys. Anal. Geom. 16 (2013) 89 [arXiv:1112.4332].
M. Passare and F. Sottile, Discriminant Coamoebas through Homology, arXiv:1201.6649.
G. Passarino and M.J.G. Veltman, One Loop Corrections for e+e− Annihilation Into μ+μ− in the Weinberg Model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].
P. Pedersen and B. Sturmfels, Product Formulas for Resultants and Chow Forms, Math. Z. 214 (1993) 377.
F. Pham, Singularities of Integrals, in Universitext, Springer, London U.K. (2011).
G. Ponzano, T. Regge, E.R. Speer and M.J. Westwater, The monodromy rings of a class of self-energy graphs, Commun. Math. Phys. 15 (1969) 83 [INSPIRE].
M. Saito, B. Sturmfels and N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, E. Becker, M. Bronstein, H. Cohen, D. Eisenbud and R. Gilman eds., in Algorithms and Computation in Mathematics 6, Springer (2000).
M. Sato, Recent development in hyperfunction theory and its application to physics (microlocal analysis of s-matrices and related quantities), in Lecture Notes in Physics 39, Springer (1975), pp. 13–29 [INSPIRE].
M. Sato, T. Miwa, M. Jimbo and T. Oshima, Holonomy Structure of Landau Singularities and Feynman Integrals, Publ. Res. Inst. Math. Sci. 12 (1976) 387.
K. Schultka, Toric geometry and regularization of Feynman integrals, arXiv:1806.01086 [INSPIRE].
M. Schulze and U. Walther, Irregularity of Hypergeometric Systems via Slopes along Coordinate Subspaces, math/0608668.
E.R. Speer and M.J. Westwater, Generic Feynman amplitudes, Ann. Inst. Henri Poincaré Phys. Theor. 14 (1971) 1.
G. Staglianò, A Package for Computations with Classical Resultants, J. Softw. Algebra Geom. 8 (2018) 21.
G. Staglianò, A Package for Computations with Sparse Resultants, arXiv:2010.00286.
G.F. Sterman, An Introduction to Quantum Field Theory, Cambridge University Press, Cambridge U.K. (1993) [https://doi.org/10.1017/CBO9780511622618].
B. Sturmfels, On the Newton Polytope of the Resultant, J. Algebr. Combinator. 3 (1994) 207.
B. Sturmfels, Introduction to Resultants, D. Cox and B. Sturmfels eds., in Proceedings of Symposia in Applied Mathematics 53, American Mathematical Society, Providence RI U.S.A. (1997), pp. 25–39.
B. Sturmfels, Solving Systems of Polynomial Equations, (2002).
O.V. Tarasov, Generalized recurrence relations for two loop propagator integrals with arbitrary masses, Nucl. Phys. B 502 (1997) 455 [hep-ph/9703319] [INSPIRE].
O.V. Tarasov, Reduction of Feynman graph amplitudes to a minimal set of basic integrals, Acta Phys. Pol. B 29 (1998) 2655 [hep-ph/9812250] [INSPIRE].
S. Weinzierl, The Art of computing loop integrals, Fields Inst. Commun. 50 (2007) 345 [hep-ph/0604068] [INSPIRE].
Wolfram Research Inc., Mathematica, Version 12.3.1 (2021).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2109.07584
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Klausen, R.P. Kinematic singularities of Feynman integrals and principal A-determinants. J. High Energ. Phys. 2022, 4 (2022). https://doi.org/10.1007/JHEP02(2022)004
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2022)004