zbMATH — the first resource for mathematics

Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems. (English) Zbl 1436.81049
Summary: We show that almost all Feynman integrals as well as their coefficients in a Laurent series in dimensional regularization can be written in terms of Horn hypergeometric functions. By applying the results of Gelfand-Kapranov-Zelevinsky (GKZ) we derive a formula for a class of hypergeometric series representations of Feynman integrals, which can be obtained by triangulations of the Newton polytope \(\Delta_G\) corresponding to the Lee-Pomeransky polynomial \(G\). Those series can be of higher dimension, but converge fast for convenient kinematics, which also allows numerical applications. Further, we discuss possible difficulties which can arise in a practical usage of this approach and give strategies to solve them.

81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81U05 \(2\)-body potential quantum scattering theory
TOPCOM; polymake; DLMF
Full Text: DOI arXiv
[1] T. Regge, Algebraic topology methods in the theory of Feynman relativistic amplitudes, in Battelle rencontres — 1967 Lectures in Mathematics and Physics, C.M. DeWitt and J.A. Wheeler eds., (1967), pg. 433 [INSPIRE]. · Zbl 0174.28202
[2] M. Kashiwara and T. Kawai, Holonomic systems of linear differential equations and Feynman integrals, Publ. Res. Inst. Math. Sci.12 (1976) 131. · Zbl 0449.35095
[3] V.A. Smirnov, Feynman integral calculus, Springer, Berlin, Heidelberg, Germany (2006).
[4] E.E. Boos and A.I. Davydychev, A method of evaluating massive Feynman integrals, Teor. Mat. Fiz.89 (1991) 56.
[5] J. Fleischer, F. Jegerlehner and O.V. Tarasov, A new hypergeometric representation of one loop scalar integrals in d dimensions, Nucl. Phys.B 672 (2003) 303 [hep-ph/0307113] [INSPIRE]. · Zbl 1058.81605
[6] R.G. Buschman and H.M. Srivastava, The H function associated with a certain class of Feynman integrals, J. Phys.A 23 (1990) 4707. · Zbl 0695.33002
[7] A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals. II. A generalisation of the H function, J. Phys.A 20 (1987) 4119. · Zbl 0634.33006
[8] A.A. Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals. I. Transformation and reduction formulae, J. Phys.A 20 (1987) 4109. · Zbl 0634.33005
[9] I. Gelfand, M. Kapranov and A. Zelevinsky, Generalized Euler integrals and A-hypergeometric functions, Adv. Math.84 (1990) 255. · Zbl 0741.33011
[10] I. Gelfand, M.I. Graev and V.S. Retakh, General hypergeometric systems of equations and series of hypergeometric type, Russ. Math. Surv.47 (1992) 1. · Zbl 0798.33010
[11] I. Gelfand, M.I. Graev and A.V. Zelevinsky, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR (1987) 14.
[12] I. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Hypergeometric functions, toric varieties and newton polyhedra, in ICM-90 satellite conference proceedings, Springer, Japan, (1991), pg. 104.
[13] I. Gelfand, Part V. General theory of hypergeometric functions, in Collected papers. Volume 3, S.G. Gindikin ed., Springer, (1989), pg. 877.
[14] I. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Modern Birkhäuser classics, Birkhäuser, Boston, MA, U.S.A. (1994). · Zbl 0827.14036
[15] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys.B 433 (1995) 501 [hep-th/9406055] [INSPIRE]. · Zbl 0908.32008
[16] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys.167 (1995) 301 [hep-th/9308122] [INSPIRE]. · Zbl 0814.53056
[17] P. Vanhove, Feynman integrals, toric geometry and mirror symmetry, in Proceedings, KMPB conference: elliptic integrals, elliptic functions and modular forms in quantum field theory, Zeuthen, Germany, 23-26 October 2017, Springer, Cham, Switzerland (2019), pg. 415 [arXiv:1807.11466] [INSPIRE].
[18] E. Nasrollahpoursamami, Periods of Feynman diagrams and GKZ D-modules, arXiv:1605.04970.
[19] L. de la Cruz, Feynman integrals as A-hypergeometric functions, JHEP12 (2019) 123 [arXiv:1907.00507] [INSPIRE]. · Zbl 1431.81061
[20] B.A. Kniehl and O.V. Tarasov, Finding new relationships between hypergeometric functions by evaluating Feynman integrals, Nucl. Phys.B 854 (2012) 841 [arXiv:1108.6019] [INSPIRE]. · Zbl 1229.81100
[21] M. Yu. Kalmykov, B.A. Kniehl, B.F.L. Ward and S.A. Yost, Hypergeometric functions, their E-expansions and Feynman diagrams, in Proceedings, 15^thinternational seminar on high energy physics (Quarks 2008), Sergiev Posad, Russia, 23-29 May 2008 [arXiv:0810.3238] [INSPIRE].
[22] V. Bytev, B. Kniehl and S. Moch, Derivatives of Horn-type hypergeometric functions with respect to their parameters, arXiv:1712.07579 [INSPIRE].
[23] T. Sadykov, Hypergeometric functions in several complex variables, Ph.D. thesis, Stockholm University, Stockholm, Sweden (2002).
[24] C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys.A 25 (2010) 2585 [arXiv:1002.3458] [INSPIRE]. · Zbl 1193.81072
[25] R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP11 (2013) 165 [arXiv:1308.6676] [INSPIRE]. · Zbl 1342.81139
[26] S. Weinzierl, The art of computing loop integrals, Fields Inst. Commun.50 (2007) 345 [hep-ph/0604068] [INSPIRE]. · Zbl 1122.81069
[27] G. Sterman, An introduction to quantum field theory, Cambridge University Press, Cambridge, U.K. (1993).
[28] E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt U., Berlin, Germany (2015) [arXiv:1506.07243] [INSPIRE]. · Zbl 1344.81024
[29] G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys.B 44 (1972) 189.
[30] E.R. Speer, Generalized Feynman amplitudes, Ann. Math. Stud.62, Princeton University Press, Princeton, NJ, U.S.A. (1969).
[31] 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]. · Zbl 1412.81141
[32] I.A. Antipova, Inversion of multidimensional Mellin transforms, Russ. Math. Surv.62 (2007) 977. · Zbl 1148.44003
[33] C. Berkesch, J. Forsgård and M. Passare, Euler-Mellin integrals and A-hypergeometric functions, arXiv:1103.6273. · Zbl 1290.32008
[34] K. Symanzik, On calculations in conformal invariant field theories, Tech. Rep. 72/6, Deutsches Elektronen Synchroton (DESY), Germany, February 1972 [Lett. Nuovo Cim.3 (1972) 734] [INSPIRE].
[35] N. Hai and H. Srivastava, The convergence problem of certain multiple Mellin-Barnes contour integrals representing H-functions in several variables, Comput. Math. Appl.29 (1995) 17. · Zbl 0816.33008
[36] R.B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes integrals, Cambridge University Press, Cambridge, U.K. (2001). · Zbl 0983.41019
[37] I. Gonzalez, V.H. Moll and I. Schmidt, A generalized Ramanujan master theorem applied to the evaluation of Feynman diagrams, arXiv:1103.0588 [INSPIRE]. · Zbl 1303.33016
[38] A. Brøndsted, An introduction to convex polytopes, Grad. Texts Math.90, Springer, New York, NY, U.S.A. (1983). · Zbl 0509.52001
[39] M. Henk, J. Richter-Gebert and G. Ziegler, Basic properties of convex polytopes, in Handbook of discrete and computational geometry, second edition, J. Goodman and J. O’Rourke eds., Chapman and Hall/CRC, (2004). · Zbl 0911.52007
[40] J. Gubeladze and W. Bruns, Polytopes, rings, and K-theory, Springer, New York, NY, U.S.A. (2009).
[41] J.A. De Loera, J. Rambau and F. Santos, Triangulations: structures for algorithms and applications, Alg. Comput. Math.25, Springer, Berlin, Heidelberg, Germany (2010). · Zbl 1207.52002
[42] S. Weinberg, High-energy behavior in quantum field theory, Phys. Rev.118 (1960) 838. · Zbl 0098.20403
[43] L. Nilsson and M. Passare, Mellin transforms of multivariate rational functions, arXiv:1010.5060. · Zbl 1271.44001
[44] K. Schultka, Toric geometry and regularization of Feynman integrals, arXiv:1806.01086 [INSPIRE].
[45] J. Horn, Über hypergeometrische Funktionen zweier Veränderlichen (in German), Math. Annalen117-117 (1940) 384. · JFM 66.0325.05
[46] K. Aomoto and M. Kita, Theory of hypergeometric functions, Springer, Japan (2011). · Zbl 1229.33001
[47] M. Saito, B. Sturmfels and N. Takayama, Gröbner deformations of hypergeometric differential equations, Springer, Berlin, Heidelberg, Germany (2000). · Zbl 0946.13021
[48] J. Stienstra, GKZ hypergeometric structures, math.AG/0511351. · Zbl 1119.14003
[49] E. Cattani, Three lectures on hypergeometric functions, (2006).
[50] N.J. Vilenkin and A.U. Klimyk, Gel’fand hypergeometric functions, in Representation of Lie groups and special functions, Springer, The Netherlands (1995), pg. 393.
[51] S.-J. Matsubara-Heo, Laplace, residue and Euler integral representations of GKZ hypergeometric functions, arXiv:1801.04075.
[52] M.-C. Fernández-Fernández, Irregular hypergeometric D-modules, arXiv:0906.3478. · Zbl 1236.14018
[53] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark eds., NIST handbook of mathematical functions, Cambridge University Press: NIST, (2010). · Zbl 1198.00002
[54] S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multiscale multiloop integrals, J. Math. Phys.43 (2002) 3363 [hep-ph/0110083] [INSPIRE]. · Zbl 1060.33007
[55] 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) 031601 [arXiv:1810.07689] [INSPIRE].
[56] F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].
[57] A.G. Grozin, Integration by parts: an introduction, Int. J. Mod. Phys.A 26 (2011) 2807 [arXiv:1104.3993] [INSPIRE]. · Zbl 1247.81138
[58] F.F. Knudsen, Construction of nice polyhedral subdivisions, in Toroidal embeddings I, Springer, Berlin, Heidelberg, Germany (1973), pg. 109.
[59] E. Gawrilow and M. Joswig, Polymake: a framework for analyzing convex polytopes, in Polytopes — combinatorics and computation, Birkhäuser, Basel, Switzerland (2000), pg. 43. · Zbl 0960.68182
[60] J. Rambau, TOPCOM: triangulations of point configurations and oriented matroids, in Mathematical software, A.M. Cohen, X.-S. Gao and N. Takayama eds., World Scientific, (2002), pg. 330. · Zbl 1057.68150
[61] A.V. Smirnov, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun.204 (2016) 189 [arXiv:1511.03614] [INSPIRE]. · Zbl 1378.65075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.