# zbMATH — the first resource for mathematics

Critical points and number of master integrals. (English) Zbl 1342.81139
Summary: We consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the critical points of the polynomials entering either of the two representations: the parametric representation and the Baikov representation. In particular, for the parametric representation the corresponding polynomial is just the sum of Symanzik polynomials. The relevant topological invariant is the sum of the Milnor numbers of the proper critical points. We present a Mathematica package Mint to automatize the counting of the master integrals for the typical case, when all critical points are isolated.

##### MSC:
 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
##### Software:
Reduze; LiteRed; Mint ; SageMath; Mathematica
Full Text:
##### References:
 [1] Chetyrkin, K.; Tkachov, F., Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys., B 192, 159, (1981) [2] Tkachov, F., A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett., B 100, 65, (1981) [3] Smirnov, A., Algorithm FIRE - Feynman integral reduction, JHEP, 10, 107, (2008) · Zbl 1245.81033 [4] A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE]. · Zbl 1219.81133 [5] Studerus, C., Reduze-Feynman integral reduction in C++, Comput. Phys. Commun., 181, 1293, (2010) · Zbl 1219.81133 [6] R. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE]. [7] Adolphson, A.; Sperber, S., On twisted de Rham cohomology, Nagoya Math. J., 146, 55, (1997) · Zbl 0915.14012 [8] Kouchnirenko, A., Polyèdres de Newton et nombres de Milnor, Inv. Math., 32, 1, (1976) · Zbl 0328.32007 [9] M. Fedoryuk, The Method of Steepest Descent (in Russian), Nauka, Moscow (1977). [10] F. Pham, Vanishing homologies and the n variable saddlepoint method, in Singularities: Proceedings of the Summer Institute on Singularities, Humboldt, California, 1981 (1983). · Zbl 0519.49026 [11] F. Pham, La descente des cols par les onglets de Lefschetz, avec vues sur Gauss- Manin, in Systèmes différentiels et singularités, Colloq. Luminy/France 1983, Astérisque 130 (1985). [12] Smirnov, A.; Petukhov, A., The number of master integrals is finite, Lett. Math. Phys., 97, 37, (2011) · Zbl 1216.81076 [13] S. Lefschetz, $$L$$’analysis situs et la géométrie algébrique, Gauthier-Villars, Paris France (1924). [14] E. Witten, Analytic Continuation Of Chern-Simons Theory, arXiv:1001.2933 [INSPIRE]. · Zbl 1337.81106 [15] V. Arnold, S. Gusein-Zade and A. Varchenko, Singularities of differentiable maps. Volume II: Monodromy and Asymptotics of Integrals, Birkhäuser (1988). · Zbl 1297.32001 [16] M. Marcolli, Motivic renormalization and singularities, in Quanta of maths. Conference on non commutative geometry in honor of Alain Connes, Paris, France, March 29-April 6, 2007, pp. 409-458. [17] M. Marcolli, Feynman Motives, World Scientific, Singapore (2010). [18] Baikov, P., Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth., A 389, 347, (1997) [19] Lee, R., Calculating multiloop integrals using dimensional recurrence relation and D-analyticity, Nucl. Phys. Proc. Suppl., 205-206, 135, (2010) [20] Baikov, P., A practical criterion of irreducibility of multi-loop Feynman integrals, Phys. Lett., B 634, 325, (2006) · Zbl 1247.81314 [21] The Mint package can be downloaded from http://www.inp.nsk.su/ lee/programs/LiteRed/#utils. [22] W. Stein et al., Sage Mathematics Software (Version 5.11), The Sage Development Team, 2013, http://www.sagemath.org.
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.