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Feynman integral relations from parametric annihilators. (English) Zbl 1412.81141
Summary: We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.

81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81T18 Feynman diagrams
58D30 Applications of manifolds of mappings to the sciences
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
57R20 Characteristic classes and numbers in differential topology
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[1] Aluffi, P., Marcolli, M.: Feynman motives and deletion-contraction relations. In: Topology of Algebraic Varieties and Singularities, vol. 538 of Contemporary Mathematics, pp. 21-64. American Mathematical Society. arXiv:0907.3225 [math-ph] (2011) · Zbl 1221.81076
[2] Anastasiou, C.; Lazopoulos, A., Automatic integral reduction for higher order perturbative calculations, J. High Energy Phys., 7, 046, (2004)
[3] Andres, D.: Algorithms for the computation of Sato’s b-functions in algebraic D-module theory. Diploma thesis, Rheinisch-Westfälische Technische Hochschule Aachen (2010)
[4] Andres, D.; Brickenstein, M.; Levandovskyy, V.; Martín, J., Constructive \(D\)-module theory with singular, Math. Comput. Sci., 4, 359-383, (2010) · Zbl 1217.13011
[5] Baikov, PA, Explicit solutions of the 3-loop vacuum integral recurrence relations, Phys. Lett. B, 385, 404-410, (1996)
[6] Baikov, P.A.: Explicit solutions of the \(n\)-loop vacuum integral recurrence relations. Preprint arXiv:hep-ph/9604254 (1996)
[7] Baikov, PA, Explicit solutions of the multi-loop integral recurrence relations and its application, Nucl. Instrum. Methods A, 389, 347-349, (1997)
[8] Baikov, PA, A practical criterion of irreducibility of multi-loop Feynman integrals, Phys. Lett. B, 634, 325-329, (2006) · Zbl 1247.81314
[9] Bardin, D.Y., Kalinovskaya, L.V., Tkachov, F.V.: New algebraic numeric methods for loop integrals: some 1-loop experience. In: High Energy Physics and Quantum Field Theory. Proceedings, 15th International Workshop, QFTHEP 2000, Tver, Russia, 14-20, 2000, pp. 230-232. arXiv:hep-ph/0012209 (2000)
[10] Bernshtein, IN, The analytic continuation of generalized functions with respect to a parameter, Funct. Anal. Appl., 6, 273-285, (1972) · Zbl 0282.46038
[11] Binosi, D.; Theußl, L., JaxoDraw: a graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun., 161, 76-86, (2004)
[12] Björk, J.-E.: Rings of Differential Operators, vol. 21. North-Holland Mathematical Library, North-Holland, Amsterdam (1979)
[13] Boels, RH; Kniehl, BA; Yang, G., Master integrals for the four-loop Sudakov form factor, Nucl. Phys. B, 902, 387-414, (2016) · Zbl 1332.81126
[14] Bogner, C.; Weinzierl, S., Feynman graph polynomials, Int. J. Mod. Phys. A, 25, 2585-2618, (2010) · Zbl 1193.81072
[15] Brown, FCS, The massless higher-loop two-point function, Commun. Math. Phys., 287, 925-958, (2009) · Zbl 1196.81130
[16] Brown, F.C.S.: On the periods of some Feynman integrals. Preprint arXiv:0910.0114 [math.AG] (2009)
[17] Brown, FCS; Schnetz, O., A K3 in \(\phi ^{4}\), Duke Math. J., 161, 1817-1862, (2012) · Zbl 1253.14024
[18] Brychkov, Y.A., Glaeske, H.J., Prudnikov, A.P., Tuan, V.K.: Multidimensional Integral Transformations. Gordon and Breach Science Publishers, Philadelphia (1992) · Zbl 0752.44004
[19] Bytev, VV; Kalmykov, MY; Kniehl, BA, Differential reduction of generalized hypergeometric functions from Feynman diagrams: one-variable case, Nucl. Phys. B, 836, 129-170, (2010) · Zbl 1206.81089
[20] Caracciolo, S., Sportiello, A., Sokal, A.D.: Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I: generalizations of the Capelli and Turnbull identities. Electron. J. Comb. 16, 103 (2009). arXiv:0809.3516 [math.CO] · Zbl 1192.15001
[21] Chetyrkin, K.G., Faisst, M., Sturm, C., Tentyukov, M.: \(\epsilon \)-finite basis of master integrals for the integration-by-parts method. Nucl. Phys. B 742, 208-229 (2006). arXiv:hep-ph/0601165
[22] Chetyrkin, KG; Tkachov, FV, Integration by parts: the algorithm to calculate \(\beta \) functions in 4 loops, Nucl. Phys. B, 192, 159-204, (1981)
[23] Collins, J.C.: Renormalization. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984)
[24] Coutinho, S.C.: A Primer of Algebraic \(D\)-Modules, Vol. 33 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1995)
[25] Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-0-2—a computer algebra system for polynomial computations (2015). http://www.singular.uni-kl.de. Accessed 1 Jan 2018
[26] Derkachov, SE; Honkonen, J.; Pis’mak, YM, Three-loop calculation of the random walk problem: an application of dimensional transformation and the uniqueness method, J. Phys. A Math. Gen., 23, 5563-5576, (1990)
[27] Ferroglia, A., Passarino, G., Passera, M., Uccirati, S.: A frontier in multi-scale multi-loop integrals: the algebraic-numerical method, nuclear instruments and methods in physics research section a: accelerators, spectrometers, detectors and associated equipment 502 (2003). In: Proceedings of the VIII International Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT), Moscow, June 24-28, pp. 391-395 (2002)
[28] Ferroglia, A.; Passera, M.; Passarino, G.; Uccirati, S., All-purpose numerical evaluation of one-loop multi-leg Feynman diagrams, Nucl. Phys. B, 650, 162-228, (2003) · Zbl 1005.81059
[29] Foata, D., Zeilberger, D.: Combinatorial proofs of Capelli’s and Turnbull’s identities from classical invariant theory. Electron. J. Comb. 1, 1 (1994). arXiv:math/9309212 · Zbl 0810.05008
[30] Fujimoto, J.; Kaneko, T., GRACE and loop integrals, PoS, LL2012, 047, (2012)
[31] Gabber, O.; Loeser, F., Faisceaux pervers \(\ell \)-adiques sur un tore, Duke Math. J., 83, 501-606, (1996) · Zbl 0896.14009
[32] Georgoudis, A., Larsen, K.J., Zhang, Y.: Azurite: an algebraic geometry based package for finding bases of loop integrals. Comput. Phys. Commun. (2017). arXiv:1612.04252 [hep-th]. https://bitbucket.org/yzhphy/azurite
[33] Ginsburg, V., Characteristic varieties and vanishing cycles, Invent. Math., 84, 327-402, (1986) · Zbl 0598.32013
[34] Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/. Accessed 1 Jan 2018
[35] Grozin, A.G.: Integration by parts: an introduction. Int. J. Mod. Phys. A 26(17), 2807-2854 (2011). arXiv:1104.3993 [hep-ph]. Extended Version of the Lectures at the School Computer Algebra and Particle Physics at DESY Zeuthen, Germany, March 21-25 (2011) · Zbl 1247.81138
[36] Gyoja, A., Bernstein-Sato’s polynomial for several analytic functions, J. Math. Kyoto Univ., 33, 399-411, (1993) · Zbl 0797.32007
[37] Hahn, T., Generating Feynman diagrams and amplitudes with FeynArts 3, Comput. Phys. Commun., 140, 418-431, (2001) · Zbl 0994.81082
[38] Helmer, M., Algorithms to compute the topological Euler characteristic. Chern-Schwartz-MacPherson class and Segre class of projective varieties, J. Symb. Comput., 73, 120-138, (2016) · Zbl 1349.14028
[39] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601, (2013)
[40] Ita, H., Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev. D, 94, 116015, (2016)
[41] Kalmykov, MY; Kniehl, BA, Counting master integrals: integration by parts vs. differential reduction, Phys. Lett. B, 702, 268-271, (2011)
[42] Kalmykov, MY; Kniehl, BA, Mellin-Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions, Phys. Lett. B, 714, 103-109, (2012)
[43] Kalmykov, MY; Kniehl, BA, Counting the number of master integrals for sunrise diagrams via the Mellin-Barnes representation, JHEP, 2017, 031, (2017) · Zbl 1380.81423
[44] Kashiwara, M., \(B\)-functions and holonomic systems. Rationality of roots of \(B\)-functions, Invent. Math., 38, 33-53, (1976) · Zbl 0354.35082
[45] Kashiwara, M., On the holonomic systems of linear differential equations, II, Invent. Math., 49, 121-135, (1978) · Zbl 0401.32005
[46] Kashiwara, M.; Kawai, T., Holonomic systems of linear differential equations and Feynman integrals, Publ. Res. Inst. Math. Sci. Kyoto, 12, 131-140, (1977) · Zbl 0449.35095
[47] Khovanskii, AG, Newton polyhedra and the genus of complete intersections, Funct. Anal. Appl., 12, 38-46, (1978) · Zbl 0406.14035
[48] Kniehl, BA; Kotikov, AV, Counting master integrals: integration-by-parts procedure with effective mass, Phys. Lett. B, 712, 233-234, (2012)
[49] Kouchnirenko, AG, Polyèdres de Newton et nombres de Milnor, Invent. Math., 32, 1-31, (1976) · Zbl 0328.32007
[50] Laporta, S., High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A, 15, 5087-5159, (2000) · Zbl 0973.81082
[51] Laumon, G.: Sur la catégorie dérivée des \({\cal{D}}\)-modules filtrés, vol. 1016 of Lecture Notes in Mathematics, pp. 151-237. Springer, Berlin (1983)
[52] Lee, RN, Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals, J. High Energy Phys., 2008, 31, (2008)
[53] Lee, R.N.: Calculating multiloop integrals using dimensional recurrence relation and \(\cal{D}\)-analyticity. Nucl. Phys. B Proc. Suppl. 205, 135-140. arXiv:1007.2256 [hep-ph]. 10th DESY Workshop on Elementary Particle Theory: Loops and Legs in Quantum Field Theory, Woerlitz, Germany, April 25-30 (2010)
[54] Lee, RN, Space-time dimensionality \(\cal{D}\) as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to \(\cal{D}\), Nucl. Phys. B, 830, 474-492, (2010) · Zbl 1203.83051
[55] Lee, R.N.: Presenting LiteRed: a tool for the Loop InTEgrals REDuction. Preprint arXiv:1212.2685 [hep-ph] (2012)
[56] Lee, R.N.: Modern techniques of multiloop calculations. In: Proceedings, 49th Rencontres de Moriond on QCD and High Energy Interactions, pp. 297-300. arXiv:1405.5616 [hep-ph] (2014)
[57] Lee, RN, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser., 523, 012059, (2014)
[58] Lee, RN; Pomeransky, AA, Critical points and number of master integrals, JHEP, 2013, 165, (2013) · Zbl 1342.81139
[59] Loeser, F.; Sabbah, C., Caractérisation des \(\cal{D}\) -modules hypergéométriques irréductibles sur le tore, C. R. Acad. Sci. Paris Sér. I Math., 312, 735-738, (1991) · Zbl 0753.14015
[60] Loeser, F.; Sabbah, C., Equations aux différences finies et déterminants d’intégrales de fonctions multiformes, Comment. Math. Helv., 66, 458-503, (1991) · Zbl 0760.39001
[61] Loeser, F.; Sabbah, C., Caractérisation des \(\cal{D}\)-modules hypergéométriques irréductibles sur le tore, II, C. R. Acad. Sci. Paris Sér. I Math., 315, 1263-1264, (1992) · Zbl 0774.14017
[62] Maierhöfer, P., Usovitsch, J., Uwer, P.: Kira—a Feynman integral reduction program. arXiv:1705.05610 [hep-ph] (2017)
[63] Malgrange, B.: Le polynôme de Bernstein d’une singularité isolée, vol. 459 of Lecture Notes in Mathematics, pp. 98-119. Springer, Berlin (1975)
[64] Nakanishi, N.: Graph theory and Feynman integrals. Mathematics and Its Applications, vol. 11. Gordon and Breach, New York (1971) · Zbl 0212.29203
[65] Oaku, T., Algorithms for the \(b\)-function and \(D\) -modules associated with a polynomial, J. Pure Appl. Algebra, 117—-118, 495-518, (1997) · Zbl 0918.32006
[66] Oaku, T.; Takayama, N., An algorithm for de Rham cohomology groups of the complement of an affine variety via \(D\)-module computation, J. Pure Appl. Algebra, 139, 201-233, (1999) · Zbl 0960.14008
[67] Oaku, T.; Takayama, N., Algorithms for \(D\)-modules—restriction, tensor product, localization, and local cohomology groups, J. Pure Appl. Algebra, 156, 267-308, (2001) · Zbl 0983.13008
[68] Passarino, G., An approach toward the numerical evaluation of multi-loop Feynman diagrams, Nucl. Phys. B, 619, 257-312, (2001) · Zbl 0991.81080
[69] Passarino, G.; Uccirati, S., Algebraic-numerical evaluation of Feynman diagrams: two-loop self-energies, Nucl. Phys. B, 629, 97-187, (2002) · Zbl 1039.81539
[70] Ruijl, B.; Ueda, T.; Vermaseren, JAM, The diamond rule for multi-loop Feynman diagrams, Phys. Lett. B, 746, 347-350, (2015) · Zbl 1343.81104
[71] Sabbah, C., Proximité évanescente. II. Équations fonctionnelles pour plusieurs fonctions analytiques, Compos. Math., 64, 213-241, (1987) · Zbl 0632.32006
[72] Saito, M., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations, Vol. 6 of Algorithms and Computation in Mathematics. Springer, Berlin (2000) · Zbl 0946.13021
[73] Sato, M.; Shintani, T.; Muro, M., Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note, Nagoya Math. J., 120, 1-34, (1990) · Zbl 0715.22014
[74] Schnetz, O., Quantum field theory over \(\mathbb{F}_{q}\), Electron. J. Combin., 18, p102, (2011) · Zbl 1217.05110
[75] Smirnov, AV, Algorithm FIRE—Feynman Integral REduction, J. High Energy Phys., 10, 107, (2008) · Zbl 1245.81033
[76] Smirnov, AV, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun., 189, 182-191, (2015) · Zbl 1344.81030
[77] Smirnov, AV; Petukhov, AV, The number of master integrals is finite, Lett. Math. Phys., 97, 37-44, (2011) · Zbl 1216.81076
[78] Smirnov, AV; Smirnov, VA, Applying Gröbner bases to solve reduction problems for Feynman integrals, J. High Energy Phys., 1, 001, (2006)
[79] Smirnov, A.V., Smirnov, V.A.: On the reduction of Feynman integrals to master integrals. In: Proceedings, 11th International Workshop on Advanced computing and analysis techniques in physics research (ACAT 2007), Vol. ACAT2007, p. 085. arXiv:0707.3993 [hep-ph] (2007)
[80] Smirnov, AV; Smirnov, VA, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun., 184, 2820-2827, (2013) · Zbl 1344.81031
[81] Smirnov, V.A.: Analytic Tools for Feynman integrals. Springer Tracts in Modern Physics, vol. 250. Springer, Berlin (2012) · Zbl 1268.81004
[82] Smirnov, VA; Steinhauser, M., Solving recurrence relations for multi-loop Feynman integrals, Nucl. Phys. B, 672, 199-221, (2003) · Zbl 1058.81606
[83] Speer, E.R.: Generalized Feynman Amplitudes. Annals of Mathematics Studies, vol. 62. Princeton University Press, Princeton (1969)
[84] Speer, ER, Ultraviolet and infrared singularity structure of generic Feynman amplitudes, Ann. Inst. H. Poincaré Sect. A, 23, 1-21, (1975)
[85] Stembridge, JR, Counting points on varieties over finite fields related to a conjecture of Kontsevich, Ann. Comb., 2, 365-385, (1998) · Zbl 0927.05002
[86] Studerus, C., Reduze—Feynman integral reduction in C++, Comput. Phys. Commun., 181, 1293-1300, (2010) · Zbl 1219.81133
[87] ’t Hooft, G.; Veltman, M., Regularization and renormalization of gauge fields, Nucl. Phys. B, 44, 189-213, (1972)
[88] Tancredi, L., Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations, Nucl. Phys. B, 901, 282-317, (2015) · Zbl 1332.81065
[89] Tarasov, OV, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D, 54, 6479-6490, (1996) · Zbl 0925.81121
[90] Tarasov, O.V.: Reduction of Feynman graph amplitudes to a minimal set of basic integrals. In: Loops and Legs in Gauge Theories. Proceedings, Zeuthen Workshop on Elementary Particle Theory, Rheinsberg, Germany, April 19-24, 1998, vol. 29, p. 2655. arXiv:hep-ph/9812250 (1998)
[91] Tarasov, O.V.: Massless on-shell box integral with arbitrary powers of propagators. Preprint arXiv:1709.07526 [hep-ph] (2017) · Zbl 1396.81096
[92] Tkachov, FV, A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett. B, 100, 65-68, (1981)
[93] Tkachov, F.V.: Algebraic algorithms for multiloop calculations The first 15 years. What’s next? Nucl. Instrum. Methods Phys. Res. Sect. A 389, 309-313 (1997). arXiv:hep-ph/9609429. New Computing Techniques in Physics Research V. Proceedings, 5th International Workshop AIHENP, Lausanne, September 2-6 (1996)
[94] Turnbull, HW, Symmetric determinants and the Cayley and Capelli operator, Proc. Edinb. Math. Soc., 8, 76-86, (1948) · Zbl 0036.15002
[95] Vermaseren, JAM, Axodraw, Comput. Phys. Commun., 83, 45-58, (1994) · Zbl 1114.68598
[96] Manteuffel, A.; Panzer, E.; Schabinger, RM, A quasi-finite basis for multi-loop Feynman integrals, J. High Energy Phys., 2015, 120, (2015) · Zbl 1388.81378
[97] Manteuffel, A.; Schabinger, RM, A novel approach to integration by parts reduction, Phys. Lett. B, 744, 101-104, (2015) · Zbl 1330.81151
[98] von Manteuffel, A., Studerus, C.: Reduze 2—distributed Feynman integral reduction. Preprint arXiv:1201.4330 [hep-ph] (2012)
[99] Zhang, Y.: Integration-by-parts identities from the viewpoint of differential geometry. In: 19th Itzykson Meeting on Amplitudes 2014 (Itzykson2014) Gif-sur-Yvette, France, June 10-13, 2014. arXiv:1408.4004 [hep-th] (2014)
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