×

zbMATH — the first resource for mathematics

The elliptic dilogarithm for the sunset graph. (English) Zbl 1319.81044
Summary: We study the sunset graph defined as the scalar two-point self-energy at two-loop order. We evaluated the sunset integral for all identical internal masses in two dimensions. We give two calculations for the sunset amplitude; one based on an interpretation of the amplitude as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the amplitude in this case is a family of periods associated to the universal family of elliptic curves over the modular curve \(X_1(6)\). We show that the integral is given by an elliptic dilogarithm evaluated at a sixth root of unity modulo periods. We explain as well how this elliptic dilogarithm value is related to the regulator of a class in the motivic cohomology of the universal elliptic family.

MSC:
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
14F42 Motivic cohomology; motivic homotopy theory
Software:
CutTools; OEIS; SageMath
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Adams, L.; Bogner, C.; Weinzierl, S., The two-loop sunrise graph with arbitrary masses · Zbl 1282.81193
[2] Bailey, David H.; Borwein, David; Borwein, Jonathan M.; Crandall, Richard, Hypergeometric forms for Ising-class integrals, Experiment. Math., 16, 3, 257-276, (2007) · Zbl 1134.33016
[3] Bailey, D. H.; Borwein, J. M.; Broadhurst, D.; Glasser, M. L., Elliptic integral evaluations of Bessel moments · Zbl 1152.33003
[4] Bauberger, S.; Bohm, M.; Weiglein, G.; Berends, F. A.; Buza, M., Calculation of two loop selfenergies in the electroweak standard model, Nuclear Phys. B Proc. Suppl., 37, 95, (1994)
[5] Beauville, A., LES familles stables de courbes elliptiques sur \(\mathbb{P}^1\) admettant quatre fibres singulières, C. R. Acad. Sci. Paris Sér I, 294, 657, (1982) · Zbl 0504.14016
[6] Bern, Z.; Dixon, L. J.; Kosower, D. A., Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci., 46, 109, (1996)
[7] Bern, Z.; Dixon, L. J.; Kosower, D. A., One loop amplitudes for \(e^+ e^-\) to four partons, Nuclear Phys. B, 513, 3, (1998)
[8] Bertin, J.; Demailly, P.-P.; Illusie, L.; Peters, C., Introduction à la théorie de Hodge, Panor. Syntheses, vol. 3, (1996), Soc. Math. France Paris · Zbl 0849.14002
[9] Bloch, Spencer J., Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, (2000), University of Chicago - AMS, CRM · Zbl 0958.19001
[10] Bloch, S.; Esnault, H.; Kreimer, D., On motives associated to graph polynomials, Comm. Math. Phys., 267, 181, (2006) · Zbl 1109.81059
[11] Bloch, S.; Kreimer, D., Feynman amplitudes and Landau singularities for 1-loop graphs, Commun. Number Theory Phys., 4, 709, (2010) · Zbl 1239.81043
[12] Britto, R., Loop amplitudes in gauge theories: modern analytic approaches, J. Phys. A, 44, 454006, (2011) · Zbl 1270.81132
[13] Britto, R.; Cachazo, F.; Feng, B., Generalized unitarity and one-loop amplitudes in \(\mathcal{N} = 4\) super-Yang-Mills, Nuclear Phys. B, 725, 275, (2005) · Zbl 1178.81202
[14] Broadhurst, D. J., The master two loop diagram with masses, Z. Phys. C, 47, 115, (1990)
[15] Broadhurst, D., Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function
[16] Broadhurst, D. J.; Fleischer, J.; Tarasov, O. V., Two loop two point functions with masses: asymptotic expansions and Taylor series, in any dimension, Z. Phys. C, 60, 287, (1993)
[17] Caffo, M.; Czyz, H.; Laporta, S.; Remiddi, E., The master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cimento A, 111, 365, (1998)
[18] Caron-Huot, S.; Larsen, K. J., Uniqueness of two-loop master contours, J. High Energy Phys., 1210, 026, (2012)
[19] Davydychev, A. I.; Delbourgo, R., A geometrical angle on Feynman integrals, J. Math. Phys., 39, 4299, (1998) · Zbl 0986.81082
[20] Davydychev, A. I.; Tausk, J. B., A magic connection between massive and massless diagrams, Phys. Rev. D, 53, 7381, (1996)
[21] Deligne, P., Formes modulaires et représentations -adiques, Séminaire Bourbaki, 21, (1968-1969), Exposé 355 · Zbl 0206.49901
[22] Deligne, P., Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci., 40, 5-58, (1971) · Zbl 0219.14007
[23] Deligne, Pierre, Théorie de Hodge III, Publ. Math. Inst. Hautes Études Sci., 44, 5-77, (1974) · Zbl 0237.14003
[24] Dieudonné, J., Calcul infinitesimal, (1980), Hermann · Zbl 0497.26004
[25] Doran, C. F.; Kerr, M., Algebraic K-theory of toric hypersurfaces, Commun. Number Theory Phys., 5, 2, 397-600, (2011) · Zbl 1274.19003
[26] Groote, S.; Korner, J. G.; Pivovarov, A. A., A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques, Eur. Phys. J. C, 72, 2085, (2012)
[27] Itzykson, C.; Zuber, J. B., Quantum field theory, Int. Ser. Pure Appl. Phys., (1980), Mcgraw-Hill New York, USA, 705 pp · Zbl 0453.05035
[28] Johansson, H.; Kosower, D. A.; Larsen, K. J., Two-loop maximal unitarity with external masses, Phys. Rev. D, 87, 025030, (2013)
[29] Johansson, H.; Kosower, D. A.; Larsen, K. J., Maximal unitarity for the four-mass double box, Phys. Rev. D, 89, 125010, (2014)
[30] Kalmykov, M. Y.; Kniehl, B. A., Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters, Nuclear Phys. B, 809, 365, (2009) · Zbl 1192.81351
[31] Katz, N.; Oda, T., On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ., 8, 199-213, (1968) · Zbl 0165.54802
[32] Kerr, M.; Lewis, J. D.; Müller-Stach, S., The Abel-Jacobi map for higher Chow groups, Compos. Math., 142, 2, 374-396, (2006) · Zbl 1123.14006
[33] Kniehl, B. A.; Kotikov, A. V.; Onishchenko, A.; Veretin, O., Two-loop sunset diagrams with three massive lines, Nuclear Phys. B, 738, 306, (2006) · Zbl 1109.81331
[34] Koblitz, N., Introduction to elliptic curves and modular forms, Grad. Texts in Math., vol. 97, (1993), Springer-Verlag · Zbl 0804.11039
[35] Kosower, D. A.; Larsen, K. J., Maximal unitarity at two loops, Phys. Rev. D, 85, 045017, (2012)
[36] Laporta, S.; Remiddi, E., Analytic treatment of the two loop equal mass sunrise graph, Nuclear Phys. B, 704, 349, (2005) · Zbl 1119.81356
[37] Maier, R. S., On rationally parametrized modular equations, J. Ramanujan Math. Soc., 24, 1-73, (2009) · Zbl 1214.11049
[38] Müller-Stach, S.; Weinzierl, S.; Zayadeh, R., A second-order differential equation for the two-loop sunrise graph with arbitrary masses, Commun. Number Theory Phys., 6, 203, (2012) · Zbl 1275.81069
[39] Müller-Stach, S.; Weinzierl, S.; Zayadeh, R., Picard-Fuchs equations for Feynman integrals, Comm. Math. Phys., 326, 237, (2014) · Zbl 1285.81029
[40] OEIS Foundation Inc., The on-line encyclopedia of integer sequences, (2011)
[41] Ossola, G.; Papadopoulos, C. G.; Pittau, R., Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nuclear Phys. B, 763, 147, (2007) · Zbl 1116.81067
[42] (Rapoport, M.; Schappacher, N.; Schneidert, P., Beilinson’s Conjectures on Special Values of L-Functions, Perspect. Math., vol. 4, (1988), Academic Press)
[43] Smirnov, V. A., Evaluating Feynman integrals, Springer Tracts Modern Phys., vol. 211, (2004), Springer · Zbl 1098.81003
[44] Soulé, C., Régulateurs, Séminaire Bourbaki, 27, (1984-1985), Exposé No. 644 · Zbl 0617.14008
[45] Stein, W. A., Sage mathematics software, (2013), (Version 5.11), The Sage Development Team
[46] Stienstra, J.; Beukers, F., On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces, Math. Ann., 271, 2, 269-304, (1985) · Zbl 0539.14006
[47] Tarasov, O. V., Hypergeometric representation of the two-loop equal mass sunrise diagram, Phys. Lett. B, 638, 195, (2006) · Zbl 1248.81136
[48] Weil, A., Elliptic functions according to Eisenstein and Kronecker, Ergeb. Math. Grenzgeb., vol. 88, (1976), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0318.33004
[49] Zagier, Don, The Bloch-Wigner-ramakrishnan polylogarithm function, Math. Ann., 286, 613-624, (1990) · Zbl 0698.33001
[50] Zagier, Don, The dilogarithm function, (Cartier, Pierre; Julia, Bernard; Moussa, Pierre; Vanhove, Pierre, Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization, (2007), Springer) · Zbl 1176.11026
[51] Zagier, D., Integral solutions of apéry-like recurrence equations, (Groups and Symmetries: From the Neolithic Scots to John McKay, CRM Proc. Lecture Notes, vol. 47, (2009), Amer. Math. Soc.), 349-366 · Zbl 1244.11042
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.