Abstract
We use the GKZ description of periods and certain classes of relative periods on families of Barth-Nieto Calabi-Yau (l − 1)-folds in order to solve the l-loop banana amplitudes with their general mass dependence. As examples we compute the mass dependencies of the banana amplitudes up to the three-loop case and check the results against the known results for special mass values.
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I.M. Gel’fand, A.V. Zelevinskiĭ and M.M. Kapranov, Hypergeometric functions and toric varieties, Funkt. Anal. Prilozhen.23 (1989) 12.
C. Bogner and S. Weinzierl, Periods and Feynman integrals, J. Math. Phys.50 (2009) 042302 [arXiv:0711.4863] [INSPIRE].
P.A. Griffiths, Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties, Amer. J. Math.90 (1968) 568.
P.A. Griffiths, Periods of integrals on algebraic manifolds. II. Local study of the period mapping, Amer. J. Math.90 (1968) 805.
I.M. Gel’fand, A. V. Zelevinskiĭ and M.M. Kapranov, Newton polyhedra of principal A-determinants, Dokl. Akad. Nauk SSSR308 (1989) 20.
I.M. Gel’fand, M.M. Kapranov and A.V. Zelevinsky, Generalized Euler integrals and A-hypergeometric functions, Adv. Math.84 (1990) 255.
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].
S. Hosono, B.H. Lian and S.-T. Yau, GKZ generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces, Commun. Math. Phys.182 (1996) 535 [alg-geom/9511001] [INSPIRE].
P.A. Griffiths, On the periods of certain rational integrals. I, II, Ann. Math.90 (1969) 460.
O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev.D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].
R.N. Lee, Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys.B 830 (2010) 474 [arXiv:0911.0252] [INSPIRE].
J.L. Bourjaily et al., Traintracks through Calabi-Yau manifolds: scattering amplitudes beyond elliptic polylogarithms, Phys. Rev. Lett.121 (2018) 071603 [arXiv:1805.09326] [INSPIRE].
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].
J.L. Bourjaily et al., Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space, JHEP01 (2020) 078 [arXiv:1910.01534] [INSPIRE].
A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys.B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE].
L. de la Cruz, Feynman integrals as A-hypergeometric functions, JHEP12 (2019) 123 [arXiv:1907.00507] [INSPIRE].
R.P. Klausen, Hypergeometric series representations of Feynman integrals by GKZ hypergeometric systems, arXiv:1910.08651 [INSPIRE].
S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, Adv. Theor. Math. Phys.21 (2017) 1373.
R. Bonciani et al., Evaluating a family of two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence, JHEP01 (2020) 132 [arXiv:1907.13156] [INSPIRE].
M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP11 (2013) 112 [arXiv:1308.0619] [INSPIRE].
S. Abreu, M. Becchetti, C. Duhr and R. Marzucca, Three-loop contributions to the ρ parameter and iterated integrals of modular forms, JHEP02 (2020) 050 [arXiv:1912.02747] [INSPIRE].
H.A. Verrill, Root lattices and pencils of varieties, J. Math. Kyoto Univ.36 (1996) 423.
P. Vanhove, The physics and the mixed Hodge structure of Feynman integrals, Proc. Symp. Pure Math.88 (2014) 161 [arXiv:1401.6438] [INSPIRE].
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom.3 (1994) 493.
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].
V.V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Comm. Math. Phys.168 (1995) 493.
V.V. Batyrev and L.A. Borisov, On Calabi-Yau complete intersections in toric varieties, in Higher-dimensional complex varieties, M. Andreatta et al. eds., de Gruyter, Berlin (1996).
B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys.173 (1995) 559 [hep-th/9402119] [INSPIRE].
P. Mayr, Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four folds, Nucl. Phys.B 494 (1997) 489 [hep-th/9610162] [INSPIRE].
A. Klemm, B. Lian, S.S. Roan and S.-T. Yau, Calabi-Yau fourfolds for M-theory and F-theory compactifications, Nucl. Phys.B 518 (1998) 515 [hep-th/9701023] [INSPIRE].
N. Cabo Bizet, A. Klemm and D. Vieira Lopes, Landscaping with fluxes and the E8 Yukawa Point in F-theory, arXiv:1404.7645 [INSPIRE].
A. Gerhardus and H. Jockers, Quantum periods of Calabi-Yau fourfolds, Nucl. Phys.B 913 (2016) 425 [arXiv:1604.05325] [INSPIRE].
A. Klemm, The B-model approach to topological string theory on Calabi-Yau n-folds, in B-model Gromov-Witten theory, E. Clader and Y.B. Ruan eds., Springer, Germany (2018).
T.-F. Feng, C.-H. Chang, J.-B. Chen and H.-B. Zhang, GKZ-hypergeometric systems for Feynman integrals, Nucl. Phys.B 953 (2020) 114952 [arXiv:1912.01726] [INSPIRE].
C. Bogner and S. Weinzierl, Feynman graph polynomials, Int. J. Mod. Phys.A 25 (2010) 2585 [arXiv:1002.3458] [INSPIRE].
R. Bott and L.W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics volume 82, Springer, Germany (1982).
D.A. Cox, J.B. Little and H.K. Schenck, Toric varieties, Graduate Studies in Mathematics volume 124, American Mathematical Society, Providence, U.S.A. (2011).
http://doc.sagemath.org/html/en/reference/schemes/sage/schemes/toric/variety.html
S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, Adv. Theor. Math. Phys.21 (2017) 1373 [arXiv:1601.08181] [INSPIRE].
V.V. Batyrev and D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Commun. Math. Phys.168 (1995) 493 [alg-geom/9307010] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys.A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
R.L. Bryant and P.A. Griffiths, Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle, in Arithmetic and geometry. Volume II, Y. Tschinkel and Y.G. Zarhin eds., Birkhäuser, Boston U.S.A. (1983).
S. Li, B.H. Lian and S.-T. Yau, Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces, arXiv:0910.4215 [INSPIRE].
C. Itzykson and J.B. Zuber, Quantum field theory, International Series In Pure and Applied Physics, McGraw-Hill, New York U.S.A. (1980).
P. Vanhove, The physics and the mixed Hodge structure of Feynman integrals, Proc. Symp. Pure Math.88 (2014) 161 [arXiv:1401.6438] [INSPIRE].
J. Broedel et al., An analytic solution for the equal-mass banana graph, JHEP09 (2019) 112 [arXiv:1907.03787] [INSPIRE].
A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph, Nucl. Phys.B 921 (2017) 316 [arXiv:1704.05465] [INSPIRE].
D. Maulik, R. Pandharipande and R.P. Thomas, Curves on K 3 surfaces and modular forms, J. Topol.3 (2010) 937.
G. Oberdieck and R. Pandharipande, Curve counting on K 3 × E, the Igusa cusp form χ10, and descendent integration, in K 3 surfaces and their moduli, C. Faber et al. eds., Springer, Germany (2016).
B.H. Lian and S.-T. Yau, Mirror maps, modular relations and hypergeometric series 1, hep-th/9507151 [INSPIRE].
J.H. Conway and S.P. Norton, Monstrous moonshine, Bull. London Math. Soc.11 (1979) 308.
A.R. Forsyth, Theory of differential equations. 1. Exact equations and Pfaff ’s problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations, Dover Publications Inc., New York U.S.A. (1959).
W. Lerche, D.J. Smit and N.P. Warner, Differential equations for periods and flat coordinates in two-dimensional topological matter theories, Nucl. Phys.B 372 (1992) 87 [hep-th/9108013] [INSPIRE].
R.E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math.132 (1998) 491.
A. Klemm and M. Mariño, Counting BPS states on the enriques Calabi-Yau, Commun. Math. Phys.280 (2008) 27 [hep-th/0512227] [INSPIRE].
T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct integration of the topological string, JHEP08 (2007) 058 [hep-th/0702187] [INSPIRE].
G. Almkvist, C. van Enckevort, D. van Straten and W. Zudilin, Tables of Calabi-Yau equations, math.AG/0507430.
D. van Straten, Calabi-Yau operators, in Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds & Picard-Fuchs equations, L. Ji and S.T. Yau eds., International Press, Somerville U.S.A. (2018).
P. Candelas, X. de la Ossa, M. Elmi and D. van Straten, A one parameter family of calabi-yau manifolds with attractor points of rank two, to appear.
A. Klemm, E. Scheidegger and D. Zagier, Periods and quasiperiods of modular forms and d-brane masses for the mirror quintic, to appear.
K. Boenisch et al., to appear.
H. Frellesvig and C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation, JHEP04 (2017) 083 [arXiv:1701.07356] [INSPIRE].
J. Bosma, M. Sogaard and Y. Zhang, Maximal cuts in arbitrary dimension, JHEP08 (2017) 051 [arXiv:1704.04255] [INSPIRE].
M. Harley, F. Moriello and R.M. Schabinger, Baikov-Lee representations of cut Feynman integrals, JHEP06 (2017) 049 [arXiv:1705.03478] [INSPIRE].
A. Gerhardus, H. Jockers and U. Ninad, The geometry of gauged linear σ-model correlation functions, Nucl. Phys.B 933 (2018) 65 [arXiv:1803.10253] [INSPIRE].
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Klemm, A., Nega, C. & Safari, R. The l-loop banana amplitude from GKZ systems and relative Calabi-Yau periods. J. High Energ. Phys. 2020, 88 (2020). https://doi.org/10.1007/JHEP04(2020)088
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DOI: https://doi.org/10.1007/JHEP04(2020)088