Abstract
We present a prescription for choosing orthogonal bases of differential n-forms belonging to quadratic twisted period integrals, with respect to the intersection number inner product. To evaluate these inner products, we additionally propose a new closed formula for intersection numbers beyond d log forms. These findings allow us to systematically construct orthonormal bases between twisted period integrals of this type. In the context of Feynman integrals, this represents all diagrams at one-loop.
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References
Y. Goto, Twisted cycles and twisted period relations for lauricella’s hypergeometric function fc, Int. J. Math. 24 (2013) 1350094 [arXiv:1308.5535] [INSPIRE].
Y. Goto, Twisted period relations for Lauricella’s hypergeometric function FA, arXiv:1310.6088.
Y. Goto and K. Matsumoto, The monodromy representation and twisted period relations for Appell’s hypergeometric function F4, arXiv:1310.4243.
K. Matsumoto, Quadratic identities for hypergeometric series of type (k,l), Kyushu J. Math. 48 (1994) 335.
K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math. J. 139 (2016) 67 [INSPIRE].
K. Matsumoto, Intersection numbers for 1-forms associated with confluent hypergeometric functions, Funkcial. Ekvac. 41 (1998) 291.
K. Matsumoto, Intersection numbers for logarithmic k-forms, Osaka J. Math. 35 (1998) 873.
K. Mimachi and M. Yoshida, Intersection numbers of twisted cycles and the correlation functions of the conformal field theory. II, Commun. Math. Phys. 234 (2003) 339 [math/0208097] [INSPIRE].
K. Mimachi and M. Yoshida, Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory, Commun. Math. Phys. 250 (2004) 23 [INSPIRE].
S.-J. Matsubara-Heo, Computing cohomology intersection numbers of GKZ hypergeometric systems, PoS MA2019 (2022) 013 [arXiv:2008.03176] [INSPIRE].
Y. Goto and S.-J. Matsubara-Heo, Homology and cohomology intersection numbers of GKZ systems, arXiv:2006.07848.
S.-J. Matsubara-Heo and N. Takayama, An algorithm of computing cohomology intersection number of hypergeometric integrals, arXiv:1904.01253 [INSPIRE].
S.-J. Matsubara-Heo, Localization formulas of cohomology intersection numbers, J. Math. Soc. Jap. 75 (2023) 909 [arXiv:2104.12584] [INSPIRE].
S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory, JHEP 02 (2019) 139 [arXiv:1810.03818] [INSPIRE].
H. Frellesvig et al., Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers, JHEP 05 (2019) 153 [arXiv:1901.11510].
H. Frellesvig et al., Vector Space of Feynman Integrals and Multivariate Intersection Numbers, Phys. Rev. Lett. 123 (2019) 201602 [arXiv:1907.02000] [INSPIRE].
A.V. Smirnov and A.V. Petukhov, The Number of Master Integrals is Finite, Lett. Math. Phys. 97 (2011) 37 [arXiv:1004.4199] [INSPIRE].
R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].
P.A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth. A 389 (1997) 347 [hep-ph/9611449] [INSPIRE].
H. Frellesvig et al., Decomposition of Feynman Integrals by Multivariate Intersection Numbers, JHEP 03 (2021) 027 [arXiv:2008.04823] [INSPIRE].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
S. Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
J. Chen, X. Jiang, X. Xu and L.L. Yang, Constructing canonical Feynman integrals with intersection theory, Phys. Lett. B 814 (2021) 136085 [arXiv:2008.03045] [INSPIRE].
J. Chen et al., Baikov representations, intersection theory, and canonical Feynman integrals, JHEP 07 (2022) 066 [arXiv:2202.08127] [INSPIRE].
J. Chen, B. Feng and L.L. Yang, Intersection theory rules symbology, Sci. China Phys. Mech. Astron. 67 (2024) 221011 [arXiv:2305.01283] [INSPIRE].
F. Gasparotto, A. Rapakoulias and S. Weinzierl, Nonperturbative computation of lattice correlation functions by differential equations, Phys. Rev. D 107 (2023) 014502 [arXiv:2210.16052] [INSPIRE].
F. Gasparotto, S. Weinzierl and X. Xu, Real time lattice correlation functions from differential equations, JHEP 06 (2023) 128 [arXiv:2305.05447] [INSPIRE].
S.L. Cacciatori and P. Mastrolia, Intersection Numbers in Quantum Mechanics and Field Theory, arXiv:2211.03729 [INSPIRE].
S. De and A. Pokraka, Cosmology meets cohomology, JHEP 03 (2024) 156 [arXiv:2308.03753] [INSPIRE].
G. Brunello et al., Fourier calculus from intersection theory, Phys. Rev. D 109 (2024) 094047 [arXiv:2311.14432] [INSPIRE].
G. Brunello and S. De Angelis, An improved framework for computing waveforms, JHEP 07 (2024) 062 [arXiv:2403.08009] [INSPIRE].
H. Frellesvig and T. Teschke, General relativity from intersection theory, Phys. Rev. D 110 (2024) 044028.
R. Bhardwaj, A. Pokraka, L. Ren and C. Rodriguez, A double copy from twisted (co)homology at genus one, JHEP 07 (2024) 040 [arXiv:2312.02148] [INSPIRE].
S. Mizera and A. Pokraka, From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals, JHEP 02 (2020) 159 [arXiv:1910.11852] [INSPIRE].
S. Weinzierl, Correlation functions on the lattice and twisted cocycles, Phys. Lett. B 805 (2020) 135449 [arXiv:2003.05839] [INSPIRE].
V. Chestnov et al., Macaulay matrix for Feynman integrals: linear relations and intersection numbers, JHEP 09 (2022) 187 [arXiv:2204.12983] [INSPIRE].
M. Giroux and A. Pokraka, Loop-by-loop differential equations for dual (elliptic) Feynman integrals, JHEP 03 (2023) 155 [arXiv:2210.09898] [INSPIRE].
X. Jiang, M. Lian and L.L. Yang, Recursive structure of Baikov representations: The top-down reduction with intersection theory, Phys. Rev. D 109 (2024) 076020 [arXiv:2312.03453] [INSPIRE].
K. Matsumoto, Relative twisted homology and cohomology groups associated with Lauricella’s FD, arXiv:1804.00366.
S. Weinzierl, On the computation of intersection numbers for twisted cocycles, J. Math. Phys. 62 (2021) 072301 [arXiv:2002.01930] [INSPIRE].
S. Caron-Huot and A. Pokraka, Duals of Feynman integrals. Part I. Differential equations, JHEP 12 (2021) 045.
S. Caron-Huot and A. Pokraka, Duals of Feynman Integrals. Part II. Generalized unitarity, JHEP 04 (2022) 078.
G. Fontana and T. Peraro, Reduction to master integrals via intersection numbers and polynomial expansions, JHEP 08 (2023) 175 [arXiv:2304.14336] [INSPIRE].
V. Chestnov et al., Intersection numbers from higher-order partial differential equations, JHEP 06 (2023) 131 [arXiv:2209.01997] [INSPIRE].
G. Brunello et al., Intersection Numbers, Polynomial Division and Relative Cohomology, arXiv:2401.01897 [INSPIRE].
S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization, Ph.D. thesis, Institute for Advanced Study (IAS), Princeton, NJ, 08540, U.S.A. (2020) [arXiv:1906.02099] [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP 07 (2019) 031 [arXiv:1905.08019] [INSPIRE].
D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun. 83 (1994) 45 [INSPIRE].
F. Gasparotto, Co-Homology and Intersection Theory for Feynman Integrals, Ph.D. thesis, Università degli Studi di Padova, Padua, Italy (2023) [INSPIRE].
S. Weinzierl, Feynman Integrals, Springer International Publishing (2022) [https://doi.org/10.1007/978-3-030-99558-4].
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser Boston (1994) [https://doi.org/10.1007/978-0-8176-4771-1].
D. Cox, J. Little and D. O’Shea, Using Algebraic Geometry, first edition, Springer New York (1998) [https://doi.org/10.1007/978-1-4757-6911-1].
C. D’Andrea and A. Dickenstein, Explicit formulas for the multivariate resultant, math/0007036.
Acknowledgments
We wish to thank Pierpaolo Mastrolia, Wojciech Flieger, Manoj Mandal, Vsevolod Chestnov, Federico Gasparotto, Mathieu Giroux, Giacomo Brunello and Sebastian Mizera for ideas, comments and feedback. We would additionally like to thank Andrzej Pokraka and Hjalte Frellesvig for invaluable discussions relating to all parts of the work. G.C. thanks Saiei-Jaeyeong Matsubara-Heo, Julian Miczajka and Francesco Calisto for stimulating conversations related to multivariate discriminants, as well as Johannes Henn and the Max Planck institute for Physics for hosting him during the development of this project.
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Crisanti, G., Smith, S. Feynman integral reductions by intersection theory with orthogonal bases and closed formulae. J. High Energ. Phys. 2024, 18 (2024). https://doi.org/10.1007/JHEP09(2024)018
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DOI: https://doi.org/10.1007/JHEP09(2024)018