Abstract
In this paper, we develop an iterative sector-level reduction strategy for Feynman integrals, which bases on module intersection in the Baikov representation and auxiliary vector for tensor structure. Using this strategy we have studied the reduction of general one-loop integrals, i.e., integrals having arbitrary tensor structures and arbitrary power for propagators. Inspired by these studies, a uniform and compact formula that iteratively reduces all one-loop integrals has been written down, where messy polynomials in integration-by-parts (IBP) relations have organized themselves to Gram determinants.
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K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [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].
M. Argeri and P. Mastrolia, Feynman Diagrams and Differential Equations, Int. J. Mod. Phys. A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
F. Moriello, Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops, JHEP 01 (2020) 150 [arXiv:1907.13234] [INSPIRE].
R. Bonciani et al., Evaluating a family of two-loop non-planar master integrals for Higgs + jet production with full heavy-quark mass dependence, JHEP 01 (2020) 132 [arXiv:1907.13156] [INSPIRE].
H. Frellesvig, M. Hidding, L. Maestri, F. Moriello and G. Salvatori, The complete set of two-loop master integrals for Higgs + jet production in QCD, JHEP 06 (2020) 093 [arXiv:1911.06308] [INSPIRE].
M. Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput. Phys. Commun. 269 (2021) 108125 [arXiv:2006.05510] [INSPIRE].
X. Liu, Y.-Q. Ma and C.-Y. Wang, A Systematic and Efficient Method to Compute Multi-loop Master Integrals, Phys. Lett. B 779 (2018) 353 [arXiv:1711.09572] [INSPIRE].
Z.-F. Liu and Y.-Q. Ma, Automatic computation of Feynman integrals containing linear propagators via auxiliary mass flow, Phys. Rev. D 105 (2022) 074003 [arXiv:2201.11636] [INSPIRE].
Z.-F. Liu and Y.-Q. Ma, Determining Feynman Integrals with Only Input from Linear Algebra, Phys. Rev. Lett. 129 (2022) 222001 [arXiv:2201.11637] [INSPIRE].
X. Liu and Y.-Q. Ma, AMFlow: A Mathematica package for Feynman integrals computation via auxiliary mass flow, Comput. Phys. Commun. 283 (2023) 108565 [arXiv:2201.11669] [INSPIRE].
X. Liu and Y.-Q. Ma, Multiloop corrections for collider processes using auxiliary mass flow, Phys. Rev. D 105 (2022) L051503 [arXiv:2107.01864] [INSPIRE].
X. Liu, Y.-Q. Ma, W. Tao and P. Zhang, Calculation of Feynman loop integration and phase-space integration via auxiliary mass flow, Chin. Phys. C 45 (2021) 013115 [arXiv:2009.07987] [INSPIRE].
T. Armadillo, R. Bonciani, S. Devoto, N. Rana and A. Vicini, Evaluation of Feynman integrals with arbitrary complex masses via series expansions, Comput. Phys. Commun. 282 (2023) 108545 [arXiv:2205.03345] [INSPIRE].
G. Passarino and M.J.G. Veltman, One Loop Corrections for e+ e- Annihilation Into mu+ mu- in the Weinberg Model, Nucl. Phys. B 160 (1979) 151 [INSPIRE].
F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [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].
A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, Comput. Phys. Commun. 247 (2020) 106877 [arXiv:1901.07808] [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, ZU-TH-01-12 (2012) [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266 (2021) 108024 [arXiv:2008.06494] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
R. Britto, E. Buchbinder, F. Cachazo and B. Feng, One-loop amplitudes of gluons in SQCD, Phys. Rev. D 72 (2005) 065012 [hep-ph/0503132] [INSPIRE].
G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147 [hep-ph/0609007] [INSPIRE].
J. Gluza, K. Kajda and D.A. Kosower, Towards a Basis for Planar Two-Loop Integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472] [INSPIRE].
T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP 07 (2019) 031 [arXiv:1905.08019] [INSPIRE].
V. Chestnov et al., Macaulay matrix for Feynman integrals: linear relations and intersection numbers, JHEP 09 (2022) 187 [arXiv:2204.12983] [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] [INSPIRE].
H. Frellesvig, F. Gasparotto, M.K. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, Vector Space of Feynman Integrals and Multivariate Intersection Numbers, Phys. Rev. Lett. 123 (2019) 201602 [arXiv:1907.02000] [INSPIRE].
S. Weinzierl, On the computation of intersection numbers for twisted cocycles, J. Math. Phys. 62 (2021) 072301 [arXiv:2002.01930] [INSPIRE].
S. Mizera, Status of Intersection Theory and Feynman Integrals, PoS MA2019 (2019) 016 [arXiv:2002.10476] [INSPIRE].
H. Frellesvig et al., Decomposition of Feynman Integrals by Multivariate Intersection Numbers, JHEP 03 (2021) 027 [arXiv:2008.04823] [INSPIRE].
X. Liu and Y.-Q. Ma, Determining arbitrary Feynman integrals by vacuum integrals, Phys. Rev. D 99 (2019) 071501 [arXiv:1801.10523] [INSPIRE].
X. Guan, X. Liu and Y.-Q. Ma, Complete reduction of integrals in two-loop five-light-parton scattering amplitudes, Chin. Phys. C 44 (2020) 093106 [arXiv:1912.09294] [INSPIRE].
K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].
K.J. Larsen and Y. Zhang, Integration-by-parts reductions from the viewpoint of computational algebraic geometry, PoS LL2016 (2016) 029 [arXiv:1606.09447] [INSPIRE].
Y. Zhang, Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry, (2016), arXiv:1612.02249 [INSPIRE].
A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: An algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun. 221 (2017) 203 [arXiv:1612.04252] [INSPIRE].
A. Georgoudis, K.J. Larsen and Y. Zhang, Cristal and Azurite: new tools for integration-by-parts reductions, PoS RADCOR2017 (2017) 020 [arXiv:1712.07510] [INSPIRE].
J. Böhm, A. Georgoudis, K.J. Larsen, M. Schulze and Y. Zhang, Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals, Phys. Rev. D 98 (2018) 025023 [arXiv:1712.09737] [INSPIRE].
J. Böhm, A. Georgoudis, K.J. Larsen, H. Schönemann and Y. Zhang, Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP 09 (2018) 024 [arXiv:1805.01873] [INSPIRE].
D. Bendle et al., Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space, JHEP 02 (2020) 079 [arXiv:1908.04301] [INSPIRE].
J. Boehm et al., Module Intersection for the Integration-by-Parts Reduction of Multi-Loop Feynman Integrals, PoS MA2019 (2022) 004 [arXiv:2010.06895] [INSPIRE].
D. Bendle et al., pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning, PCFT-21-08 (2021), arXiv:2104.06866 [INSPIRE].
B. Feng and T. Li, PV-reduction of sunset topology with auxiliary vector, Commun. Theor. Phys. 74 (2022) 095201 [arXiv:2203.16881] [INSPIRE].
J.M. Henn, A. Matijašić and J. Miczajka, One-loop hexagon integral to higher orders in the dimensional regulator, JHEP 01 (2023) 096 [arXiv:2210.13505] [INSPIRE].
R.M. Schabinger, A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations, JHEP 01 (2012) 077 [arXiv:1111.4220] [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].
B. Feng, T. Li, H. Wang and Y. Zhang, Reduction of general one-loop integrals using auxiliary vector, JHEP 05 (2022) 065 [arXiv:2203.14449] [INSPIRE].
J. Fleischer, F. Jegerlehner and O.V. Tarasov, Algebraic reduction of one loop Feynman graph amplitudes, Nucl. Phys. B 566 (2000) 423 [hep-ph/9907327] [INSPIRE].
S. Laporta, Calculation of master integrals by difference equations, Phys. Lett. B 504 (2001) 188 [hep-ph/0102032] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240] [INSPIRE].
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].
W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-3-0 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, (2022).
B. Feng, T. Li and X. Li, Analytic tadpole coefficients of one-loop integrals, JHEP 09 (2021) 081 [arXiv:2107.03744] [INSPIRE].
C. Hu, T. Li and X. Li, One-loop Feynman integral reduction by differential operators, Phys. Rev. D 104 (2021) 116014 [arXiv:2108.00772] [INSPIRE].
B. Feng, J. Gong and T. Li, Universal treatment of the reduction for one-loop integrals in a projective space, Phys. Rev. D 106 (2022) 056025 [arXiv:2204.03190] [INSPIRE].
B. Feng, C. Hu, T. Li and Y. Song, Reduction with degenerate Gram matrix for one-loop integrals, JHEP 08 (2022) 110 [arXiv:2205.03000] [INSPIRE].
R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].
J. Chen, X. Jiang, C. Ma, X. Xu and L.L. Yang, Baikov representations, intersection theory, and canonical Feynman integrals, JHEP 07 (2022) 066 [arXiv:2202.08127] [INSPIRE].
E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph, Nucl. Phys. B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE].
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Chen, J., Feng, B. Module intersection and uniform formula for iterative reduction of one-loop integrals. J. High Energ. Phys. 2023, 178 (2023). https://doi.org/10.1007/JHEP02(2023)178
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DOI: https://doi.org/10.1007/JHEP02(2023)178