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IBIS: Inverse BInomial sum Solver

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arxiv 2506.19904 v1 pith:LVIOB42H submitted 2025-06-24 hep-ph

Paul A.J.W. van Hoegaerden , Coenraad B. Marinissen , Wouter J. Waalewijn This is my paper
classification hep-ph
keywords sumsbinomialinversecalculationshigher-loopibisrepresentationsresults
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abstract

In higher-loop calculations, Mellin-Barnes representations are used to simplify the denominators encountered in Feynman parameter integrals. The contour integral of these representations yield sums over residues. We extend the classes of such sums that can be calculated, to include those involving inverse binomials. Our results are expressed in terms of so-called $S$-sums, where the dependence on the upper limit of the sum is analytic. This is accomplished by deriving several new recursion relations, obtained from telescoping series and repeated synchronization. We make our results available through IBIS ("Inverse BInomial sum Solver"), a FORM program to perform such inverse binomial sums. It expresses them in terms of $S$-sums, which can be handled by XSUMMER. To illustrate the efficiency of our code: sums up to weight 6 can be carried out in less than a second. We show in an example how inverse binomial sums arise, though in many instances these are nested with binomial sums, beyond the cases studied here. Our work thus provides a starting point for studying such sums, that will open up new avenues in higher-loop calculations.

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Works this paper leans on

81 extracted references · 9 canonical work pages

  1. [1]

    Ellis, Z

    R.K. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi,One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts,Phys. Rept.518(2012) 141 [1105.4319]

    arXiv 2012

  2. [2]

    Berger, Z

    C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde, H. Ita et al.,An Automated Implementation of On-Shell Methods for One-Loop Amplitudes,Phys. Rev. D78(2008) 036003 [0803.4180]. [3]GoSamcollaboration,GOSAM-2.0: a tool for automated one-loop calculations within the Standard Model and beyond,Eur. Phys. J. C74(2014) 3001 [1404.7096]

    arXiv 2008

  3. [4]

    Bevilacqua, M

    G. Bevilacqua, M. Czakon, M.V. Garzelli, A. van Hameren, A. Kardos, C.G. Papadopoulos et al.,HELAC-NLO,Comput. Phys. Commun.184(2013) 986 [1110.1499]

    arXiv 2013

  4. [5]

    Cascioli, P

    F. Cascioli, P. Maierhofer and S. Pozzorini,Scattering Amplitudes with Open Loops,Phys. Rev. Lett.108(2012) 111601 [1111.5206]

    arXiv 2012

  5. [6]

    Badger, B

    S. Badger, B. Biedermann, P. Uwer and V. Yundin,Numerical evaluation of virtual corrections to multi-jet production in massless QCD,Comput. Phys. Commun.184(2013) 1981 [1209.0100]. – 23 –

    arXiv 2013

  6. [7]

    Gehrmann-De Ridder, T

    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and J. Pires,Triple Differential Dijet Cross Section at the LHC,Phys. Rev. Lett.123(2019) 102001 [1905.09047]

    arXiv 2019

  7. [8]

    Czakon, A

    M. Czakon, A. van Hameren, A. Mitov and R. Poncelet,Single-jet inclusive rates with exact color atO(α 4 s),JHEP10(2019) 262 [1907.12911]

    arXiv 2019

  8. [9]

    Boughezal, C

    R. Boughezal, C. Focke, X. Liu and F. Petriello,W-boson production in association with a jet at next-to-next-to-leading order in perturbative QCD,Phys. Rev. Lett.115(2015) 062002 [1504.02131]

    arXiv 2015

Show all 81 references
  1. [10]

    Gehrmann-De Ridder, T

    A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan,Precise QCD predictions for the production of a Z boson in association with a hadronic jet,Phys. Rev. Lett.117(2016) 022001 [1507.02850]

    2016 arXiv

  2. [11]

    Campbell, R.K

    J.M. Campbell, R.K. Ellis and C. Williams,Direct Photon Production at Next-to–Next-to-Leading Order,Phys. Rev. Lett.118(2017) 222001 [1612.04333]

    2017 arXiv

  3. [12]

    X. Chen, T. Gehrmann, N. Glover, M. H¨ ofer and A. Huss,Isolated photon and photon+jet production at NNLO QCD accuracy,JHEP04(2020) 166 [1904.01044]

    2020 arXiv

  4. [13]

    Czakon, D

    M. Czakon, D. Heymes and A. Mitov,High-precision differential predictions for top-quark pairs at the LHC,Phys. Rev. Lett.116(2016) 082003 [1511.00549]

    2016 arXiv

  5. [14]

    Catani, S

    S. Catani, S. Devoto, M. Grazzini, S. Kallweit and J. Mazzitelli,Top-quark pair production at the LHC: Fully differential QCD predictions at NNLO,JHEP07(2019) 100 [1906.06535]

    2019 arXiv

  6. [15]

    Czakon, A

    M. Czakon, A. Mitov and R. Poncelet,Next-to-Next-to-Leading Order Study of Three-Jet Production at the LHC,Phys. Rev. Lett.127(2021) 152001 [2106.05331]

    2021 arXiv

  7. [16]

    Anastasiou, C

    C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger,Higgs Boson Gluon-Fusion Production in QCD at Three Loops,Phys. Rev. Lett.114(2015) 212001 [1503.06056]

    2015 arXiv

  8. [17]

    Dulat, B

    F. Dulat, B. Mistlberger and A. Pelloni,Precision predictions at N 3LO for the Higgs boson rapidity distribution at the LHC,Phys. Rev. D99(2019) 034004 [1810.09462]

    2019 arXiv

  9. [18]

    C. Duhr, F. Dulat and B. Mistlberger,Drell-Yan Cross Section to Third Order in the Strong Coupling Constant,Phys. Rev. Lett.125(2020) 172001 [2001.07717]

    2020 arXiv

  10. [19]

    X. Chen, T. Gehrmann, N. Glover, A. Huss, T.-Z. Yang and H.X. Zhu,Dilepton Rapidity Distribution in Drell-Yan Production to Third Order in QCD,Phys. Rev. Lett.128(2022) 052001 [2107.09085]

    2022 arXiv

  11. [20]

    Laporta,High-precision calculation of multiloop Feynman integrals by difference equations, Int

    S. Laporta,High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A15(2000) 5087 [hep-ph/0102033]

    2000 arXiv

  12. [21]

    Gehrmann and E

    T. Gehrmann and E. Remiddi,Differential equations for two-loop four-point functions,Nucl. Phys. B580(2000) 485 [hep-ph/9912329]

    2000 arXiv

  13. [22]

    Henn,Multiloop integrals in dimensional regularization made simple,Phys

    J.M. Henn,Multiloop integrals in dimensional regularization made simple,Phys. Rev. Lett. 110(2013) 251601 [1304.1806]

    2013 arXiv

  14. [23]

    Binoth and G

    T. Binoth and G. Heinrich,Numerical evaluation of phase space integrals by sector decomposition,Nucl. Phys. B693(2004) 134 [hep-ph/0402265]

    2004 arXiv

  15. [24]

    Anastasiou, K

    C. Anastasiou, K. Melnikov and F. Petriello,A new method for real radiation at NNLO, Phys. Rev. D69(2004) 076010 [hep-ph/0311311]. – 24 –

    2004 arXiv

  16. [25]

    Catani and M

    S. Catani and M. Grazzini,An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC,Phys. Rev. Lett.98(2007) 222002 [hep-ph/0703012]

    2007 arXiv

  17. [26]

    Gehrmann-De Ridder, T

    A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover,Antenna subtraction at NNLO, JHEP09(2005) 056 [hep-ph/0505111]

    2005 arXiv

  18. [27]

    Currie, E.W.N

    J. Currie, E.W.N. Glover and S. Wells,Infrared Structure at NNLO Using Antenna Subtraction,JHEP04(2013) 066 [1301.4693]

    2013 arXiv

  19. [28]

    Czakon,A novel subtraction scheme for double-real radiation at NNLO,Phys

    M. Czakon,A novel subtraction scheme for double-real radiation at NNLO,Phys. Lett. B 693(2010) 259 [1005.0274]

    2010 arXiv

  20. [29]

    Boughezal, K

    R. Boughezal, K. Melnikov and F. Petriello,A subtraction scheme for NNLO computations, Phys. Rev. D85(2012) 034025 [1111.7041]

    2012 arXiv

  21. [30]

    Boughezal, X

    R. Boughezal, X. Liu and F. Petriello,N-jettiness soft function at next-to-next-to-leading order,Phys. Rev. D91(2015) 094035 [1504.02540]

    2015 arXiv

  22. [31]

    Gaunt, M

    J. Gaunt, M. Stahlhofen, F.J. Tackmann and J.R. Walsh,N-jettiness Subtractions for NNLO QCD Calculations,JHEP09(2015) 058 [1505.04794]

    2015 arXiv

  23. [32]

    Somogyi and Z

    G. Somogyi and Z. Trocsanyi,A Subtraction scheme for computing QCD jet cross sections at NNLO: Integrating the subtraction terms. I.,JHEP08(2008) 042 [0807.0509]

    2008 arXiv

  24. [33]

    Del Duca, C

    V. Del Duca, C. Duhr, A. Kardos, G. Somogyi, Z. Sz˝ or, Z. Tr´ ocs´ anyi et al.,Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions,Phys. Rev. D94 (2016) 074019 [1606.03453]

    2016 arXiv

  25. [34]

    Banik and S

    S. Banik and S. Friot,Multiple Mellin-Barnes integrals and triangulations of point configurations,Phys. Rev. D110(2024) 036002 [2309.00409]

    2024 arXiv

  26. [35]

    Smirnov,Analytical result for dimensionally regularized massless on shell double box, Phys

    V.A. Smirnov,Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B460(1999) 397 [hep-ph/9905323]

    1999 arXiv

  27. [36]

    Tausk,Nonplanar massless two loop Feynman diagrams with four on-shell legs,Phys

    J.B. Tausk,Nonplanar massless two loop Feynman diagrams with four on-shell legs,Phys. Lett. B469(1999) 225 [hep-ph/9909506]

    1999 arXiv

  28. [37]

    Smirnov and V.A

    A.V. Smirnov and V.A. Smirnov,On the Resolution of Singularities of Multiple Mellin-Barnes Integrals,Eur. Phys. J. C62(2009) 445 [0901.0386]

    2009 arXiv

  29. [38]

    Kalmykov, V

    M. Kalmykov, V. Bytev, B.A. Kniehl, S.-O. Moch, B.F.L. Ward and S.A. Yost, Hypergeometric Functions and Feynman Diagrams, inAntidifferentiation and the Calculation of Feynman Amplitudes, 12, 2020, DOI [2012.14492]

    2020 arXiv

  30. [39]

    Vollinga and S

    J. Vollinga and S. Weinzierl,Numerical evaluation of multiple polylogarithms,Comput. Phys. Commun.167(2005) 177 [hep-ph/0410259]

    2005 arXiv

  31. [40]

    Dubovyk, J

    I. Dubovyk, J. Gluza and T. Riemann,Optimizing the Mellin-Barnes Approach to Numerical Multiloop Calculations,Acta Phys. Polon. B50(2019) 1993 [1912.11326]

    2019 arXiv

  32. [41]

    Smirnov,Analytic tools for Feynman integrals, vol

    V.A. Smirnov,Analytic tools for Feynman integrals, vol. 250 (2012), 10.1007/978-3-642-34886-0

    2012 doi

  33. [42]

    Dubovyk, J

    I. Dubovyk, J. Gluza and G. Somogyi,Mellin-Barnes Integrals: A Primer on Particle Physics Applications,Lect. Notes Phys.1008(2022) pp. [2211.13733]

    2022 arXiv

  34. [43]

    Czakon,Automatized analytic continuation of Mellin-Barnes integrals,Comput

    M. Czakon,Automatized analytic continuation of Mellin-Barnes integrals,Comput. Phys. Commun.175(2006) 559 [hep-ph/0511200]. – 25 –

    2006 arXiv

  35. [44]

    Gluza, K

    J. Gluza, K. Kajda and T. Riemann,AMBRE: A Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals,Comput. Phys. Commun.177 (2007) 879 [0704.2423]

    2007 arXiv

  36. [45]

    Ochman and T

    M. Ochman and T. Riemann,MBsums - a Mathematica package for the representation of Mellin-Barnes integrals by multiple sums,Acta Phys. Polon. B46(2015) 2117 [1511.01323]

    2015 arXiv

  37. [46]

    Belitsky, A.V

    A.V. Belitsky, A.V. Smirnov and V.A. Smirnov,MB tools reloaded,Nucl. Phys. B986 (2023) 116067 [2211.00009]

    2023 arXiv

  38. [47]

    Vermaseren,Harmonic sums, Mellin transforms and integrals,Int

    J.A.M. Vermaseren,Harmonic sums, Mellin transforms and integrals,Int. J. Mod. Phys. A 14(1999) 2037 [hep-ph/9806280]

    1999 arXiv

  39. [48]

    Vermaseren,New features of FORM,math-ph/0010025

    J.A.M. Vermaseren,New features of FORM,math-ph/0010025

    arXiv

  40. [49]

    Ruijl, T

    B. Ruijl, T. Ueda and J. Vermaseren,FORM version 4.2,1707.06453

    arXiv

  41. [50]

    Ablinger,The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums,PoSLL2014(2014) 019 [1407.6180]

    J. Ablinger,The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums,PoSLL2014(2014) 019 [1407.6180]

    2014 arXiv

  42. [51]

    Moch and P

    S. Moch and P. Uwer,XSummer: Transcendental functions and symbolic summation in form,Comput. Phys. Commun.174(2006) 759 [math-ph/0508008]

    2006 arXiv

  43. [52]

    Bauer, A

    C.W. Bauer, A. Frink and R. Kreckel,Introduction to the GiNaC framework for symbolic computation within the C++ programming language,J. Symb. Comput.33(2002) 1 [cs/0004015]

    2002 arXiv

  44. [53]

    Vollinga,GiNaC: Symbolic computation with C++,Nucl

    J. Vollinga,GiNaC: Symbolic computation with C++,Nucl. Instrum. Meth. A559(2006) 282 [hep-ph/0510057]

    2006 arXiv

  45. [54]

    Gould,Combinatorial identities: A standardized set of tables listing 500 binomial coefficient summations,The Fibonacci Quarterly10(1972) 662

    H.W. Gould,Combinatorial identities: A standardized set of tables listing 500 binomial coefficient summations,The Fibonacci Quarterly10(1972) 662

    1972

  46. [55]

    Rockett,Sums of the inverses of binomial coefficients,The Fibonacci Quarterly19(1981) 433

    A. Rockett,Sums of the inverses of binomial coefficients,The Fibonacci Quarterly19(1981) 433

    1981

  47. [56]

    Pla,The sum of inverses of binomial coefficients revisited,The Fibonacci Quarterly35 (1997)

    J. Pla,The sum of inverses of binomial coefficients revisited,The Fibonacci Quarterly35 (1997)

    1997

  48. [57]

    Trif,Combinatorial sums and series involving inverses of binomial coefficients,The Fibonacci Quarterly38(2000) 79

    T. Trif,Combinatorial sums and series involving inverses of binomial coefficients,The Fibonacci Quarterly38(2000) 79

    2000

  49. [58]

    Sury,Sum of the reciprocals of the binomial coefficients,European Journal of Combinatorics14(1993) 351

    B. Sury,Sum of the reciprocals of the binomial coefficients,European Journal of Combinatorics14(1993) 351

    1993

  50. [59]

    Mansour,Combinatorial identities and inverse binomial coefficients,Advances in Applied Mathematics28(2002) 196

    T. Mansour,Combinatorial identities and inverse binomial coefficients,Advances in Applied Mathematics28(2002) 196

    2002

  51. [60]

    B. Sury, T. Wang and F.-Z. Zhao,Identities involving reciprocals of binomial coefficients, Journal of Integer Sequences7(2004) Art. 04.2.8, 12 p., electronic only

    2004

  52. [61]

    Batir and A

    N. Batir and A. Sofo,Finite sums involving reciprocals of the binomial and central binomial coefficients and harmonic numbers,Symmetry13(2021)

    2021

  53. [62]

    Jin and D.K

    H.-T. Jin and D.K. Du,Abel’s Lemma and Identities on Harmonic Numbers,arXiv e-prints (2014) [1411.6916]

    2014 arXiv

  54. [63]

    Batir,Combinatorial identities involving harmonic numbers,arXiv e-prints(2018) [1806.03022]

    N. Batir,Combinatorial identities involving harmonic numbers,arXiv e-prints(2018) [1806.03022]. – 26 –

    2018 arXiv

  55. [64]

    Ablinger, J

    J. Ablinger, J. Bl¨ umlein, C.G. Raab and C. Schneider,Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams,PoSLL2014(2014) 020 [1407.4721]

    2014 arXiv

  56. [65]

    Ablinger, J

    J. Ablinger, J. Bl¨ umlein, C.G. Raab and C. Schneider,Iterated Binomial Sums and their Associated Iterated Integrals,J. Math. Phys.55(2014) 112301 [1407.1822]

    2014 arXiv

  57. [66]

    Weinzierl,Expansion around half integer values, binomial sums and inverse binomial sums,J

    S. Weinzierl,Expansion around half integer values, binomial sums and inverse binomial sums,J. Math. Phys.45(2004) 2656 [hep-ph/0402131]

    2004 arXiv

  58. [67]

    Jegerlehner, M.Y

    F. Jegerlehner, M.Y. Kalmykov and O. Veretin,MS-bar versus pole masses of gauge bosons

  59. [68]

    Two loop electroweak fermion corrections,Nucl. Phys. B658(2003) 49 [hep-ph/0212319]

    2003 arXiv

  60. [69]

    Fleischer, A.V

    J. Fleischer, A.V. Kotikov and O.L. Veretin,Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass,Nucl. Phys. B547(1999) 343 [hep-ph/9808242]

    1999 arXiv

  61. [70]

    S. Moch, P. Uwer and S. Weinzierl,Nested sums, expansion of transcendental functions and multiscale multiloop integrals,J. Math. Phys.43(2002) 3363 [hep-ph/0110083]

    2002 arXiv

  62. [71]

    Hoffman,The Hopf algebra structure of multiple harmonic sums,Nucl

    M.E. Hoffman,The Hopf algebra structure of multiple harmonic sums,Nucl. Phys. B Proc. Suppl.135(2004) 215 [math/0406589]

    2004 arXiv

  63. [72]

    Weinzierl,The Art of computing loop integrals,Fields Inst

    S. Weinzierl,The Art of computing loop integrals,Fields Inst. Commun.50(2007) 345 [hep-ph/0604068]

    2007 arXiv

  64. [73]

    Ablinger,Computer Algebra Algorithms for Special Functions in Particle Physics, Ph.D

    J. Ablinger,Computer Algebra Algorithms for Special Functions in Particle Physics, Ph.D. thesis, Linz U., 4, 2012.1305.0687

    2012

  65. [74]

    Euler,Meditationes circa singulare serierum genus,Novi Commentarii academiae scientiarum Petropolitanae20(1776) 140

    L. Euler,Meditationes circa singulare serierum genus,Novi Commentarii academiae scientiarum Petropolitanae20(1776) 140

  66. [75]

    Zagier,Values of zeta functions and their applications, inFirst European Congress of Mathematics Paris, July 6–10, 1992: Vol

    D. Zagier,Values of zeta functions and their applications, inFirst European Congress of Mathematics Paris, July 6–10, 1992: Vol. II: Invited Lectures (Part 2), A. Joseph, F. Mignot, F. Murat, B. Prum and R. Rentschler, eds., (Basel), pp. 497–512, Birkh¨ auser Basel (1994), DOI

    1994

  67. [76]

    Goncharov,Multiple polylogarithms, cyclotomy and modular complexes,Mathematical Research Letters5(2011) 497 [1105.2076]

    A.B. Goncharov,Multiple polylogarithms, cyclotomy and modular complexes,Mathematical Research Letters5(2011) 497 [1105.2076]

    2011 arXiv

  68. [77]

    Lewin,Polylogarithms and Associated Functions, North Holland (1981)

    L. Lewin,Polylogarithms and Associated Functions, North Holland (1981)

    1981

  69. [78]

    Nielsen,Der Eulersche Dilogarithmus und seine Verallgemeinerungen: eine Monographie, Nova Acta

    N. Nielsen,Der Eulersche Dilogarithmus und seine Verallgemeinerungen: eine Monographie, Nova Acta. Abh. der Kaiserl. Leop.-Carol. Deutschen Akademie der Naturforscher, Druck Ehrhardt Karras (1909)

    1909

  70. [79]

    Remiddi and J.A.M

    E. Remiddi and J.A.M. Vermaseren,Harmonic polylogarithms,Int. J. Mod. Phys. A15 (2000) 725 [hep-ph/9905237]

    2000 arXiv

  71. [80]

    Borwein, D.M

    J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisonek,Special values of multiple polylogarithms,Transactions of the American Mathematical Society353(2001) 907 [math/9910045]

    2001 arXiv

  72. [81]

    Schneider,Simplifying multiple sums in difference fields,Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions(2013) 325 [1304.4134]

    C. Schneider,Simplifying multiple sums in difference fields,Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions(2013) 325 [1304.4134]

    2013 arXiv

  73. [82]

    Bierenbaum and S

    I. Bierenbaum and S. Weinzierl,The Massless two loop two point function,Eur. Phys. J. C 32(2003) 67 [hep-ph/0308311]. – 27 –

    2003 arXiv

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