Abstract
We apply the method of regions to the massive two-loop integrals appearing in the Higgs pair production cross section at the next-to-leading order, in the high energy limit. For the non-planar integrals, a subtle problem arises because of the indefinite sign of the second Symanzik polynomial. We solve this problem by performing an analytic continuation of the Mandelstam variables such that the second Symanzik polynomial has a definite sign. Furthermore, we formulate the procedure of applying the method of regions systematically. As a by-product of the analytic continuation of the Mandelstam variables, we obtain crossing relations between integrals in a simple and systematic way. In our formulation, a concept of “template integral” is introduced, which represents and controls the contribution of each region. All of the template integrals needed in the computation of the Higgs pair production at the next-to-leading order are given explicitly. We also develop techniques to solve Mellin-Barnes integrals, and show them in detail. Although most of the calculation is shown for the concrete example of the Higgs pair production process, the application to other similar processes is straightforward, and we anticipate that our method can be useful also for other cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Two-loop planar master integrals for Higgs→ 3 partons with full heavy-quark mass dependence, JHEP 12 (2016) 096 [arXiv:1609.06685] [INSPIRE].
S.P. Jones, M. Kerner and G. Luisoni, Next-to-Leading-Order QCD Corrections to Higgs Boson Plus Jet Production with Full Top-Quark Mass Dependence, Phys. Rev. Lett. 120 (2018) 162001 [arXiv:1802.00349] [INSPIRE].
S. Borowka et al., Higgs Boson Pair Production in Gluon Fusion at Next-to-Leading Order with Full Top-Quark Mass Dependence, Phys. Rev. Lett. 117 (2016) 012001 [Erratum ibid. 117 (2016) 079901] [arXiv:1604.06447] [INSPIRE].
S. Borowka et al., Full top quark mass dependence in Higgs boson pair production at NLO, JHEP 10 (2016) 107 [arXiv:1608.04798] [INSPIRE].
S. Borowka, G. Heinrich, S.P. Jones, M. Kerner, J. Schlenk and T. Zirke, SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470 [arXiv:1502.06595] [INSPIRE].
S. Borowka et al., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222 (2018) 313 [arXiv:1703.09692] [INSPIRE].
J. Baglio, F. Campanario, S. Glaus, M. Mühlleitner, M. Spira and J. Streicher, Gluon fusion into Higgs pairs at NLO QCD and the top mass scheme, arXiv:1811.05692 [INSPIRE].
P. Mastrolia, M. Passera, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs, JHEP 11 (2017) 198 [arXiv:1709.07435] [INSPIRE].
S. Di Vita, S. Laporta, P. Mastrolia, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs, JHEP 09 (2018) 016 [arXiv:1806.08241] [INSPIRE].
L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP 10 (2018) 206 [arXiv:1806.04981] [INSPIRE].
M. Becchetti and R. Bonciani, Two-Loop Master Integrals for the Planar QCD Massive Corrections to Di-photon and Di-jet Hadro-production, JHEP 01 (2018) 048 [arXiv:1712.02537] [INSPIRE].
X. Xu and L.L. Yang, Towards a new approximation for pair-production and associated-production of the Higgs boson, JHEP 01 (2019) 211 [arXiv:1810.12002] [INSPIRE].
V.A. Smirnov, Applied Asymptotic Expansions in Momenta and Masses, Springer Tracts Mod. Phys. 177 (2002) 1.
R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, JHEP 06 (2013) 072 [arXiv:1302.6216] [INSPIRE].
X. Chen, T. Gehrmann, E.W.N. Glover and M. Jaquier, Precise QCD predictions for the production of Higgs + jet final states, Phys. Lett. B 740 (2015) 147 [arXiv:1408.5325] [INSPIRE].
R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order, Phys. Rev. Lett. 115 (2015) 082003 [arXiv:1504.07922] [INSPIRE].
R. Boughezal, C. Focke, W. Giele, X. Liu and F. Petriello, Higgs boson production in association with a jet at NNLO using jettiness subtraction, Phys. Lett. B 748 (2015) 5 [arXiv:1505.03893] [INSPIRE].
F. Caola, K. Melnikov and M. Schulze, Fiducial cross sections for Higgs boson production in association with a jet at next-to-next-to-leading order in QCD, Phys. Rev. D 92 (2015) 074032 [arXiv:1508.02684] [INSPIRE].
X. Chen, J. Cruz-Martinez, T. Gehrmann, E.W.N. Glover and M. Jaquier, NNLO QCD corrections to Higgs boson production at large transverse momentum, JHEP 10 (2016) 066 [arXiv:1607.08817] [INSPIRE].
T. Neumann and C. Williams, The Higgs boson at high p T, Phys. Rev. D 95 (2017) 014004 [arXiv:1609.00367] [INSPIRE].
D. de Florian et al., Differential Higgs Boson Pair Production at Next-to-Next-to-Leading Order in QCD, JHEP 09 (2016) 151 [arXiv:1606.09519] [INSPIRE].
D. de Florian and J. Mazzitelli, Two-loop virtual corrections to Higgs pair production, Phys. Lett. B 724 (2013) 306 [arXiv:1305.5206] [INSPIRE].
J. Grigo, J. Hoff, K. Melnikov and M. Steinhauser, On the Higgs boson pair production at the LHC, Nucl. Phys. B 875 (2013) 1 [arXiv:1305.7340] [INSPIRE].
D. de Florian and J. Mazzitelli, Higgs Boson Pair Production at Next-to-Next-to-Leading Order in QCD, Phys. Rev. Lett. 111 (2013) 201801 [arXiv:1309.6594] [INSPIRE].
F. Maltoni, E. Vryonidou and M. Zaro, Top-quark mass effects in double and triple Higgs production in gluon-gluon fusion at NLO, JHEP 11 (2014) 079 [arXiv:1408.6542] [INSPIRE].
J. Grigo, K. Melnikov and M. Steinhauser, Virtual corrections to Higgs boson pair production in the large top quark mass limit, Nucl. Phys. B 888 (2014) 17 [arXiv:1408.2422] [INSPIRE].
J. Grigo, J. Hoff and M. Steinhauser, Higgs boson pair production: top quark mass effects at NLO and NNLO, Nucl. Phys. B 900 (2015) 412 [arXiv:1508.00909] [INSPIRE].
G. Degrassi, P.P. Giardino and R. Gröber, On the two-loop virtual QCD corrections to Higgs boson pair production in the Standard Model, Eur. Phys. J. C 76 (2016) 411 [arXiv:1603.00385] [INSPIRE].
K. Melnikov, L. Tancredi and C. Wever, Two-loop gg → Hg amplitude mediated by a nearly massless quark, JHEP 11 (2016) 104 [arXiv:1610.03747] [INSPIRE].
K. Melnikov, L. Tancredi and C. Wever, Two-loop amplitudes for qg → Hq and \( q\overline{q}\to Hg \) mediated by a nearly massless quark, Phys. Rev. D 95 (2017) 054012 [arXiv:1702.00426] [INSPIRE].
K. Kudashkin, K. Melnikov and C. Wever, Two-loop amplitudes for processes gg → Hg, qg → Hq and \( q\overline{q}\to Hg \) at large Higgs transverse momentum, JHEP 02 (2018) 135 [arXiv:1712.06549] [INSPIRE].
J. Davies, G. Mishima, M. Steinhauser and D. Wellmann, Double-Higgs boson production in the high-energy limit: planar master integrals, JHEP 03 (2018) 048 [arXiv:1801.09696] [INSPIRE].
R. Bonciani, G. Degrassi, P.P. Giardino and R. Gröber, Analytical Method for Next-to-Leading-Order QCD Corrections to Double-Higgs Production, Phys. Rev. Lett. 121 (2018) 162003 [arXiv:1806.11564] [INSPIRE].
J. Davies, G. Mishima, M. Steinhauser and D. Wellmann, Double Higgs boson production at NLO in the high-energy limit: complete analytic results, JHEP 01 (2019) 176 [arXiv:1811.05489] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
V.A. Smirnov, Analytic Tools for Feynman Integrals, Springer Tracts Mod. Phys. 250 (2012) 1.
V.A. Smirnov, Asymptotic expansions in limits of large momenta and masses, Commun. Math. Phys. 134 (1990) 109 [INSPIRE].
R. Harlander, T. Seidensticker and M. Steinhauser, Corrections of O(αα s) to the decay of the Z boson into bottom quarks, Phys. Lett. B 426 (1998) 125 [hep-ph/9712228] [INSPIRE].
T. Seidensticker, Automatic application of successive asymptotic expansions of Feynman diagrams, in 6th International Workshop on New Computing Techniques in Physics Research: Software Engineering, Artificial Intelligence Neural Nets, Genetic Algorithms, Symbolic Algebra, Automatic Calculation (AIHENP 99), Heraklion, Crete, Greece, April 12–16, 1999 [hep-ph/9905298] [INSPIRE].
B. Jantzen, Foundation and generalization of the expansion by regions, JHEP 12 (2011) 076 [arXiv:1111.2589] [INSPIRE].
T.Y. Semenova, A.V. Smirnov and V.A. Smirnov, On the status of expansion by regions, arXiv:1809.04325 [INSPIRE].
A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].
B. Jantzen, A.V. Smirnov and V.A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [INSPIRE].
B. Ananthanarayan, A. Pal, S. Ramanan and R. Sarkar, Unveiling Regions in multi-scale Feynman Integrals using Singularities and Power Geometry, Eur. Phys. J. C 79 (2019) 57 [arXiv:1810.06270] [INSPIRE].
V.A. Smirnov, Problems of the strategy of regions, Phys. Lett. B 465 (1999) 226 [hep-ph/9907471] [INSPIRE].
R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].
C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi and J.B. Tausk, The Tensor reduction and master integrals of the two loop massless crossed box with lightlike legs, Nucl. Phys. B 580 (2000) 577 [hep-ph/0003261] [INSPIRE].
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].
D. Kosower, https://mbtools.hepforge.org.
L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press (1966).
A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and series, volume 3, Gordon and Breach Science Publishers (1990).
H.R.P. Ferguson and D.H. Bailey, A Polynomial Time, Numerically Stable Integer Relation Algorithm, RNR Technical Report, RNR-91-032 (1992).
H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of PSLQ, an Integer Relation Finding Algorithm, NAS Technical Report, NAS-96-005 (1996).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1812.04373
Electronic supplementary material
Below is the link to the electronic supplementary material.
ESM 1
(ZIP 4 kb)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mishima, G. High-energy expansion of two-loop massive four-point diagrams. J. High Energ. Phys. 2019, 80 (2019). https://doi.org/10.1007/JHEP02(2019)080
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2019)080