Abstract
We present a general method to account for full colour dependence Yang-Mills amplitudes at loop level. The method fits most naturally into the framework of multi-loop integrand reduction and in a nutshell amounts to consistently retaining the colour structures of the unitarity cuts from which the integrand is gradually constructed. This technique has already been used in the recent calculation of the two-loop five-gluon amplitude in pure Yang-Mills theory with all positive helicities, JHEP 10 (2015) 064. In this note, we give a careful exposition of the method and discuss its connection to looplevel Kleiss-Kuijf relations. We also explore its implications for cancellation of nontrivial symmetry factors at two loops. As an example of its generality, we show how it applies to the three-loop case in supersymmetric Yang-Mills case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two-loop QCD corrections to the scattering of massless distinct quarks, Nucl. Phys. B 601 (2001) 318 [hep-ph/0010212] [INSPIRE].
C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to massless identical quark scattering, Nucl. Phys. B 601 (2001) 341 [hep-ph/0011094] [INSPIRE].
C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to massless quark gluon scattering, Nucl. Phys. B 605 (2001) 486 [hep-ph/0101304] [INSPIRE].
E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to gluon-gluon scattering, Nucl. Phys. B 605 (2001) 467 [hep-ph/0102201] [INSPIRE].
L.W. Garland, T. Gehrmann, E.W.N. Glover, A. Koukoutsakis and E. Remiddi, The Two loop QCD matrix element for e + e − → 3 jets, Nucl. Phys. B 627 (2002) 107 [hep-ph/0112081] [INSPIRE].
L.W. Garland, T. Gehrmann, E.W.N. Glover, A. Koukoutsakis and E. Remiddi, Two loop QCD helicity amplitudes for e + e − → 3 jets, Nucl. Phys. B 642 (2002) 227 [hep-ph/0206067] [INSPIRE].
S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Diphoton production at hadron colliders: a fully-differential QCD calculation at NNLO, Phys. Rev. Lett. 108 (2012) 072001 [arXiv:1110.2375] [INSPIRE].
T. Gehrmann, M. Jaquier, E.W.N. Glover and A. Koukoutsakis, Two-Loop QCD Corrections to the Helicity Amplitudes for H → 3 partons, JHEP 02 (2012) 056 [arXiv:1112.3554] [INSPIRE].
M. Czakon, P. Fiedler and A. Mitov, Total Top-Quark Pair-Production Cross section at Hadron Colliders Through O(α 4 S ), Phys. Rev. Lett. 110 (2013) 252004 [arXiv:1303.6254] [INSPIRE].
M. Grazzini, S. Kallweit, D. Rathlev and A. Torre, Zγ production at hadron colliders in NNLO QCD, Phys. Lett. B 731 (2014) 204 [arXiv:1309.7000] [INSPIRE].
F. Cascioli et al., ZZ production at hadron colliders in NNLO QCD, Phys. Lett. B 735 (2014) 311 [arXiv:1405.2219] [INSPIRE].
T. Gehrmann et al., W + W − Production at Hadron Colliders in Next to Next to Leading Order QCD, Phys. Rev. Lett. 113 (2014) 212001 [arXiv:1408.5243] [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].
F. Caola, J.M. Henn, K. Melnikov, A.V. Smirnov and V.A. Smirnov, Two-loop helicity amplitudes for the production of two off-shell electroweak bosons in quark-antiquark collisions, JHEP 11 (2014) 041 [arXiv:1408.6409] [INSPIRE].
M. Czakon, P. Fiedler and A. Mitov, Resolving the Tevatron Top Quark Forward-Backward Asymmetry Puzzle: Fully Differential Next-to-Next-to-Leading-Order Calculation, Phys. Rev. Lett. 115 (2015) 052001 [arXiv:1411.3007] [INSPIRE].
T. Gehrmann, A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for \( q{\overline{q}}^{\prime}\to {V}_1{V}_2\to\ 4 \) leptons, JHEP 09 (2015) 128 [arXiv:1503.04812] [INSPIRE].
F. Caola, J.M. Henn, K. Melnikov, A.V. Smirnov and V.A. Smirnov, Two-loop helicity amplitudes for the production of two off-shell electroweak bosons in gluon fusion, JHEP 06 (2015) 129 [arXiv:1503.08759] [INSPIRE].
A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for gg → V 1 V 2 → 4 leptons, JHEP 06 (2015) 197 [arXiv:1503.08835] [INSPIRE].
M. Grazzini, S. Kallweit and D. Rathlev, W γ and Zγ production at the LHC in NNLO QCD, JHEP 07 (2015) 085 [arXiv:1504.01330] [INSPIRE].
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 [arXiv:1504.02131] [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].
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 [arXiv:1507.02850] [INSPIRE].
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 [arXiv:1503.06056] [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].
D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev. D 75 (2007) 125019 [arXiv:0704.1835] [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].
C. Anastasiou, R. Britto, B. Feng, Z. Kunszt and P. Mastrolia, D-dimensional unitarity cut method, Phys. Lett. B 645 (2007) 213 [hep-ph/0609191] [INSPIRE].
W.T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes, JHEP 04 (2008) 049 [arXiv:0801.2237] [INSPIRE].
Z. Bern et al., Next-to-Leading Order W + 5-Jet Production at the LHC, Phys. Rev. D 88 (2013) 014025 [arXiv:1304.1253] [INSPIRE].
S. Badger, B. Biedermann, P. Uwer and V. Yundin, Next-to-leading order QCD corrections to five jet production at the LHC, Phys. Rev. D 89 (2014) 034019 [arXiv:1309.6585] [INSPIRE].
D.A. Kosower and K.J. Larsen, Maximal Unitarity at Two Loops, Phys. Rev. D 85 (2012) 045017 [arXiv:1108.1180] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
H. Johansson, D.A. Kosower and K.J. Larsen, Two-Loop Maximal Unitarity with External Masses, Phys. Rev. D 87 (2013) 025030 [arXiv:1208.1754] [INSPIRE].
H. Johansson, D.A. Kosower and K.J. Larsen, Maximal Unitarity for the Four-Mass Double Box, Phys. Rev. D 89 (2014) 125010 [arXiv:1308.4632] [INSPIRE].
M. Søgaard, Global Residues and Two-Loop Hepta-Cuts, JHEP 09 (2013) 116 [arXiv:1306.1496] [INSPIRE].
M. Søgaard and Y. Zhang, Multivariate Residues and Maximal Unitarity, JHEP 12 (2013) 008 [arXiv:1310.6006] [INSPIRE].
M. Søgaard and Y. Zhang, Unitarity Cuts of Integrals with Doubled Propagators, JHEP 07 (2014) 112 [arXiv:1403.2463] [INSPIRE].
M. Søgaard and Y. Zhang, Massive Nonplanar Two-Loop Maximal Unitarity, JHEP 12 (2014) 006 [arXiv:1406.5044] [INSPIRE].
M. Søgaard and Y. Zhang, Elliptic Functions and Maximal Unitarity, Phys. Rev. D 91 (2015) 081701 [arXiv:1412.5577] [INSPIRE].
H. Johansson, D.A. Kosower, K.J. Larsen and M. Søgaard, Cross-Order Integral Relations from Maximal Cuts, Phys. Rev. D 92 (2015) 025015 [arXiv:1503.06711] [INSPIRE].
P. Mastrolia and G. Ossola, On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes, JHEP 11 (2011) 014 [arXiv:1107.6041] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, Hepta-Cuts of Two-Loop Scattering Amplitudes, JHEP 04 (2012) 055 [arXiv:1202.2019] [INSPIRE].
H. Ita, Two-loop Integrand Decomposition into Master Integrals and Surface Terms, Phys. Rev. D 94 (2016) 116015 [arXiv:1510.05626] [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].
Y. Zhang, Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods, JHEP 09 (2012) 042 [arXiv:1205.5707] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Scattering Amplitudes from Multivariate Polynomial Division, Phys. Lett. B 718 (2012) 173 [arXiv:1205.7087] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, An Integrand Reconstruction Method for Three-Loop Amplitudes, JHEP 08 (2012) 065 [arXiv:1207.2976] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Integrand-Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division, Phys. Rev. D 87 (2013) 085026 [arXiv:1209.4319] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes, Phys. Lett. B 727 (2013) 532 [arXiv:1307.5832] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, A Two-Loop Five-Gluon Helicity Amplitude in QCD, JHEP 12 (2013) 045 [arXiv:1310.1051] [INSPIRE].
S. Badger, G. Mogull, A. Ochirov and D. O’Connell, A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory, JHEP 10 (2015) 064 [arXiv:1507.08797] [INSPIRE].
S. Badger, G. Mogull and T. Peraro, Local integrands for two-loop all-plus Yang-Mills amplitudes, JHEP 08 (2016) 063 [arXiv:1606.02244] [INSPIRE].
T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett. 116 (2016) 062001 [Erratum ibid. 116 (2016) 189903] [arXiv:1511.05409] [INSPIRE].
D.C. Dunbar and W.B. Perkins, Two-loop five-point all plus helicity Yang-Mills amplitude, Phys. Rev. D 93 (2016) 085029 [arXiv:1603.07514] [INSPIRE].
D.C. Dunbar, G.R. Jehu and W.B. Perkins, The two-loop n-point all-plus helicity amplitude, Phys. Rev. D 93 (2016) 125006 [arXiv:1604.06631] [INSPIRE].
D.C. Dunbar, G.R. Jehu and W.B. Perkins, Two-loop six gluon all plus helicity amplitude, Phys. Rev. Lett. 117 (2016) 061602 [arXiv:1605.06351] [INSPIRE].
Z. Bern, J.S. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 super Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].
Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for gluon-gluon scattering in QCD and supersymmetric Yang-Mills theory, JHEP 03 (2002) 018 [hep-ph/0201161] [INSPIRE].
S.G. Naculich, All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes, Phys. Lett. B 707 (2012) 191 [arXiv:1110.1859] [INSPIRE].
A.C. Edison and S.G. Naculich, SU(N) group-theory constraints on color-ordered five-point amplitudes at all loop orders, Nucl. Phys. B 858 (2012) 488 [arXiv:1111.3821] [INSPIRE].
R. Kleiss and H. Kuijf, Multi-Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
V. Del Duca, A. Frizzo and F. Maltoni, Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit, Nucl. Phys. B 568 (2000) 211 [hep-ph/9909464] [INSPIRE].
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
Z. Bern and Y.-t. Huang, Basics of Generalized Unitarity, J. Phys. A 44 (2011) 454003 [arXiv:1103.1869] [INSPIRE].
F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, Multi-loop Integrand Reduction with Computational Algebraic Geometry, J. Phys. Conf. Ser. 523 (2014) 012061 [arXiv:1310.4445] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, A two loop four gluon helicity amplitude in QCD, JHEP 01 (2000) 027 [hep-ph/0001001] [INSPIRE].
Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for quark gluon scattering in QCD and gluino gluon scattering in supersymmetric Yang-Mills theory, JHEP 06 (2003) 028 [Erratum ibid. 04 (2014) 112] [hep-ph/0304168] [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
Z. Bern, A. De Freitas, L.J. Dixon and H.L. Wong, Supersymmetric regularization, two loop QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hep-ph/0202271] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson, D.A. Kosower and R. Roiban, Three-Loop Superfiniteness of N = 8 Supergravity, Phys. Rev. Lett. 98 (2007) 161303 [hep-th/0702112] [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N = 8 Supergravity, Phys. Rev. D 78 (2008) 105019 [arXiv:0808.4112] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, The Complete Four-Loop Four-Point Amplitude in N = 4 super-Yang-Mills Theory, Phys. Rev. D 82 (2010) 125040 [arXiv:1008.3327] [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Logarithmic Singularities and Maximally Supersymmetric Amplitudes, JHEP 06 (2015) 202 [arXiv:1412.8584] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].
J.M. Henn and B. Mistlberger, Four-Gluon Scattering at Three Loops, Infrared Structure and the Regge Limit, Phys. Rev. Lett. 117 (2016) 171601 [arXiv:1608.00850] [INSPIRE].
S. Franco, D. Galloni and A. Mariotti, The Geometry of On-Shell Diagrams, JHEP 08 (2014) 038 [arXiv:1310.3820] [INSPIRE].
B. Chen, G. Chen, Y.-K.E. Cheung, Y. Li, R. Xie and Y. Xin, Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes, Eur. Phys. J. C 77 (2017) 80 [arXiv:1411.3889] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [INSPIRE].
S. Franco, D. Galloni, B. Penante and C. Wen, Non-Planar On-Shell Diagrams, JHEP 06 (2015) 199 [arXiv:1502.02034] [INSPIRE].
B. Chen, G. Chen, Y.-K.E. Cheung, R. Xie and Y. Xin, Top-forms of Leading Singularities in Nonplanar Multi-loop Amplitudes, arXiv:1506.02880 [INSPIRE].
J.L. Bourjaily, S. Franco, D. Galloni and C. Wen, Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams, JHEP 10 (2016) 003 [arXiv:1607.01781] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, Cambridge University Press (2012).
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
M.L. Mangano, The Color Structure of Gluon Emission, Nucl. Phys. B 309 (1988) 461 [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
F. Maltoni, K. Paul, T. Stelzer and S. Willenbrock, Color flow decomposition of QCD amplitudes, Phys. Rev. D 67 (2003) 014026 [hep-ph/0209271] [INSPIRE].
H. Ita and K. Ozeren, Colour Decompositions of Multi-quark One-loop QCD Amplitudes, JHEP 02 (2012) 118 [arXiv:1111.4193] [INSPIRE].
C. Reuschle and S. Weinzierl, Decomposition of one-loop QCD amplitudes into primitive amplitudes based on shuffle relations, Phys. Rev. D 88 (2013) 105020 [arXiv:1310.0413] [INSPIRE].
T. Schuster, Color ordering in QCD, Phys. Rev. D 89 (2014) 105022 [arXiv:1311.6296] [INSPIRE].
T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88 (2013) 014020 [arXiv:1304.7809] [INSPIRE].
T. Melia, Getting more flavor out of one-flavor QCD, Phys. Rev. D 89 (2014) 074012 [arXiv:1312.0599] [INSPIRE].
T. Melia, Dyck words and multi-quark amplitudes, PoS (RADCOR 2013) 031.
H. Johansson and A. Ochirov, Color-Kinematics Duality for QCD Amplitudes, JHEP 01 (2016) 170 [arXiv:1507.00332] [INSPIRE].
T. Melia, Proof of a new colour decomposition for QCD amplitudes, JHEP 12 (2015) 107 [arXiv:1509.03297] [INSPIRE].
T. Ohl, Drawing Feynman diagrams with Latex and Metafont, Comput. Phys. Commun. 90 (1995) 340 [hep-ph/9505351] [INSPIRE].
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: 1612.04366
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Ochirov, A., Page, B. Full colour for loop amplitudes in Yang-Mills theory. J. High Energ. Phys. 2017, 100 (2017). https://doi.org/10.1007/JHEP02(2017)100
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2017)100