Abstract
We show that color-kinematics duality is present in tree-level amplitudes of quantum chromodynamics with massive flavored quarks. Starting with the color structure of QCD, we work out a new color decomposition for n-point tree amplitudes in a reduced basis of primitive amplitudes. These primitives, with k quark-antiquark pairs and (n − 2k) gluons, are taken in the (n − 2)!/k! Melia basis, and are independent under the color-algebra Kleiss-Kuijf relations. This generalizes the color decomposition of Del Duca, Dixon, and Maltoni to an arbitrary number of quarks. The color coefficients in the new decomposition are given by compact expressions valid for arbitrary gauge group and representation. Considering the kinematic structure, we show through explicit calculations that color-kinematics duality holds for amplitudes with general configurations of gluons and massive quarks. The new (massive) amplitude relations that follow from the duality can be mapped to a well-defined subset of the familiar BCJ relations for gluons. They restrict the amplitude basis further down to (n − 3)!(2k − 2)/k! primitives, for two or more quark lines. We give a decomposition of the full amplitude in that basis. The presented results provide strong evidence that QCD obeys the color-kinematics duality, at least at tree level. The results are also applicable to supersymmetric and D-dimensional extensions of QCD.
Article PDF
Similar content being viewed by others
References
Z. Bern, L.J. Dixon, F. Febres Cordero, S. Höche, H. Ita, D.A. Kosower et al., Next-to-Leading Order W + 5-Jet Production at the LHC, Phys. Rev. D 88 (2013) 014025 [arXiv:1304.1253] [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].
Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280] [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].
R.K. Ellis, W.T. Giele and Z. Kunszt, A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes, JHEP 03 (2008) 003 [arXiv:0708.2398] [INSPIRE].
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. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].
G. Ossola, C.G. Papadopoulos and R. Pittau, CutTools: A Program implementing the OPP reduction method to compute one-loop amplitudes, JHEP 03 (2008) 042 [arXiv:0711.3596] [INSPIRE].
P. Mastrolia, G. Ossola, C.G. Papadopoulos and R. Pittau, Optimizing the Reduction of One-Loop Amplitudes, JHEP 06 (2008) 030 [arXiv:0803.3964] [INSPIRE].
W.T. Giele and G. Zanderighi, On the Numerical Evaluation of One-Loop Amplitudes: The Gluonic Case, JHEP 06 (2008) 038 [arXiv:0805.2152] [INSPIRE].
R.K. Ellis, W.T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop amplitudes for W + 3 jet production in hadron collisions, JHEP 01 (2009) 012 [arXiv:0810.2762] [INSPIRE].
C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg et al., Precise Predictions for W + 3 Jet Production at Hadron Colliders, Phys. Rev. Lett. 102 (2009) 222001 [arXiv:0902.2760] [INSPIRE].
G. Bevilacqua, M. Czakon, C.G. Papadopoulos, R. Pittau and M. Worek, Assault on the NLO Wishlist: \( pp\to t\overline{t}b\overline{b} \), JHEP 09 (2009) 109 [arXiv:0907.4723] [INSPIRE].
P. Mastrolia, G. Ossola, T. Reiter and F. Tramontano, Scattering AMplitudes from Unitarity-based Reduction Algorithm at the Integrand-level, JHEP 08 (2010) 080 [arXiv:1006.0710] [INSPIRE].
C.F. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde, T. Gleisberg et al., Precise Predictions for W + 4 Jet Production at the Large Hadron Collider, Phys. Rev. Lett. 106 (2011) 092001 [arXiv:1009.2338] [INSPIRE].
S. Badger, B. Biedermann and P. Uwer, NGluon: A Package to Calculate One-loop Multi-gluon Amplitudes, Comput. Phys. Commun. 182 (2011) 1674 [arXiv:1011.2900] [INSPIRE].
V. Hirschi, R. Frederix, S. Frixione, M.V. Garzelli, F. Maltoni and R. Pittau, Automation of one-loop QCD corrections, JHEP 05 (2011) 044 [arXiv:1103.0621] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD, Phys. Rev. Lett. 91 (2003) 182002 [hep-ph/0306192] [INSPIRE].
V. A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1.
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].
F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].
F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
J.M. Drummond and J.M. Henn, All tree-level amplitudes in N = 4 SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].
L.J. Dixon, J.M. Henn, J. Plefka and T. Schuster, All tree-level amplitudes in massless QCD, JHEP 01 (2011) 035 [arXiv:1010.3991] [INSPIRE].
R. Kleiss and H. Kuijf, Multi - Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [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, 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].
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].
D. Kosower, B.-H. Lee and V.P. Nair, Multi gluon scattering: a string based calculation, Phys. Lett. B 201 (1988) 85 [INSPIRE].
M.L. Mangano, The Color Structure of Gluon Emission, Nucl. Phys. B 309 (1988) 461 [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].
Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the Square of Gauge Theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].
J.J. Carrasco and H. Johansson, Five-Point Amplitudes in N = 4 super-Yang-Mills Theory and N = 8 Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
Z. Bern, C. Boucher-Veronneau and H. Johansson, N ≥ 4 Supergravity Amplitudes from Gauge Theory at One Loop, Phys. Rev. D 84 (2011) 105035 [arXiv:1107.1935] [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].
J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled \( \mathcal{N}\le 4 \) supergravity, JHEP 03 (2013) 056 [arXiv:1212.1146] [INSPIRE].
R.H. Boels, R.S. Isermann, R. Monteiro and D. O’Connell, Colour-Kinematics Duality for One-Loop Rational Amplitudes, JHEP 04 (2013) 107 [arXiv:1301.4165] [INSPIRE].
N.E.J. Bjerrum-Bohr, T. Dennen, R. Monteiro and D. O’Connell, Integrand Oxidation and One-Loop Colour-Dual Numerators in N = 4 Gauge Theory, JHEP 07 (2013) 092 [arXiv:1303.2913] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, Y.-t. Huang and J. Nohle, Color-Kinematics Duality for Pure Yang-Mills and Gravity at One and Two Loops, Phys. Rev. D 92 (2015) 045041 [arXiv:1303.6605] [INSPIRE].
J. Nohle, Color-Kinematics Duality in One-Loop Four-Gluon Amplitudes with Matter, Phys. Rev. D 90 (2014) 025020 [arXiv:1309.7416] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].
M. Chiodaroli, Q. Jin and R. Roiban, Color/kinematics duality for general abelian orbifolds of N = 4 super Yang-Mills theory, JHEP 01 (2014) 152 [arXiv:1311.3600] [INSPIRE].
H. Johansson and A. Ochirov, Pure Gravities via Color-Kinematics Duality for Fundamental Matter, JHEP 11 (2015) 046 [arXiv:1407.4772] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in \( \mathcal{N}=5 \) supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].
T. Sondergaard, New Relations for Gauge-Theory Amplitudes with Matter, Nucl. Phys. B 821 (2009) 417 [arXiv:0903.5453] [INSPIRE].
S. Weinzierl, Fermions and the scattering equations, JHEP 03 (2015) 141 [arXiv:1412.5993] [INSPIRE].
S.G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].
S.G. Naculich, CHY representations for gauge theory and gravity amplitudes with up to three massive particles, JHEP 05 (2015) 050 [arXiv:1501.03500] [INSPIRE].
L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].
F.A. Berends and W. Giele, The Six Gluon Process as an Example of Weyl-Van Der Waerden Spinor Calculus, Nucl. Phys. B 294 (1987) 700 [INSPIRE].
M.L. Mangano, S.J. Parke and Z. Xu, Duality and Multi - Gluon Scattering, Nucl. Phys. B 298 (1988) 653 [INSPIRE].
Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].
P. Cvitanovic, P.G. Lauwers and P.N. Scharbach, Gauge Invariance Structure of Quantum Chromodynamics, Nucl. Phys. B 186 (1981) 165 [INSPIRE].
D.A. Kosower, Color Factorization for Fermionic Amplitudes, Nucl. Phys. B 315 (1989) 391 [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
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 [arXiv:1105.4319] [INSPIRE].
H. Ita and K. Ozeren, Colour Decompositions of Multi-quark One-loop QCD Amplitudes, JHEP 02 (2012) 118 [arXiv:1111.4193] [INSPIRE].
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 [arXiv:1209.0100] [INSPIRE].
T. Schuster, Color ordering in QCD, Phys. Rev. D 89 (2014) 105022 [arXiv:1311.6296] [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].
R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP 07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Scattering amplitudes in \( \mathcal{N}=2 \) Maxwell-Einstein and Yang-Mills/Einstein supergravity, JHEP 01 (2015) 081 [arXiv:1408.0764] [INSPIRE].
H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].
C.R. Mafra and O. Schlotterer, The Structure of n-Point One-Loop Open Superstring Amplitudes, JHEP 08 (2014) 099 [arXiv:1203.6215] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
S. Stieberger and T.R. Taylor, Closed String Amplitudes as Single-Valued Open String Amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].
C.R. Mafra and O. Schlotterer, Multiparticle SYM equations of motion and pure spinor BRST blocks, JHEP 07 (2014) 153 [arXiv:1404.4986] [INSPIRE].
S.H. Henry Tye and Y. Zhang, Dual Identities inside the Gluon and the Graviton Scattering Amplitudes, JHEP 06 (2010) 071 [Erratum ibid. 1104 (2011) 114] [arXiv:1003.1732] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
L.A. Barreiro and R. Medina, RNS derivation of N-point disk amplitudes from the revisited S-matrix approach, Nucl. Phys. B 886 (2014) 870 [arXiv:1310.5942] [INSPIRE].
A. Ochirov and P. Tourkine, BCJ duality and double copy in the closed string sector, JHEP 05 (2014) 136 [arXiv:1312.1326] [INSPIRE].
C.R. Mafra and O. Schlotterer, Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace, JHEP 10 (2015) 124 [arXiv:1505.02746] [INSPIRE].
R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [INSPIRE].
B. Feng, R. Huang and Y. Jia, Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program, Phys. Lett. B 695 (2011) 350 [arXiv:1004.3417] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering in Three Dimensions from Rational Maps, JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
T. Melia, Dyck words and multi-quark amplitudes, PoS(RADCOR 2013)031.
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
M. Kiermaier, Gravity as the Square of Gauge Theory talk at Amplitudes 2010, 4–7 May 2010 http://strings.ph.qmul.ac.uk/˜theory/Amplitudes2010/.
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The Momentum Kernel of Gauge and Gravity Theories, JHEP 01 (2011) 001 [arXiv:1010.3933] [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].
Y.-J. Du, M. Sjödahl and J. Thorén, Recursion in multiplet bases for tree-level MHV gluon amplitudes, JHEP 05 (2015) 119 [arXiv:1503.00530] [INSPIRE].
L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [INSPIRE].
E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [arXiv:1008.0098] [INSPIRE].
E. Gardi, J.M. Smillie and C.D. White, On the renormalization of multiparton webs, JHEP 09 (2011) 114 [arXiv:1108.1357] [INSPIRE].
E. Gardi, J.M. Smillie and C.D. White, The Non-Abelian Exponentiation theorem for multiple Wilson lines, JHEP 06 (2013) 088 [arXiv:1304.7040] [INSPIRE].
M. Dukes, E. Gardi, H. McAslan, D.J. Scott and C.D. White, Webs and Posets, JHEP 01 (2014) 024 [arXiv:1310.3127] [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: 1507.00332
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
Johansson, H., Ochirov, A. Color-kinematics duality for QCD amplitudes. J. High Energ. Phys. 2016, 170 (2016). https://doi.org/10.1007/JHEP01(2016)170
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)170