Abstract
The duality between color and kinematics was originally observed for purely adjoint massless gauge theories, and later found to hold even after introducing massive fermionic and scalar matter in arbitrary gauge-group representations. Such a generalization was critical for obtaining both loop amplitudes in pure Einstein gravity and realistic gravitational matter from the double copy. In this paper we elaborate on the double copy that yields amplitudes in gravitational theories coupled to flavored massive matter with spin, which is relevant to the problems of black-hole scattering and gravitational waves. Our construction benefits from making the little group explicit for the massive particles, as shown on lower-point examples. For concreteness, we focus on the double copy of QCD with massive quarks, for which we work out the gravitational Lagrangian up to quartic scalar and vector-scalar couplings. We find new gauge-invariant double-copy formulae for tree-level amplitudes with two distinct-flavor pairs of matter and any number of gravitons. These are similar to, but inherently different from, the well-known Kawai-Lewellen-Tye formulae, since the latter only hold for the double copy of purely adjoint gauge theories.
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References
LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett.116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
LIGO Scientific and Virgo collaborations, GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett.119 (2017) 161101 [arXiv:1710.05832] [INSPIRE].
D. Neill and I.Z. Rothstein, Classical Space-Times from the S Matrix, Nucl. Phys.B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
V. Vaidya, Gravitational spin Hamiltonians from the S matrix, Phys. Rev.D 91 (2015) 024017 [arXiv:1410.5348] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, G. Festuccia, L. Planté and P. Vanhove, General Relativity from Scattering Amplitudes, Phys. Rev. Lett.121 (2018) 171601 [arXiv:1806.04920] [INSPIRE].
C. Cheung, I.Z. Rothstein and M.P. Solon, From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion, Phys. Rev. Lett.121 (2018) 251101 [arXiv:1808.02489] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables and Classical Scattering, JHEP02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
A. Guevara, A. Ochirov and J. Vines, Scattering of Spinning Black Holes from Exponentiated Soft Factors, arXiv:1812.06895 [INSPIRE].
Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order, Phys. Rev. Lett.122 (2019) 201603 [arXiv:1901.04424] [INSPIRE].
A. Brandhuber and G. Travaglini, On higher-derivative effects on the gravitational potential and particle bending, arXiv:1905.05657 [INSPIRE].
A. Cristofoli, N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, On Post-Minkowskian Hamiltonians in General Relativity, arXiv:1906.01579 [INSPIRE].
B. Maybee, D. O’Connell and J. Vines, Observables and amplitudes for spinning particles and black holes, arXiv:1906.09260 [INSPIRE].
A. Guevara, A. Ochirov and J. Vines, Black-hole scattering with general spin directions from minimal-coupling amplitudes, arXiv:1906.10071 [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].
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].
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].
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].
H. Johansson and A. Ochirov, Pure Gravities via Color-Kinematics Duality for Fundamental Matter, JHEP11 (2015) 046 [arXiv:1407.4772] [INSPIRE].
H. Johansson and A. Ochirov, Color-Kinematics Duality for QCD Amplitudes, JHEP01 (2016) 170 [arXiv:1507.00332] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Proof of the fundamental BCJ relations for QCD amplitudes, JHEP09 (2015) 197 [arXiv:1508.01432] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Spontaneously Broken Yang-Mills-Einstein Supergravities as Double Copies, JHEP06 (2017) 064 [arXiv:1511.01740] [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].
C. Boucher-Veronneau and L.J. Dixon, N ≥ 4 Supergravity Amplitudes from Gauge Theory at Two Loops, JHEP12 (2011) 046 [arXiv:1110.1132] [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, S. Davies, T. Dennen and Y.-t. Huang, Absence of Three-Loop Four-Point Divergences in N = 4 Supergravity, Phys. Rev. Lett.108 (2012) 201301 [arXiv:1202.3423] [INSPIRE].
H. Johansson, G. Kälin and G. Mogull, Two-loop supersymmetric QCD and half-maximal supergravity amplitudes, JHEP09 (2017) 019 [arXiv:1706.09381] [INSPIRE].
R. Monteiro and D. O’Connell, The Kinematic Algebra From the Self-Dual Sector, JHEP07 (2011) 007 [arXiv:1105.2565] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy, JHEP07 (2017) 002 [arXiv:1703.00421] [INSPIRE].
M. Chiodaroli, Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities, in Space-Time-Matter: Analytic and Geometric Structures, J. Brüning and M. Staudacher eds., De Gruyter, Berlin, Boston, pp. 266–287 (2018) [DOI:10.1515/9783110452150-011] [arXiv:1607.04129] [INSPIRE].
N. Arkani-Hamed, L. Rodina and J. Trnka, Locality and Unitarity of Scattering Amplitudes from Singularities and Gauge Invariance, Phys. Rev. Lett.120 (2018) 231602 [arXiv:1612.02797] [INSPIRE].
A. Anastasiou, L. Borsten, M.J. Duff, S. Nagy and M. Zoccali, Gravity as Gauge Theory Squared: A Ghost Story, Phys. Rev. Lett.121 (2018) 211601 [arXiv:1807.02486] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Complete construction of magical, symmetric and homogeneous N = 2 supergravities as double copies of gauge theories, Phys. Rev. Lett.117 (2016) 011603 [arXiv:1512.09130] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Gauged Supergravities and Spontaneous Supersymmetry Breaking from the Double Copy Construction, Phys. Rev. Lett.120 (2018) 171601 [arXiv:1710.08796] [INSPIRE].
J.J.M. Carrasco, M. Chiodaroli, M. Günaydin and R. Roiban, One-loop four-point amplitudes in pure and matter-coupled \( \mathcal{N} \) ≤ 4 supergravity, JHEP03 (2013) 056 [arXiv:1212.1146] [INSPIRE].
M. Ben-Shahar and M. Chiodaroli, One-loop amplitudes for \( \mathcal{N} \) = 2 homogeneous supergravities, JHEP03 (2019) 153 [arXiv:1812.00402] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Scattering amplitudes in \( \mathcal{N} \) = 2 Maxwell-Einstein and Yang-Mills/Einstein supergravity, JHEP01 (2015) 081 [arXiv:1408.0764] [INSPIRE].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Plane wave backgrounds and colour-kinematics duality, JHEP02 (2019) 198 [arXiv:1810.05115] [INSPIRE].
J.A. Farrow, A.E. Lipstein and P. McFadden, Double copy structure of CFT correlators, JHEP02 (2019) 130 [arXiv:1812.11129] [INSPIRE].
R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP12 (2014) 056 [arXiv:1410.0239] [INSPIRE].
A. Luna, R. Monteiro, D. O’Connell and C.D. White, The classical double copy for Taub-NUT spacetime, Phys. Lett.B 750 (2015) 272 [arXiv:1507.01869] [INSPIRE].
A. Luna, R. Monteiro, I. Nicholson, A. Ochirov, D. O’Connell, N. Westerberg et al., Perturbative spacetimes from Yang-Mills theory, JHEP04 (2017) 069 [arXiv:1611.07508] [INSPIRE].
A. Luna, R. Monteiro, I. Nicholson and D. O’Connell, Type D Spacetimes and the Weyl Double Copy, Class. Quant. Grav.36 (2019) 065003 [arXiv:1810.08183] [INSPIRE].
D.S. Berman, E. Chacón, A. Luna and C.D. White, The self-dual classical double copy and the Eguchi-Hanson instanton, JHEP01 (2019) 107 [arXiv:1809.04063] [INSPIRE].
K. Lee, Kerr-Schild Double Field Theory and Classical Double Copy, JHEP10 (2018) 027 [arXiv:1807.08443] [INSPIRE].
M. Gurses and B. Tekin, Classical Double Copy: Kerr-Schild-Kundt metrics from Yang-Mills Theory, Phys. Rev.D 98 (2018) 126017 [arXiv:1810.03411] [INSPIRE].
W. Cho and K. Lee, Heterotic Kerr-Schild Double Field Theory and Classical Double Copy, JHEP07 (2019) 030 [arXiv:1904.11650] [INSPIRE].
N. Arkani-Hamed, Y.-t. Huang and D. O’Connell, Kerr Black Holes as Elementary Particles, arXiv:1906.10100 [INSPIRE].
A. Luna, R. Monteiro, I. Nicholson, D. O’Connell and C.D. White, The double copy: Bremsstrahlung and accelerating black holes, JHEP06 (2016) 023 [arXiv:1603.05737] [INSPIRE].
W.D. Goldberger and A.K. Ridgway, Radiation and the classical double copy for color charges, Phys. Rev.D 95 (2017) 125010 [arXiv:1611.03493] [INSPIRE].
N. Bahjat-Abbas, A. Luna and C.D. White, The Kerr-Schild double copy in curved spacetime, JHEP12 (2017) 004 [arXiv:1710.01953] [INSPIRE].
M. Carrillo-González, R. Penco and M. Trodden, The classical double copy in maximally symmetric spacetimes, JHEP04 (2018) 028 [arXiv:1711.01296] [INSPIRE].
A. Luna, I. Nicholson, D. O’Connell and C.D. White, Inelastic Black Hole Scattering from Charged Scalar Amplitudes, JHEP03 (2018) 044 [arXiv:1711.03901] [INSPIRE].
W.D. Goldberger and A.K. Ridgway, Bound states and the classical double copy, Phys. Rev.D 97 (2018) 085019 [arXiv:1711.09493] [INSPIRE].
W.D. Goldberger, J. Li and S.G. Prabhu, Spinning particles, axion radiation and the classical double copy, Phys. Rev.D 97 (2018) 105018 [arXiv:1712.09250] [INSPIRE].
C.-H. Shen, Gravitational Radiation from Color-Kinematics Duality, JHEP11 (2018) 162 [arXiv:1806.07388] [INSPIRE].
J. Plefka, J. Steinhoff and W. Wormsbecher, Effective action of dilaton gravity as the classical double copy of Yang-Mills theory, Phys. Rev.D 99 (2019) 024021 [arXiv:1807.09859] [INSPIRE].
M. Carrillo González, R. Penco and M. Trodden, Radiation of scalar modes and the classical double copy, JHEP11 (2018) 065 [arXiv:1809.04611] [INSPIRE].
N. Bahjat-Abbas, R. Stark-Muchão and C.D. White, Biadjoint wires, Phys. Lett.B 788 (2019) 274 [arXiv:1810.08118] [INSPIRE].
M. Carrillo González, B. Melcher, K. Ratliff, S. Watson and C.D. White, The classical double copy in three spacetime dimensions, JHEP07 (2019) 167 [arXiv:1904.11001] [INSPIRE].
Y.F. Bautista and A. Guevara, From Scattering Amplitudes to Classical Physics: Universality, Double Copy and Soft Theorems, arXiv:1903.12419 [INSPIRE].
J. Plefka, C. Shi, J. Steinhoff and T. Wang, Breakdown of the classical double copy for the effective action of dilaton-gravity at NNLO, arXiv:1906.05875 [INSPIRE].
Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys.B 546 (1999) 423 [hep-th/9811140] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, The Momentum Kernel of Gauge and Gravity Theories, JHEP01 (2011) 001 [arXiv:1010.3933] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Double Copies of Fermions as Matter that Interacts Only Gravitationally, Phys. Rev. Lett.116 (2016) 201601 [arXiv:1601.04523] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory, JHEP08 (2011) 129 [arXiv:1105.3503] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, R. Monteiro and D. O’Connell, Algebras for Amplitudes, JHEP06 (2012) 061 [arXiv:1203.0944] [INSPIRE].
N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, JHEP05 (2018) 096 [arXiv:1711.09102] [INSPIRE].
R.W. Brown and S.G. Naculich, KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry, JHEP03 (2018) 057 [arXiv:1802.01620] [INSPIRE].
C.R. Mafra, Berends-Giele recursion for double-color-ordered amplitudes, JHEP07 (2016) 080 [arXiv:1603.09731] [INSPIRE].
M. Chiodaroli, Q. Jin and R. Roiban, Color/kinematics duality for general abelian orbifolds of N = 4 super Yang-Mills theory, JHEP01 (2014) 152 [arXiv:1311.3600] [INSPIRE].
M.M. Anber, Self-conjugate QCD, arXiv:1906.10315 [INSPIRE].
M. Günaydin, G. Sierra and P.K. Townsend, The Geometry of N = 2 Maxwell-Einstein Supergravity and Jordan Algebras, Nucl. Phys.B 242 (1984) 244 [INSPIRE].
M. Günaydin, G. Sierra and P.K. Townsend, More on d = 5 Maxwell-Einstein Supergravity: Symmetric Spaces and Kinks, Class. Quant. Grav.3 (1986) 763 [INSPIRE].
J.F. Luciani, Coupling of O(2) Supergravity with Several Vector Multiplets, Nucl. Phys.B 132 (1978) 325 [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, Scattering Amplitudes For All Masses and Spins, arXiv:1709.04891 [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Non-Abelian gauged supergravities as double copies, JHEP06 (2019) 099 [arXiv:1812.10434] [INSPIRE].
A. Ochirov, Helicity amplitudes for QCD with massive quarks, JHEP04 (2018) 089 [arXiv:1802.06730] [INSPIRE].
M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, The simplest massive S-matrix: from minimal coupling to Black Holes, JHEP04 (2019) 156 [arXiv:1812.08752] [INSPIRE].
Y.M. Zinoviev, Massive supermultiplets with spin 3/2, JHEP05 (2007) 092 [hep-th/0703118] [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge, U.K. (2012) [INSPIRE].
M.E. Peskin and D.V. Schroeder, An Introduction to quantum field theory, Addison-Wesley (1995) [INSPIRE].
Z. Bern, D.C. Dunbar and T. Shimada, String based methods in perturbative gravity, Phys. Lett.B 312 (1993) 277 [hep-th/9307001] [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.D. Badger, E.W.N. Glover, V.V. Khoze and P. Svrček, Recursion relations for gauge theory amplitudes with massive particles, JHEP07 (2005) 025 [hep-th/0504159] [INSPIRE].
S.D. Badger, E.W.N. Glover and V.V. Khoze, Recursion relations for gauge theory amplitudes with massive vector bosons and fermions, JHEP01 (2006) 066 [hep-th/0507161] [INSPIRE].
R. Britto and A. Ochirov, On-shell recursion for massive fermion currents, JHEP01 (2013) 002 [arXiv:1210.1755] [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].
M.L. Mangano, The Color Structure of Gluon Emission, Nucl. Phys.B 309 (1988) 461 [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, 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].
L.P.S. Singh and C.R. Hagen, Lagrangian formulation for arbitrary spin. 1. The boson case, Phys. Rev.D 9 (1974) 898 [INSPIRE].
L.P.S. Singh and C.R. Hagen, Lagrangian formulation for arbitrary spin. 2. The fermion case, Phys. Rev.D 9 (1974) 910 [INSPIRE].
Y.M. Zinoviev, Massive spin two supermultiplets, hep-th/0206209 [INSPIRE].
N.A. Ondo and A.J. Tolley, Deconstructing Supergravity: Massive Supermultiplets, JHEP11 (2018) 082 [arXiv:1612.08752] [INSPIRE].
I.L. Buchbinder, M.V. Khabarov, T.V. Snegirev and Y.M. Zinoviev, Lagrangian formulation of the massive higher spin N = 1 supermultiplets in AdS4space, Nucl. Phys.B 942 (2019) 1 [arXiv:1901.09637] [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].
A. Ochirov and B. Page, Multi-Quark Colour Decompositions from Unitarity, arXiv:1908.02695 [INSPIRE].
D. Vaman and Y.-P. Yao, Constraints and Generalized Gauge Transformations on Tree-Level Gluon and Graviton Amplitudes, JHEP11 (2010) 028 [arXiv:1007.3475] [INSPIRE].
J.J.M. Carrasco, C.R. Mafra and O. Schlotterer, Abelian Z-theory: NLSM amplitudes and α′-corrections from the open string, JHEP06 (2017) 093 [arXiv:1608.02569] [INSPIRE].
W.D. Goldberger, S.G. Prabhu and J.O. Thompson, Classical gluon and graviton radiation from the bi-adjoint scalar double copy, Phys. Rev.D 96 (2017) 065009 [arXiv:1705.09263] [INSPIRE].
J. Li and S.G. Prabhu, Gravitational radiation from the classical spinning double copy, Phys. Rev.D 97 (2018) 105019 [arXiv:1803.02405] [INSPIRE].
T. Azevedo, M. Chiodaroli, H. Johansson and O. Schlotterer, Heterotic and bosonic string amplitudes via field theory, JHEP10 (2018) 012 [arXiv:1803.05452] [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].
O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys.A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys.61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
A. Ochirov and P. Tourkine, BCJ duality and double copy in the closed string sector, JHEP05 (2014) 136 [arXiv:1312.1326] [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].
Y.-t. Huang, O. Schlotterer and C. Wen, Universality in string interactions, JHEP09 (2016) 155 [arXiv:1602.01674] [INSPIRE].
Y. Geyer and R. Monteiro, Gluons and gravitons at one loop from ambitwistor strings, JHEP03 (2018) 068 [arXiv:1711.09923] [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press (2007) [INSPIRE].
M. de Roo, Matter Coupling in N = 4 Supergravity, Nucl. Phys.B 255 (1985) 515 [INSPIRE].
E. Bergshoeff, I.G. Koh and E. Sezgin, Coupling of Yang-Mills to N = 4, D = 4 Supergravity, Phys. Lett.155B (1985) 71 [INSPIRE].
J.H. Schwarz, Dilaton-axion symmetry, in International Workshop on String Theory, Quantum Gravity and the Unification of Fundamental Interactions, Rome, Italy, 21-26 September 1992, pp. 503–520 (1992) [hep-th/9209125] [INSPIRE].
P.B. Pal, Dirac, Majorana and Weyl fermions, Am. J. Phys.79 (2011) 485 [arXiv:1006.1718] [INSPIRE].
F.J. Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields, Physica7 (1940) 449.
L. Rosenfeld, Sur le tenseur D’Impulsion-Energie, Acad. Roy. Belg. Memoirs de Classes de Science18 (1940) (fasc. 6).
C. Cheung and G.N. Remmen, Hidden Simplicity of the Gravity Action, JHEP09 (2017) 002 [arXiv:1705.00626] [INSPIRE].
L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics, vol. 2, Pergamon Press, Oxford (1975) [INSPIRE].
E. Poisson and C.M. Will, Gravity: Newtonian, Post-Newtonian, Relativistic, Cambridge University Press (2014).
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Johansson, H., Ochirov, A. Double copy for massive quantum particles with spin. J. High Energ. Phys. 2019, 40 (2019). https://doi.org/10.1007/JHEP09(2019)040
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DOI: https://doi.org/10.1007/JHEP09(2019)040