Abstract
Using the CHY-formalism and its extension to a double cover we provide covariant expressions for tree-level amplitudes with two massive scalar legs and an arbitrary number of gravitons in D dimensions. Using unitarity methods, such amplitudes are needed inputs for the computation of post-Newtonian and post-Minkowskian expansions in classical general relativity.
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References
D. Neill and I.Z. Rothstein, Classical Space-Times from the S Matrix, Nucl. Phys. B 877 (2013) 177 [arXiv:1304.7263] [INSPIRE].
B.R. Holstein and A. Ross, Spin Effects in Long Range Gravitational Scattering, arXiv:0802.0716 [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and P. Vanhove, On-shell Techniques and Universal Results in Quantum Gravity, JHEP 02 (2014) 111 [arXiv:1309.0804] [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, J.F. Donoghue, B.R. Holstein, L. Planté and P. Vanhove, Bending of Light in Quantum Gravity, Phys. Rev. Lett. 114 (2015) 061301 [arXiv:1410.7590] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein, L. Plante and P. Vanhove, Light-like Scattering in Quantum Gravity, JHEP 11 (2016) 117 [arXiv:1609.07477] [INSPIRE].
R.A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [arXiv:1601.04914] [INSPIRE].
T. Damour, Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory, Phys. Rev. D 94 (2016) 104015 [arXiv:1609.00354] [INSPIRE].
F. Cachazo and A. Guevara, Leading Singularities and Classical Gravitational Scattering, arXiv:1705.10262 [INSPIRE].
A. Guevara, Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering, JHEP 04 (2019) 033 [arXiv:1706.02314] [INSPIRE].
T. Damour, High-energy gravitational scattering and the general relativistic two-body problem, Phys. Rev. D 97 (2018) 044038 [arXiv:1710.10599] [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].
M. Levi, Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, arXiv:1807.01699 [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].
M.-Z. Chung, Y.-T. Huang, J.-W. Kim and S. Lee, The simplest massive S-matrix: from minimal coupling to Black Holes, JHEP 04 (2019) 156 [arXiv:1812.08752] [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].
Z. Bern, C. Cheung, R. Roiban, C.-H. Shen, M.P. Solon and M. Zeng, Black Hole Binary Dynamics from the Double Copy and Effective Theory, JHEP 10 (2019) 206 [arXiv:1908.01493] [INSPIRE].
A. Antonelli, A. Buonanno, J. Steinhoff, M. van de Meent and J. Vines, Energetics of two-body Hamiltonians in post-Minkowskian gravity, Phys. Rev. D 99 (2019) 104004 [arXiv:1901.07102] [INSPIRE].
A. Cristofoli, N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Post-Minkowskian Hamiltonians in general relativity, Phys. Rev. D 100 (2019) 084040 [arXiv:1906.01579] [INSPIRE].
A. Koemans Collado, P. Di Vecchia and R. Russo, Revisiting the second post-Minkowskian eikonal and the dynamics of binary black holes, Phys. Rev. D 100 (2019) 066028 [arXiv:1904.02667] [INSPIRE].
B. Maybee, D. O’Connell and J. Vines, Observables and amplitudes for spinning particles and black holes, arXiv:1906.09260 [INSPIRE].
Y. Iwasaki, Quantum theory of gravitation vs. classical theory: fourth-order potential, Prog. Theor. Phys. 46 (1971) 1587 [INSPIRE].
B.R. Holstein and J.F. Donoghue, Classical physics and quantum loops, Phys. Rev. Lett. 93 (2004) 201602 [hep-th/0405239] [INSPIRE].
D.A. Kosower, B. Maybee and D. O’Connell, Amplitudes, Observables and Classical Scattering, JHEP 02 (2019) 137 [arXiv:1811.10950] [INSPIRE].
J.F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum gravitational corrections to the nonrelativistic scattering potential of two masses, Phys. Rev. D 67 (2003) 084033 [Erratum ibid. D 71 (2005) 069903] [hep-th/0211072] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein, Quantum corrections to the Schwarzschild and Kerr metrics, Phys. Rev. D 68 (2003) 084005 [Erratum ibid. D 71 (2005) 069904] [hep-th/0211071] [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, 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, B. Feng and T. Sondergaard, Proof of Gravity and Yang-Mills Amplitude Relations, JHEP 09 (2010) 067 [arXiv:1007.3111] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, B. Feng and T. Sondergaard, Gravity and Yang-Mills Amplitude Relations, Phys. Rev. D 82 (2010) 107702 [arXiv:1005.4367] [INSPIRE].
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.D. Badger, E.W.N. Glover, V.V. Khoze and P. Svřcek, Recursion relations for gauge theory amplitudes with massive particles, JHEP 07 (2005) 025 [hep-th/0504159] [INSPIRE].
D. Forde and D.A. Kosower, All-multiplicity amplitudes with massive scalars, Phys. Rev. D 73 (2006) 065007 [hep-th/0507292] [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].
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 Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
H. Gomez, Λ scattering equations, JHEP 06 (2016) 101 [arXiv:1604.05373] [INSPIRE].
C. Cardona and H. Gomez, Elliptic scattering equations, JHEP 06 (2016) 094 [arXiv:1605.01446] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and H. Gomez, New Factorization Relations for Yang-Mills Amplitudes, Phys. Rev. D 99 (2019) 025014 [arXiv:1810.05023] [INSPIRE].
N.E.J. Bjerrum-Bohr, H. Gomez and A. Helset, New factorization relations for nonlinear σ-model amplitudes, Phys. Rev. D 99 (2019) 045009 [arXiv:1811.06024] [INSPIRE].
H. Gomez, Scattering equations and a new factorization for amplitudes. Part I. Gauge theories, JHEP 05 (2019) 128 [arXiv:1810.05407] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Analytic representations of Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 964 [arXiv:1605.06501] [INSPIRE].
C. Cardona, B. Feng, H. Gomez and R. Huang, Cross-ratio Identities and Higher-order Poles of CHY-integrand, JHEP 09 (2016) 133 [arXiv:1606.00670] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Manifesting Color-Kinematics Duality in the Scattering Equation Formalism, JHEP 09 (2016) 094 [arXiv:1608.00006] [INSPIRE].
T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].
E. Casali and P. Tourkine, Infrared behaviour of the one-loop scattering equations and supergravity integrands, JHEP 04 (2015) 013 [arXiv:1412.3787] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [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].
L. Dolan and P. Goddard, Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].
H. Gomez and A. Helset, Scattering equations and a new factorization for amplitudes. Part II. Effective field theories, JHEP 05 (2019) 129 [arXiv:1902.02633] [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].
C.S. Lam and Y.-P. Yao, Evaluation of the Cachazo-He-Yuan gauge amplitude, Phys. Rev. D 93 (2016) 105008 [arXiv:1602.06419] [INSPIRE].
C.-H. Fu, Y.-J. Du, R. Huang and B. Feng, Expansion of Einstein-Yang-Mills Amplitude, JHEP 09 (2017) 021 [arXiv:1702.08158] [INSPIRE].
Y. Geyer and R. Monteiro, Gluons and gravitons at one loop from ambitwistor strings, JHEP 03 (2018) 068 [arXiv:1711.09923] [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 and H. Gomez, Computation of Contour Integrals on \( {\mathcal{M}}_{0,n} \), JHEP 04 (2016) 108 [arXiv:1505.03571] [INSPIRE].
J. Plefka and W. Wormsbecher, New relations for graviton-matter amplitudes, Phys. Rev. D 98 (2018) 026011 [arXiv:1804.09651] [INSPIRE].
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Bjerrum-Bohr, N.E.J., Cristofoli, A., Damgaard, P.H. et al. Scalar-graviton amplitudes. J. High Energ. Phys. 2019, 148 (2019). https://doi.org/10.1007/JHEP11(2019)148
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DOI: https://doi.org/10.1007/JHEP11(2019)148