Abstract
In this work we propose to use leading singularities to obtain the classical pieces of amplitudes of two massive particles whose only interaction is gravitational. Leading singularities are generalizations of unitarity cuts. At one-loop we find that leading singularities obtained by multiple discontinuities in the t-channel contain all the classical information. As the main example, we show how to obtain a compact formula for the fully relativistic classical one-loop contribution to the scattering of two particles with different masses. The non-relativistic limit of the leading singularity agrees with known results in the post-Newtonian expansion. We also compute a variety of higher loop leading singularities including some all-loop families and study some of their properties.
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References
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [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].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of Residues and Grassmannian Dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The All-Loop Integrand For Scattering Amplitudes in Planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [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 (2016).
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge (1966) [INSPIRE].
LIGO Scientific and Virgo collaborations, Tests of general relativity with GW150914, Phys. Rev. Lett. 116 (2016) 221101 [Erratum ibid. 121 (2018) 129902] [arXiv:1602.03841] [INSPIRE].
N. Yunes, K. Yagi and F. Pretorius, Theoretical Physics Implications of the Binary Black-Hole Mergers GW150914 and GW151226, Phys. Rev. D 94 (2016) 084002 [arXiv:1603.08955] [INSPIRE].
M.J. Duff, Quantum Tree Graphs and the Schwarzschild Solution, Phys. Rev. D 7 (1973) 2317 [INSPIRE].
H.W. Hamber and S. Liu, On the quantum corrections to the Newtonian potential, Phys. Lett. B 357 (1995) 51 [hep-th/9505182] [INSPIRE].
A.A. Akhundov, S. Bellucci and A. Shiekh, Gravitational interaction to one loop in effective quantum gravity, Phys. Lett. B 395 (1997) 16 [gr-qc/9611018] [INSPIRE].
I.J. Muzinich and S. Vokos, Long range forces in quantum gravity, Phys. Rev. D 52 (1995) 3472 [hep-th/9501083] [INSPIRE].
J.F. Donoghue, Dispersion relations and effective field theory, in Advanced School on Effective Theories, Almunecar, Spain, 25 June–1 July 1995 (1996) [hep-ph/9607351] [INSPIRE].
S. Scherer, Introduction to chiral perturbation theory, Adv. Nucl. Phys. 27 (2003) 277 [hep-ph/0210398] [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].
L. Blanchet, Gravitational radiation from postNewtonian sources and inspiraling compact binaries, Living Rev. Rel. 5 (2002) 3 [gr-qc/0202016] [INSPIRE].
W.D. Goldberger and I.Z. Rothstein, An Effective field theory of gravity for extended objects, Phys. Rev. D 73 (2006) 104029 [hep-th/0409156] [INSPIRE].
T. Futamase and Y. Itoh, The post-Newtonian approximation for relativistic compact binaries, Living Rev. Rel. 10 (2007) 2 [INSPIRE].
B.R. Holstein and A. Ross, Spin Effects in Long Range Gravitational Scattering, arXiv:0802.0716 [INSPIRE].
R.A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [arXiv:1601.04914] [INSPIRE].
J.F. Donoghue, M.M. Ivanov and A. Shkerin, EPFL Lectures on General Relativity as a Quantum Field Theory, arXiv:1702.00319 [INSPIRE].
A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D 59 (1999) 084006 [gr-qc/9811091] [INSPIRE].
A. Buonanno and T. Damour, Transition from inspiral to plunge in binary black hole coalescences, Phys. Rev. D 62 (2000) 064015 [gr-qc/0001013] [INSPIRE].
T. Damour, Coalescence of two spinning black holes: an effective one-body approach, Phys. Rev. D 64 (2001) 124013 [gr-qc/0103018] [INSPIRE].
T. Damour, P. Jaranowski and G. Schaefer, On the determination of the last stable orbit for circular general relativistic binaries at the third postNewtonian approximation, Phys. Rev. D 62 (2000) 084011 [gr-qc/0005034] [INSPIRE].
N.E.J. Bjerrum-Bohr, B.R. Holstein, L. Planté and P. Vanhove, Graviton-Photon Scattering, Phys. Rev. D 91 (2015) 064008 [arXiv:1410.4148] [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].
B.R. Holstein, Analytical On-shell Calculation of Higher Order Scattering: Massless Particles, arXiv:1611.03074 [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].
D. Bai and Y. Huang, More on the Bending of Light in Quantum Gravity, Phys. Rev. D 95 (2017) 064045 [arXiv:1612.07629] [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].
B.R. Holstein, Analytical On-shell Calculation of Higher Order Scattering: Massive Particles, arXiv:1610.07957 [INSPIRE].
D.J. Burger, R. Carballo-Rubio, N. Moynihan, J. Murugan and A. Weltman, Amplitudes for astrophysicists: known knowns, Gen. Rel. Grav. 50 (2018) 156 [arXiv:1704.05067] [INSPIRE].
B.R. Holstein and J.F. Donoghue, Classical physics and quantum loops, Phys. Rev. Lett. 93 (2004) 201602 [hep-th/0405239] [INSPIRE].
J.F. Donoghue and T. Torma, On the power counting of loop diagrams in general relativity, Phys. Rev. D 54 (1996) 4963 [hep-th/9602121] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
B.A. Kniehl, A.A. Penin, V.A. Smirnov and M. Steinhauser, Potential NRQCD and heavy quarkonium spectrum at next-to-next-to-next-to-leading order, Nucl. Phys. B 635 (2002) 357 [hep-ph/0203166] [INSPIRE].
G. Feinberg and J. Sucher, The Two Photon Exchange Force Between Charged Systems. 1. Spinless Particles, Phys. Rev. D 38 (1988) 3763 [Erratum ibid. D 44 (1991) 3997] [INSPIRE].
E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [INSPIRE].
V.P. Nair, A Current Algebra for Some Gauge Theory Amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
A. Brandhuber, B.J. Spence and G. Travaglini, One-loop gauge theory amplitudes in N = 4 super Yang-Mills from MHV vertices, Nucl. Phys. B 706 (2005) 150 [hep-th/0407214] [INSPIRE].
F. Cachazo, P. Svrček and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [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, 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].
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].
J.B. Gilmore and A. Ross, Effective field theory calculation of second post-Newtonian binary dynamics, Phys. Rev. D 78 (2008) 124021 [arXiv:0810.1328] [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].
W.D. Goldberger, Les Houches lectures on effective field theories and gravitational radiation, in Les Houches Summer School — Session 86: Particle Physics and Cosmology: The Fabric of Spacetime, Les Houches, France, 31 July–25 August 2006 (2007) [hep-ph/0701129] [INSPIRE].
J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [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].
R.A. Porto, Post-Newtonian corrections to the motion of spinning bodies in NRGR, Phys. Rev. D 73 (2006) 104031 [gr-qc/0511061] [INSPIRE].
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Cachazo, F., Guevara, A. Leading singularities and classical gravitational scattering. J. High Energ. Phys. 2020, 181 (2020). https://doi.org/10.1007/JHEP02(2020)181
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DOI: https://doi.org/10.1007/JHEP02(2020)181