Abstract
Using the helicity-spinor language we explore the non-perturbative constraints that Lorentz symmetry imposes on three-point amplitudes where the asymptotic states can be massive. As it is well known, in the case of only massless states the three-point amplitude is fixed up to a coupling constant by these constraints plus some physical requirements. We find that a similar statement can be made when some of the particles have mass. We derive the generic functional form of the three-point amplitude by virtue of Lorentz symmetry, which displays several functional structures accompanied by arbitrary constants. These constants can be related to the coupling constants of the theory, but in an unambiguous fashion only in the case of one massive particle. Constraints on these constants are obtained by imposing that in the UV limit the massive amplitude matches the massless one. In particular, there is a certain Lorentz frame, which corresponds to projecting all the massive momenta along the same null momentum, where the three-point massive amplitude is fully fixed, and has a universal form.
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References
S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
L.J. Dixon, Calculating scattering amplitudes efficiently, in QCD and beyond. Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder, U.S.A., 4-30 June 1995 [hep-ph/9601359] [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
L.J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. A 44 (2011) 454001 [arXiv:1105.0771] [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
M. Srednicki, Quantum field theory, Cambridge University Press (2007).
J.M. Henn and J.C. Plefka, Scattering Amplitudes in Gauge Theories, Lect. Notes Phys. 883 (2014) 1.
S. Weinberg, The quantum theory of fields. Vol. 1: foundations, Cambridge University Press (2005).
P. Schuster and N. Toro, On the Theory of Continuous-Spin Particles: Wavefunctions and Soft-Factor Scattering Amplitudes, JHEP 09 (2013) 104 [arXiv:1302.1198] [INSPIRE].
S. Weinberg, photons and gravitons in s matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. 135 (1964) B1049 [INSPIRE].
P. Benincasa and F. Cachazo, Consistency Conditions on the S-matrix of Massless Particles, arXiv:0705.4305 [INSPIRE].
J.A. Strathdee, J.F. Boyce, R. Delbourgo and A. Salam, Partial wave analysis. I, ICTP, Trieste, preprint IC/67/21 (1967).
R.H. Boels and C. Schwinn, On-shell supersymmetry for massive multiplets, Phys. Rev. D 84 (2011) 065006 [arXiv:1104.2280] [INSPIRE].
R.H. Boels, Three particle superstring amplitudes with massive legs, JHEP 06 (2012) 026 [arXiv:1201.2655] [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].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605.
S.D. Badger, E.W.N. Glover, V.V. Khoze and P. Svrček, Recursion relations for gauge theory amplitudes with massive particles, JHEP 07 (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, JHEP 01 (2006) 066 [hep-th/0507161] [INSPIRE].
D. Forde and D.A. Kosower, All-multiplicity amplitudes with massive scalars, Phys. Rev. D 73 (2006) 065007 [hep-th/0507292] [INSPIRE].
P. Ferrario, G. Rodrigo and P. Talavera, Compact multigluonic scattering amplitudes with heavy scalars and fermions, Phys. Rev. Lett. 96 (2006) 182001 [hep-th/0602043] [INSPIRE].
R.H. Boels, No triangles on the moduli space of maximally supersymmetric gauge theory, JHEP 05 (2010) 046 [arXiv:1003.2989] [INSPIRE].
P. Benincasa, On-shell diagrammatics and the perturbative structure of planar gauge theories, arXiv:1510.03642 [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].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].
S. He and E.Y. Yuan, One-loop Scattering Equations and Amplitudes from Forward Limit, Phys. Rev. D 92 (2015) 105004 [arXiv:1508.06027] [INSPIRE].
G. Chen and K.G. Savvidy, Spinor formalism for massive fields with half-integral spin, Eur. Phys. J. C 72 (2012) 1952 [arXiv:1105.3851] [INSPIRE].
S. Dittmaier, Weyl-van der Waerden formalism for helicity amplitudes of massive particles, Phys. Rev. D 59 (1998) 016007 [hep-ph/9805445] [INSPIRE].
R. Boels, Covariant representation theory of the Poincaré algebra and some of its extensions, JHEP 01 (2010) 010 [arXiv:0908.0738] [INSPIRE].
T. Cohen, H. Elvang and M. Kiermaier, On-shell constructibility of tree amplitudes in general field theories, JHEP 04 (2011) 053 [arXiv:1010.0257] [INSPIRE].
P. Schuster and N. Toro, On the Theory of Continuous-Spin Particles: Helicity Correspondence in Radiation and Forces, JHEP 09 (2013) 105 [arXiv:1302.1577] [INSPIRE].
P. Schuster and N. Toro, A Gauge Field Theory of Continuous-Spin Particles, JHEP 10 (2013) 061 [arXiv:1302.3225] [INSPIRE].
P. Schuster and N. Toro, Continuous-spin particle field theory with helicity correspondence, Phys. Rev. D 91 (2015) 025023 [arXiv:1404.0675] [INSPIRE].
S.L. Adler, Collinearity constraints for on-shell massless particle three-point functions and implications for allowed-forbidden n + 1-point functions, Phys. Rev. D 93 (2016) 065028 [arXiv:1602.05060] [INSPIRE].
L.-A. Douxchamps, Récursion bcfw et amplitudes de diffusion, MSc Thesis, Université Libre de Bruxelles (2013).
D.A. McGady and L. Rodina, Higher-spin massless S-matrices in four-dimensions, Phys. Rev. D 90 (2014) 084048 [arXiv:1311.2938] [INSPIRE].
Z. Bern and G. Chalmers, Factorization in one loop gauge theory, Nucl. Phys. B 447 (1995) 465 [hep-ph/9503236] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, On-shell recurrence relations for one-loop QCD amplitudes, Phys. Rev. D 71 (2005) 105013 [hep-th/0501240] [INSPIRE].
E. Conde, E. Joung and K. Mkrtchyan, Spinor-Helicity Three-Point Amplitudes from Local Cubic Interactions, JHEP 08 (2016) 040 [arXiv:1605.07402] [INSPIRE].
N. Craig, H. Elvang, M. Kiermaier and T. Slatyer, Massive amplitudes on the Coulomb branch of N = 4 SYM, JHEP 12 (2011) 097 [arXiv:1104.2050] [INSPIRE].
L.J. Dixon, E.W.N. Glover and V.V. Khoze, MHV rules for Higgs plus multi-gluon amplitudes, JHEP 12 (2004) 015 [hep-th/0411092] [INSPIRE].
L.D. Landau, On the angular momentum of a system of two photons, Dokl. Akad. Nauk Ser. Fiz. 60 (1948) 207.
C.-N. Yang, Selection Rules for the Dematerialization of a Particle Into Two Photons, Phys. Rev. 77 (1950) 242 [INSPIRE].
M. Kiermaier, The Coulomb-branch S-matrix from massless amplitudes, arXiv:1105.5385 [INSPIRE].
P. Benincasa and E. Conde, Exploring the S-matrix of Massless Particles, Phys. Rev. D 86 (2012) 025007 [arXiv:1108.3078] [INSPIRE].
Z. Bern, J.J. Carrasco, T. Dennen, Y.-t. Huang and H. Ita, Generalized Unitarity and Six-Dimensional Helicity, Phys. Rev. D 83 (2011) 085022 [arXiv:1010.0494] [INSPIRE].
H. Elvang, Y.-t. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].
S. Davies, One-Loop QCD and Higgs to Partons Processes Using Six-Dimensional Helicity and Generalized Unitarity, Phys. Rev. D 84 (2011) 094016 [arXiv:1108.0398] [INSPIRE].
C. Cheung and D. O’Connell, Amplitudes and Spinor-Helicity in Six Dimensions, JHEP 07 (2009) 075 [arXiv:0902.0981] [INSPIRE].
B. Czech, Y.-t. Huang and M. Rozali, Chiral three-point interactions in 5 and 6 dimensions, JHEP 10 (2012) 143 [arXiv:1110.2791] [INSPIRE].
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Conde, E., Marzolla, A. Lorentz constraints on massive three-point amplitudes. J. High Energ. Phys. 2016, 41 (2016). https://doi.org/10.1007/JHEP09(2016)041
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DOI: https://doi.org/10.1007/JHEP09(2016)041