A minimal approach to the scattering of physical massless bosons

  • Rutger H. Boels
  • Hui Luo
Open Access
Regular Article - Theoretical Physics


Tree and loop level scattering amplitudes which involve physical massless bosons are derived directly from physical constraints such as locality, symmetry and unitarity, bypassing path integral constructions. Amplitudes can be projected onto a minimal basis of kinematic factors through linear algebra, by employing four dimensional spinor helicity methods or at its most general using projection techniques. The linear algebra analysis is closely related to amplitude relations, especially the Bern-Carrasco-Johansson relations for gluon amplitudes and the Kawai-Lewellen-Tye relations between gluons and graviton amplitudes. Projection techniques are known to reduce the computation of loop amplitudes with spinning particles to scalar integrals. Unitarity, locality and integration-by-parts identities can then be used to fix complete tree and loop amplitudes efficiently. The loop amplitudes follow algorithmically from the trees. A number of proof-of-concept examples are presented. These include the planar four point two-loop amplitude in pure Yang-Mills theory as well as a range of one loop amplitudes with internal and external scalars, gluons and gravitons. Several interesting features of the results are highlighted, such as the vanishing of certain basis coefficients for gluon and graviton amplitudes. Effective field theories are naturally and efficiently included into the framework. Dimensional regularisation is employed throughout; different regularisation schemes are worked out explicitly. The presented methods appear most powerful in non-supersymmetric theories in cases with relatively few legs, but with potentially many loops. For instance, in the introduced approach iterated unitarity cuts of four point amplitudes for non-supersymmetric gauge and gravity theories can be computed by matrix multiplication, generalising the so-called rung-rule of maximally supersymmetric theories. The philosophy of the approach to kinematics also leads to a technique to control colour quantum numbers of scattering amplitudes with matter, especially efficient in the adjoint and fundamental representations.


Scattering Amplitudes Space-Time Symmetries 


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.

Supplementary material

13130_2018_8214_MOESM1_ESM.nb (2.1 mb)
ESM1 Results in different schemes (NB 2179 kb)
13130_2018_8214_MOESM2_ESM.txt (21 kb)
ESM2 Coefficient tables cij of ten gluon basis elements and master integrals (TXT 21 kb)
13130_2018_8214_MOESM3_ESM.txt (18 kb)
ESM3 Coefficient tables of eight independent graviton basis elements and three master integrals (TXT 18 kb)


  1. [1]
    H. Elvang and Y.-T. Huang, Scattering amplitudes, arXiv:1308.1697 [INSPIRE].
  2. [2]
    J.M. Henn and J.C. Plefka, Scattering amplitudes in gauge theories, Lect. Notes Phys. 883 (2014) 1 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    S. Dittmaier, Weyl-van der Waerden formalism for helicity amplitudes of massive particles, Phys. Rev. D 59 (1998) 016007 [hep-ph/9805445] [INSPIRE].
  5. [5]
    C. Cheung and D. O’Connell, Amplitudes and spinor-helicity in six dimensions, JHEP 07 (2009) 075 [arXiv:0902.0981] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    R.H. Boels and D. O’Connell, Simple superamplitudes in higher dimensions, JHEP 06 (2012) 163 [arXiv:1201.2653] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, Scattering amplitudes for all masses and spins, arXiv:1709.04891 [INSPIRE].
  8. [8]
    Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    E.W.N. Glover and M.E. Tejeda-Yeomans, Two loop QCD helicity amplitudes for massless quark massless gauge boson scattering, JHEP 06 (2003) 033 [hep-ph/0304169] [INSPIRE].
  10. [10]
    T. Gehrmann, M. Jaquier, E.W.N. Glover and A. Koukoutsakis, Two-loop QCD corrections to the helicity amplitudes for H → 3 partons, JHEP 02 (2012) 056 [arXiv:1112.3554] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  11. [11]
    M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, Reading U.S.A., (1995) [INSPIRE].Google Scholar
  12. [12]
    R. Kleiss and H. Kuijf, Multi-gluon cross-sections and five jet production at hadron colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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].ADSMathSciNetGoogle Scholar
  14. [14]
    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].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    S. Stieberger, Open & closed vs. pure open string disk amplitudes, arXiv:0907.2211 [INSPIRE].
  16. [16]
    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].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    H. Kawai, D.C. Lewellen and S.-H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    D. Lancaster and P. Mansfield, Relations between disk diagrams, Phys. Lett. B 217 (1989) 416 [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    Y.-J. Du, B. Feng and F. Teng, Expansion of all multitrace tree level EYM amplitudes, JHEP 12 (2017) 038 [arXiv:1708.04514] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    R.H. Boels and R. Medina, Graviton and gluon scattering from first principles, Phys. Rev. Lett. 118 (2017) 061602 [arXiv:1607.08246] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    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].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    Z. Bern, J.J. Carrasco, W.-M. Chen, H. Johansson and R. Roiban, Gravity amplitudes as generalized double copies of gauge-theory amplitudes, Phys. Rev. Lett. 118 (2017) 181602 [arXiv:1701.02519] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    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].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    N. Arkani-Hamed and J. Kaplan, On tree amplitudes in gauge theory and gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    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].
  29. [29]
    N. Arkani-Hamed, L. Rodina and J. Trnka, Locality and unitarity from singularities and gauge invariance, arXiv:1612.02797 [INSPIRE].
  30. [30]
    L. Rodina, Uniqueness from locality and BCFW shifts, arXiv:1612.03885 [INSPIRE].
  31. [31]
    S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page and M. Zeng, Two-loop four-gluon amplitudes from numerical unitarity, Phys. Rev. Lett. 119 (2017) 142001 [arXiv:1703.05273] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    Z. Bern, A. Edison, D. Kosower and J. Parra-Martinez, Curvature-squared multiplets, evanescent effects and the U(1) anomaly in N = 4 supergravity, Phys. Rev. D 96 (2017) 066004 [arXiv:1706.01486] [INSPIRE].ADSGoogle Scholar
  33. [33]
    E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals Math. 40 (1939) 149 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S.R. Coleman and J. Mandula, All possible symmetries of the S matrix, Phys. Rev. 159 (1967) 1251 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    E. Noether, Invariant variation problems, Gott. Nachr. 1918 (1918) 235 [physics/0503066] [INSPIRE].
  36. [36]
    F. Bartelmann, Scattering amplitudes from first principles: parity-odd and fermionic case, master’s thesis, Universität Hamburg, Hamburg Germany, (2017).Google Scholar
  37. [37]
    W.-M. Chen, Y.-T. Huang and D.A. McGady, Anomalies without an action, arXiv:1402.7062 [INSPIRE].
  38. [38]
    R.H. Boels, Three particle superstring amplitudes with massive legs, JHEP 06 (2012) 026 [arXiv:1201.2655] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    P.C. Schuster and N. Toro, Constructing the tree-level Yang-Mills S-matrix using complex factorization, JHEP 06 (2009) 079 [arXiv:0811.3207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    R.H. Boels, On the field theory expansion of superstring five point amplitudes, Nucl. Phys. B 876 (2013) 215 [arXiv:1304.7918] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    P. Kravchuk and D. Simmons-Duffin, Counting conformal correlators, JHEP 02 (2018) 096 [arXiv:1612.08987] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    R. van Damme and G. ’t Hooft, Breakdown of unitarity in the dimensional reduction scheme, Phys. Lett. B 150 (1985) 133 [INSPIRE].
  43. [43]
    I. Jack, D.R.T. Jones and K.L. Roberts, Dimensional reduction in nonsupersymmetric theories, Z. Phys. C 62 (1994) 161 [hep-ph/9310301] [INSPIRE].
  44. [44]
    R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
  46. [46]
    R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
  47. [47]
    S. Badger, C. Brønnum-Hansen, F. Buciuni and D. O’Connell, A unitarity compatible approach to one-loop amplitudes with massive fermions, JHEP 06 (2017) 141 [arXiv:1703.05734] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    R.H. Boels, Q. Jin and H. Lüo, Efficient integrand reduction for particles with spin, arXiv:1802.06761 [INSPIRE].
  49. [49]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  51. [51]
    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].
  52. [52]
    K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].ADSMathSciNetGoogle Scholar
  53. [53]
    H. Ita, Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev. D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].ADSMathSciNetGoogle Scholar
  54. [54]
    A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: an algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun. 221 (2017) 203 [arXiv:1612.04252] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
  57. [57]
    N.E.J. Bjerrum-Bohr, D.C. Dunbar and W.B. Perkins, Analytic structure of three-mass triangle coefficients, JHEP 04 (2008) 038 [arXiv:0709.2086] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336] [INSPIRE].
  59. [59]
    R.H. Boels, B.A. Kniehl and G. Yang, On a four-loop form factor in N = 4, PoS(LL2016)039 [arXiv:1607.00172] [INSPIRE].
  60. [60]
    O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  61. [61]
    N.E.J. Bjerrum-Bohr, P.H. Damgaard, H. Johansson and T. Sondergaard, Monodromy-like relations for finite loop amplitudes, JHEP 05 (2011) 039 [arXiv:1103.6190] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    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, pg. 539 [hep-ph/9601359] [INSPIRE].
  63. [63]
    L.V. Bork, D.I. Kazakov, M.V. Kompaniets, D.M. Tolkachev and D.E. Vlasenko, Divergences in maximal supersymmetric Yang-Mills theories in diverse dimensions, JHEP 11 (2015) 059 [arXiv:1508.05570] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  64. [64]
    D.C. Dunbar and P.S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for gluon-gluon scattering in QCD and supersymmetric Yang-Mills theory, JHEP 03 (2002) 018 [hep-ph/0201161] [INSPIRE].
  67. [67]
    Z. Bern, L.J. Dixon and D.A. Kosower, A two loop four gluon helicity amplitude in QCD, JHEP 01 (2000) 027 [hep-ph/0001001] [INSPIRE].
  68. [68]
    S.G. Naculich, All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes, Phys. Lett. B 707 (2012) 191 [arXiv:1110.1859] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    A.C. Edison and S.G. Naculich, Symmetric-group decomposition of SU(N) group-theory constraints on four-, five- and six-point color-ordered amplitudes, JHEP 09 (2012) 069 [arXiv:1207.5511] [INSPIRE].ADSCrossRefGoogle Scholar
  70. [70]
    T. Reichenbächer, Relations among color-ordered gauge theory scattering amplitudes at higher loop-order, master’s thesis, Universität Hamburg, Hamburg Germany, (2013).Google Scholar
  71. [71]
    A. Ochirov and B. Page, Full colour for loop amplitudes in Yang-Mills theory, JHEP 02 (2017) 100 [arXiv:1612.04366] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376] [INSPIRE].
  73. [73]
    R.H. Boels, B.A. Kniehl, O.V. Tarasov and G. Yang, Color-kinematic duality for form factors, JHEP 02 (2013) 063 [arXiv:1211.7028] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    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].ADSCrossRefGoogle Scholar
  75. [75]
    R. Roiban and A.A. Tseytlin, On four-point interactions in massless higher spin theory in flat space, JHEP 04 (2017) 139 [arXiv:1701.05773] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany

Personalised recommendations