The analytic structure of amplitudes on backgrounds from gauge invariance and the infra-red


Gauge invariance and soft limits can be enough to determine the analytic structure of scattering amplitudes in certain theories. This prompts the question of how gauge invariance is connected to analytic structure in more general theories. Here we focus on QED in background plane waves. We show that imposing gauge invariance introduces new virtuality poles into internal momenta on which amplitudes factorise into a series of terms. Each term is gauge invariant, has a different analytic structure in external momenta, and exhibits a hard/soft factorisation. The introduced poles are dictated by infra-red behaviour, which allows us to extend our results to scalar Yukawa theory. The background is treated non-perturbatively throughout.

A preprint version of the article is available at ArXiv.


  1. [1]

    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].

  2. [2]

    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].

  3. [3]

    R.H. Boels and R. Medina, Graviton and gluon scattering from first principles, Phys. Rev. Lett. 118 (2017) 061602 [arXiv:1607.08246] [INSPIRE].

  4. [4]

    M. Berg, I. Buchberger and O. Schlotterer, String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry, JHEP 07 (2017) 138 [arXiv:1611.03459] [INSPIRE].

  5. [5]

    R.H. Boels and H. Lüo, A minimal approach to the scattering of physical massless bosons, JHEP 05 (2018) 063 [arXiv:1710.10208] [INSPIRE].

  6. [6]

    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].

  7. [7]

    L.A. Barreiro and R. Medina, The origin of the KLT relations and nonlinear relations for Yang-Mills amplitudes, Phys. Lett. B 803 (2020) 135299 [arXiv:1910.13519] [INSPIRE].

  8. [8]

    L. Rodina, Uniqueness from gauge invariance and the Adler zero, JHEP 09 (2019) 084 [arXiv:1612.06342] [INSPIRE].

  9. [9]

    C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, On-Shell Recursion Relations for Effective Field Theories, Phys. Rev. Lett. 116 (2016) 041601 [arXiv:1509.03309] [INSPIRE].

  10. [10]

    L. Rodina, Scattering Amplitudes from Soft Theorems and Infrared Behavior, Phys. Rev. Lett. 122 (2019) 071601 [arXiv:1807.09738] [INSPIRE].

  11. [11]

    C. Cheung, K. Kampf, J. Novotny and J. Trnka, Effective Field Theories from Soft Limits of Scattering Amplitudes, Phys. Rev. Lett. 114 (2015) 221602 [arXiv:1412.4095] [INSPIRE].

  12. [12]

    A. Padilla, D. Stefanyszyn and T. Wilson, Probing Scalar Effective Field Theories with the Soft Limits of Scattering Amplitudes, JHEP 04 (2017) 015 [arXiv:1612.04283] [INSPIRE].

  13. [13]

    C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, J. Trnka and C. Wen, Vector Effective Field Theories from Soft Limits, Phys. Rev. Lett. 120 (2018) 261602 [arXiv:1801.01496] [INSPIRE].

  14. [14]

    C. Cheung, C.-H. Shen and C. Wen, Unifying Relations for Scattering Amplitudes, JHEP 02 (2018) 095 [arXiv:1705.03025] [INSPIRE].

  15. [15]

    J.J.M. Carrasco and L. Rodina, UV considerations on scattering amplitudes in a web of theories, Phys. Rev. D 100 (2019) 125007 [arXiv:1908.08033] [INSPIRE].

  16. [16]

    J. Bonifacio, K. Hinterbichler, L.A. Johnson, A. Joyce and R.A. Rosen, Matter Couplings and Equivalence Principles for Soft Scalars, arXiv:1911.04490 [INSPIRE].

  17. [17]

    G. Durieux, T. Kitahara, Y. Shadmi and Y. Weiss, The electroweak effective field theory from on-shell amplitudes, JHEP 01 (2020) 119 [arXiv:1909.10551] [INSPIRE].

  18. [18]

    B. Bachu and A. Yelleshpur, On-Shell Electroweak Sector and the Higgs Mechanism, arXiv:1912.04334 [INSPIRE].

  19. [19]

    D.M. Wolkow, Uber eine Klasse von Losungen der Diracschen Gleichung, Z. Phys. 94 (1935) 250 [INSPIRE].

  20. [20]

    J.-M. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de Poincaŕe, Ann. I.H.P. Phys. Th́eor. 3 (1965) 1.

  21. [21]

    C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll symmetry of plane gravitational waves, Class. Quant. Grav. 34 (2017) 175003 [arXiv:1702.08284] [INSPIRE].

  22. [22]

    T. Adamo, E. Casali, L. Mason and S. Nekovar, Amplitudes on plane waves from ambitwistor strings, JHEP 11 (2017) 160 [arXiv:1708.09249] [INSPIRE].

  23. [23]

    Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications, arXiv:1909.01358 [INSPIRE].

  24. [24]

    T. Adamo, E. Casali, L. Mason and S. Nekovar, Scattering on plane waves and the double copy, Class. Quant. Grav. 35 (2018) 015004 [arXiv:1706.08925] [INSPIRE].

  25. [25]

    T. Adamo, E. Casali, L. Mason and S. Nekovar, Plane wave backgrounds and colour-kinematics duality, JHEP 02 (2019) 198 [arXiv:1810.05115] [INSPIRE].

  26. [26]

    V.I. Ritus, Vacuum polarization correction to elastic electron and muon scattering in an intense field and pair electro- and muoproduction, Nucl. Phys. B 44 (1972) 236 [INSPIRE].

  27. [27]

    A. Ilderton, Trident pair production in strong laser pulses, Phys. Rev. Lett. 106 (2011) 020404 [arXiv:1011.4072] [INSPIRE].

  28. [28]

    D. Seipt and B. Kampfer, Two-photon Compton process in pulsed intense laser fields, Phys. Rev. D 85 (2012) 101701 [arXiv:1201.4045] [INSPIRE].

  29. [29]

    B. King and H. Ruhl, Trident pair production in a constant crossed field, Phys. Rev. D 88 (2013) 013005 [arXiv:1303.1356] [INSPIRE].

  30. [30]

    V. Dinu and G. Torgrimsson, Trident pair production in plane waves: Coherence, exchange and spacetime inhomogeneity, Phys. Rev. D 97 (2018) 036021 [arXiv:1711.04344] [INSPIRE].

  31. [31]

    F. Mackenroth and A. Di Piazza, Nonlinear trident pair production in an arbitrary plane wave: a focus on the properties of the transition amplitude, Phys. Rev. D 98 (2018) 116002 [arXiv:1805.01731] [INSPIRE].

  32. [32]

    P.M. Zhang, M. Cariglia, M. Elbistan and P.A. Horvathy, Scaling and conformal symmetries for plane gravitational waves, J. Math. Phys. 61 (2020) 022502 [arXiv:1905.08661] [INSPIRE].

  33. [33]

    H. Bondi, F.A.E. Pirani and I. Robinson, Gravitational waves in general relativity. 3. Exact plane waves, Proc. Roy. Soc. Lond. A 251 (1959) 519 [INSPIRE].

  34. [34]

    R. Monteiro, D. O’Connell and C.D. White, Black holes and the double copy, JHEP 12 (2014) 056 [arXiv:1410.0239] [INSPIRE].

  35. [35]

    J. Ehlers and W. Kundt, Exact solutions of the gravitational field equations, in Gravitation: An Introduction to Current Research, John Wiley & Sons, pp. 49–101 (1962) [INSPIRE].

  36. [36]

    V. Dinu, T. Heinzl and A. Ilderton, Infra-Red Divergences in Plane Wave Backgrounds, Phys. Rev. D 86 (2012) 085037 [arXiv:1206.3957] [INSPIRE].

  37. [37]

    P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, The Memory Effect for Plane Gravitational Waves, Phys. Lett. B 772 (2017) 743 [arXiv:1704.05997] [INSPIRE].

  38. [38]

    Y. Hamada and S. Sugishita, Notes on the gravitational, electromagnetic and axion memory effects, JHEP 07 (2018) 017 [arXiv:1803.00738] [INSPIRE].

  39. [39]

    G.M. Shore, Memory, Penrose Limits and the Geometry of Gravitational Shockwaves and Gyratons, JHEP 12 (2018) 133 [arXiv:1811.08827] [INSPIRE].

  40. [40]

    T.W.B. Kibble, Frequency Shift in High-Intensity Compton Scattering, Phys. Rev. 138 (1965) B740 [INSPIRE].

  41. [41]

    B.S. DeWitt, Quantum Theory of Gravity. 2. The Manifestly Covariant Theory, Phys. Rev. 162 (1967) 1195 [INSPIRE].

  42. [42]

    G. ’t Hooft, The Background Field Method in Gauge Field Theories, in Functional and Probabilistic Methods in Quantum Field Theory. 1. Proceedings, 12th Winter School of Theoretical Physics, Karpacz, 17 February–2 March 1975, pp. 345–369 (1975) [INSPIRE].

  43. [43]

    D.G. Boulware, Gauge Dependence of the Effective Action, Phys. Rev. D 23 (1981) 389 [INSPIRE].

  44. [44]

    L.F. Abbott, Introduction to the Background Field Method, Acta Phys. Polon. B 13 (1982) 33 [INSPIRE].

  45. [45]

    W.H. Furry, On Bound States and Scattering in Positron Theory, Phys. Rev. 81 (1951) 115 [INSPIRE].

  46. [46]

    T. Heinzl, Light cone quantization: Foundations and applications, Lect. Notes Phys. 572 (2001) 55 [hep-th/0008096] [INSPIRE].

  47. [47]

    A. Di Piazza, C. Muller, K.Z. Hatsagortsyan and C.H. Keitel, Extremely high-intensity laser interactions with fundamental quantum systems, Rev. Mod. Phys. 84 (2012) 1177 [arXiv:1111.3886] [INSPIRE].

  48. [48]

    B. King and T. Heinzl, Measuring Vacuum Polarisation with High Power Lasers, arXiv:1510.08456 [INSPIRE].

  49. [49]

    D. Seipt, Volkov States and Non-linear Compton Scattering in Short and Intense Laser Pulses, in Proceedings, Quantum Field Theory at the Limits: from Strong Fields to Heavy Quarks (HQ 2016), Dubna, Russia, 18–30 July 2016, pp. 24–43 (2017) [DOI] [arXiv:1701.03692] [INSPIRE].

  50. [50]

    A. Ilderton, B. King and A.J. Macleod, Absorption cross section in an intense plane wave background, Phys. Rev. D 100 (2019) 076002 [arXiv:1907.12835] [INSPIRE].

  51. [51]

    S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge University Press (2005) [INSPIRE].

  52. [52]

    M.E. Peskin and D.V. Schroeder, An Introduction to quantum field theory, Addison-Wesley, Reading, U.S.A. (1995).

  53. [53]

    S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rept. 301 (1998) 299 [hep-ph/9705477] [INSPIRE].

  54. [54]

    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].

  55. [55]

    D. Nguyen, M. Spradlin, A. Volovich and C. Wen, The Tree Formula for MHV Graviton Amplitudes, JHEP 07 (2010) 045 [arXiv:0907.2276] [INSPIRE].

  56. [56]

    C. Boucher-Veronneau and A.J. Larkoski, Constructing Amplitudes from Their Soft Limits, JHEP 09 (2011) 130 [arXiv:1108.5385] [INSPIRE].

  57. [57]

    D. Nandan and C. Wen, Generating All Tree Amplitudes in N = 4 SYM by Inverse Soft Limit, JHEP 08 (2012) 040 [arXiv:1204.4841] [INSPIRE].

  58. [58]

    J.M. Cornwall, Confinement and Infrared Properties of Yang-Mills Theory, in US-Japan Seminar on geometric models of the elementary particles, Osaka, 7–11 June 1976 [INSPIRE].

  59. [59]

    M. Lavelle, Gauge invariant effective gluon mass from the operator product expansion, Phys. Rev. D 44 (1991) 26 [INSPIRE].

  60. [60]

    D. Binosi and J. Papavassiliou, Pinch Technique: Theory and Applications, Phys. Rept. 479 (2009) 1 [arXiv:0909.2536] [INSPIRE].

  61. [61]

    A. Denner, G. Weiglein and S. Dittmaier, Gauge invariance of Green functions: Background field method versus pinch technique, Phys. Lett. B 333 (1994) 420 [hep-ph/9406204] [INSPIRE].

  62. [62]

    J.J.M. Carrasco, L. Rodina, Z. Yin and S. Zekioglu, Simple encoding of higher derivative gauge and gravity counterterms, arXiv:1910.12850 [INSPIRE].

  63. [63]

    T. Adamo and A. Ilderton, Gluon helicity flip in a plane wave background, JHEP 06 (2019) 015 [arXiv:1903.01491] [INSPIRE].

  64. [64]

    M. Boca and V. Florescu, Nonlinear Compton scattering with a laser pulse, Phys. Rev. A 80 (2009) 053403 [INSPIRE].

  65. [65]

    H. Hu, C. Muller and C.H. Keitel, Complete QED theory of multiphoton trident pair production in strong laser fields, Phys. Rev. Lett. 105 (2010) 080401 [arXiv:1002.2596] [INSPIRE].

  66. [66]

    U.H. Acosta and B. K¨ampfer, Laser pulse-length effects in trident pair production, Plasma Phys. Control. Fusion 61 (2019) 084011.

Download references

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

Author information



Corresponding author

Correspondence to Alexander J. MacLeod.

Additional information

ArXiv ePrint: 2001.10553

Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ilderton, A., MacLeod, A.J. The analytic structure of amplitudes on backgrounds from gauge invariance and the infra-red. J. High Energ. Phys. 2020, 78 (2020).

Download citation


  • Gauge Symmetry
  • Scattering Amplitudes