Gluon helicity flip in a plane wave background


We compute the leading probability for a gluon to flip helicity state upon traversing a background plane wave gauge field in pure Yang-Mills theory and QCD, with an arbitrary number of colours and flavours. This is a one-loop calculation in perturbative gauge theory around the gluonic plane wave background, which is treated without approximation (i.e., to all orders in the coupling). We introduce a background-dressed version of the spinor helicity formalism and use it to obtain simple formulae for the flip amplitude with pure external gluon polarizations. We also give in-depth examples for gauge group SU(2), and evaluate both the high- and low-energy limits. Throughout, we compare and contrast with the calculation of photon helicity flip in strong-field QED.

A preprint version of the article is available at ArXiv.


  1. [1]

    J. Jaeckel and A. Ringwald, The Low-Energy Frontier of Particle Physics, Ann. Rev. Nucl. Part. Sci. 60 (2010) 405 [arXiv:1002.0329] [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    B. Dobrich and H. Gies, Axion-like-particle search with high-intensity lasers, JHEP 10 (2010) 022 [arXiv:1006.5579] [INSPIRE].

    ADS  Article  Google Scholar 

  3. [3]

    J. Redondo and A. Ringwald, Light shining through walls, Contemp. Phys. 52 (2011) 211 [arXiv:1011.3741] [INSPIRE].

    ADS  Article  Google Scholar 

  4. [4]

    G.V. Dunne, New Strong-Field QED Effects at ELI: Nonperturbative Vacuum Pair Production, Eur. Phys. J. D 55 (2009) 327 [arXiv:0812.3163] [INSPIRE].

    ADS  Google Scholar 

  5. [5]

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

  6. [6]

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

  7. [7]

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

    ADS  MathSciNet  Google Scholar 

  8. [8]

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

    MathSciNet  Google Scholar 

  9. [9]

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

    ADS  MathSciNet  Article  Google Scholar 

  10. [10]

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

    ADS  Article  Google Scholar 

  11. [11]

    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, July 18–30, 2016, pp. 24–43 (2017) [] [arXiv:1701.03692] [INSPIRE].

  12. [12]

    V.I. Ritus, Quantum effects of the interaction of elementary particles with an intense electromagnetic field, J. Russ. Laser Res. 6 (1985) 497.

    Article  Google Scholar 

  13. [13]

    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].

    ADS  Article  Google Scholar 

  14. [14]

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

  15. [15]

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

    ADS  MathSciNet  Article  Google Scholar 

  16. [16]

    M. Basler and A. Hadicke, ON nonabelian SU(2) plane waves, Phys. Lett. 144B (1984) 83 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  18. [18]

    M. Srednicki, Quantum field theory, Cambridge University Press (2007) [INSPIRE].

  19. [19]

    H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].

  20. [20]

    L.J. Dixon, A brief introduction to modern amplitude methods, in Proceedings, 2012 European School of High-Energy Physics (ESHEP 2012), La Pommeraye, Anjou, France, June 06–19, 2012, pp. 31–67 (2014) [] [arXiv:1310.5353] [INSPIRE].

  21. [21]

    C. Cheung, TASI Lectures on Scattering Amplitudes, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics (TASI 2016), Boulder, CO, U.S.A., June 6–July 1, 2016, pp. 571–623 (2018) [] [arXiv:1708.03872] [INSPIRE].

  22. [22]

    J.S. Toll, The Dispersion relation for light and its application to problems involving electron pairs, Ph.D. Thesis, Princeton U. (1952) [INSPIRE].

  23. [23]

    N.B. Narozhny, Propagation of plane electomagnetic waves in a constant field, JETP 28 (1969) 371.

    ADS  Google Scholar 

  24. [24]

    V.I. Ritus, Radiative corrections in quantum electrodynamics with intense field and their analytical properties, Annals Phys. 69 (1972) 555 [INSPIRE].

    ADS  Article  Google Scholar 

  25. [25]

    G.M. Shore, Superluminality and UV completion, Nucl. Phys. B 778 (2007) 219 [hep-th/0701185] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  26. [26]

    W. Becker and H. Mitter, Vacuum polarization in laser fields, J. Phys. A 8 (1975) 1638 [INSPIRE].

    ADS  Google Scholar 

  27. [27]

    V.N. Baier, A.I. Milshtein and V.M. Strakhovenko, Interaction Between a Photon and a High Intensity Electromagnetic Wave, Zh. Eksp. Teor. Fiz. 69 (1975) 1893 [INSPIRE].

    Google Scholar 

  28. [28]

    E. Iancu, A. Leonidov and L. McLerran, The Color glass condensate: An Introduction, in QCD perspectives on hot and dense matter. Proceedings, NATO Advanced Study Institute, Summer School, Cargese, France, August 6–18, 2001, pp. 73–145 (2002) [hep-ph/0202270] [INSPIRE].

  29. [29]

    E. Iancu and R. Venugopalan, The Color glass condensate and high-energy scattering in QCD, in Quark-gluon plasma 4, R.C. Hwa and X.-N. Wang eds., pp. 249–3363 (2003) [] [hep-ph/0303204] [INSPIRE].

    Chapter  Google Scholar 

  30. [30]

    F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, The Color Glass Condensate, Ann. Rev. Nucl. Part. Sci. 60 (2010) 463 [arXiv:1002.0333] [INSPIRE].

    ADS  Article  Google Scholar 

  31. [31]

    Y.V. Kovchegov and E. Levin, Quantum chromodynamics at high energy, vol. 33, Cambridge University Press (2012) [INSPIRE].

  32. [32]

    J.-P. Blaizot, High gluon densities in heavy ion collisions, Rept. Prog. Phys. 80 (2017) 032301 [arXiv:1607.04448] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  33. [33]

    V. Dinu, T. Heinzl, A. Ilderton, M. Marklund and G. Torgrimsson, Vacuum refractive indices and helicity flip in strong-field QED, Phys. Rev. D 89 (2014) 125003 [arXiv:1312.6419] [INSPIRE].

    ADS  Google Scholar 

  34. [34]

    J.-M. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de poincaré, Ann. Inst. H. Poincare Phys. Theor. 3 (1965) 1.

    MATH  Google Scholar 

  35. [35]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  36. [36]

    S.R. Coleman, Nonabelian Plane Waves, Phys. Lett. 70B (1977) 59 [INSPIRE].

    ADS  Article  Google Scholar 

  37. [37]

    E. Kovacs and S.-y. Lo, Selfdual Propagating Wave Solutions in Yang-Mills Gauge Theory, Phys. Rev. D 19 (1979) 3649 [INSPIRE].

    ADS  Google Scholar 

  38. [38]

    S.-Y. Lo, P. Desmond and E. Kovacs, General selfdual nonabelian plane waves, Phys. Lett. 90B (1980) 419 [INSPIRE].

    ADS  Article  Google Scholar 

  39. [39]

    A. Trautman, A class of null solutions to Yang-Mills equations, J. Phys. A 13 (1980) L1 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  40. [40]

    J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  41. [41]

    T. Heinzl and A. Ilderton, Superintegrable relativistic systems in spacetime-dependent background fields, J. Phys. A 50 (2017) 345204 [arXiv:1701.09168] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  42. [42]

    G.W. Gibbons, Quantized Fields Propagating in Plane Wave Space-Times, Commun. Math. Phys. 45 (1975) 191 [INSPIRE].

    ADS  Article  Google Scholar 

  43. [43]

    S. Deser, Plane waves do not polarize the vacuum, J. Phys. A 8 (1975) 1972 [INSPIRE].

    ADS  Google Scholar 

  44. [44]

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

    ADS  Google Scholar 

  45. [45]

    A. Ilderton and G. Torgrimsson, Scattering in plane-wave backgrounds: infra-red effects and pole structure, Phys. Rev. D 87 (2013) 085040 [arXiv:1210.6840] [INSPIRE].

    ADS  Google Scholar 

  46. [46]

    T. Heinzl, A. Ilderton and M. Marklund, Laser intensity effects in noncommutative QED, Phys. Rev. D 81 (2010) 051902 [arXiv:0909.0656] [INSPIRE].

    ADS  Google Scholar 

  47. [47]

    S. Villalba-Chavez and C. Muller, Photo-production of scalar particles in the field of a circularly polarized laser beam, Phys. Lett. B 718 (2013) 992 [arXiv:1208.3595] [INSPIRE].

    ADS  Article  Google Scholar 

  48. [48]

    B.M. Dillon and B. King, ALP production through non-linear Compton scattering in intense fields, Eur. Phys. J. C 78 (2018) 775 [arXiv:1802.07498] [INSPIRE].

    ADS  Article  Google Scholar 

  49. [49]

    B. King, Electron-seeded ALP production and ALP decay in an oscillating electromagnetic field, Phys. Lett. B 782 (2018) 737 [arXiv:1802.07507] [INSPIRE].

    ADS  Article  Google Scholar 

  50. [50]

    S. Meuren, C.H. Keitel and A. Di Piazza, Nonlinear neutrino-photon interactions inside strong laser pulses, JHEP 06 (2015) 127 [arXiv:1504.02722] [INSPIRE].

    ADS  Article  Google Scholar 

  51. [51]

    T. Heinzl, B. Liesfeld, K.-U. Amthor, H. Schwoerer, R. Sauerbrey and A. Wipf, On the observation of vacuum birefringence, Opt. Commun. 267 (2006) 318 [hep-ph/0601076] [INSPIRE].

  52. [52]

    F. Karbstein, H. Gies, M. Reuter and M. Zepf, Vacuum birefringence in strong inhomogeneous electromagnetic fields, Phys. Rev. D 92 (2015) 071301 [arXiv:1507.01084] [INSPIRE].

    ADS  Google Scholar 

  53. [53]

    H.-P. Schlenvoigt, T. Heinzl, U. Schramm, T.E. Cowan and R. Sauerbrey, Detecting vacuum birefringence with x-ray free electron lasers and high-power optical lasers: a feasibility study, Phys. Scripta 91 (2016) 023010.

    ADS  Article  Google Scholar 

  54. [54]

    Y.V. Kovchegov and M.D. Sievert, Small-x Helicity Evolution: an Operator Treatment, Phys. Rev. D 99 (2019) 054032 [arXiv:1808.09010] [INSPIRE].

    ADS  Google Scholar 

  55. [55]

    Y.V. Kovchegov and M.D. Sievert, Valence Quark Transversity at Small x, Phys. Rev. D 99 (2019) 054033 [arXiv:1808.10354] [INSPIRE].

    ADS  Google Scholar 

  56. [56]

    D. Mustaki, S. Pinsky, J. Shigemitsu and K. Wilson, Perturbative renormalization of null plane QED, Phys. Rev. D 43 (1991) 3411 [INSPIRE].

    ADS  Google Scholar 

  57. [57]

    N.C.J. Schoonderwoerd and B.L.G. Bakker, Equivalence of renormalized covariant and light front perturbation theory. 1. Longitudinal divergences in the Yukawa model, Phys. Rev. D 57 (1998) 4965 [INSPIRE].

  58. [58]

    P.P. Srivastava and S.J. Brodsky, Light front quantized QCD in light cone gauge, Phys. Rev. D 64 (2001) 045006 [hep-ph/0011372] [INSPIRE].

  59. [59]

    L. Mantovani, B. Pasquini, X. Xiong and A. Bacchetta, Revisiting the equivalence of light-front and covariant QED in the light-cone gauge, Phys. Rev. D 94 (2016) 116005 [arXiv:1609.00746] [INSPIRE].

    ADS  Google Scholar 

  60. [60]

    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].

  61. [61]

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

    ADS  Article  Google Scholar 

  62. [62]

    T. Heinzl, Light cone zero modes revisited, in Light cone physics: Hadrons and beyond: Proceedings. 2003, 2003, hep-th/0310165 [INSPIRE].

  63. [63]

    A. Casher, Gauge Fields on the Null Plane, Phys. Rev. D 14 (1976) 452 [INSPIRE].

    ADS  Google Scholar 

  64. [64]

    W. Dittrich and H. Gies, Probing the quantum vacuum. Perturbative effective action approach in quantum electrodynamics and its application, Springer Tracts Mod. Phys. 166 (2000) 1.

  65. [65]

    T. Adamo, Lectures on twistor theory, PoS(Modave2017)003 (2018) [arXiv:1712.02196] [INSPIRE].

  66. [66]

    E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  67. [67]

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

    ADS  MathSciNet  Article  Google Scholar 

  68. [68]

    T.W.B. Kibble, A. Salam and J.A. Strathdee, Intensity Dependent Mass Shift and Symmetry Breaking, Nucl. Phys. B 96 (1975) 255 [INSPIRE].

    ADS  Article  Google Scholar 

  69. [69]

    F. Hebenstreit, A. Ilderton, M. Marklund and J. Zamanian, Strong field effects in laser pulses: the Wigner formalism, Phys. Rev. D 83 (2011) 065007 [arXiv:1011.1923] [INSPIRE].

    ADS  Google Scholar 

  70. [70]

    C. Harvey, T. Heinzl, A. Ilderton and M. Marklund, Intensity-Dependent Electron Mass Shift in a Laser Field: Existence, Universality and Detection, Phys. Rev. Lett. 109 (2012) 100402 [arXiv:1203.6077] [INSPIRE].

    ADS  Article  Google Scholar 

  71. [71]

    V. Dinu, Exact final state integrals for strong field QED, Phys. Rev. A 87 (2013) 052101 [arXiv:1302.1513] [INSPIRE].

    ADS  Article  Google Scholar 

  72. [72]

    T. Heinzl, A. Ilderton and M. Marklund, Finite size effects in stimulated laser pair production, Phys. Lett. B 692 (2010) 250 [arXiv:1002.4018] [INSPIRE].

    ADS  Article  Google Scholar 

  73. [73]

    T. Podszus and A. Di Piazza, High-energy behavior of strong-field QED in an intense plane wave, Phys. Rev. D 99 (2019) 076004 [arXiv:1812.08673] [INSPIRE].

    ADS  Google Scholar 

  74. [74]

    A. Ilderton, Note on the conjectured breakdown of QED perturbation theory in strong fields, Phys. Rev. D 99 (2019) 085002 [arXiv:1901.00317] [INSPIRE].

    ADS  Google Scholar 

  75. [75]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  76. [76]

    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].

    ADS  MathSciNet  Google Scholar 

  77. [77]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  78. [78]

    Z. Bern, T. Dennen, Y.-t. Huang and M. Kiermaier, Gravity as the Square of Gauge Theory, Phys. Rev. D 82 (2010) 065003 [arXiv:1004.0693] [INSPIRE].

    ADS  Google Scholar 

  79. [79]

    Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].

    ADS  Article  Google Scholar 

  80. [80]

    Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in \( \mathcal{N} \) = 5 supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].

    ADS  Google Scholar 

  81. [81]

    Z. Bern, J.J.M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev. D 96 (2017) 126012 [arXiv:1708.06807] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  82. [82]

    Z. Bern et al., Ultraviolet Properties of \( \mathcal{N} \) = 8 Supergravity at Five Loops, Phys. Rev. D 98 (2018) 086021 [arXiv:1804.09311] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  83. [83]

    W.D. Goldberger and A.K. Ridgway, Radiation and the classical double copy for color charges, Phys. Rev. D 95 (2017) 125010 [arXiv:1611.03493] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  84. [84]

    A. Luna, I. Nicholson, D. O’Connell and C.D. White, Inelastic Black Hole Scattering from Charged Scalar Amplitudes, JHEP 03 (2018) 044 [arXiv:1711.03901] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  85. [85]

    W.D. Goldberger, J. Li and S.G. Prabhu, Spinning particles, axion radiation and the classical double copy, Phys. Rev. D 97 (2018) 105018 [arXiv:1712.09250] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  86. [86]

    C.-H. Shen, Gravitational Radiation from Color-Kinematics Duality, JHEP 11 (2018) 162 [arXiv:1806.07388] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  87. [87]

    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].

    ADS  Article  Google Scholar 

  88. [88]

    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].

    ADS  Article  Google Scholar 

  89. [89]

    N. Bahjat-Abbas, A. Luna and C.D. White, The Kerr-Schild double copy in curved spacetime, JHEP 12 (2017) 004 [arXiv:1710.01953] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  90. [90]

    M. Carrillo-González, R. Penco and M. Trodden, The classical double copy in maximally symmetric spacetimes, JHEP 04 (2018) 028 [arXiv:1711.01296] [INSPIRE].

    MathSciNet  Article  Google Scholar 

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Adamo, T., Ilderton, A. Gluon helicity flip in a plane wave background. J. High Energ. Phys. 2019, 15 (2019).

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  • Perturbative QCD
  • Scattering Amplitudes