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String-inspired representations of photon/gluon amplitudes

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Abstract

The string-based Bern-Kosower rules provide an efficient way for obtaining parameter integral representations of the one-loop N -photon/gluon amplitudes involving a scalar, spinor or gluon loop, starting from a master formula and using a certain integration-by-parts (“IBP”) procedure. Strassler observed that this algorithm also relates to gauge invariance, since it leads to the absorption of polarization vectors into field strength tensors. Here we present a systematic IBP algorithm that works for arbitrary N and leads to an integrand that is not only suitable for the application of the Bern-Kosower rules but also optimized with respect to gauge invariance. In the photon case this means manifest transversality at the integrand level, in the gluon case that a form factor decomposition of the amplitude into transversal and longitudinal parts is generated naturally by the IBP, without the necessity to consider the nonabelian Ward identities. Our algorithm is valid off-shell, and provides an extremely efficient way of calculating the one-loop one-particle-irreducible off-shell Green’s functions (“vertices”) in QCD. It can also be applied essentially unchanged to the one-loop gauge boson amplitudes in open string theory. In the abelian case, we study the systematics of the IBP also for the practically important case of the one-loop N -photon amplitudes in a constant field.

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References

  1. J.-L. Gervais and A. Neveu, Feynman rules for massive gauge fields with dual diagram topology, Nucl. Phys. B 46 (1972) 381 [INSPIRE].

    Article  ADS  Google Scholar 

  2. M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].

    Article  ADS  Google Scholar 

  3. J.A. Minahan, One loop amplitudes on orbifolds and the renormalization of coupling constants, Nucl. Phys. B 298 (1988) 36 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. V.S. Kaplunovsky, One Loop Threshold Effects in String Unification, Nucl. Phys. B 307 (1988) 145 [Erratum ibid. B 382 (1992) 436] [hep-th/9205068] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. R. Metsaev and A.A. Tseytlin, On loop corrections to string theory effective actions, Nucl. Phys. B 298 (1988) 109 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. Z. Bern and D.A. Kosower, Efficient calculation of one loop QCD amplitudes, Phys. Rev. Lett. 66 (1991) 1669 [INSPIRE].

    Article  ADS  Google Scholar 

  7. Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. Z. Bern and D.A. Kosower, The Computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70 (1993) 2677 [hep-ph/9302280] [INSPIRE].

    Article  ADS  Google Scholar 

  10. Z. Bern and D.C. Dunbar, A Mapping between Feynman and string motivated one loop rules in gauge theories, Nucl. Phys. B 379 (1992) 562 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. M.J. Strassler, Field theory without Feynman diagrams: one loop effective actions, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. R. Feynman, Mathematical formulation of the quantum theory of electromagnetic interaction, Phys. Rev. 80 (1950) 440 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. M. Mondragón, L. Nellen, M.G. Schmidt and C. Schubert, Yukawa couplings for the spinning particle and the worldline formalism, Phys. Lett. B 351 (1995) 200 [hep-th/9502125] [INSPIRE].

    ADS  Google Scholar 

  14. M. Mondragón, L. Nellen, M.G. Schmidt and C. Schubert, Axial couplings on the worldline, Phys. Lett. B 366 (1996) 212 [hep-th/9510036] [INSPIRE].

    ADS  Google Scholar 

  15. E. D’Hoker and D.G. Gagné, Worldline path integrals for fermions with scalar, pseudoscalar and vector couplings, Nucl. Phys. B 467 (1996) 272 [hep-th/9508131] [INSPIRE].

    Article  ADS  Google Scholar 

  16. E. D’Hoker and D.G. Gagné, Worldline path integrals for fermions with general couplings, Nucl. Phys. B 467 (1996) 297 [hep-th/9512080] [INSPIRE].

    Article  ADS  Google Scholar 

  17. M.G. Schmidt and C. Schubert, On the calculation of effective actions by string methods, Phys. Lett. B 318 (1993) 438 [hep-th/9309055] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  18. M.G. Schmidt and C. Schubert, Multiloop calculations in the string inspired formalism: the Single spinor loop in QED, Phys. Rev. D 53 (1996) 2150 [hep-th/9410100] [INSPIRE].

    ADS  Google Scholar 

  19. H.-T. Sato and M.G. Schmidt, Exact combinatorics of Bern-Kosower type amplitudes for two loop φ 3 theory, Nucl. Phys. B 524 (1998) 742 [hep-th/9802127] [INSPIRE].

    Article  ADS  Google Scholar 

  20. H.-T. Sato and M.G. Schmidt, Worldline approach to the Bern-Kosower formalism in two loop Yang-Mills theory, Nucl. Phys. B 560 (1999) 551 [hep-th/9812229] [INSPIRE].

    Article  ADS  Google Scholar 

  21. H.-T. Sato, M.G. Schmidt and C. Zahlten, Two loop Yang-Mills theory in the worldline formalism and an Euler-Heisenberg type action, Nucl. Phys. B 579 (2000) 492 [hep-th/0003070] [INSPIRE].

    Article  ADS  Google Scholar 

  22. J.M. Pawlowski, M.G. Schmidt and J.-H. Zhang, On the Yang-Mills two-loop effective action with wordline methods, Phys. Lett. B 677 (2009) 100 [arXiv:0812.0008] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  23. P. Dai and W. Siegel, Worldline Green Functions for Arbitrary Feynman Diagrams, Nucl. Phys. B 770 (2007) 107 [hep-th/0608062] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. R. Shaisultanov, On the string inspired approach to QED in external field, Phys. Lett. B 378 (1996) 354 [hep-th/9512142] [INSPIRE].

    ADS  Google Scholar 

  25. M. Reuter, M.G. Schmidt and C. Schubert, Constant external fields in gauge theory and the spin 0, 1/2, 1 path integrals, Annals Phys. 259 (1997) 313 [hep-th/9610191] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. C. Schubert, Vacuum polarization tensors in constant electromagnetic fields. Part 1., Nucl. Phys. B 585 (2000) 407 [hep-ph/0001288] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. F. Bastianelli and P. van Nieuwenhuizen, Path Integrals and Anomalies in Curved Space, Cambridge University Press, Cambridge, U.K. (2006).

    Book  MATH  Google Scholar 

  28. F. Bastianelli and A. Zirotti, Worldline formalism in a gravitational background, Nucl. Phys. B 642 (2002) 372 [hep-th/0205182] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. F. Bastianelli, O. Corradini and A. Zirotti, dimensional regularization for N = 1 supersymmetric σ-models and the worldline formalism, Phys. Rev. D 67 (2003) 104009 [hep-th/0211134] [INSPIRE].

    ADS  Google Scholar 

  30. F. Bastianelli, O. Corradini and A. Zirotti, BRST treatment of zero modes for the worldline formalism in curved space, JHEP 01 (2004) 023 [hep-th/0312064] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  31. F. Bastianelli, P. Benincasa and S. Giombi, Worldline approach to vector and antisymmetric tensor fields, JHEP 04 (2005) 010 [hep-th/0503155] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. F. Bastianelli, P. Benincasa and S. Giombi, Worldline approach to vector and antisymmetric tensor fields. II., JHEP 10 (2005) 114 [hep-th/0510010] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. F. Bastianelli and C. Schubert, One loop photon-graviton mixing in an electromagnetic field: part 1, JHEP 02 (2005) 069 [gr-qc/0412095] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. F. Bastianelli, U. Nucamendi, C. Schubert and V. Villanueva, One loop photon-graviton mixing in an electromagnetic field: part 2, JHEP 11 (2007) 099 [arXiv:0710.5572] [INSPIRE].

    Article  ADS  Google Scholar 

  35. F. Bastianelli, U. Nucamendi, C. Schubert and V. Villanueva, Photon-graviton mixing in an electromagnetic field, J. Phys. A 41 (2008) 164048 [arXiv:0711.0992] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  36. D. McKeon and A. Rebhan, Thermal Green’s functions from quantum mechanical path integrals, Phys. Rev. D 47 (1993) 5487 [hep-th/9211076] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  37. D. McKeon and A. Rebhan, Thermal Green’s functions from quantum mechanical path integrals 2: inclusion of fermions, Phys. Rev. D 49 (1994) 1047 [hep-th/9306148] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  38. M. Haack and M.G. Schmidt, Axion decay in a constant electromagnetic background field and at finite temperature using worldline methods, Eur. Phys. J. C 7 (1999) 149 [hep-th/9806138] [INSPIRE].

    ADS  Google Scholar 

  39. H.-T. Sato, Integral representations of thermodynamic 1PI Green functions in the worldline formalism, J. Math. Phys. 40 (1999) 6407 [hep-th/9809053] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. H.-T. Sato, Integral representations of thermodynamic 1PI Green functions in the worldline formalism, J. Math. Phys. 40 (1999) 6407 [hep-th/9809053] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. S. Weinberg, The Quantum Theory of Fields, Cambridge University Press, Cambridge, U.K. (1995).

    Google Scholar 

  42. Z. Bern, A Compact representation of the one loop N gluon amplitude, Phys. Lett. B 296 (1992) 85 [INSPIRE].

    ADS  Google Scholar 

  43. Z. Bern, String based perturbative methods for gauge theories, hep-ph/9304249 [INSPIRE].

  44. M.J. Strassler, Field theory without Feynman diagrams: a Demonstration using actions induced by heavy particles., SLAC-PUB-5978 (1992) [INSPIRE].

  45. C. Schubert, The Structure of the Bern-Kosower integrand for the N gluon amplitude, Eur. Phys. J. C 5 (1998) 693 [hep-th/9710067] [INSPIRE].

    ADS  Google Scholar 

  46. N. Bjerrum-Bohr and P. Vanhove, Absence of Triangles in Maximal Supergravity Amplitudes, JHEP 10 (2008) 006 [arXiv:0805.3682] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. J.S. Ball and T.-W. Chiu, Analytic properties of the vertex function in gauge theories. 2., Phys. Rev. D 22 (1980) 2550 [Erratum ibid. D 23 (1981) 3085] [INSPIRE].

    ADS  Google Scholar 

  48. M.J. Strassler, The Bern-Kosower Rules and their Relation to Quantum Field Theory, Ph.D. Thesis, Stanford University (1993).

  49. S. Badger, N. Bjerrum-Bohr and P. Vanhove, Simplicity in the Structure of QED and Gravity Amplitudes, JHEP 02 (2009) 038 [arXiv:0811.3405] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. S. Kim and M. Baker, Consequences of gauge invariance for the interacting vertices in nonabelian gauge theories, Nucl. Phys. B 164 (1980) 152 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. J.M. Cornwall and J. Papavassiliou, Gauge Invariant Three Gluon Vertex in QCD, Phys. Rev. D 40 (1989) 3474 [INSPIRE].

    ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  53. N. Ahmadiniaz and C. Schubert, A covariant representation of the Ball-Chiu vertex, arXiv:1210.2331 [INSPIRE].

  54. L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].

  55. 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].

    Article  ADS  Google Scholar 

  56. 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].

    ADS  Google Scholar 

  57. S. Hashimoto, J. Kodaira, Y. Yasui and K. Sasaki, The Background field method: alternative way of deriving the pinch technique’s results, Phys. Rev. D 50 (1994) 7066 [hep-ph/9406271] [INSPIRE].

    ADS  Google Scholar 

  58. A. Pilaftsis, Generalized pinch technique and the background field method in general gauges, Nucl. Phys. B 487 (1997) 467 [hep-ph/9607451] [INSPIRE].

    Article  ADS  Google Scholar 

  59. M. Binger and S.J. Brodsky, The Form-factors of the gauge-invariant three-gluon vertex, Phys. Rev. D 74 (2006) 054016 [hep-ph/0602199] [INSPIRE].

    ADS  Google Scholar 

  60. D. McKeon and T. Sherry, Radiative effects in a constant magnetic field using the quantum mechanical path integral, Mod. Phys. Lett. A 9 (1994) 2167 [INSPIRE].

    ADS  Google Scholar 

  61. D. Cangemi, E. D’Hoker and G.V. Dunne, Derivative expansion of the effective action and vacuum instability for QED in (2 + 1)-dimensions, Phys. Rev. D 51 (1995) 2513 [hep-th/9409113] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  62. V. Gusynin and I. Shovkovy, Derivative expansion for the one loop effective Lagrangian in QED, Can. J. Phys. 74 (1996) 282 [hep-ph/9509383] [INSPIRE].

    Article  ADS  Google Scholar 

  63. S.L. Adler and C. Schubert, Photon splitting in a strong magnetic field: recalculation and comparison with previous calculations, Phys. Rev. Lett. 77 (1996) 1695 [hep-th/9605035] [INSPIRE].

    Article  ADS  Google Scholar 

  64. D. Fliegner, M. Reuter, M. Schmidt and C. Schubert, The Two loop Euler-Heisenberg Lagrangian in dimensional renormalization, Theor. Math. Phys. 113 (1997) 1442 [hep-th/9704194] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  65. B. Körs and M.G. Schmidt, The Effective two loop Euler-Heisenberg action for scalar and spinor QED in a general constant background field, Eur. Phys. J. C 6 (1999) 175 [hep-th/9803144] [INSPIRE].

    ADS  Google Scholar 

  66. G.V. Dunne and C. Schubert, Two loop selfdual Euler-Heisenberg Lagrangians. 1. Real part and helicity amplitudes, JHEP 08 (2002) 053 [hep-th/0205004] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  67. G.V. Dunne and C. Schubert, Two loop selfdual Euler-Heisenberg Lagrangians. 2. Imaginary part and Borel analysis, JHEP 06 (2002) 042 [hep-th/0205005] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  68. L.C. Martin, C. Schubert and V.M. Villanueva Sandoval, On the low-energy limit of the QED N photon amplitudes, Nucl. Phys. B 668 (2003) 335 [hep-th/0301022] [INSPIRE].

    Article  ADS  Google Scholar 

  69. R. Karplus and M. Neuman, The scattering of light by light, Phys. Rev. 83 (1951) 776 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  70. V. Costantini, B. de Tollis and G. Pistoni, Nonlinear effects in quantum electrodynamics, Nuov. Cim. A2 (1971) 733.

    Article  ADS  Google Scholar 

  71. T. Binoth, J.P. Guillet, G. Heinrich, E. Pilon and C. Schubert, An Algebraic/numerical formalism for one-loop multi-leg amplitudes, JHEP 10 (2005) 015 [hep-ph/0504267] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  72. N. Ahmadiniaz, C. Lopez and C. Schubert, The four-photon amplitudes fully off-shell, in preparation.

  73. Z. Bern, L.J. Dixon and D.A. Kosower, On-Shell Methods in Perturbative QCD, Annals Phys. 322 (2007) 1587 [arXiv:0704.2798] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  74. L.J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. A 44 (2011) 454001 [arXiv:1105.0771] [INSPIRE].

    ADS  Google Scholar 

  75. A.I. Davydychev, P. Osland and L. Saks, Quark mass dependence of the one loop three gluon vertex in arbitrary dimension, JHEP 08 (2001) 050 [hep-ph/0105072] [INSPIRE].

    Article  ADS  Google Scholar 

  76. N. Ahmadiniaz and C. Schubert, A Compact Representation of the Three-Gluon Vertex, Frascati Phys. Ser. 55 (2012) 1 [arXiv:1206.1843] [INSPIRE].

    Google Scholar 

  77. G.V. Dunne and C. Schubert, Two loop Euler-Heisenberg QED pair production rate, Nucl. Phys. B 564 (2000) 591 [hep-th/9907190] [INSPIRE].

    Article  ADS  Google Scholar 

  78. G.V. Dunne and C. Schubert, Multiloop information from the QED effective Lagrangian, J. Phys. Conf. Ser. 37 (2006) 59 [hep-th/0409021] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  79. I. Huet, D. McKeon and C. Schubert, Euler-Heisenberg lagrangians and asymptotic analysis in 1 + 1 QED, part 1: two-loop, JHEP 12 (2010) 036 [arXiv:1010.5315] [INSPIRE].

    Article  ADS  Google Scholar 

  80. I. Huet, M. Rausch de Traubenberg and C. Schubert, The Euler-Heisenberg Lagrangian Beyond One Loop, in proceedings of Eleventh Conference on Quantum field theory under the influence of external conditions (QFEXT ’11), M. Asorey, M. Bordag and E. Elizalde eds., Int. J. Mod. Phys. Conf. Ser. 14 (2012) 383 [arXiv:1112.1049] [INSPIRE].

  81. Z. Bern, D.C. Dunbar and T. Shimada, String based methods in perturbative gravity, Phys. Lett. B 312 (1993) 277 [hep-th/9307001] [INSPIRE].

    ADS  Google Scholar 

  82. 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].

    Article  ADS  Google Scholar 

  83. M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, Cambridge, U.K. (1987).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Edificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, México

    Naser Ahmadiniaz, Christian Schubert & Victor M. Villanueva

  2. Dipartimento di Fisica, Università di Bologna and INFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, Italy

    Naser Ahmadiniaz & Christian Schubert

  3. Institutes of Physics and Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Christian Schubert

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  1. Naser Ahmadiniaz
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Ahmadiniaz, N., Schubert, C. & Villanueva, V.M. String-inspired representations of photon/gluon amplitudes. J. High Energ. Phys. 2013, 132 (2013). https://doi.org/10.1007/JHEP01(2013)132

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