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All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes

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Abstract

The infrared singularities of gravitational amplitudes are one-loop exact, in that higher-loop divergences are characterized by the exponential of the one-loop divergence. We show that the contributions to SU(N) gauge-theory amplitudes that are mostsubleading in the 1/N expansion are also one-loop exact, provided that the dipole conjecture holds. Possible corrections to the dipole conjecture, beginning at three loops, could violate one-loop-exactness, though would still maintain the absence of collinear divergences. We also demonstrate a relation between L-loop four-point \( \mathcal{N}=8 \) supergravity and mostsubleading-color \( \mathcal{N}=4 \) SYM amplitudes that holds for the two leading IR divergences, \( \mathcal{O}\left( {{1 \left/ {{{\in^L}}} \right.}} \right) \) and \( \mathcal{O}\left( {{1 \left/ {{{\in^{L-1 }}}} \right.}} \right) \), but breaks down at \( \mathcal{O}\left( {{1 \left/ {{{\in^{L-2 }}}} \right.}} \right) \).

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References

  1. S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516.

    Article  MathSciNet  ADS  Google Scholar 

  2. D.C. Dunbar and P.S. Norridge, Infinities within graviton scattering amplitudes, Class. Quant. Grav. 14 (1997) 351 [hep-th/9512084] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. S.G. Naculich, H. Nastase and H.J. Schnitzer, Two-loop graviton scattering relation and IR behavior in N = 8 supergravity, Nucl. Phys. B 805 (2008) 40 [arXiv:0805.2347] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. S.G. Naculich and H.J. Schnitzer, Eikonal methods applied to gravitational scattering amplitudes, JHEP 05 (2011) 087 [arXiv:1101.1524] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. C.D. White, Factorization properties of soft graviton amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [INSPIRE].

    Article  ADS  Google Scholar 

  6. R. Akhoury, R. Saotome and G. Sterman, Collinear and soft divergences in perturbative quantum gravity, Phys. Rev. D 84 (2011) 104040 [arXiv:1109.0270] [INSPIRE].

    ADS  Google Scholar 

  7. G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].

    Article  ADS  Google Scholar 

  8. S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].

    ADS  Google Scholar 

  9. T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [arXiv:0901.0722] [INSPIRE].

    Article  ADS  Google Scholar 

  10. E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].

    Article  ADS  Google Scholar 

  11. T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [arXiv:0903.1126] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. Z. Bern, J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Manifest ultraviolet behavior for the three-loop four-point amplitude of N = 8 supergravity, Phys. Rev. D 78 (2008) 105019 [arXiv:0808.4112] [INSPIRE].

    ADS  Google Scholar 

  13. L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  14. V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, The infrared structure of gauge theory amplitudes in the high-energy limit, JHEP 12 (2011) 021 [arXiv:1109.3581] [INSPIRE].

    Article  ADS  Google Scholar 

  15. V. Ahrens, M. Neubert and L. Vernazza, Structure of infrared singularities of gauge-theory amplitudes at three and four loops, JHEP 09 (2012) 138 [arXiv:1208.4847] [INSPIRE].

    Article  ADS  Google Scholar 

  16. S.G. Naculich, H. Nastase and H.J. Schnitzer, Subleading-color contributions to gluon-gluon scattering in N = 4 SYM theory and relations to N = 8 supergravity, JHEP 11 (2008) 018 [arXiv:0809.0376] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. S.G. Naculich and H.J. Schnitzer, IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in N = 4 SYM theory, JHEP 10 (2009) 048 [arXiv:0907.1895] [INSPIRE].

    Article  ADS  Google Scholar 

  18. H. Nastase and H.J. Schnitzer, Twistor and polytope interpretations for subleading color one-loop amplitudes, Nucl. Phys. B 855 (2012) 901 [arXiv:1104.2752] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. S.G. Naculich, H. Nastase and H.J. Schnitzer, Linear relations between N ≥ 4 supergravity and subleading-color SYM amplitudes, JHEP 01 (2012) 041 [arXiv:1111.1675] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. D. Miller and C. White, The gravitational cusp anomalous dimension from AdS space, Phys. Rev. D 85 (2012) 104034 [arXiv:1201.2358] [INSPIRE].

    ADS  Google Scholar 

  21. M. Beneke and G. Kirilin, Soft-collinear gravity, JHEP 09 (2012) 066 [arXiv:1207.4926] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. Z. Bern, L.J. Dixon, D. Dunbar, M. Perelstein and J. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162] [INSPIRE].

    Article  ADS  Google Scholar 

  23. Z. Bern, L.J. Dixon, M. Perelstein and J. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. S. Catani and M. Seymour, The dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B 378 (1996) 287 [hep-ph/9602277] [INSPIRE].

    Article  ADS  Google Scholar 

  25. S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].

    Article  ADS  Google Scholar 

  26. Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  27. 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].

    Article  MathSciNet  ADS  Google Scholar 

  28. 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  MathSciNet  ADS  Google Scholar 

  29. 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 

  30. A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  31. V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].

    Article  ADS  Google Scholar 

  32. V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].

    Article  ADS  Google Scholar 

  33. A.C. Edison and S.G. Naculich, SU(N) group-theory constraints on color-ordered five-point amplitudes at all loop orders, Nucl. Phys. B 858 (2012) 488 [arXiv:1111.3821] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. S. Oxburgh and C. White, BCJ duality and the double copy in the soft limit, JHEP 02 (2013) 127 [arXiv:1210.1110] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. Z. Bern, J. Rozowsky and B. Yan, Two loop four gluon amplitudes in N = 4 super Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [INSPIRE].

    Article  ADS  Google Scholar 

  36. J. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  37. K. Kolbig, J. Mignoco and E. Remiddi, On Nielsen’s generalized polylogarithms and their numerical calculation, BIT Num. Math. 10 (1970) 38.

    Article  Google Scholar 

  38. C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Bowdoin College, Brunswick, ME, 04011, U.S.A.

    Stephen G. Naculich

  2. Instituto de Física Teórica, UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo, 01140-070, SP, Brazil

    Horatiu Nastase

  3. Theoretical Physics Group, Martin Fisher School of Physics, Brandeis University, Waltham, MA, 02454, U.S.A.

    Howard J. Schnitzer

Authors
  1. Stephen G. Naculich
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  2. Horatiu Nastase
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  3. Howard J. Schnitzer
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Correspondence to Stephen G. Naculich.

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ArXiv ePrint: 1301.2234

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Naculich, S.G., Nastase, H. & Schnitzer, H.J. All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes. J. High Energ. Phys. 2013, 114 (2013). https://doi.org/10.1007/JHEP04(2013)114

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