Abstract
We extend the analysis of Higgs-regulated planar amplitudes of \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory to four loops for the four-gluon amplitude and to two loops for the five-gluon amplitude. Our calculations are consistent with a proposed all-loop ansatz for planar MHV n-gluon amplitudes that is the analog of the BDS ansatz in dimensional regularization. In all cases considered, we have verified that the IR-finite parts of the logarithm of the amplitudes have the same dependence on kinematic variables as the corresponding functions in dimensionally-regulated amplitudes (up to overall additive constants, which we determine).
We also study various Regge limits of \( \mathcal{N} = 4 \) SYM planar n-gluon amplitudes. Euclidean Regge limits of Higgs-regulated n ≥ 4 amplitudes yield results similar in form to those found using dimensional regularization, but with different expressions for the gluon trajectory and Regge vertices resulting from the different regulator scheme. We also show that the Regge limit of the four-gluon amplitude is dominated at next-to-leading-log order by vertical ladder diagrams together with the class of vertical ladder diagrams with a single H-shaped insertion.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [SPIRES].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One-loop n-point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [SPIRES].
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] [SPIRES].
E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [SPIRES].
Z. Bern, J.J.M. Carrasco, H. Johansson and D.A. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [SPIRES].
F. Cachazo and D. Skinner, On the structure of scattering amplitudes in N = 4 super Yang-Mills and N = 8 supergravity, arXiv:0801.4574 [SPIRES].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].
J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [SPIRES].
A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [SPIRES].
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] [SPIRES].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [SPIRES].
J. Bartels, L.N. Lipatov and A. Sabio Vera, BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [SPIRES].
Z. Bern et al., The two-loop six-gluon MHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [SPIRES].
L.F. Alday and R. Roiban, Scattering amplitudes, Wilson loops and the string/gauge theory correspondence, Phys. Rept. 468 (2008) 153 [arXiv:0807.1889] [SPIRES].
J.M. Henn, Duality between Wilson loops and gluon amplitudes, Fortsch. Phys. 57 (2009) 729 [arXiv:0903.0522] [SPIRES].
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] [SPIRES].
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] [SPIRES].
J.-H. Zhang, On the two-loop hexagon Wilson loop remainder function in N = 4 SYM, arXiv:1004.1606 [SPIRES].
N. Beisert, J. Henn, T. McLoughlin and J. Plefka, One-loop superconformal and Yangian symmetries of scattering amplitudes in N = 4 super Yang-Mills, JHEP 04 (2010) 085 [arXiv:1002.1733] [SPIRES].
L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N = 4 super Yang- Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [SPIRES].
G.P. Korchemsky, Double logarithmic asymptotics in QCD, Phys. Lett. B 217 (1989) 330 [SPIRES].
H. Kawai and T. Suyama, Some implications of perturbative approach to AdS/CFT correspondence, Nucl. Phys. B 794 (2008) 1 [arXiv:0708.2463] [SPIRES].
R.M. Schabinger, Scattering on the moduli space of N = 4 super Yang-Mills, arXiv:0801.1542 [SPIRES].
J. McGreevy and A. Sever, Planar scattering amplitudes from Wilson loops, JHEP 08 (2008) 078 [arXiv:0806.0668] [SPIRES].
A. Gorsky and A. Zhiboedov, Aspects of the N = 4 SYM amplitude – Wilson polygon duality, Nucl. Phys. B 835 (2010) 343 [arXiv:0911.3626] [SPIRES].
R.H. Boels, No triangles on the moduli space of maximally supersymmetric gauge theory, JHEP 05 (2010) 046 [arXiv:1003.2989] [SPIRES].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, Higgs-regularized three-loop four-gluon amplitude in N = 4 SYM: exponentiation and Regge limits, JHEP 04 (2010) 038 [arXiv:1001.1358] [SPIRES].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [SPIRES].
Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [SPIRES].
F. Cachazo, M. Spradlin and A. Volovich, Four-loop cusp anomalous dimension from obstructions, Phys. Rev. D 75 (2007) 105011, [hep-th/0612309] [SPIRES].
V.S. Fadin, R. Fiore and M.I. Kotsky, Gluon regge trajectory in the two-loop approximation, Phys. Lett. B 387 (1996) 593 [hep-ph/9605357] [SPIRES].
I.A. Korchemskaya and G.P. Korchemsky, Evolution equation for gluon Regge trajectory, Phys. Lett. B 387 (1996) 346 [hep-ph/9607229] [SPIRES].
A.V. Kotikov and L.N. Lipatov, NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories, Nucl. Phys. B 582 (2000) 19 [hep-ph/0004008] [SPIRES].
S.G. Naculich and H.J. Schnitzer, Regge behavior of gluon scattering amplitudes in N = 4 SYM theory, Nucl. Phys. B 794 (2008) 189 [arXiv:0708.3069] [SPIRES].
V. Del Duca and E.W.N. Glover, Testing high-energy factorization beyond the next-toleading-logarithmic accuracy, JHEP 05 (2008) 056 [arXiv:0802.4445] [SPIRES].
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] [SPIRES].
R.C. Brower, H. Nastase, H.J. Schnitzer and C.-I. Tan, Implications of multi-Regge limits for the Bern-Dixon-Smirnov conjecture, Nucl. Phys. B 814 (2009) 293 [arXiv:0801.3891] [SPIRES].
R.C. Brower, H. Nastase, H.J. Schnitzer and C.-I. Tan, Analyticity for multi-Regge limits of the Bern-Dixon-Smirnov amplitudes, Nucl. Phys. B 822 (2009) 301 [arXiv:0809.1632] [SPIRES].
S. Catani, The singular behaviour of QCD amplitudes at two-loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [SPIRES].
G. Sterman and M.E. Tejeda-Yeomans, Multi-loop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [SPIRES].
H.R.P. Ferguson and D.H. Bailey, A polynomial time, numerically stable integer relation algorithm http://crd.lbl.gov/∼dhbailey/dhbpapers/pslq.pdf
P. Bertok, PSLQ integer relation algorithm implementation, http://library.wolfram.com/infocenter/MathSource/4263/.
V. A. Smirnov, Feynman integral calculus, Springer, Berlin Germany (2006).
M. Czakon, MBasymptotics, http://projects.hepforge.org/mbtools/.
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [SPIRES].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [SPIRES].
D. Nguyen, M. Spradlin and A. Volovich, New dual conformally invariant off-shell integrals, Phys. Rev. D 77 (2008) 025018 [arXiv:0709.4665] [SPIRES].
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-Matrix, Cambridge University Press, Cambridge U.K. (1966).
P.D.B. Collins, An introduction to regge theory and high-energy physics, Cambridge University Press, Cambridge U.K. (1977).
I.G. Halliday, High-energy behaviour in perturbation theory, Nuovo Cim. 30 (1963) 177 [SPIRES].
G. Tiktopoulos, High-energy behavior of Feynman amplitudes, Phys. Rev. 131 (1963) 480 [SPIRES].
F. Cachazo, M. Spradlin and A. Volovich, Iterative structure within the five-particle two-loop amplitude, Phys. Rev. D 74 (2006) 045020 [hep-th/0602228] [SPIRES].
Z. Bern, M. Czakon, D.A. Kosower, R. Roiban and V.A. Smirnov, Two-loop iteration of five-point N = 4 super-Yang-Mills amplitudes, Phys. Rev. Lett. 97 (2006) 181601 [hep-th/0604074] [SPIRES].
V. Del Duca, C. Duhr and E.W. Nigel Glover, The five-gluon amplitude in the high-energy limit, JHEP 12 (2009) 023 [arXiv:0905.0100] [SPIRES].
A. Hodges, The box integrals in momentum-twistor geometry, arXiv:1004.3323 [SPIRES].
L. Mason and D. Skinner, Amplitudes at weak coupling as polytopes in AdS 5, arXiv:1004.3498 [SPIRES].
C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [SPIRES].
A. Mitov and S. Moch, The singular behavior of massive QCD amplitudes, JHEP 05 (2007) 001 [hep-ph/0612149] [SPIRES].
C. Anastasiou et al., Two-loop polygon Wilson loops in N = 4 SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [SPIRES].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, The hexagon Wilson loop and the BDS ansatz for the six- gluon amplitude, Phys. Lett. B 662 (2008) 456 [arXiv:0712.4138] [SPIRES].
J. Bartels, L.N. Lipatov and A. Sabio Vera, N=4 supersymmetric Yang-Mills scattering amplitudes at high energies: the Regge cut contribution, Eur. Phys. J. C 65 (2010) 587 [arXiv:0807.0894] [SPIRES].
V. Gribov, The theory of complex angular momenta, Cambridge University Press, Cambridge U.K. (2003).
P.G. Federbush and M.T. Grisaru, The high energy behavior of scattering amplitudes in perturbation theory, Ann. Phys. 22 (1963) 263.
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1004.5381
Research supported in part by the NSF under grant PHY-0756518
Research supported in part by the DOE under grant DE–FG02–92ER40706
Research supported in part by the DOE under grant DE–FG02-91ER40688
Rights and permissions
About this article
Cite this article
Henn, J.M., Naculich, S.G., Schnitzer, H.J. et al. More loops and legs in Higgs-regulated \( \mathcal{N} = 4 \) SYM amplitudes. J. High Energ. Phys. 2010, 2 (2010). https://doi.org/10.1007/JHEP08(2010)002
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2010)002