Abstract
We propose a simple geometric algorithm for determining the complete set of branch points of amplitudes in planar \( \mathcal{N} \) = 4 super-Yang-Mills theory directly from the amplituhedron, without resorting to any particular representation in terms of local Feynman integrals. This represents a step towards translating integrands directly into integrals. In particular, the algorithm provides information about the symbol alphabets of general amplitudes. We illustrate the algorithm applied to the one- and two-loop MHV amplitudes.
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L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].
N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed and J. Trnka, Into the amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].
Y. Bai and S. He, The amplituhedron from momentum twistor diagrams, JHEP 02 (2015) 065 [arXiv:1408.2459] [INSPIRE].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the amplituhedron, JHEP 03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
T. Lam, Amplituhedron cells and Stanley symmetric functions, Commun. Math. Phys. 343 (2016) 1025 [arXiv:1408.5531] [INSPIRE].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive amplitudes in the amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
Y. Bai, S. He and T. Lam, The amplituhedron and the one-loop Grassmannian measure, JHEP 01 (2016) 112 [arXiv:1510.03553] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Towards the amplituhedron volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a nonplanar amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].
D. Galloni, Positivity sectors and the amplituhedron, arXiv:1601.02639 [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr, M. von Hippel and J. Pennington, Bootstrapping six-gluon scattering in planar N = 4 super-Yang-Mills theory, PoS(LL2014)077 [arXiv:1407.4724] [INSPIRE].
J. Golden and M. Spradlin, A cluster bootstrap for two-loop MHV amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
T. Dennen, M. Spradlin and A. Volovich, Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory, JHEP 03 (2016) 069 [arXiv:1512.07909] [INSPIRE].
S. Mandelstam, Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. General theory, Phys. Rev. 112 (1958) 1344 [INSPIRE].
S. Mandelstam, Analytic properties of transition amplitudes in perturbation theory, Phys. Rev. 115 (1959) 1741 [INSPIRE].
L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181 [INSPIRE].
R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429 [INSPIRE].
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] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280] [INSPIRE].
S. Abreu, R. Britto, C. Duhr and E. Gardi, From multiple unitarity cuts to the coproduct of Feynman integrals, JHEP 10 (2014) 125 [arXiv:1401.3546] [INSPIRE].
S. Abreu, R. Britto and H. Grönqvist, Cuts and coproducts of massive triangle diagrams, JHEP 07 (2015) 111 [arXiv:1504.00206] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge U.K., (1966) [INSPIRE].
S. Coleman and R.E. Norton, Singularities in the physical region, Nuovo Cim. 38 (1965) 438 [INSPIRE].
S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
D.B. Fairlie, P.V. Landshoff, J. Nuttall and J.C. Polkinghorne, Singularities of the second type, J. Math. Phys. 3 (1962) 594.
D.B. Fairlie, P.V. Landshoff, J. Nuttall and J.C. Polkinghorne, Physical sheet properties of second type singularities, Phys. Lett. 3 (1962) 55.
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] [INSPIRE].
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].
J.L. Bourjaily and J. Trnka, Local integrand representations of all two-loop amplitudes in planar SYM, JHEP 08 (2015) 119 [arXiv:1505.05886] [INSPIRE].
T. Dennen, MHV Landau, unpublished notes.
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].
A.E. Lipstein and L. Mason, From d logs to dilogs the super Yang-Mills MHV amplitude revisited, JHEP 01 (2014) 169 [arXiv:1307.1443] [INSPIRE].
K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831 [INSPIRE].
A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
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].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].
L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].
S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a five-loop amplitude using Steinmann relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].
J.M. Drummond, G. Papathanasiou and M. Spradlin, A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
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Dennen, T., Prlina, I., Spradlin, M. et al. Landau singularities from the amplituhedron. J. High Energ. Phys. 2017, 152 (2017). https://doi.org/10.1007/JHEP06(2017)152
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DOI: https://doi.org/10.1007/JHEP06(2017)152