Abstract
In this paper, by treating massive loop momenta as massless momenta in higher dimensions, we are able to treat all-loop scattering equations as tree ones. As an application of the new perspective, we consider the CHY-construction of bi-adjoint ϕ 3 theory. We present the explicit formula for two-loop planar integrands. We discuss in details how to subtract various forward singularities in the construction. We count the number of terms obtained by our formula and by direct Feynman diagram calculation and find the perfect match, thus provide a strong support for our results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
D. Fairlie and D. Roberts, Dual Models without Tachyons — a New Approach, unpublished, Durham preprint PRINT-72-2440 (1972).
D. Roberts, Mathematical Structure of Dual Amplitudes, Ph.D. Thesis, Durham University, Durham U.K. (1972).
D.B. Fairlie, A Coding of Real Null Four-Momenta into World-Sheet Co-ordinates, Adv. Math. Phys. 2009 (2009) 284689 [arXiv:0805.2263] [INSPIRE].
D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].
E. Witten, Parity invariance for strings in twistor space, Adv. Theor. Math. Phys. 8 (2004) 779 [hep-th/0403199] [INSPIRE].
P. Caputa and S. Hirano, Observations on Open and Closed String Scattering Amplitudes at High Energies, JHEP 02 (2012) 111 [arXiv:1108.2381] [INSPIRE].
P. Caputa, Lightlike contours with fermions, Phys. Lett. B 716 (2012) 475 [arXiv:1205.6369] [INSPIRE].
Y. Makeenko and P. Olesen, The QCD scattering amplitude from area behaved Wilson loops, Phys. Lett. B 709 (2012) 285 [arXiv:1111.5606] [INSPIRE].
F. Cachazo, Fundamental BCJ Relation in N = 4 SYM From The Connected Formulation, arXiv:1206.5970 [INSPIRE].
L. Dolan and P. Goddard, Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [INSPIRE].
N. Berkovits, Infinite Tension Limit of the Pure Spinor Superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].
H. Gomez and E.Y. Yuan, N-point tree-level scattering amplitude in the new Berkovits‘ string, JHEP 04 (2014) 046 [arXiv:1312.5485] [INSPIRE].
T. Adamo, E. Casali and D. Skinner, Ambitwistor strings and the scattering equations at one loop, JHEP 04 (2014) 104 [arXiv:1312.3828] [INSPIRE].
Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor Strings in Four Dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].
Y. Geyer, A.E. Lipstein and L. Mason, Ambitwistor strings at null infinity and (subleading) soft limits, Class. Quant. Grav. 32 (2015) 055003 [arXiv:1406.1462] [INSPIRE].
E. Casali and P. Tourkine, Infrared behaviour of the one-loop scattering equations and supergravity integrands, JHEP 04 (2015) 013 [arXiv:1412.3787] [INSPIRE].
T. Adamo and E. Casali, Scattering equations, supergravity integrands and pure spinors, JHEP 05 (2015) 120 [arXiv:1502.06826] [INSPIRE].
K. Ohmori, Worldsheet Geometries of Ambitwistor String, JHEP 06 (2015) 075 [arXiv:1504.02675] [INSPIRE].
E. Casali, Y. Geyer, L. Mason, R. Monteiro and K.A. Roehrig, New Ambitwistor String Theories, JHEP 11 (2015) 038 [arXiv:1506.08771] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop Integrands for Scattering Amplitudes from the Riemann Sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, P. Tourkine and P. Vanhove, Scattering Equations and String Theory Amplitudes, Phys. Rev. D 90 (2014) 106002 [arXiv:1403.4553] [INSPIRE].
C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Integration Rules for Scattering Equations, JHEP 09 (2015) 129 [arXiv:1506.06137] [INSPIRE].
C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Scattering Equations and Feynman Diagrams, JHEP 09 (2015) 136 [arXiv:1507.00997] [INSPIRE].
C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Integration Rules for Loop Scattering Equations, JHEP 11 (2015) 080 [arXiv:1508.03627] [INSPIRE].
L. Dolan and P. Goddard, The Polynomial Form of the Scattering Equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].
C. Kalousios, Scattering equations, generating functions and all massless five point tree amplitudes, JHEP 05 (2015) 054 [arXiv:1502.07711] [INSPIRE].
R. Huang, J. Rao, B. Feng and Y.-H. He, An Algebraic Approach to the Scattering Equations, JHEP 12 (2015) 056 [arXiv:1509.04483] [INSPIRE].
M. Sogaard and Y. Zhang, Scattering Equations and Global Duality of Residues, arXiv:1509.08897 [INSPIRE].
L. Dolan and P. Goddard, General Solution of the Scattering Equations, arXiv:1511.09441 [INSPIRE].
C. Cardona and C. Kalousios, Elimination and recursions in the scattering equations, Phys. Lett. B 756 (2016) 180 [arXiv:1511.05915] [INSPIRE].
C. Cardona and C. Kalousios, Comments on the evaluation of massless scattering, JHEP 01 (2016) 178 [arXiv:1509.08908] [INSPIRE].
F. Cachazo and H. Gomez, Computation of Contour Integrals on \( \mathrm{\mathcal{M}} \) 0,n , JHEP 04 (2016) 108 [arXiv:1505.03571] [INSPIRE].
S. He and E.Y. Yuan, One-loop Scattering Equations and Amplitudes from Forward Limit, Phys. Rev. D 92 (2015) 105004 [arXiv:1508.06027] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, One-Loop Corrections from Higher Dimensional Tree Amplitudes, arXiv:1512.05001 [INSPIRE].
C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily, S. Caron-Huot, P.H. Damgaard and B. Feng, New Representations of the Perturbative S-matrix, Phys. Rev. Lett. 116 (2016) 061601 [arXiv:1509.02169] [INSPIRE].
R. Huang, Q. Jin, J. Rao, K. Zhou and B. Feng, The \( \mathcal{Q} \) -cut Representation of One-loop Integrands and Unitarity Cut Method, JHEP 03 (2016) 057 [arXiv:1512.02860] [INSPIRE].
S.G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].
S.G. Naculich, CHY representations for gauge theory and gravity amplitudes with up to three massive particles, JHEP 05 (2015) 050 [arXiv:1501.03500] [INSPIRE].
S.G. Naculich, Amplitudes for massive vector and scalar bosons in spontaneously-broken gauge theory from the CHY representation, JHEP 09 (2015) 122 [arXiv:1506.06134] [INSPIRE].
C.S. Lam and Y.-P. Yao, Off-Shell CHY Amplitudes, Nucl. Phys. B 907 (2016) 678 [arXiv:1511.05050] [INSPIRE].
S. Caron-Huot, Loops and trees, JHEP 05 (2011) 080 [arXiv:1007.3224] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1601.05864
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Feng, B. CHY-construction of planar loop integrands of cubic scalar theory. J. High Energ. Phys. 2016, 61 (2016). https://doi.org/10.1007/JHEP05(2016)061
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2016)061