Abstract
In our recent work, we proposed a differential operator for the evaluation of the multi-dimensional residues on isolated (zero-dimensional) poles. In this paper we discuss some new insight on evaluating the (generalized) Cachazo-He-Yuan (CHY) forms of the scattering amplitudes using this differential operator. We introduce a tableau represen-tation for the coefficients appearing in the proposed differential operator. Combining the tableaux with the polynomial form of the scattering equations, the evaluation of the gen-eralized CHY form becomes a simple combinatoric problem. It is thus possible to obtain the coefficients arising in the differential operator in a straightforward way. We present the procedure for a complete solution of the n-gon amplitudes at one-loop level in a generalized CHY form. We also apply our method to fully evaluate the one-loop five-point amplitude in the maximally supersymmetric Yang-Mills theory; the final result is identical to the one obtained by Q-Cut.
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References
D.J. Gross and J.L. Manes, The high-energy behavior of open string scattering, Nucl. Phys. B 326 (1989) 73 [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].
E. Witten, Parity invariance for strings in twistor space, Adv. Theor. Math. Phys. 8 (2004) 779 [hep-th/0403199] [INSPIRE].
F. Cachazo, Fundamental BCJ relation in \( \mathcal{N}=4 \) SYM from the connected formulation, arXiv:1206.5970 [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].
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].
L. Dolan and P. Goddard, The polynomial form of the scattering equations, JHEP 07 (2014) 029 [arXiv:1402.7374] [INSPIRE].
N. Berkovits, Infinite tension limit of the pure spinor superstring, JHEP 03 (2014) 017 [arXiv:1311.4156] [INSPIRE].
L. Mason and D. Skinner, Ambitwistor strings and the scattering equations, JHEP 07 (2014) 048 [arXiv:1311.2564] [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].
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].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-loop scattering amplitudes from the Riemann sphere, Phys. Rev. D 94 (2016) 125029 [arXiv:1607.08887] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, One-loop corrections from higher dimensional tree amplitudes, JHEP 08 (2016) 008 [arXiv:1512.05001] [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].
B. Feng, CHY-construction of planar loop integrands of cubic scalar theory, JHEP 05 (2016) 061 [arXiv:1601.05864] [INSPIRE].
F. Cachazo and H. Gomez, Computation of contour integrals on ℳ0,n , JHEP 04 (2016) 108 [arXiv:1505.03571] [INSPIRE].
H. Gomez, Λ scattering equations, JHEP 06 (2016) 101 [arXiv:1604.05373] [INSPIRE].
C. Cardona and H. Gomez, Elliptic scattering equations, JHEP 06 (2016) 094 [arXiv:1605.01446] [INSPIRE].
H. Gomez, S. Mizera and G. Zhang, CHY loop integrands from holomorphic forms, JHEP 03 (2017) 092 [arXiv:1612.06854] [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.S. Lam and Y.-P. Yao, Role of Möbius constants and scattering functions in Cachazo-He-Yuan scalar amplitudes, Phys. Rev. D 93 (2016) 105004 [arXiv:1512.05387] [INSPIRE].
C.S. Lam and Y.-P. Yao, Evaluation of the Cachazo-He-Yuan gauge amplitude, Phys. Rev. D 93 (2016) 105008 [arXiv:1602.06419] [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].
C.R. Mafra, Berends-Giele recursion for double-color-ordered amplitudes, JHEP 07 (2016) 080 [arXiv:1603.09731] [INSPIRE].
R. Huang, B. Feng, M.-x. Luo and C.-J. Zhu, Feynman rules of higher-order poles in CHY construction, JHEP 06 (2016) 013 [arXiv:1604.07314] [INSPIRE].
N.E.J. Bjerrum-Bohr, J.L. Bourjaily, P.H. Damgaard and B. Feng, Analytic representations of Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 964 [arXiv:1605.06501] [INSPIRE].
C. Cardona, B. Feng, H. Gomez and R. Huang, Cross-ratio identities and higher-order poles of CHY-integrand, JHEP 09 (2016) 133 [arXiv:1606.00670] [INSPIRE].
C. Kalousios, Massless scattering at special kinematics as Jacobi polynomials, J. Phys. A 47 (2014) 215402 [arXiv:1312.7743] [INSPIRE].
S. Weinzierl, On the solutions of the scattering equations, JHEP 04 (2014) 092 [arXiv:1402.2516] [INSPIRE].
C.S. Lam, Permutation symmetry of the scattering equations, Phys. Rev. D 91 (2015) 045019 [arXiv:1410.8184] [INSPIRE].
C. Kalousios, Scattering equations, generating functions and all massless five point tree amplitudes, JHEP 05 (2015) 054 [arXiv:1502.07711] [INSPIRE].
C. Cardona and C. Kalousios, Comments on the evaluation of massless scattering, JHEP 01 (2016) 178 [arXiv:1509.08908] [INSPIRE].
C. Cardona and C. Kalousios, Elimination and recursions in the scattering equations, Phys. Lett. B 756 (2016) 180 [arXiv:1511.05915] [INSPIRE].
L. Dolan and P. Goddard, General solution of the scattering equations, JHEP 10 (2016) 149 [arXiv:1511.09441] [INSPIRE].
Y.-j. Du, F. Teng and Y.-s. Wu, CHY formula and MHV amplitudes, JHEP 05 (2016) 086 [arXiv:1603.08158] [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. Søgaard and Y. Zhang, Scattering equations and global duality of residues, Phys. Rev. D 93 (2016) 105009 [arXiv:1509.08897] [INSPIRE].
J. Bosma, M. Søgaard and Y. Zhang, The polynomial form of the scattering equations is an H-basis, Phys. Rev. D 94 (2016) 041701 [arXiv:1605.08431] [INSPIRE].
M. Zlotnikov, Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes, JHEP 08 (2016) 143 [arXiv:1605.08758] [INSPIRE].
T. Wang, G. Chen, Y.-K.E. Cheung and F. Xu, A differential operator for integrating one-loop scattering equations, JHEP 01 (2017) 028 [arXiv:1609.07621] [INSPIRE].
R. Hartshorne, Algebraic geometry, vol. 52, Springer Science & Business Media (2013).
P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons (2014).
J.J. Carrasco and H. Johansson, Five-point amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills theory and \( \mathcal{N}=8 \) supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
C. Baadsgaard et al., New representations of the perturbative S-matrix, Phys. Rev. Lett. 116 (2016) 061601 [arXiv:1509.02169] [INSPIRE].
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].
J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].
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Wang, T., Chen, G., Cheung, YK.E. et al. A combinatoric shortcut to evaluate CHY-forms. J. High Energ. Phys. 2017, 15 (2017). https://doi.org/10.1007/JHEP06(2017)015
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DOI: https://doi.org/10.1007/JHEP06(2017)015