Abstract
We reveal a new mechanism of conformal symmetry breaking at Born level. It occurs in generalized form factors with several local operators and an on-shell state of massless particles. The effect is due to hidden singularities on collinear configurations of the momenta. This conformal anomaly is different from the holomorphic anomaly of amplitudes. We present a number of examples in four and six dimensions. We find an application of the new conformal anomaly to finite loop momentum integrals with one or more massless legs. The collinear region around a massless leg creates a contact anomaly, made visible by the loop integration. The anomalous conformal Ward identity for an ℓ−loop integral is a 2nd-order differential equation whose right-hand side is an (ℓ − 1)−loop integral. It could serve as a new useful tool to find/test analytic expressions for conformal integrals. We illustrate this point with several examples of known integrals. We propose a new differential equation for the four-dimensional scalar double box.
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L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, JHEP 09 (2011) 123 [arXiv:1007.3243] [INSPIRE].
B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, JHEP 12 (2011) 002 [arXiv:1007.3246] [INSPIRE].
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] [INSPIRE].
F. Cachazo, P. Svrček and E. Witten, Gauge theory amplitudes in twistor space and holomorphic anomaly, JHEP 10 (2004) 077 [hep-th/0409245] [INSPIRE].
T. Bargheer, N. Beisert, W. Galleas, F. Loebbert and T. McLoughlin, Exacting N = 4 Superconformal Symmetry, JHEP 11 (2009) 056 [arXiv:0905.3738] [INSPIRE].
G.P. Korchemsky and E. Sokatchev, Symmetries and analytic properties of scattering amplitudes in N = 4 SYM theory, Nucl. Phys. B 832 (2010) 1 [arXiv:0906.1737] [INSPIRE].
T. Bargheer, N. Beisert and F. Loebbert, Exact Superconformal and Yangian Symmetry of Scattering Amplitudes, J. Phys. A 44 (2011) 454012 [arXiv:1104.0700] [INSPIRE].
O.T. Engelund and R. Roiban, Correlation functions of local composite operators from generalized unitarity, JHEP 03 (2013) 172 [arXiv:1209.0227] [INSPIRE].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
S. Caron-Huot and S. He, Jumpstarting the All-Loop S-matrix of Planar N = 4 Super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].
M. Bullimore and D. Skinner, Descent Equations for Superamplitudes, arXiv:1112.1056 [INSPIRE].
D. Chicherin and E. Sokatchev, \( \mathcal{N} \) = 4 super-Yang-Mills in LHC superspace part I: classical and quantum theory, JHEP 02 (2017) 062 [arXiv:1601.06803] [INSPIRE].
D. Chicherin and E. Sokatchev, \( \mathcal{N} \) = 4 super-Yang-Mills in LHC superspace part II: non-chiral correlation functions of the stress-tensor multiplet, JHEP 03 (2017) 048 [arXiv:1601.06804] [INSPIRE].
D.J. Broadhurst, Summation of an infinite series of ladder diagrams, Phys. Lett. B 307 (1993) 132 [INSPIRE].
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] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett. B 302 (1993) 299 [Erratum ibid. B 318 (1993) 649] [hep-ph/9212308] [INSPIRE].
C. Anastasiou, E.W.N. Glover and C. Oleari, Application of the negative dimension approach to massless scalar box integrals, Nucl. Phys. B 565 (2000) 445 [hep-ph/9907523] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N = 4 SYM, JHEP 06 (2011) 100 [arXiv:1104.2787] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, The massless hexagon integral in D = 6 dimensions, Phys. Lett. B 703 (2011) 363 [arXiv:1104.2781] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
D. Nandan, M.F. Paulos, M. Spradlin and A. Volovich, Star Integrals, Convolutions and Simplices, JHEP 05 (2013) 105 [arXiv:1301.2500] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
D. Chicherin, V. Kazakov, F. Loebbert, D. Müller and D.-l. Zhong, Yangian Symmetry for Fishnet Feynman Graphs, Phys. Rev. D 96 (2017) 121901 [arXiv:1708.00007] [INSPIRE].
C. Cheung and D. O’Connell, Amplitudes and Spinor-Helicity in Six Dimensions, JHEP 07 (2009) 075 [arXiv:0902.0981] [INSPIRE].
I.M. Gelfand and G.E. Shilov, Generalized functions. Vol. 1. Properties and operations, Academic Press. San Diego, U.S.A., (1964).
J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].
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Chicherin, D., Sokatchev, E. Conformal anomaly of generalized form factors and finite loop integrals. J. High Energ. Phys. 2018, 82 (2018). https://doi.org/10.1007/JHEP04(2018)082
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DOI: https://doi.org/10.1007/JHEP04(2018)082