Abstract
We find a new duality for form factors of lightlike Wilson loops in planar \(\mathcal N=4\) super-Yang-Mills theory. The duality maps a form factor involving an n-sided lightlike polygonal super-Wilson loop together with m external on-shell states, to the same type of object but with the edges of the Wilson loop and the external states swapping roles. This relation can essentially be seen graphically in Lorentz harmonic chiral (LHC) superspace where it is equivalent to planar graph duality. However there are some crucial subtleties with the cancellation of spurious poles due to the gauge fixing. They are resolved by finding the correct formulation of the Wilson loop and by careful analytic continuation from Minkowski to Euclidean space. We illustrate all of these subtleties explicitly in the simplest non-trivial NMHV-like case.
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Notes
- 1.
We use two-component spinor notation for vectors, e.g., \(x^{{\dot{\alpha }}\alpha } =(\sigma _\mu )^{{\dot{\alpha }}\alpha } x^\mu \). The Lorentz and R symmetry indices take values \(\alpha =1,2\), \({\dot{\alpha }}=1,2\) and \(A=1,2,3,4\), respectively.
- 2.
This does not follow from chiral supersymmetry (32) and it would be interesting to understand the symmetry leading to such a structure.
- 3.
Notice that the divergent part of the Wilson loop form factor automatically satisfies the duality relation (15). Namely, the IR divergencies of \(W_{n,m}\) match the UV divergences of \(W_{m,n}\) and vice versa.
- 4.
We cannot fully justify applicability of the Fourier transform in Euclidean space until we have checked for integrand level cancellation of spurious poles, but we assume this here.
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Acknowledgements
We profited from numerous discussions with Simon Caron-Huot and Ömer Gürdoğan. G.K. would like to thank Ömer Gürdoğan for collaboration at the early stage of this project. We acknowledge partial support by the French National Agency for Research (ANR) under contract StrongInt (BLANC-SIMI-4-2011). The work of D.C. has been partially supported by the RFBR grant 14-01-00341. The work of P.H. has been partially supported by an STFC Consolidated Grant ST/L000407/1. P.H. would also like to thank the CNRS for financial support and LAPTh for hospitality where part of this work was done.
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Chicherin, D., Heslop, P., Korchemsky, G.P., Sokatchev, E. (2018). Wilson Loop Form Factors: A New Duality. In: Dobrev, V. (eds) Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics Volume 2. LT-XII/QTS-X 2017. Springer Proceedings in Mathematics & Statistics, vol 255. Springer, Singapore. https://doi.org/10.1007/978-981-13-2179-5_8
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