Top-forms of leading singularities in nonplanar multi-loop amplitudes

The European Physical Journal C, Feb 2018

The on-shell diagram is a very important tool in studying scattering amplitudes. In this paper we discuss the on-shell diagrams without external BCFW bridges. We introduce an extra step of adding an auxiliary external momentum line. Then we can decompose the on-shell diagrams by removing external BCFW bridges to a planar diagram whose top-form is well known now. The top-form of the on-shell diagram with the auxiliary line can be obtained by adding the BCFW bridges in an inverse order as discussed in our former paper (Chen et al. in Eur Phys J C 77(2):80 2017). To get the top-form of the original diagram, the soft limit of the auxiliary line is needed. We obtain the evolution rule for the Grassmannian integral and the geometry constraint in the soft limit. This completes the top-form description of leading singularities in nonplanar scattering amplitudes of \(\mathcal {N}=4\) Super Yang–Mills (SYM), which is valid for arbitrary higher-loops and beyond the Maximally-Helicity-Violation (MHV) amplitudes.

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Top-forms of leading singularities in nonplanar multi-loop amplitudes

Eur. Phys. J. C Top-forms of leading singularities in nonplanar multi-loop amplitudes Baoyi Chen 0 Gang Chen 0 1 Yeuk-Kwan E. Cheung 0 Ruofei Xie 0 Yuan Xin 0 0 Department of Physics, Nanjing University , 22 Hankou Road, Nanjing 210093 , People's Republic of China 1 Department of Physics, Zhejiang Normal University , Jinhua, Zhejiang Province , People's Republic of China The on-shell diagram is a very important tool in studying scattering amplitudes. In this paper we discuss the on-shell diagrams without external BCFW bridges. We introduce an extra step of adding an auxiliary external momentum line. Then we can decompose the on-shell diagrams by removing external BCFW bridges to a planar diagram whose top-form is well known now. The top-form of the on-shell diagram with the auxiliary line can be obtained by adding the BCFW bridges in an inverse order as discussed in our former paper (Chen et al. in Eur Phys J C 77(2):80 2017). To get the top-form of the original diagram, the soft limit of the auxiliary line is needed. We obtain the evolution rule for the Grassmannian integral and the geometry constraint in the soft limit. This completes the top-form description of leading singularities in nonplanar scattering amplitudes of N = 4 Super Yang-Mills (SYM), which is valid for arbitrary higher-loops and beyond the Maximally-Helicity-Violation (MHV) amplitudes. 1 Introduction Bipartite diagrams and the associated Grassmannian geometry [2,3] have recently found their way into the scattering amplitude studies. An amazing discovery was to exploit them in computing scattering amplitudes in N = 4 SYM theory [4–11]. Planar scattering amplitudes are represented by on-shell bipartite diagrams and expressed in “top-form” as contour integrations over the Grassmannian submanifolds. Planar loop integrands in N = 4 SYM have recently been constructed in [4,12] along with the introduction of the Grassmannian and on-shell method. As a result, the “dlog” form and the Yangian symmetry [13–17] of the scattering amplitudes are made manifest in the planar limit. It is natural to extend the construction to non-planar scattering amplitudes [1,18–20], and theories of reduced (super-) symmetries [21–23]. The leading singularities are represented in the top-form of Grassmannian integrals in which the integrands are comprised of rational functions of minors {R(MC )} of the Grassmannian C matrices. The top-form is elegant in that the amplitude structures are simple and compact; and the Yangian symmetry is manifest in the positive diffeomorphisms of positive Grassmannian geometry [4]. It is therefore crucial to express the scattering amplitudes in top-form in order to explore the power to further uncover hidden symmetries and dualities of the scattering amplitudes. We present in this letter our successful construction of top-forms for non-planar scattering amplitudes. Our method applies to multi-loop, beyondMHV leading singularities. Recently, exciting progress in N = 4 SYM scattering amplitude computation (by the on-shell method) was reported by many research groups in [1,3,18,19,24–28]. Together we have made a step forward in the computation of nonplanar N = 4 SYM scattering amplitudes, and hopefully in the formulation of the AdS/CFT correspondence at finite N . 2 BCFW-bridge decompositions of leading singularities The aim of this work is to obtain a simple and compact analytical expression of leading singularities of scattering amplitudes, valid for arbitrary number of loops, beyond the planar limit. A general leading singularity can be represented by a reduced on-shell diagram. BCFW-bridge decomposition provides an efficient way of constructing on-shell diagrams in the planar limit. In non-planar cases, we can obtain the BCFW-bridge decomposition chain by extracting planar subdiagrams and computing them recursively [1] as shown in Fig. 1. For the sub-diagrams that are BCFW-decomposible, we follow the recipe presented in [1]. There exist, however, a b c Fig. 1 a Obtaining the Lth loop amplitude recursively. b Utilizing the U ( 1 ) decoupling relation to turn a nonplanar diagram into a planar one. c Introduction of an auxiliary external momentum line to form the BCFW bridge “No Bridge” (NB) diagrams which do not contain any BCFW bridges [1,19]. We have presented a method in [1] to transform some NB diagrams, schematically depicted in Fig. 1b, by applying U ( 1 )-decoupling relations [29]. In this work we present a general method applicable to any NB diagrams. The key is to add an auxiliary external momentum line to form an auxiliary BCFW bridge, shown in Fig. 1c. To regain the original NB diagram we take the soft limit [30–33], setting the auxiliary momentum to zero. This way the BCFW-bridge decomposition chain of the reduced on-shell diagrams beyond the planar limit can be obtained. In the rest of this letter we present a recipe for constructing an analytical expression, the top-form, for a nonplanar leading singularity usin (...truncated)


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Baoyi Chen, Gang Chen, Yeuk-Kwan E. Cheung, Ruofei Xie, Yuan Xin. Top-forms of leading singularities in nonplanar multi-loop amplitudes, The European Physical Journal C, 2018, pp. 164, Volume 78, Issue 2, DOI: 10.1140/epjc/s10052-018-5629-z