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)