Abstract
Generalised bi-adjoint scalar amplitudes, obtained from integrations over moduli space of punctured ℂℙk − 1, are novel extensions of the CHY formalism. These amplitudes have realisations in terms of Grassmannian cluster algebras. Recently connections between one-loop integrands for bi-adjoint cubic scalar theory and {mathcal{D}}_n cluster polytope have been established. In this paper using the Gr (3, 6) cluster algebra, we relate the singularities of (3, 6) amplitude to four-point one-loop integrand in the bi-adjoint cubic scalar theory through the {mathcal{D}}_4 cluster polytope. We also study factorisation properties of the (3, 6) amplitude at various boundaries in the worldsheet.
Highlights
Are of interest as they offer novel insights into quantum field theories
CEGM formalism is a novel extension of the CHY formalism from the moduli space of npunctured CP1 to that of n-punctured CPk−1
We have used the equivalence between Gr(3, 6) and D4 cluster algebras to relate the (k = 3, n = 6) CEGM amplitude to the cluster polytope for four-point one-loop integrand described in [72]
Summary
CEGM amplitudes for arbitrary values of k and n are beautiful mathematical constructions. There are speculations in the literature that Gr (4, n) amplitudes are related to singularities of loop-level amplitudes in N = 4 SYM theory. This motivates us to explore the first non-trivial example of CEGM amplitudes, the Gr (3, 6) amplitude. In [79], cluster string integral corresponding to the D4 cluster algebra has been considered, which in the α → 0 limit produces the four-point one-loop integrand for planar cubic scalar theory [72]. We will review how these tropical hypersurfaces can be obtained from cluster algebras and their relation with the polytope in the kinematic space. We will take the example of Gr(2, 6) throughout to illustrate various features
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