A combinatoric shortcut to evaluate CHY-forms

Wang, Tianheng; Chen, Gang; Cheung, Yeuk-Kwan E.; Xu, Feng

doi:10.1007/jhep06(2017)015

Cited by 10 publications

(9 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reduced Pfaffian has been studied quite extensively in the literature [32][33][34][35][36][37][38][39][40][41]. The expression given by the reduced Pfaffian is not very well suited to generalise towards off-shell momenta, unphysical polarisation or away from the solutions of the scattering equations.…”

Section: The Polarisation Scattering Formmentioning

confidence: 99%

Properties of scattering forms and their relation to associahedra

Cruz

Kniss

Weinzierl

2018

J. High Energ. Phys.

View full text Add to dashboard Cite

We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms -the scattering forms -on the moduli space of a Riemann sphere with n marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor M 0,n \M 0,n . Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.In this article we bring three things together, which really should be viewed together: (i) the Cachazo-He-Yuan (CHY) representation of tree-level n-point scattering amplitudes [1-3], (ii) the moduli space M 0,n of n marked points on a Riemann surface of genus zero and (iii) "positive" geometries / "canonical" forms, as recently discussed by Bai and Lam [4]. The integrand of the CHY representation for bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes is constructed from two factors, a cyclic factor (or Parke-Taylor factor) and a polarisation factor (also known as reduced Pfaffian). We show that the cyclic factor and the polarisation factor lead to differential (n − 3)-forms Ω cyclic scattering and Ω pol scattering , respectively, on the compactification M 0,n of M 0,n , such that the only singularities of the differential forms Ω scattering are on the divisor M 0,n \M 0,n . Each singularity is logarithmic and the residue factorises into two differential forms of lower points. These scattering forms figure prominently in the recent work by Mizera [5,6]. The scattering forms are cocycles and Mizera has shown that the amplitudes are intersection numbers of these cocycles, twisted by a one-form derived from the scattering equations.We put "positive" geometry into quotes. The reason is the following: The solutions of the scattering equations are in general complex and correspond to points in M 0,n . Only for very special external momenta p are the solutions of the scattering equations real [7][8][9]. If the solutions are real, we may limit ourselves to the space of real points M 0,n (R). This is a positive space in the sense of Arkani-Hamed, Bai and Lam [4], with boundary M 0,n (R)\M 0,n (R). However, we are interested in the general situation. This forces us to work throughout the paper with the complex numbers C instead of the real numbers R. For simplicity we write M 0,n instead of M 0,n (C). We find that the notion of "positivity" is not essential, what is essential is the structure of the divisor

show abstract

Section: The Polarisation Scattering Formmentioning

confidence: 99%

Properties of scattering forms and their relation to associahedra

Cruz

Kniss

Weinzierl

2018

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…For details, see [23]. In particular, when h 0 = σ i , the CHY-like integral is studied in [24] and the corresponding differential operator is worked out analytically.…”

Section: Preliminary: Review Of Differential Operator Methodsmentioning

confidence: 99%

“…In [23,24] Cheung, Xu and the current authors proposed a method for evaluating the CHY forms using a differential operator and studied the combinatoric properties of the scattering equations. This method bypasses the need for solving the scattering equations and leads to the analytic evaluation of a particular class of CHY forms, called the prepared forms.…”

Section: Introductionmentioning

confidence: 99%

BCJ numerators from differential operator of multidimensional residue

Chen

Wang

2020

Eur. Phys. J. C

Self Cite

View full text Add to dashboard Cite

In previous works, we devised a differential operator for evaluating typical integrals appearing in the Cachazo-He-Yuan (CHY) forms and in this paper we further streamline this method. We observe that at tree level, the number of free parameters controlling the differential operator depends solely on the number of external lines, after solving the constraints arising from the scattering equations. This allows us to construct a reduction matrix that relates the parameters of a higher-order differential operator to those of a lower-order one. The reduction matrix is theory-independent and can be obtained by solving a set of explicitly given linear conditions. The repeated application of such reduction matrices eventually transforms a given tree-level CHY-like integral to a prepared form. We also provide analytic expressions for the parameters associated with any such prepared form at tree level. We finally give a compact expression for the multidimensional residue for any CHY-like integral in terms of the reduction matrices. We adopt a dual basis projector which leads to the CHY-like representation for the non-local Bern-Carrasco-Johansson (BCJ) numerators at tree level in Yang-Mills theory. These BCJ numerators are efficiently computed by the improved method involving the reduction matrix.

show abstract

“…differs from the reduced Pfaffian of the CHY formalism. The reduced Pfaffian has been extensively studied in the literature [23][24][25][26][27][28][29][30][31][32]. The reduced Pfaffian…”

Section: The Invariant Measurementioning

confidence: 99%