BCJ numerators from differential operator of multidimensional residue

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

Section: The Polarisation Scattering Formmentioning

confidence: 99%

Properties of scattering forms and their relation to associahedra

Cruz

Kniss

Weinzierl

2018

“…It is curious to note that the objects that the kinematic Lie algebra should compute, the kinematic BCJ numerators, are under better control than the algebra itself. Numerators that manifest the color-kinematics duality have been constructed at any multiplicity for tree-level amplitudes in pure Yang-Mills theory [20][21][22][23][24][25][26][27]. For supersymmetric Yang-Mills theory, it has been observed that a recursive construction similar to Berends-Giele recursion [28] can be used to generate BCJ numerators after using appropriate non-linear gauge transformations [23].…”

Section: Introductionmentioning

confidence: 99%

On the kinematic algebra for BCJ numerators beyond the MHV sector

Chen

Johansson

Teng

et al. 2019

Self Cite

106

The duality between color and kinematics present in scattering amplitudes of Yang-Mills theory strongly suggest the existence of a hidden kinematic Lie algebra that controls the gauge theory. While associated BCJ numerators are known on closed forms to any multiplicity at tree level, the kinematic algebra has only been partially explored for the simplest of four-dimensional amplitudes: up to the MHV sector. In this paper we introduce a framework that allows us to characterize the algebra beyond the MHV sector. This allows us to both constrain some of the ambiguities of the kinematic algebra, and better control the generalized gauge freedom that is associated with the BCJ numerators. Specifically, in this paper, we work in dimension-agnostic notation and determine the kinematic algebra valid up to certain O (ε i ·ε j ) 2 terms that in four dimensions compute the next-to-MHV sector involving two scalars. The kinematic algebra in this sector is simple, given that we introduce tensor currents that generalize standard Yang-Mills vector currents. These tensor currents controls the generalized gauge freedom, allowing us to generate multiple different versions of BCJ numerators from the same kinematic algebra. The framework should generalize to other sectors in Yang-Mills theory.1 Cubic interactions are obtained by the replacement Tr([A µ , A ν ] 2 ) → − 1 2 (B µν ) 2 +Tr([A µ , A ν ]B µν ).

“…The duality provides a rich structure to tree-level amplitudes [20,24,[35][36][37][38][39][40][41][42][43][44][45][46], most notably through the color-ordered n-point gluon amplitudes, which are constrained by the so-called BCJ relations [13,47,48] -these can be used to eliminate all but (n − 3)! independent amplitudes.…”

Section: Introductionmentioning

confidence: 99%

Unraveling conformal gravity amplitudes

Johansson

Mogull

Teng

2018

100

Conformal supergravity amplitudes are obtained from the double-copy construction using gauge-theory amplitudes, and compared to direct calculations starting from conformal supergravity Lagrangians. We consider several different theories: minimal N = 4 conformal supergravity, non-minimal N = 4 Berkovits-Witten conformal supergravity, mass-deformed versions of these theories, as well as supersymmetry truncations thereof. Coupling the theories to a Yang-Mills sector is also considered. For all cases we give the gravity Lagrangians that the double copy implicitly generates. The two main results are: we determine a Lagrangian for the non-minimal Berkovits-Witten theory, and we uncover the double-copy prescription for the minimal N = 4 conformal supergravity.