Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2012.00713

On the Lagrangian formulation of the double copy to cubic order

Pietro Ferrero·Dario Francia
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the Lagrangian formulation of the double-copy correspondence between gauge theories and gravity, up to the cubic order. Building on the definition of the double-copy field as a convolution of two vectors, we obtain free gravitational Lagrangians as products of two Yang-Mills Lagrangians, in a form amenable to be easily extended to the massive case. We discuss the origin of these results from tensionless strings and show the existence of gauge fixings that mix the two spin-one sectors and lead to an alternative, especially simple, version of the free Lagrangian. We then construct cubic vertices for the full double-copy multiplet, comprising a graviton, a two-form and a scalar particle, by means of the Noether procedure. Both at the free and at the cubic level the result gets uniquely fixed only upon imposing, on top of gauge invariance, a left-right Lorentz symmetry ruling contraction of indices among double-copy fields. Whereas the outcome nicely matches the cubic interactions of $\mathcal{N}=0$ supergravity, including the gauge-invariant coupling between the scalar particle and the two-form, such a twofold Lorentz symmetry seems to conflict with the perturbative reconstruction of space-time geometry.

Explore related subjects

hep-th

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pietro Ferrero, Dario Francia. 2021-03-01. On the Lagrangian formulation of the double copy to cubic order. https://doi.org/10.1007/jhep02(2021)213

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Introduction to Generalized Symmetries

These notes were prepared for a series of intensive lectures delivered at Hokkaido University, Nagoya University, Kyoto University, and Kyushu University. We begin with a brief review of higher-form symmetries, anomalies, and discrete gauge theories, before introducing non-invertible symmetries in $(1+1)$-dimensional systems. The basic structure of fusion categories is then discussed, including a discussion of categorical analogs of discrete gauging and representation theory. We subsequently turn to $(3+1)$-dimensional theories, where several physical applications of non-invertible symmetries are discussed. These notes are intended to be largely self-contained, and require no prior familiarity with subjects such as conformal field theory or lattice models.

hep-th↗
arXiv2026

Planar loop integrands from cuts in $D$ dimensions

We present a direct reconstruction formula for planar loop integrands from $D$-dimensional generalized unitarity cuts in any colored theory. The reconstruction combinatorics is separated from the theory-dependent tree amplitudes entering the cuts: for the $L$-loop $n$-point color-ordered amplitude, the integrand is expressed as a sum over admissible non-scaleless scalar graphs dressed by corresponding cuts in $D$ dimensions; the coefficients are given by the universal Möbius-inversion formula of the refinement poset, or equivalently one minus the Euler characteristics of associated complexes. As an application we write down closed-formulas for loop integrands in pure Yang--Mills theory, where the required cuts are generated by gluing $D$-dimensional tree amplitudes and summing over internal gluon states. We also use the two-loop five-point case as a validation, comparing with known integrand data and after integration-by-parts reduction, with known integrated helicity amplitudes. The same framework also produces compact cut-organized data for larger examples, including the two-loop six-point and three-loop four-point cases. We also describe the corresponding simplification in maximally supersymmetric Yang--Mills theory, where the absence of bubble and triangle subgraphs reduces the relevant cut poset substantially.

hep-th↗
arXiv2026

Free Field Realization of $\mathcal{W}$-Algebra Associated with Exceptional Lie Algebras

We study the free field realization of the $\mathcal{W}$-algebra associated with the exceptional Lie algebras $E_6$, $E_7$, $E_8$, and $F_4$. We develop a recursive construction in which a $\mathcal{W}$-algebra of rank $r$ is obtained from a $\mathcal{W}$-algebra of rank $r-1$ together with a free boson. The $\mathcal{W}$-currents are constructed from the zero commutation relation with the screening charges. The $\mathcal{W}E_6/\mathcal{W}E_7$ algebra is constructed from the $\mathcal{W}D_5/\mathcal{W}D_6$ algebra and is shown to be the same as that realized from the $\mathcal{W}A_5/\mathcal{W}E_6$ algebra, up to a change of the free field basis. The spin-$8$ generator of the $\mathcal{W}E_8$ algebra is built from the $\mathcal{W}D_7$ algebra. The recursive construction of the $\mathcal{W}BC_r$ algebras is also studied. We then realize the $\mathcal{W}F_4$ algebra based on the $\mathcal{W}BC_3$ algebra. Furthermore, the $\mathcal{W}$-charges of the generators of the $\mathcal{W}E_{6,7}$, $\mathcal{W}BC_{2,3}$, and $\mathcal{W}F_4$ algebras are calculated and expressed in terms of the Casimir invariants.

hep-th↗