Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms

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 ).

“…ref. [70]), that yields local Yang-Mills numerators from a closed algebra. For non-local numerators there exists an algebraic construction using string vertex operators [27].…”

Section: Introductionmentioning

confidence: 99%

On the kinematic algebra for BCJ numerators beyond the MHV sector

Chen

Johansson

Teng

et al. 2019

106

“…Considering the relation to heterotic string theory and to its N = 4 supersymmetric version at tree level, we feel it is reasonable to suspect that certain weaker version of the Hopf algebraic structure survives in Yang-Mills. A hint that may be related to this structure was recently observed in [57], where it was demonstrated that the Yang-Mills cubic vertex can be obtained as a projected bracket of the Drinfeld double constructed naturally by regarding gauge fields as vector fields supplemented with dual one-forms. The projection broke the Jacobi identity which was shown to be restored once the quartic vertex contribution is included.…”

Section: Introductionmentioning

confidence: 79%

“…This strengthens the idea of an underlying principle responsible for the colour-kinematics duality. Another piece of evidence supporting this idea is that the Lie algebra of diffeomorphism has been identified to give rise to the kinematic numerators at least for special helicity configurations and for small multiplicity in [54][55][56][57].…”

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

A vertex operator algebra construction of the colour-kinematics dual numerator

Vanhove

Wang

2018

Self Cite

We derive a vertex operator based expression for the kinematic numerators of Yang-Mills amplitudes by applying the momentum kernel formalism to open string amplitudes. The expression involves an α -weighted commutator induced by the monodromy relations between the colour ordered Yang-Mills amplitudes, which mirrors the α deformed colour structure observed in open string and semi-abelian Ztheory. The kinematic algebra given by this construction contains the Lie algebra of diffeomorphism as an obvious sub-algebra.where |f is the modified external state defined in (3.6) and the generators T i here are vertex operators, being properly analytic continued and integrated,1.2) originally introduced in [60] to express the string theory generalization of the Del Duca-Dixon-Maltoni [61] colour decomposition of Yang-Mills amplitude M open nderived from string monodromy relation, and recently again observed in semi-abelian Z-theory in [62]. The construction of the present paper gives a kinematic analogue of the colour traces in (1.3) and provides an

“…See[46,47] for earlier Lagrangian-based approaches to the BCJ duality and[48] for a connection with the Drinfeld double of the Lie algebra of vector fields. Also see[49] for the kinematic algebra in the self-dual sectors of D = 4 YM theory and gravity.7 See for instance[57][58][59][60] for the use of tree-level Berends-Giele currents in D > 4-dimensional loop amplitudes of gauge theories with maximal and half-maximal supersymmetry.…”

mentioning

confidence: 99%

Berends-Giele currents in Bern-Carrasco-Johansson gauge for F3- and F4-deformed Yang-Mills amplitudes

Garozzo

Queimada

Schlotterer

2019

We construct new representations of tree-level amplitudes in D-dimensional gauge theories with deformations via higher-mass-dimension operators α F 3 and α 2 F 4 . Based on Berends-Giele recursions, the tensor structure of these amplitudes is compactly organized via off-shell currents. On the one hand, we present manifestly cyclic representations, where the complexity of the currents is systematically reduced. On the other hand, the duality between color and kinematics due to Bern, Carrasco and Johansson is manifested by means of non-linear gauge transformations of the currents. We exploit the resulting notion of Bern-Carrasco-Johansson gauge to provide explicit and manifestly local double-copy representations for gravitational amplitudes involving α R 2 and α 2 R 3 operators. arXiv:1809.08103v1 [hep-th] 21 Sep 2018 45 A.3 Gauge algebra 47 B The explicit form of gauge scalars towards BCJ gauge 50 B.1 The local building block h 12345 50 B.2 An alternative expression for H 1234 51 B.3 The Berends-Giele version H 12345 52 C Deriving a BCJ representation for (YM+F 3 +F 4 ) amplitudes 52 -1 -1 The low-energy effective action of the open bosonic string involves another operator ∼ ζ2α 2 F 4 at the mass dimensions in (1.1) which will not be discussed in this article. Said ζ2α 2 F 4 -operator is also known from the superstring and cannot be reconciled with the BCJ duality [18]. 2 See [ 23,24] for earlier work on the interplay of the KLT relations at the three-and four-point level with gravitational matrix elements of R 2 , R 3 operators and F 3 , F 4 -deformed gauge-theory amplitudes.3 In slight abuse of terminology, we will usually refer to the matrix elements from higher-mass-dimension operators as "amplitudes". In the case at hand, we will be interested in contributions from single-or doubleinsertions of α R 2 operators and single-insertions of α 2 R 3 4 See [27,28] for generating series of Berends-Giele currents, their non-linear gauge transformations and BCJ gauge in ten-dimensional SYM.5 Said higher-derivative extension of the NLSM is defined by the ζ2α 2 -order of abelian Z-theory [44].