Abstract:In this paper, we study from various perspectives the expansion of tree level single trace Einstein-Yang-Mills amplitudes into linear combination of color-ordered YangMills amplitudes. By applying the gauge invariance principle, a programable recursive construction is devised to expand EYM amplitude with arbitrary number of gravitons into EYM amplitudes with fewer gravitons. Based on this recursive technique we write down the complete expansion of any single trace EYM amplitude in the basis of color-order Yang-Mills amplitude. As a byproduct, an algorithm for constructing a polynomial form of the BCJ numerator for Yang-Mills amplitudes is also outlined in this paper. In addition, by applying BCFW recursion relation we show how to arrive at the same EYM amplitude expansion from the on-shell perspective. And we examine the EYM expansion using KLT relations and show how to evaluate the expansion coefficients efficiently.

We demonstrate that the canonical change of variables that yields the MHV lagrangian, also provides contributions to scattering amplitudes that evade the equivalence theorem. This 'ET evasion' in particular provides the tree-level (−++) amplitude, which is non-vanishing off shell, or on shell with complex momenta or in (2, 2) signature, and is missing from the MHV (a.k.a. CSW) rules. At one loop there are ET-evading diagrammatic contributions to the amplitudes with all positive helicities. We supply the necessary regularisation in order to define these contributions (and quantum MHV methods in general) by starting from the light-cone Yang-Mills lagrangian in D dimensions and making a canonical change of variables for all D − 2 transverse degrees of freedom of the gauge field. In this way, we obtain dimensionally regularised three-and four-point MHV amplitudes. Returning to the one-loop (++++) amplitude, we demonstrate that its quadruple cut coincides with the known result, and show how the original light-cone Yang-Mills contributions can in fact be algebraically recovered from the ET-evading contributions. We conclude that the canonical MHV lagrangian, supplemented with the extra terms brought to correlation functions by the non-linear field transformation, provide contributions which are just a rearrangement of those from light-cone Yang-Mills and thus coincide with them both on and off shell.

In this paper, we investigate the color-kinematics duality in nonlinear sigma model (NLSM). We present explicit polynomial expressions for the kinematic numerators (BCJ numerators). The calculation is done separately in two parametrization schemes of the theory using Kawai-Lewellen-Tye relation inspired technique, both lead to polynomial numerators. We summarize the calculation in each case into a set of rules that generates BCJ numerators for all multilplicities. In Cayley parametrization we find the numerator is described by a particularly simple formula solely in terms of momentum kernel.

We present a field theoretical proof of the conjectured KLT relation which states that the full tree-level scattering amplitude of gluons can be written as a product of color-ordered amplitude of gluons and color-ordered amplitude of scalars with only cubic vertex. To give a proof we establish the KK relation and BCJ relation of color-ordered scalar amplitude using BCFW recursion relation with nonzero boundary contributions. As a byproduct, an off-shell version of fundamental BCJ relation is proved, which plays an important role in our work. since on one side of the equation, the full supermultiplet of N = 8 SUGRA theory labeled by eight Grassmann variables η i , i = 1, ..., 8 contains both graviton and gluon, while one the other side of the equation the full supermultiplet of N = 4 SYM theory labeld by four Grassmann variables (η i , i = 1, ..., 4 for A and η i , i = 5, ..., 8 forÃ) contains both gluon and scalar. Specifically, we obtain expressions of amplitudes of the desired particle types by reading off the corresponding Grassmann variable η i 1 ...η ir 1 Recently, KK and BCJ like relations for the remaining rational function of one-loop gluon amplitudes have also been investigated in [27].

One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.

Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work.

Recently, a BCJ dual of the color-ordered formula for Yang-Mills amplitude was proposed, where the dual-trace factor satisfies cyclic symmetry and KK-relation. In this paper, we present a systematic construction of the dual-trace factor based on its proposed relations to kinematic numerators in dual-DDM form. We show that the construction presented respects relabeling symmetry. In addition, we show that using relabeling symmetry as conditions, the same construction can be solved independently.

Recently, new soft graviton theorem proposed by Cachazo and Strominger has inspired a lot of works. In this note, we use the KLT-formula to investigate the theorem. We have shown how the soft behavior of color ordered Yang-Mills amplitudes can be combined with KLT relation to give the soft behavior of gravity amplitudes. As a byproduct, we find two nontrivial identities of the KLT momentum kernel must hold.