Abstract
Abstract We investigate color-kinematics duality for gauge-theory amplitudes produced by the pure nonabelian Yang-Mills action deformed by higher-dimension operators. For the operator denoted by F 3, the product of three field strengths, the existence of color-kinematic dual representations follows from string-theory monodromy relations. We provide explicit dual representations, and show how the double-copy construction of gravity amplitudes based on them is consistent with the Kawai-Lewellen-Tye relations. It leads to the amplitudes produced by Einstein gravity coupled to a dilaton field ϕ, and deformed by operators of the form ϕR 2 and R 3. For operators with higher dimensions than F 3, such as F 4-type operators appearing at the next order in the low-energy expansion of bosonic and superstring theory, the situation is more complex. The color structure of some of the F 4 operators is incompatible with a simple color-kinematics duality based on structure constants f abc, but even the color-compatible F 4 operators do not admit the duality. In contrast, the next term in the α′ expansion of the superstring effective action — a particular linear combination of D 2 F 4 and F 5-type operators — does admit the duality, at least for amplitudes with up to six external gluons.
Highlights
We find that the bosonic and superstring amplitudes individually correspond to operators whose color structure is incompatible with the usual color-kinematics duality, because they cannot be expressed in terms of structure constants f abc alone. (In this paper we do not consider extensions of color-kinematics duality to four-index antisymmetric structure constants [17].) The difference between the bosonic string and superstring amplitudes has a colorcompatible representation, and even obeys a BCJ-like monodromy relation
Starting from the decomposition in eq (2.1), it takes only a couple of steps to demonstrate the idea behind color-kinematic dual representations: the color and the kinematic part of the amplitude are brought into a form in which they behave identically under certain symmetry operations
Because dual representations are available for all gluon tree amplitudes in pure Yang-Mills theory, we can combine YM with F 3 amplitudes in order to obtain pure-graviton amplitudes which would be at O(α′)
Summary
For a general m-point scattering amplitude at tree level, the correspondence between color and kinematics [2] begins with the cubic-graph representation, Am =. A color-kinematic dual representation does imply the existence of BCJ amplitude relations. Concerning the reverse direction, explicit (albeit non-local) representations of dual numerators ng in terms of color-ordered amplitudes have been obtained for any number of external legs in (renormalizable) gauge theory, based on string theory arguments [41,42,43]. At the same time we will test the BCJ amplitude relations explicitly, and provide a general argument for their validity These results further support the equivalence of BCJ relations and color-kinematic dual representations, at least in the F 3 case
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