Abstract
We construct the on-shell amplitude basis and the corresponding effective operators for generic modified gravity theory, such as pure gravity with higher derivatives, scalar-tensor gravity, Einstein-Yang-Mills, etc. Taking the Weyl tensor as the building block, we utilize the Young tensor technique to obtain independent operators, without equation of motion and total derivative redundancies. We update our algorithm and vastly increase the speed for finding the monomial basis (m-basis) of effective operators expressed in terms of Weyl tensors with Lorentz indices, the familiar form for the General Relativity community. Finally, we obtain the complete and independent amplitude and operator basis for GRSMEFT and GRLEFT up to mass dimension 10.
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Acknowledgments
J.H.Y. is supported by the National Science Foundation of China under Grants No. 12022514, No. 12375099 and No.12047503, and National Key Research and Development Program of China Grant No. 2020YFC2201501, No. 2021YFA0718304, and CAS Project for Young Scientists in Basic Research YSBR-006, the Key Research Program of the CAS Grant No. XDPB15. M.-L.X. is supported by the U.S. Department of Energy under contracts No. DE-AC02-06CH11357 at Argonne. H.-L.L is supported by the 4.4517.08 IISN-F.N.R.S convention.
Note added. During the preparation of this work, ref. [44] appeared, which implements our algorithm for enumeration of the SMEFT operators up to dimension-12 and GRSMEFT up to dimension-11. However, in their work, they express the operators in terms of the fields with purely spinor indices for the Lorentz group, and with purely fundamental indices for the gauge group, basically the y-basis in our notation. In contrast, in our work, we convert the y-basis operator into the form of m-basis, where the field strength tensor and Weyl tensors are written in terms of ordinary four-component Lorentz indices, and the corresponding gauge indices are expressed in a more commonly used notation. This conversion is a non-trivial task and is one of our new updates to our previous algorithm. In addition, we also provide the independent amplitudes for each operator type in the bracket notation.
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Li, HL., Ren, Z., Xiao, ML. et al. On-shell operator construction in the effective field theory of gravity. J. High Energ. Phys. 2023, 19 (2023). https://doi.org/10.1007/JHEP10(2023)019
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DOI: https://doi.org/10.1007/JHEP10(2023)019