Abstract
We present a complete list of the dimension 8 operator basis in the standard model effective field theory using group theoretic techniques in a systematic and automated way. We adopt a new form of operators in terms of the irreducible representations of the Lorentz group, and identify the Lorentz structures as states in a $SU(N)$ group. In this way, redundancy from equations of motion is absent and that from integration-by-part is treated using the fact that the independent Lorentz basis forms an invariant subspace of the $SU(N)$ group. We also decompose operators into the ones with definite permutation symmetries among flavor indices to deal with subtlety from repeated fields. For the first time, we provide the explicit form of independent flavor-specified operators in a systematic way. Our algorithm can be easily applied to higher dimensional standard model effective field theory and other effective field theories, making these studies more approachable.
Highlights
The standard model (SM) of particle physics is a great triumph of modern physics
We provided the full result of the independent dimension-eight operator basis in the standard model effective field theory
What is more important is that the form of the operators we provide here has definite symmetry over the flavor indices, making it possible to identify independent flavor-specified operators
Summary
The standard model (SM) of particle physics is a great triumph of modern physics. It has successfully explained almost all experimental results and predicted a wide variety of phenomena with unprecedented accuracy. This task is highly nontrivial because different structured operators may be related by the equation of motion (EOM), integration by parts (IBP), and Fierz identities These redundancies could be avoided by imposing the EOMs and IBPs explicitly, the independent dimension-six operators in the Warsaw basis [3] were constructed based on this principle, and the complete renormalization group equations are written in Refs. Reference [16] generates an overcomplete list of operators at first, and reduces it to an independent basis by putting all the redundant relations into a matrix, which has been applied [24] to write down the partial list of the dimensioneight operators involving only the bosonic fields Another difficulty is how to obtain independent flavor structures when repeated fields are present. In Appendix B, we introduce some basics of symmetric groups Sm and a few group theory tools we used in the paper, including the basis symmetrizer bλ and the projection operator involved in the inner product decomposition
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