Abstract
Until recently little was known about the high-dimensional operators of the standard model effective field theory (SMEFT). However, in the past few years the number of these operators has been counted up to mass dimension 15 using techniques involving the Hilbert series. In this work I will show how to perform the same counting with a different method. This alternative approach makes it possible to cross-check results (it confirms the SMEFT numbers), but it also provides some more information on the operators beyond just counting their number. The considerations made here apply equally well to any other model besides SMEFT and, with this purpose in mind, they were implemented in a computer code.
Highlights
It is sometimes useful to consider interactions which are allowed by symmetry, even if they are not renormalizable
This construction is often called the standard model effective field theory (SMEFT), and it has been studied for a long time
One might try to compute the irreducible Lorentz representations associated with the components f∂igF and remove those which appear in the equations of motion f∂i−1gh∂F i
Summary
It is sometimes useful to consider interactions which are allowed by symmetry, even if they are not renormalizable. The authors of [18] computed the number of SMEFT operators up to dimension 15 Despite these advantages, the computations performed in the Hilbert series method are very different from those performed in what I will call the traditional method, which consists of multiplying all fields in all possible ways and retaining those combinations which are invariant under the action of all relevant symmetry groups—usually the Lorentz and gauge groups. While the dimension of these terms (six) is still quite small, it is nontrivial to derive the number of independent couplings associated with each, assuming nX flavors of the field type X 1⁄4 Nc; Q; L.1 These two complications can be handled in a systematic way. This makes it possible to count operators in models such as SMEFT, up to high mass dimensions, using the traditional approach mentioned earlier.
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