Abstract
We investigate how fields transform under the Poincar\'e group in nonrelativistic effective field theories of QCD. In constructing these transformations, we rely only on symmetries and field redefinitions to limit the number of allowed terms. By requiring invariance of the action under these transformations, nontrivial relations between Wilson coefficients for both nonrelativistic QCD and potential nonrelativistic QCD are derived. We show explicitly how the Poincar\'e algebra is satisfied, and how this gives complementary information on the Wilson coefficients. We also briefly discuss the implications of our results, as well as the possibility of applying this method to other types of effective field theories.
Highlights
AND OUTLINEEffective field theories (EFTs) are a standard tool for particle and nuclear physics and have been for at least forty years [1]
Low energy EFTs have been constructed for different sectors of the Standard Model to describe specific low energy systems: for example, EFTs of quantum electrodynamics (QED) to describe atomic physics, or EFTs of quantum chromodynamics (QCD) to describe hadronic and nuclear physics
We have investigated boost transformations of nonrelativistic fields in low energy EFTs for heavy quarks, by starting from the general form allowed by charge conjugation, parity, and time reversal, while exploiting the freedom to remove redundant terms through field redefinitions
Summary
Effective field theories (EFTs) are a standard tool for particle and nuclear physics and have been for at least forty years [1]. Demanding that all generators satisfy the commutation relations of the Poincarealgebra provides some exact constraints on the Wilson coefficients of the EFTs. Reparametrization invariance is a symmetry found in low energy EFTs of QCD, like the HQET or soft collinear effective theory (SCET) [20,21,22,23]. We take from [28] that the boost transformation of the nonrelativistic field is realized in a nonlinear way and that requiring the invariance of the Lagrangian under this boost leads to constraints on the Wilson coefficients, but apart from that we will not refer to the induced representation. IV with a summary and a short outlook on possible applications to other effective field theories
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