Conformal structure of the heavy particle EFT operator basis

Abstract

An operator basis of an effective theory with a heavy particle, subject to external gauge fields, is spanned by a particular kind of neutral scalar primary of the non-relativistic conformal group. We calculate the characters that can be used for generating the operators in a non-relativistic effective field theory, which accounts for redundancies from the equations of motion and integration by parts. I. INTRODUCTIONIf one can say that a particle, and not its antiparticle, exists in the laboratory, then the length scale of its spatial wave function ∆x is parametrically larger than its Compton wavelength 1/M . This hierarchy of scales leads to heavy particle effective field theories (heavy particle EFTs), where one can systematically include higher powers of 1/(∆x M ). Such systems are, in fact, fairly common. For example, the b quark can be located anywhere within a B meson, which has a spatial size of ∼ 1/Λ QCD , and heavy quark effective field theory is an expansion in powers of Λ QCD /m b ∼ 0.3. A more dramatic example is the electron in a hydrogen atom, whose wave function has a size ∆x ∼ 10 −10 m, and 1/(∆x m e ) ∼ 10 −3 , which is why the Schrödinger equation for single-partial quantum mechanics works so well in describing this system, using only the first-order expansion in 1/M . Sometimes these theories are called non-relativistic effective field theories, insofar as there is a inertial frame in which there are non-relativistic particles.Even though heavy particle EFTs describe common physical scenarios, enumerating the independent operators that appear in the Lagrangian, i.e., defining the operator basis, takes considerable effort. The reason for this is that defining the operator basis is more than just requiring that the operators preserve certain symmetries -it also involves accounting for non-trivial redundancies between operators from the classical equations of motion and integration by parts [1,2]. There are popular EFTs with Lagrangians containing heavy fields, e.g., NRQED (external abelian gauge fields), and HQET and NRQCD (external color gauge fields). The operator basis for NRQED was written to order O(1/M 3 ) in Ref. [3] and to O(1/M 4 ) in Ref. [4]. The HQET/NRQCD operator basis was enumerated up to O(1/M 3 ) in Ref. [5], and to O(1/M 4 ) by Ref. [6], which was later confirmed in Ref. [7].A huge stride was taken recently by the authors of Refs. [8][9][10], where they noticed that the operator basis for a relativistic EFT can be organized according to the representations of the conformal group. In particular, accounting for the redundancies from the classical equations of motion can be mapped to the null conditions that saturate unitarity in the conformal group, and choosing the operator basis to be spanned only by primaries of the conformal algebra removes any redundancies associated with integration by parts. By embedding operators into representations of the conformal group, one can use characters as inputs into a Hilbert series, which then can generate the operator basis, counting the numbe...

Reparameterization invariant operator basis for NRQED and HQET

“…In Ref. [2] we showed that the operator basis in Ref. [1] is spanned by scalar primaries of the non-relativistic conformal group.…”

“…for a general velocity. 2 The Lagrangian in Eq. (3) can be rewritten now in terms of Q v and Q v , where now we will let Λ → M , as is conventional in heavy particle effective theory:…”

“…Interestingly, we found that this operator basis can be organized according to irreducible representations of non-relativistic conformal group[2].…”

The spinful large charge sector of non-relativistic CFTs: from phonons to vortex crystals

“…Using the effective field theory of the Goldstone mode, we compute the dimensions of operators to leading order in the angular momentum L and charge Q. We find a diverse set of scaling behaviors for NRCFTs in both d = 2 and d = 3.1 It is important to mention that Schrödinger symmetry is not simply the non-relativistic limit of the conformal symmetry but rather an entirely distinct algebra [15,16].2 It should be emphasized that this is not the only possibility. Ultimately the question "Given this NRCFT, what state of matter describes its large charge sector?"…”

“…1 It is important to mention that Schrödinger symmetry is not simply the non-relativistic limit of the conformal symmetry but rather an entirely distinct algebra [15,16].…”