Abstract
The Born--Oppenheimer approximation is the standard tool for the study of molecular systems. It is founded on the observation that the energy scale of the electron dynamics in a molecule is larger than that of the nuclei. A very similar physical picture can be used to describe QCD states containing heavy quarks as well as light-quarks or gluonic excitations. In this work, we derive the Born--Oppenheimer approximation for QED molecular systems in an effective field theory framework by sequentially integrating out degrees of freedom living at energies above the typical energy scale where the dynamics of the heavy degrees of freedom occurs. In particular, we compute the matching coefficients of the effective field theory for the case of the $H^+_2$ diatomic molecule that are relevant to compute its spectrum up to ${\cal O}(m\alpha^5)$. Ultrasoft photon loops contribute at this order, being ultimately responsible for the molecular Lamb shift. In the effective field theory the scaling of all the operators is homogeneous, which facilitates the determination of all the relevant contributions, an observation that may become useful for high-precision calculations. Using the above case as a guidance, we construct under some conditions an effective field theory for QCD states formed by a color-octet heavy quark-antiquark pair bound with a color-octet light-quark pair or excited gluonic state, highlighting the similarities and differences between the QED and QCD systems. Assuming that the multipole expansion is applicable, we construct the heavy-quark potential up to next-to-leading order in the multipole expansion in terms of nonperturbative matching coefficients to be obtained from lattice QCD.
Highlights
AND MOTIVATIONThe discovery in the last decade of the XYZ mesons has brought into QCD challenges enduring since the early days of molecular physics in QED—for a recent overview, see Ref. [1]
A way to see this is by noticing that mixing terms in the energy levels of the Born–Oppenheimer EFT (BOEFT) would count like mα2, a fact that would prevent the separation of the electron from the nuclei dynamics
Our initial assumption was that the kinetic energy associated with the relative motion of the nuclei is small compared to the ultrasoft scale, from there we integrated out the latter and matched potential NRQED (pNRQED) to the BOEFT
Summary
The discovery in the last decade of the XYZ mesons has brought into QCD challenges enduring since the early days of molecular physics in QED—for a recent overview, see Ref. [1]. We project the Lagrangian in Eq (1) on the subspace of one electron and two nuclei This is similar to the pNRQED bound state calculations for the hydrogen atom [16,17], but since the projection for one light and two heavy particles with different charges has not been done so far in the literature, we present the procedure with some detail. This implies that the kinetic energy associated witph tffihffiffiffieffiffiffirffieffiffilative motion of the nuclei is −∇2r=M ∼ mα m=M ∼ mα11=4 Using this counting, and disregarding operators that produce emission or absorption of photons that contribute only in loops, the leading-order operators in Eq (21) are h0ðr; zÞ þ VLZOZ ðrÞ, which are of Oðmα2Þ. In this paper, following the logic of EFTs, we will integrate out from pNRQED the ultrasoft degrees of freedom to obtain an EFT at the energy scale of the two-nuclei dynamics.
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