The Born-Oppenheimer approximation in an effective field theory language

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 $\mathcal{O}(m{\alpha }^5)$. 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.

arXiv commentJournal Version. Figures 4 and 5 added, bibliography extended. 16 pages, 5 figures

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• Are the XYZ states unconventional states or conventional states with unconventional properties?
• The nature of ${\chi }_{c1}\left(3872\right)$ and $T_{cc}^{+}\left(3875\right)$
• Unravelling Pentaquarks with Born--Oppenheimer effective theory
• Born-Oppenheimer Potentials for $SU(3)$ Gauge Theory
• Substructure of Multiquark Hadrons (Snowmass 2021 White Paper)
• The Direct Coupling of Light Quarks to Heavy Di-quarks
• Quantum field theory for multipolar composite bosons with mass defect and relativistic corrections
• Baryons, multi-hadron systems, and composite dark matter in non-relativistic QCD
• Magnetic field induced hair structure in charmonium gluon-dissociation
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• Misconceptions on Effective Field Theories and spontaneous symmetry breaking: Response to Ellis' article
• Analysis of the exotic resonances in the systems $QQ\bar{q}\bar{q}$ and $Qq\bar{Q}\bar{q}$ with the extended recoupling model