Аннотация
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 $O(m\alpha^5)$.
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.
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