Effective field theories for QED bound states: extending Nonrelativistic QED to study retardation effects

TL;DR

The paper develops an effective field theory framework to tackle QED bound states with multiple dynamical scales. By refining NRQED to separate soft and ultra-soft photon effects and introducing MQED via a multipole/taylor expansion, it provides systematic counting rules that assign a unique α-order to each diagram, including retardation effects. It extends the formalism to arbitrary masses and charges and demonstrates how ultra-soft contributions can be organized to yield physically relevant corrections such as Lamb shifts. This approach clarifies scale separations and offers a practical pathway for high-precision bound-state calculations in QED and related EFTs.

Abstract

Nonrelativistic QED bound states are difficult to study because of the presence of at least three widely different scales: the masses, three-momenta ($p_i$) and kinetic energies ($K_i$) of the constituents. Nonrelativistic QED (NRQED), an effective field theory developed by Caswell and Lepage, simplifies greatly bound state calculations by eliminating the masses as dynamical scales. As we demonstrate, NRQED diagrams involving only photons of energy $E_γ\simeq p_i$ contribute, in any calculation, to a unique order in $α$. This is not the case, however, for diagrams involving photons with energies $E_γ\simeq K_i$ (``retardation effects"), for which no simple counting counting rules can be given. We present a new effective field theory in which the contribution of those ultra-soft photons can be isolated order by order in $α$. This is effectively accomplished by performing a multipole expansion of the NRQED vertices.

Figures (11)

• Figure 1: NRQED Feynman rules
• Figure 2: Time-ordered propagators for two fermions or two fermions plus one transverse photon.
• Figure 3: The two time-ordered diagrams corresponding to the exchange of a transverse photon (the vertical lines indicate the intermediate states used for the time-ordered propagators).
• Figure 4: Separation of a transverse photon into a soft, instantaneous contribution (represented by a vertical line) and an ultra-soft propagator (represented by the broken wavy line).
• Figure 5: Generic bound state potential; the dependence on the charges and on the masses of each vertex is indicated.