Bound-State QED and NRQED

Bound-state QED is a multiscale problem even when $\alpha$ is small. In a Coulombic two-body system, the hard masses, inverse Bohr radius, binding energy, recoil scale, nuclear size, and nuclear excitation scales are parametrically distinct. NRQED makes that separation explicit: hard fluctuations determine Wilson coefficients, Coulomb exchange is solved nonperturbatively, and relativistic, radiative, recoil, and structure effects are inserted in a controlled expansion.

Required background. Vacuum polarization and the running charge supplies a representative hard-loop correction and its subtraction dependence. Nonrelativistic and potential effective theories supplies matching, power counting, and potential-region methods.

Helpful background. Evolution kernels and resummation explains how coefficients are transported when logarithms between bound-state scales become large.

The hydrogenic scale hierarchy

Let the constituents have masses $m$ and $M$, charges $-e$ and $Ze$, and reduced mass

$$m_r=\frac{mM}{m+M}.$$

For a weakly coupled Coulombic state,

$$v\sim Z\alpha, \qquad |\mathbf p|\sim m_rv, \qquad E_{\mathrm{bind}}\sim m_rv^2.$$

The momentum regions have different scalings:

Region or input	Energy scaling	Momentum scaling	Physical role
Hard lepton	$m$	$m$	Lepton form factors and local NRQED coefficients
Hard second constituent	$M$	$M$	Its magnetic, recoil, and contact coefficients
Soft	$m_rv$	$m_rv$	Matching from dynamical nonrelativistic modes to potentials
Potential	$m_rv^2$	$m_rv$	Coulomb ladder and instantaneous subleading potentials
Ultrasoft	$m_rv^2$	$m_rv^2$	Retardation and radiation at the binding-energy scale
Nuclear size	set by nuclear excitation and $r_N^{-1}$	set by $r_N^{-1}$	Form factors, polarizability, and contact operators

When $m$ and $M$ are widely separated, there is no single “hard scale”: each species is matched near its own mass, followed by evolution and threshold matching as needed. A point-nucleus approximation additionally requires $m_rv\,r_N\ll1$. If that product is not small, a local radius expansion is inadequate and the full form factor or a more detailed nuclear EFT is required.

NRQED operators and matching

For a spin-$1/2$ field $\psi$ of signed charge $q$, take $D_\mu=\partial_\mu+iqA_\mu$. Through $1/m^3$, the one-fermion terms relevant to many spectroscopic applications include

$$\begin{aligned} \mathcal L_\psi=\psi^\dagger\Bigg(&iD_t +\frac{\mathbf D^2}{2m} +\frac{\mathbf D^4}{8m^3} +c_F\frac{q\,\boldsymbol\sigma\cdot\mathbf B}{2m}\\ &+c_D\frac{q\,[\boldsymbol\nabla\cdot\mathbf E]}{8m^2} +ic_S\frac{q\,\boldsymbol\sigma\cdot (\mathbf D\times\mathbf E-\mathbf E\times\mathbf D)}{8m^2} +\cdots\Bigg)\psi . \end{aligned}$$

The brackets in the Darwin term mean that the derivative acts only on the electric field. A second spinful constituent has its own copy of these operators with $m\to M$ and independent coefficients. Gauge-invariant four-fermion operators such as

$$\frac{d_s}{mM}(\psi^\dagger\psi)(N^\dagger N) +\frac{d_v}{mM} (\psi^\dagger\boldsymbol\sigma\psi) \cdot(N^\dagger\boldsymbol\sigma N)$$

encode hard two-body scattering, annihilation where allowed, and short-distance spin dependence. A composite nucleus also requires charge-radius, magnetic-radius, polarizability, and higher-moment operators. The displayed basis is therefore a hierarchy, not a complete all-orders Lagrangian.

Matching equates on-shell amplitudes in QED and NRQED at a scale $\mu$ below the integrated-out hard modes. With the relativistic vertex normalized as on the lepton magnetic-moment page, one finds for a point spin-$1/2$ constituent

$$\begin{aligned} c_F(\mu)&=F_1(0;\mu)+F_2(0;\mu),\\ c_D(\mu)&=F_1(0;\mu)+2F_2(0;\mu) +8m^2F_1'(0;\mu),\\ c_S(\mu)&=F_1(0;\mu)+2F_2(0;\mu), \end{aligned}$$

where $F_1'=dF_1/dq^2$. In the physical-charge normalization $F_1(0)=1$, so

$$c_S=2c_F-1.$$

At tree level $c_F=c_D=c_S=1$. Beyond tree level, individual coefficients depend on the renormalization scheme and matching scale; their dependence cancels against potential and matrix-element contributions in an energy level. The operator construction and matching logic are developed in Caswell and Lepage 1986, pp. 437–442 and Burgess 2020, Chapter 12, pp. 296–326.

These are matching relations, not statements that every full on-shell form-factor derivative is separately observable. Use the same infrared regulator in QED and NRQED, subtract the EFT amplitude from the full-theory amplitude, and assign the infrared-finite hard remainder to the coefficient. In particular, $c_D(\mu)$ can carry scale and scheme dependence that cancels only after contact potentials and bound-state matrix elements are included.

Coulomb exchange is leading order

The static Coulomb potential is

$$V_C(r)=-\frac{Z\alpha}{r}.$$

Although every exchange carries $\alpha$, a potential propagator and nearly on-shell intermediate state enhance repeated exchanges. With $r^{-1}\sim m_rv$ and $v\sim Z\alpha$,

$$V_C\sim m_rv^2\sim\frac{\mathbf p^2}{2m_r}.$$

The potential is therefore the same order as the kinetic energy and must be iterated to all orders. Solving the Schrödinger problem gives the reference spectrum

$$E_n^{(0)}=-\frac{m_r(Z\alpha)^2}{2n^2}, \qquad |\psi_{nS}(0)|^2=\frac{(m_rZ\alpha)^3}{\pi n^3}.$$

All subsequent terms are perturbations around Coulomb eigenstates unless a further near-degeneracy requires degenerate perturbation theory. Treating Coulomb exchange as a single perturbative insertion destroys the bound-state poles one is trying to predict.

Organizing radiative, recoil, and structure effects

A durable prediction labels every term by both its physical origin and its power counting. The following hierarchy is schematic—some coefficients contain logarithms, mass-ratio enhancements, or selection-rule zeros—but it identifies the necessary separation.

Contribution	Representative size	EFT origin and check
Coulomb binding	$m_r(Z\alpha)^2$	Schrödinger eigenvalue; reproduce the reduced-mass spectrum
Relativistic and fine structure	$m_r(Z\alpha)^4$	$\mathbf D^4$, spin–orbit, Darwin, Breit potentials; recover the Dirac limit as $M\to\infty$
Radiative self-energy and vacuum polarization	$\alpha m_r(Z\alpha)^4$ and higher	Hard coefficients plus potential/ultrasoft matrix elements; cancel $\mu$ and regulator dependence
Recoil	powers of $m_r/M$ multiplying binding corrections	Dynamical second constituent and two-body potentials; vanish in the static-source limit
Hyperfine structure	$m_r^3(Z\alpha)^4/(mM)$ times magnetic coefficients	$c_F^{(m)}c_F^{(M)}$ and spin-dependent contacts; track magnetic-moment conventions
Finite charge radius for an $S$ state	$Z\alpha\,r_E^2\lvert\psi_{nS}(0)\rvert^2$	Local charge-radius contact; vanish for a point source and at leading order for $\ell>0$
Nuclear polarizability	set by inelastic nuclear response	Two-photon contact or response function; do not fold into an elastic radius twice
Ultrasoft radiation	begins at radiative binding orders	Retardation and Bethe-logarithm matrix elements; cancel the matching-scale split from harder regions

For example, the leading elastic finite-size shift of an $nS$ level in the small-radius expansion is

$$\Delta E_{nS}^{(r_E^2)} =\frac{2\pi Z\alpha}{3}r_E^2|\psi_{nS}(0)|^2.$$

Its coefficient depends on a precisely defined charge radius, and higher moments and two-photon exchange must be kept separate. For antiparticle systems, annihilation contacts can have imaginary parts; their real parts shift levels and their imaginary parts determine widths.

Bethe’s 1947 calculation is a dated historical anchor for separating a relativistic mass scale from atomic momenta in the Lamb shift Bethe 1947, pp. 339–341. Modern EFT matching turns that separation into a systematic operator expansion rather than a cutoff prescription. Bound-state poles and their perturbative treatment are also developed in Weinberg 1995, §§ 14.1–14.3, pp. 559–589.

Scale and double-counting checks

Introduce factorization scales only to separate regions. A physical transition frequency obeys

$$\frac{d}{d\ln\mu} \left(E_{\mathrm{hard}}+E_{\mathrm{soft/potential}}+E_{\mathrm{us}}\right)=0$$

through the claimed order. A residual $\mu$ dependence estimates missing orders only after every term required at that order has been included.

Expansion by regions supplies a practical overlap test: expand each region according to its scaling, integrate it over the full loop domain, and subtract any EFT overlap required by the regulator. Simply adding a full-theory form-factor potential to a separately matched Darwin coefficient can count the same hard momentum twice. Similarly, inserting a phenomenological nuclear radius correction and a two-photon term derived from the same elastic form factor requires an explicit subtraction convention.

Four limits expose many mistakes:

$$M\to\infty, \qquad r_N\to0, \qquad \alpha\to0, \qquad \mu\frac{dE}{d\mu}\to0.$$

They test, respectively, recoil, nuclear structure, interaction dependence, and matching consistency. Dimensional analysis provides a fifth: every energy shift must have mass dimension one.

Comparing spectroscopy with data

For transitions $\nu_a=(E_{u,a}-E_{l,a})/h$, collect all external inputs into a vector $x_i$ and write

$$\nu_a^{\mathrm{th}}=\nu_a^{\mathrm{th}}(x_i;\mu,\text{scheme})+\delta_a^{\mathrm{trunc}}.$$

The input covariance propagates as

$$C_{ab}^{\mathrm{param}} =\sum_{ij} \frac{\partial\nu_a}{\partial x_i} C_{ij}^{(x)} \frac{\partial\nu_b}{\partial x_j}.$$

Truncation, numerical, nuclear-model, and experimental covariances are then added with their actual correlations, not automatically in quadrature. Before treating a theory–experiment difference as evidence, record:

Item	Required information	Failure it prevents
Constants	Numerical values, units, defining release, covariance	Mixing incompatible $\alpha$, masses, radii, or conversion factors
Matching	Operator basis, perturbative order, scheme, scale, running	Combining coefficients and matrix elements from different conventions
Bound-state theory	Included orders by $\alpha$, $Z\alpha$, and $m/M$; numerical method	Hidden omissions or double counting between regions
Nuclear structure	Radius definition, elastic/inelastic split, isotope correlations	Treating model spread as independent data or counting polarizability twice
Experiment	Transition definition, line-shape model, corrections, full covariance	Comparing different observables or correlated lines as independent
Source history	Publication/release date, correction or replacement history	Retaining a superseded value
Independence	Which constants or radii were fitted from which transitions	Circularly “predicting” data used to determine the inputs

Circularity is especially consequential. If a charge radius or Rydberg-like constant is fitted using the same transition being tested, the residual must be formed from the joint fit or from an independent input set. Removing the input’s marginal uncertainty while ignoring its covariance with the transition produces an artificially precise comparison.

The separation between a durable NRQED derivation and a mutable spectroscopy comparison is shown below. The lower route is part of the scientific result: source versions, correlations, corrections, and supersession rules determine which numerical comparison was actually made.

Evergreen theory and dated evidence meet only through a declared observable contract. Numerical constants, measurements, likelihoods, covariances, and later corrections remain versioned; a new evidence snapshot can supersede a comparison without changing the underlying NRQED derivation. The workflow is schematic.

No current recommended constants, measured transition values, or present discrepancies are quoted here. Such comparisons require a dated, versioned source set with the information above.

Common pitfalls

Expanding the Coulomb potential perturbatively. In a bound state, $V_C$ and the kinetic energy have the same scaling. Solve their joint leading-order problem, then perturb with suppressed operators.

Using one scale for every mode. Hard, soft, potential, and ultrasoft contributions contain different physics. A common numerical scale may conceal large logarithms and does not remove the need for matching.

Calling every nuclear effect a radius correction. Elastic size, two-photon exchange, polarizability, and higher moments have distinct operators and correlations. State the convention that prevents overlap.

Using a fitted constant as independent input. A spectacular residual can be generated by dropping covariance or by predicting a transition with a constant extracted from it. Trace each external input back to the measurements in its fit.

References

• Bethe, Hans A. “The Electromagnetic Shift of Energy Levels.” Physical Review 72 (1947): 339–341. DOI.
• Burgess, C. P. Introduction to Effective Field Theory: Thinking Effectively about Hierarchies of Scale. Cambridge: Cambridge University Press, 2020. DOI.
• Caswell, William E., and G. Peter Lepage. “Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field Theories.” Physics Letters B 167 (1986): 437–442. DOI.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.