Bound-State QED and NRQED

Bound-state QED is a multiscale problem even when $\alpha$ is small. A Coulombic two-body system involves the hard masses, inverse Bohr radius, binding energy, recoil, and any nuclear size or excitation scales; their ordering determines which EFT expansion applies. 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, $v$ counts the relative motion rather than either constituent’s velocity separately:

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 charged spin- $1/2$ field $\psi$ , use the global $D_\mu=\partial_\mu-iqA_\mu$ . Write $A_\mu=(A_0,-\mathbf A)$ , so

\begin{aligned} D_t&=\partial_t-iqA_0, &\mathbf D&=\boldsymbol\nabla+iq\mathbf A,\\ \mathbf E&=-\boldsymbol\nabla A_0-\partial_t\mathbf A, &\mathbf B&=\boldsymbol\nabla\times\mathbf A. \end{aligned}

These fields are constructed from the site potential. The conventional electromagnetic potential is $A^{\rm em}=-A$ , with $\mathbf E^{\rm em}=-\mathbf E$ and $\mathbf B^{\rm em}=-\mathbf B$ . The signs can be checked directly from the relativistic Dirac equation:

H_D=\boldsymbol\alpha\cdot\boldsymbol\pi+\beta m-qA_0, \qquad \boldsymbol\pi=\mathbf p+q\mathbf A=-i\mathbf D.

Eliminating the small spinor component to leading order in $1/m$ gives $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2/(2m)$ . Since

[\pi_i,\pi_j]=-iq\epsilon_{ijk}B_k, \qquad (\boldsymbol\sigma\cdot\boldsymbol\pi)^2 =\boldsymbol\pi^2+q\boldsymbol\sigma\cdot\mathbf B,

the nonrelativistic Hamiltonian contains $+q\boldsymbol\sigma\cdot\mathbf B/(2m)=-q\boldsymbol\sigma\cdot\mathbf B^{\rm em}/(2m)$ . This is the usual $g=2$ magnetic energy. 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. Expressed using $A^{\rm em}$ , all three displayed terms linear in $\mathbf E$ or $\mathbf B$ instead have the usual positive Lagrangian coefficients. A second spinful constituent has its own copy of these operators with $m\to M$ , its own charge, 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 $F_1'=dF_1/dk^2$ at $k^2=0$ , with $k$ the transferred momentum, distinct from the signed charge $q$ . 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, form-factor matching, and cancellation of infrared contributions between the full and effective theories are developed in Burgess 2021, §§ 12.1.1–12.1.2, pp. 296–305.

For an explicit operator and form-factor cross-check, see Hill, Lee, Paz, and Solon 2013, Eqs. (1), (7), and (15), manuscript pp. 2, 4, and 6–7 (PDF). Their derivative is $\partial+igZA^{\rm H}$ ; set $A^{\rm H}=-A$ and $gZ=q$ . Their form factors and one-photon coefficients include the charge label $Z$ ; for a charged constituent, dividing each by $Z$ gives the normalization used here. The field reversal supplies the minus signs above without changing the normalized matching relations.

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. The same separation between an external Coulomb interaction summed to all orders and subsequent radiative corrections is explained in Weinberg 1995, § 14.1, p. 565.

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	Typically $\alpha m_r^3(Z\alpha)^4/m^2$ for the lepton contribution, with logarithms	Hard coefficients plus potential/ultrasoft matrix elements; the familiar $\alpha m_r(Z\alpha)^4$ estimate assumes $m_r\sim m$
Vacuum polarization from a loop of mass $m_f$	$\alpha m_r^3(Z\alpha)^4/m_f^2$ when $m_f\gg m_rZ\alpha$	A local contact only in this heavy-loop regime; retain the finite-range potential when the loop mass is of order the Bohr momentum
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

The loop mass in the vacuum-polarization row is independent of both constituent masses. Its effect is derived below; it cannot in general be assigned the same mass counting as the lepton self-energy.

For comparison, define the normalized elastic electric form factor of the nucleus at small spacelike momentum transfer by $F_N(-Q^2)=1-r_E^2Q^2/6+\cdots$ . This radius uses a specified subtraction of point-particle QED radiative effects. Expanding its attractive potential gives

\begin{aligned} \widetilde V_N(\mathbf Q)&=-\frac{4\pi Z\alpha}{Q^2}F_N(-Q^2),\\ \delta V_{r_E^2}(\mathbf r)&=\frac{2\pi Z\alpha}{3}r_E^2\delta^{(3)}(\mathbf r). \end{aligned}

The constant in momentum space is a contact in position space. Consequently, 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.

For positive $r_E^2$ this shift is positive: spreading the source weakens the attraction at the origin. Its coefficient depends on a precisely defined charge radius, and higher moments and two-photon exchange must be kept separate. The same elastic contribution may already be included in a matched Darwin coefficient; do not add it again as a separate form-factor potential. For antiparticle systems, annihilation contacts can have imaginary parts; their real parts shift levels and their imaginary parts determine widths.

Vacuum polarization: contact or finite-range potential

Consider one unit-charge Dirac species of mass $m_f$ in the photon loop, at first order in its vacuum polarization. Use the Thomson coupling and the leading static interaction between point constituents. The loop mass is kept well above the binding-energy scale, so its leading effect is an instantaneous potential. Write $p_B=m_rZ\alpha$ for the Bohr momentum and $Q=|\mathbf Q|$ for the exchanged spatial momentum. The subtracted Euclidean polarization gives

\begin{aligned} \delta\widetilde V(\mathbf Q) &=-\frac{4\pi Z\alpha}{Q^2}\Pi_E(Q^2),\\ \Pi_E(Q^2) &=\frac{\alpha Q^2}{15\pi m_f^2} +O\!\left(\frac{\alpha Q^4}{m_f^4}\right). \end{aligned}

Use $\delta V(\mathbf r)=\int \mathrm d^3Q\,e^{i\mathbf Q\cdot\mathbf r}\delta\widetilde V(\mathbf Q)/(2\pi)^3$ . When $m_f\gg p_B$ , the leading local term therefore yields

\begin{aligned} \delta V_{\mathrm{VP}}(\mathbf r) &=-\frac{4Z\alpha^2}{15m_f^2}\delta^{(3)}(\mathbf r),\\ \Delta E_{nS}^{\mathrm{VP}} &=-\frac{4\alpha(Z\alpha)^4m_r^3}{15\pi n^3m_f^2}. \end{aligned}

The sign is negative: vacuum polarization strengthens the attraction relative to the long-distance Thomson normalization. The familiar order $\alpha m_r(Z\alpha)^4$ follows only when the loop mass is of order $m_r$ . As a normalization check, when the loop and constituent masses coincide and $Z=1$ , the coefficient is the contact $-16\pi\alpha d_2/m_f^2$ with $d_2=\alpha/(60\pi)$ in Pineda and Soto 1999, Eqs. (2.2) and (2.8), manuscript pp. 5 and 8 (PDF).

When $m_f$ is of order $p_B$ , expanding the loop in $Q^2/m_f^2$ fails at the momentum sampled by the atom. Retain its finite range instead. The same subtracted dispersion relation, with the one-loop spectral density, gives

\begin{aligned} \rho_f(s)&=\frac{\alpha}{3\pi} \left(1+\frac{2m_f^2}{s}\right) \sqrt{1-\frac{4m_f^2}{s}}, \qquad s\geq4m_f^2,\\ \delta V_{\mathrm{VP}}(r) &=-\frac{Z\alpha}{r}\int_{4m_f^2}^{\infty} \frac{\mathrm ds}{s}\,\rho_f(s)e^{-\sqrt{s}\,r}. \end{aligned}

This follows by Fourier transforming $4\pi/(Q^2+s)$ into $e^{-\sqrt{s}r}/r$ . Positivity of $\rho_f$ makes the attractive sign manifest. The separation between a local heavy-species contribution and a nontrivial potential from a species near the Bohr scale is also explicit in Pineda and Soto 1999, Appendix B, manuscript p. 17 (PDF).

The Coulomb ground state provides a direct finite-range check. With $|\psi_{1S}(r)|^2=p_B^3e^{-2p_Br}/\pi$ , the radial integral gives

\begin{aligned} \left\langle\frac{e^{-\sqrt{s}r}}{r}\right\rangle_{1S} &=4p_B^3\int_0^\infty r\,e^{-(\sqrt{s}+2p_B)r}\,\mathrm dr =\frac{4p_B^3}{(\sqrt{s}+2p_B)^2},\\ \Delta E_{1S}^{\mathrm{VP}} &=-4Z\alpha p_B^3\int_{4m_f^2}^{\infty} \frac{\mathrm ds\,\rho_f(s)}{s(\sqrt{s}+2p_B)^2}. \end{aligned}

For a heavy loop, the leading term uses $\int\mathrm ds\,\rho_f(s)/s^2=\alpha/(15\pi m_f^2)$ and recovers the contact shift above. The exact expression also explains why the bound-state error cannot simply be read as a relative $p_B^2/m_f^2$ from the low-momentum Taylor series: the factor $(\sqrt{s}+2p_B)^{-2}$ already has a correction linear in $p_B/m_f$ . Coulomb wavefunctions probe arbitrarily short distances.

For $m_f\sim p_B$ , rescaling $s$ by $p_B^2$ instead gives a shift of order $\alpha m_r(Z\alpha)^2$ times a dimensionless function of $m_f/p_B$ . This remains a first-order radiative perturbation to Coulomb binding, but it is not represented by the heavy-loop contact. These formulas isolate vacuum polarization at leading nonrelativistic order; recoil, relativistic potentials, nuclear structure, and higher-loop terms still require their own matching and matrix elements.

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.

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.

Exercises

The Darwin sign in hydrogen. Take an infinitely heavy point nucleus with charge $Ze$ , an electron with $q=-e$ , and $c_D=c_S=1$ . Use the static site potential $A_0=-Ze/(4\pi r)$ to find the electron Darwin contact interaction and its first-order shift of an $nS$ level. Also recover the sign of the spin–orbit interaction outside the origin.

Solution

The site electric field obeys $\boldsymbol\nabla\cdot\mathbf E=-Ze\delta^{(3)}(\mathbf r)$ . Reversing the Lagrangian sign to obtain the static Hamiltonian gives

H_{\rm Darwin}=\frac{q}{8m^2}[\boldsymbol\nabla\cdot\mathbf E] =\frac{\pi Z\alpha}{2m^2}\delta^{(3)}(\mathbf r), \qquad \Delta E_{nS}^{\rm Darwin}=\frac{m(Z\alpha)^4}{2n^3}.

The shift is positive and has energy dimension one. It vanishes at this order for states whose wavefunction is zero at the origin; it is only one contribution to fine structure.

For the electrostatic field, $\boldsymbol\nabla\times\mathbf E=0$ , so $\mathbf D\times\mathbf E=-\mathbf E\times\mathbf D$ when acting on the electron wavefunction. With $\mathbf A=0$ , the spin–orbit Hamiltonian becomes

H_{\rm SO}=\frac{q}{4m^2}\boldsymbol\sigma\cdot(\mathbf E\times\mathbf p) =\frac{Z\alpha}{2m^2r^3}\mathbf L\cdot\mathbf S, \qquad r>0.

Both signs agree with the Dirac Coulomb limit. The simultaneous replacement $\mathbf E^{\rm em}=-\mathbf E$ and the corresponding operator-sign translation produces exactly the same shifts.

References

Burgess, C. P. Introduction to Effective Field Theory: Thinking Effectively about Hierarchies of Scale. Cambridge: Cambridge University Press, 2021. DOI.
Hill, Richard J., Gabriel Lee, Gil Paz, and Mikhail P. Solon. “The NRQED Lagrangian at Order $1/M^4$ .” Physical Review D 87 (2013): 053017. DOI. Open PDF, arXiv:1212.4508v2.
Pineda, Antonio, and Joan Soto. “Potential NRQED: The Positronium Case.” Physical Review D 59 (1999): 016005. DOI. Open PDF, arXiv:hep-ph/9805424v2.
Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.