Completing the two-particle relativistic QED treatment of electromagnetic processes on hydrogen: recoil, retardation, near-pole instability, and error bounds beyond the Eden–Rickayzen (1953) strong-interaction approximation

The 1953 paper by R. J. Eden and G. Rickayzen, 'Quantum field theory of bound states III. Bound states in strong interaction', gave a relativistic two-particle (Bethe–Salpeter) account of electromagnetic processes involving the hydrogen atom — coherent (Rayleigh) scattering of light, photo-electric emission, and electron–hydrogen scattering with exchange — and showed how the one-particle fixed-nucleus results of Franz and of Brown & Woodward emerge when the proton mass $M$ tends to infinity. The treatment explicitly deferred its own quantitative completion: the near-pole region where the bound state is unstable; the marginal-case corrections and boundaries of validity of the scattering ('strong-interaction') approximation; retardation and transverse-photon corrections to the instantaneous-Coulomb amplitudes; the error analysis of neglected subleading residues; and the finite-$M$ (recoil) corrections, with a statement of how large $M$ must be at given photon energies and momentum transfers.

Background

In quantum electrodynamics (QED) a bound state such as the hydrogen atom cannot be described as a small perturbation of free particles. The Bethe–Salpeter equation (Salpeter & Bethe 1951) is the covariant two-body bound-state equation of QED: it determines the bound-state amplitude (the 'wave function', a function of the relative four-momentum, including a relative energy $\epsilon$) from an integral equation whose kernel $G$ is built from exchanged-photon diagrams; in the lowest approximation $G$ is single-photon exchange. The central object is the two-particle propagator $K$ — the four-point function describing the propagation of an electron of mass $m$ and a proton of mass $M$ from one space-time point pair to another as a function of the total energy–momentum four-vector $P$ (the paper uses the metric convention in which a state of mass $\mu$ sits at $P^{2}+\mu^{2}=0$). $K$ decomposes into simple poles at $P^{2}+M_n^{2}=0$, one for each hydrogen bound-state mass $M_n$, whose residues (the contour-integral coefficients of the poles, which fix the bound-state wave functions) carry the bound-state physics, plus a non-singular remainder describing the electron and proton propagating freely with successive interactions. Two approximations dominated 1950s applications. The first is the instantaneous approximation: the exchanged Coulomb photon is taken to act at equal times, $G(\mathbf{k})=\gamma_4^e\gamma_4^P/\mathbf{k}^{2}$ with reduced kernel $G'(\mathbf{k})=1/\mathbf{k}^{2}$, which reduces the Bethe–Salpeter equation to Salpeter's three-dimensional equation and, for infinite proton mass, to the ordinary Dirac equation for an electron in an external Coulomb field; retardation (the dependence of the exchanged interaction on the relative time or energy) and transverse (magnetic, non-Coulomb) photons are then dropped. The second is the fixed-nucleus approximation: the proton is an infinitely heavy source, so the atom does not recoil. Under these assumptions the relativistic theory of light scattering by hydrogen had been computed by Franz (1935) for Rayleigh (elastic, coherent) scattering of hard radiation and by Brown & Woodward (1952) including first-order intermediate-state interactions, while Salpeter (1952) had computed the leading finite-proton-mass ('recoil') corrections — but for the hydrogen fine structure (energy levels), not for scattering amplitudes. R. J. Eden's series 'Quantum field theory of bound states' (Parts I–IV, 1952–53) organized this territory through the competition between the pole part of $K$ ('steady' or compound propagation of the atom as a whole) and the remainder ('unsteady' propagation through free intermediate states): Part I showed that the two-body propagator acquires a complex rest mass, the relativistic analogue of the complex energy of an unstable decaying state; Part II built a relativistic resonance theory from it; Part III (with G. Rickayzen) established when, for processes involving hydrogen, the pole part may be neglected and a scattering expansion about free propagators (the 'strong-interaction approximation') is valid, and computed the matrix elements for coherent scattering of light, photo-electric emission, and electron–hydrogen scattering with exchange in that framework, reproducing the fixed-nucleus results of Franz and Brown & Woodward in the limit $M\to\infty$ and identifying the conditions (internal momenta $|\mathbf{p}|,|\mathbf{k}|\ll m$) under which the Bethe–Salpeter internal wave function reduces to the Dirac wave function. Part IV (Eden 1953) then developed the relativistic theory of the unstable excited states. The Part III paper was explicit about what its calculation did not do; these deferrals define the open quantitative problem.

Problem Statement

Complete, with quantitative control, the relativistic two-particle quantum-electrodynamic description of electromagnetic processes involving the hydrogen atom — coherent (Rayleigh) scattering of light, photo-electric emission, and electron–hydrogen scattering with exchange — beyond the approximations of Eden & Rickayzen (1953), who used the instantaneous Coulomb kernel $G'(\mathbf{k})=1/\mathbf{k}^{2}$, neglected angular-momentum linear factors, evaluated the internal-energy integrals by retaining only the dominant residues, and took the proton mass $M\to\infty$ at the end. Throughout, $m$ and $M$ are the electron and proton masses, $M_n$ is the mass of the $n$-th hydrogen bound state, and $P$ is the total energy–momentum four-vector of the electron–proton system, in the metric convention where a state of mass $\mu$ has $P^{2}+\mu^{2}=0$. Beyond these approximations, the open core of the account comprises, for the stated amplitudes: (1) the behavior of the two-particle amplitudes in the near-pole region, where the total four-momentum approaches a bound-state pole ($P^{2}+M_n^{2}=0$), in which the stable (steady) description fails and a finite-width (complex rest mass) treatment must supply the resonance behavior of these processes, as developed by the unstable-state methods of Eden's Parts I and II; (2) the quantitative domain of validity of the scattering ('strong-interaction') approximation, i.e. the boundaries and marginal-case corrections of the conditions under which the bound-state (pole) part of the two-particle propagator may be neglected: that no virtual bound state — a pole of the propagator with mass above the electron–proton threshold — has $M_n>m+M$, and that $P^{2}+(m+M)^{2}<0$, which in the stated metric convention is equivalent to a total invariant mass $W>m+M$ above that threshold; (3) the retardation and transverse-photon corrections to the instantaneous-Coulomb amplitudes, including the omitted angular-momentum factors; (4) a systematic evaluation or error bound for the subleading residues neglected in the residue evaluation of the internal-energy integrals; and (5) the finite-proton-mass (recoil) corrections to the amplitudes, with a quantitative statement of how large $M$ must be, as a function of photon energy and momentum transfer, for the one-particle (infinite-nuclear-mass) approximation to hold. The problem concerns the stated amplitudes and corrections and is not tied to any single framework. Its five parts are jointly posed, and each must be settled by an explicit result accurate enough to fix its magnitude and the regime in which it matters: the near-pole resonance behavior of the amplitudes in finite-width form; the boundary of the strong-interaction approximation together with the marginal case at the threshold $W=m+M$; the explicit retardation and transverse-photon corrections to the stated amplitudes, with the order in $\alpha$ and the momentum/energy range for which each is given; a bound on the subleading residues stating which residues it covers; and the explicit recoil corrections together with the quantitative statement, over the stated photon-energy and momentum-transfer ranges, of how large $M$ must be for the one-particle approximation to hold.

References

1. R. J. Eden and G. Rickayzen, 'Quantum field theory of bound states III. Bound states in strong interaction', Proc. R. Soc. Lond. A 219, 109–119 (1953). doi:10.1098/rspa.1953.0134, https://doi.org/10.1098/rspa.1953.0134
2. E. E. Salpeter and H. A. Bethe, 'A Relativistic Equation for Bound-State Problems', Phys. Rev. 84, 1232–1242 (1951). doi:10.1103/PhysRev.84.1232, https://doi.org/10.1103/PhysRev.84.1232
3. R. J. Eden, 'Quantum field theory of bound states. I. Bound states in weak interaction', Proc. R. Soc. Lond. A 215, 133–146 (1952). doi:10.1098/rspa.1952.0202, https://doi.org/10.1098/rspa.1952.0202
4. R. J. Eden, 'Quantum field theory of bound states II. Relativistic theory of resonance reactions', Proc. R. Soc. Lond. A 217, 390–408 (1953). doi:10.1098/rspa.1953.0070, https://doi.org/10.1098/rspa.1953.0070
5. R. J. Eden, 'Quantum field theory of bound states IV. Relativistic theory of the excited states of hydrogen', Proc. R. Soc. Lond. A 219, 516–526 (1953). doi:10.1098/rspa.1953.0164, https://doi.org/10.1098/rspa.1953.0164
6. E. E. Salpeter, 'Mass Corrections to the Fine Structure of Hydrogen-Like Atoms', Phys. Rev. 87, 328–343 (1952). doi:10.1103/PhysRev.87.328, https://doi.org/10.1103/PhysRev.87.328
7. G. E. Brown and J. B. Woodward, 'Coherent Scattering of Gamma-Rays by Bound Electrons', Proc. Phys. Soc. A 65, 977–980 (1952). doi:10.1088/0370-1298/65/12/303, https://doi.org/10.1088/0370-1298/65/12/303
8. W. Franz, 'Rayleighsche Streuung harter Strahlung an schweren Atomen', Z. Phys. 98, 314–320 (1935). doi:10.1007/BF01331074, https://doi.org/10.1007/BF01331074
9. A. Klein and C. Zemach, 'Many-Body Problem in Quantum Field Theory', Phys. Rev. 108, 126–138 (1957). doi:10.1103/PhysRev.108.126, https://doi.org/10.1103/PhysRev.108.126
10. H. Kita and M. Wakano, 'On the Bethe-Salpeter Equation and the Excited States of the Hydrogen Atom', Prog. Theor. Phys. 17, 63–79 (1957). doi:10.1143/ptp.17.63, https://doi.org/10.1143/ptp.17.63
11. M. Gavrila, 'Elastic Scattering of Photons by a Hydrogen Atom', Phys. Rev. 163, 147–155 (1967). doi:10.1103/PhysRev.163.147, https://doi.org/10.1103/PhysRev.163.147
12. M. Gavrila and A. Costescu, 'Retardation in the Elastic Scattering of Photons by Atomic Hydrogen', Phys. Rev. A 2, 1752–1758 (1970). doi:10.1103/PhysRevA.2.1752, https://doi.org/10.1103/PhysRevA.2.1752
13. W. E. Caswell and G. P. Lepage, 'Effective lagrangians for bound state problems in QED, QCD, and other field theories', Phys. Lett. B 167, 437–442 (1986). doi:10.1016/0370-2693(86)91297-9, https://doi.org/10.1016/0370-2693(86)91297-9
14. P. Labelle, 'Effective field theories for QED bound states: Extending nonrelativistic QED to study retardation effects', Phys. Rev. D 58, 093013 (1998). doi:10.1103/PhysRevD.58.093013, arXiv:hep-ph/9608491, https://arxiv.org/abs/hep-ph/9608491
15. V. A. Yerokhin and V. M. Shabaev, 'Nuclear Recoil Effect in the Lamb Shift of Light Hydrogenlike Atoms', Phys. Rev. Lett. 115, 233002 (2015). doi:10.1103/PhysRevLett.115.233002, https://doi.org/10.1103/PhysRevLett.115.233002
16. V. A. Yerokhin and V. M. Shabaev, 'Nuclear recoil corrections to the Lamb shift of hydrogen and light hydrogenlike ions', Phys. Rev. A 93, 062514 (2016). doi:10.1103/PhysRevA.93.062514, https://doi.org/10.1103/PhysRevA.93.062514
17. U. D. Jentschura, 'Relativistic reduced-mass and recoil corrections to vacuum polarization in muonic hydrogen, muonic deuterium, and muonic helium ions', Phys. Rev. A 84, 012505 (2011). doi:10.1103/PhysRevA.84.012505, https://doi.org/10.1103/PhysRevA.84.012505
18. V. Florescu, M. Marinescu and R. H. Pratt, 'Relativistic effects on low-frequency Rayleigh scattering including spin flip for hydrogenic atoms', Phys. Rev. A 42, 3844–3851 (1990). doi:10.1103/PhysRevA.42.3844, https://doi.org/10.1103/PhysRevA.42.3844
19. A. A. Peshkov, A. V. Volotka, A. Surzhykov and S. Fritzsche, 'Rayleigh scattering of twisted light by hydrogenlike ions', Phys. Rev. A 97, 023802 (2018). doi:10.1103/PhysRevA.97.023802, arXiv:1805.08274, https://arxiv.org/abs/1805.08274
20. J. Sommerfeldt, N. M. Mayer, A. Maiorova, W. Middents, S. Fritzsche, T. Stöhlker and A. Surzhykov, 'Bound-state Compton scattering of linearly polarized photons', arXiv:2604.14748 (2026). https://arxiv.org/abs/2604.14748