Controlled irreducible kernels and an intrinsic bound-state criterion for the covariant two-body equation: the open core of the 1951 Salpeter–Bethe program

In 1951 Salpeter and Bethe derived the covariant two-body bound-state equation (the Bethe–Salpeter equation) and listed what it could not yet do: no closed form for the full irreducible interaction kernel, so the method was inapplicable when the dimensionless coupling $g^2$ is not small (e.g. the deuteron with pseudo-scalar mesons); no intrinsic criterion for deciding which solutions of the derived four-dimensional differential-integral equation are physical bound states (the proposed adiabatic criterion being ambiguous at zero binding energy); higher-order irreducible graphs (crossed, Lamb-shift-type, self-energy, vacuum-polarization) left uncalculated, with cancellations beyond $O((\mu/M)^2)$ unverified; no convergence-controlled scheme for the full four-dimensional equation; and the hydrogen fine/hyperfine and Fermi–Yang applications deferred. Subsequent work completed the hydrogen application (Salpeter 1952; modern bound-state quantum electrodynamics (QED)), produced practical four-dimensional solvers, and superseded the pseudo-scalar deuteron and Fermi–Yang programs by chiral effective field theory, lattice QCD and the quark model. The surviving open core, active in 2015–2024 research, is twofold: (i) a systematically improvable, quantitatively controlled truncation of the irreducible kernel beyond the ladder/rainbow-ladder term for strongly coupled bound states, and (ii) an intrinsic criterion selecting the physical solutions, including the status of the anomalous “abnormal” solutions. The problem remains open.

Background

By 1951 the techniques of relativistic quantum field theory — Dyson's and Feynman's diagrammatic expansion of scattering amplitudes — were mature, but bound states had no comparably systematic treatment. Salpeter and Bethe (Phys. Rev. 84, 1232 (1951)), with the rigorous companion derivation of Gell-Mann and Low (Phys. Rev. 84, 350 (1951)), filled this gap for two fermions interacting through a quantized field. They described the bound state by an amplitude depending on a separate time for each constituent, or equivalently on the total energy-momentum $P$ and a relative four-momentum $p$ whose time component $p_0$ (the “relative energy”, conjugate to the relative time between the particles) is an unavoidable extra variable with no analogue in the Schrödinger equation. The amplitude obeys an eigenvalue integral equation of the form $\Phi = S_1 S_2\, K\, \Phi$, where $S_{1,2}$ are the exact propagators of the two constituents and the kernel $K$ is the sum of all Feynman graphs that are “irreducible”, i.e. cannot be split into two simpler graphs by a cut crossing both constituent lines only once. Graphs of this kind include, beyond the single-exchange graph, the crossed exchange (two quanta passing each other), vacuum-polarization insertions (virtual particle–antiparticle loops screening the exchanged charge), Lamb-shift-type terms (radiative corrections acting on one constituent), and self-energy/mass-renormalization insertions (a constituent emitting and reabsorbing a quantum, absorbed into its physical mass). No closed form for the full $K$ is known, and each order of its perturbative expansion is meaningful only when the dimensionless coupling $g^2$ (the analogue of the fine-structure constant) is small.

Replacing $K$ by its lowest term resums the infinitely repeated single exchanges (the “ladder” graphs) and in the nonrelativistic limit reduces exactly to the Schrödinger equation with a Yukawa potential $-g^2 e^{-\mu r}/4\pi r$, where $\mu$ is the exchanged quantum's mass and $M$ the constituent mass. Salpeter and Bethe applied this ladder equation to the deuteron (the loosely bound proton–neutron system) with scalar meson exchange, computed the relativistic retardation correction to the coupling needed for the observed binding energy, and argued that for neutral exchange the two-quantum irreducible graphs cancel that retardation correction to relative order $(\mu/M)^2$ — an argument made through a field transformation rather than explicit calculation. For charged exchange the correction is of order $\mu/M$ (about 15 percent).

The paper then derived a more practical four-dimensional differential-integral equation (its Eq. (19)) by applying two Dirac-type operators to the integral equation. Solutions of this differential equation are only necessary, not sufficient: some solve Eq. (19) but not the original integral equation with its boundary condition. The authors wanted an intrinsic criterion of “good behavior” selecting the physical solutions, and observed that their best candidate — following an adiabatic (infinitely slow) switching of the coupling to zero, where physical bound states must reduce to free-particle pairs — is ambiguous precisely at zero binding energy.

The subsequent history defined the modern landscape. Wick (Phys. Rev. 96, 1124 (1954)) rotated the relative energy to Euclidean values (the “Wick rotation”) and, with Cutkosky (Phys. Rev. 96, 1135 (1954)), solved the massless-exchange ladder equation exactly (the Wick–Cutkosky model: two massive scalar constituents exchanging a massless scalar). Their spectra split into “normal” states, which tend to the Coulomb spectrum as the binding vanishes, and “abnormal” states, which have arbitrarily small binding energies yet no nonrelativistic counterpart at all. Nakanishi later established the normalization condition for Bethe–Salpeter amplitudes and gave the standard classification and review of normal and abnormal solutions (Phys. Rev. 138, B1182 (1965); Prog. Theor. Phys. Suppl. 43, 1 (1969)).

In parallel, the strong-coupling goal migrated to quantum chromodynamics (QCD), where hadrons are strongly bound quark–antiquark or three-quark systems. Two first-principles routes now dominate: lattice QCD, which computes the hadron spectrum ab initio by Euclidean Monte Carlo simulation (Dürr et al., Science 322, 1224 (2008)), and the functional method, which couples the Bethe–Salpeter equation to propagators obtained from the Dyson–Schwinger equations (the infinite tower of exact functional equations relating a field theory's Green functions, solvable only with truncations); its workhorse is the symmetry-preserving “rainbow-ladder” truncation, the direct analogue of the 1951 ladder kernel (Maris and Roberts, Int. J. Mod. Phys. E 12, 297 (2003); Eichmann et al., Prog. Part. Nucl. Phys. 91, 1 (2016)). The original hadronic applications were meanwhile transformed beyond recognition: nuclear forces are derived systematically from chiral effective field theory, a low-energy expansion of QCD organized by a systematic power counting in momenta and masses (Machleidt and Entem, Phys. Rep. 503, 1 (2011)), rather than from pseudo-scalar meson theory, and the Fermi–Yang model — the 1949 conjecture of Fermi and Yang (Phys. Rev. 76, 1739 (1949)) that the pion is a nucleon–antinucleon bound state — was superseded by the quark model. The hydrogen fine and hyperfine program grew into precision bound-state QED, completed in its systematic form by Salpeter's 1952 recoil corrections and reviewed in final form by Eides, Grotch and Shelyuto (Phys. Rep. 342, 63 (2001)).

What has survived to the present as genuinely open, with active work in 2015–2024, is exactly the 1951 paper's central unfinished business: controlling the kernel beyond the ladder term when no small coupling is available, and deciding intrinsically which solutions of the four-dimensional equation are physical. Crossed-ladder contributions in relativistic scalar models are known to be “very large and attractive” (Carbonell and Karmanov, Eur. Phys. J. A 27, 11 (2006)), beyond-rainbow-ladder graphs can qualitatively change spectra (Mishima, Jinno and Kitahara, Phys. Rev. D 91, 076011 (2015)), and the abnormal solutions are still being characterized rather than decided upon (Carbonell, Karmanov and Sazdjian, Eur. Phys. J. C 81, 50 (2021); Carbonell, Karmanov, Kupriyanova and Sazdjian, Few Body Syst. 65, 56 (2024); Karmanov, Eur. Phys. J. C 84, 58 (2024)).

Problem Statement

The problem is the unfinished core of the program announced in Salpeter and Bethe (1951): making the covariant two-body bound-state equation quantitatively reliable and physically interpretable when the dimensionless coupling $g^2$ is not small. Two sub-questions, serving this single objective, remain open.

(1) Kernel control. For the Bethe–Salpeter equation $\Phi = S_1 S_2\, K\, \Phi$ (with exact constituent propagators $S_{1,2}$ and irreducible kernel $K$), provide a systematically improvable truncation of $K$ that goes beyond the ladder (single-exchange) term — including at least the crossed exchange and the one-loop vertex, self-energy and vacuum-polarization insertions (graphs 2A–2D of the 1951 paper) — together with a quantitative error or convergence statement for bound-state observables (masses and amplitudes) that remains meaningful at strong coupling. The construction must preserve the exact symmetries and Ward-type identities (exact relations among Green functions imposed by the underlying symmetries, such as the gauge relation between a constituent propagator and its interaction vertex) of the underlying theory, and must verify explicitly, beyond the transformation-based argument of 1951, the cancellation structure of the two-quantum graphs against retardation effects for neutral exchange to $O((\mu/M)^2)$ and determine the corrections of higher order.

(2) Physical-solution criterion. Provide an intrinsic criterion — a condition on the Bethe–Salpeter amplitude alone, without reference to the inhomogeneous integral equation or its boundary conditions — that determines which solutions of the four-dimensional differential-integral equation (Eq. (19) of the 1951 paper) correspond to physical bound states. The criterion must resolve the ambiguity of the adiabatic argument at zero binding energy, and must state its verdict for the abnormal solutions of the Wick–Cutkosky model, for both massless and massive exchange.

Both sub-questions address the covariant two-body bound-state equation at non-small coupling: the ladder-only kernel and instantaneous three-dimensional reductions, which eliminate the relative-energy variable, do not realize this target.

References

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