Completing Glauber's exponential-coupling nucleon–meson model: relativistic two-nucleon bound states, angular distributions of multiple-meson production, and the weak-coupling expansion at the physical mass ratio

Roy J. Glauber's 1951 paper 'Some Notes on Multiple-Boson Processes' developed an operator calculus for nonlinear functions of quantized boson fields, exact wherever the field's self-commutator is a c-number; it used the calculus to rederive the Bloch–Nordsieck infrared result for radiation by classical currents and applied it to nucleon–meson theories whose interaction Hamiltonian depends exponentially on the meson field. The paper acknowledged six unresolved limitations. Two were settled by subsequent literature: fermionic analogues of the operator calculus now exist (Grassmann-variable functional methods and fermionic coherent states), and the multiple-soft-photon problem was solved with recoil included to all orders (Yennie–Frautschi–Suura; Grammer–Yennie), while the pseudoscalar–pseudovector correspondence was clarified by the field-redefinition equivalence theorem. The precise surviving open core concerns the exponential-coupling model itself: derive its two-nucleon interaction including recoil and retardation and decide whether it admits bound states; compute the angular distribution of multiply produced mesons with the interaction kernel's momentum dependence retained, including interference; and establish the domain of validity of the weak-coupling expansion at the physical meson–nucleon mass ratio. These questions are unanswered in the literature, but the model class was superseded by QCD-based pion physics, so the problem's significance is historical and methodological rather than phenomenological.

Background

Quantum field theory describes matter — here nucleons, represented by a spinor field $\psi(x)$ whose quanta obey Fermi–Dirac statistics — interacting with quantized boson fields (fields such as the photon field or a meson field $\varphi(x)$ , whose quanta obey Bose–Einstein statistics and can be emitted and absorbed in arbitrary numbers). In 1951 Roy J. Glauber observed that existing technique handled nonlinear functions of boson fields poorly, and that a simple structural fact permits exact manipulation: for a boson field, the commutator of the field with itself is a c-number (an ordinary number-valued function rather than an operator), $[\varphi(x),\varphi(x')]=i\Delta(x-x')$ in units $\hbar=c=1$ . Operators whose commutator is a c-number satisfy closed exponential identities, for example $e^a e^b = e^{a+b+[a,b]}$ when $[a,b]$ commutes with both $a$ and $b$ , and vacuum expectation values (expectation values in the state containing no real quanta, written $\langle\cdots\rangle_0$ ) of exponentials reduce to the exponential of a self-contraction. These identities separate, at the operator level, the creation of real quanta from virtual (vacuum-fluctuation) corrections, without expanding in powers of a coupling constant. The method does not apply to spinor fields, whose self-commutators are not c-numbers. Glauber illustrated the method on two problems. First, the radiation of photons by a prescribed classical current $j_\mu(x)$ (a source treated as a known function rather than an operator): the scattering operator $S=\exp(i\int j_\mu A^\mu\,dx)$ gives, without any power-series expansion, a Poisson distribution $w_n = W^n e^{-W}/n!$ for emitting $n$ photons, with mean $W$ equal to the classically radiated energy, so that successive emissions are statistically independent. This compactly rederives the result of Bloch and Nordsieck for the infrared catastrophe — the infinite accumulation of undetectably soft photons accompanying any sudden acceleration of charge, which makes naive finite-order perturbation theory diverge. The Bloch–Nordsieck approximations, neglecting the recoil of the accelerated charge and the creation of electron–positron pairs, are justified only for very soft photons; Glauber's derivation shares them, and he notes that the statistical independence is destroyed when recoil is not negligible. The second problem belonged to the nuclear physics of the time: the nuclear force and the multiple production of mesons (pion-like particles) in collisions. Instead of the usual interaction Hamiltonians linear in the meson field, Glauber studied exponential coupling,

H(x)=\lambda\,f(x)\,e^{i\beta\varphi(x)},

where $f(x)$ is the nucleon density built from the spinor field ( $\bar\psi\psi$ in the scalar case; $\bar\psi i\gamma_5\psi$ in the pseudoscalar case, with the same exponential $e^{i\beta\varphi}$ in both cases), $\lambda$ is the overall coupling strength, and $\beta$ fixes the mean number of mesons taking part in an elementary event. In the pseudoscalar case this exponential form is exactly what a contact transformation (a canonical redefinition of the state vector) produces from the standard pseudovector coupling (a coupling of the nucleon to the gradient of the meson field) — a correspondence shown by Dyson — so the exponential theory packages many higher-order effects of the familiar theory. However, treating $\beta$ (equivalently the derivative-coupling parameter) as small is controlled only when the meson mass $\mu$ is much larger than the nucleon mass $M$ ( $\mu\gg M$ ), the opposite of the physical pion–nucleon situation. Working to lowest order in $\lambda$ but to all orders in $\beta$ , Glauber computed the nucleon self-energy (its divergent part absorbed by renormalization, a redefinition of $\lambda$ ), the multiple production of mesons in meson–nucleon collisions (spherically symmetric emission with a computable most-probable multiplicity growing with collision energy), and the two-nucleon interaction. In the static limit — neglecting both the recoil of the nucleons and the retardation (the propagation delay of the meson field between the two nucleons) — the scalar potential emerges as a series of increasingly singular inverse powers of the separation $r$ near the origin, far too singular to support a two-nucleon bound state; in the pseudoscalar case Glauber could decide nothing rigorously but judged bound states very unlikely. For nucleon–nucleon collisions he employed the common approximation of Lewis, Oppenheimer, and Wouthuysen, which replaces the interaction kernel $V(x-x')$ by a four-dimensional delta function and thereby discards its momentum dependence; he noted that this greatly exaggerates the interference between mesons emitted by the two nucleons, whereas the experimentally observed angular distributions of produced mesons tend to cluster around the collision axis, and that understanding such clustering requires interactions less singular than delta functions.

Problem Statement

The question is the surviving core of the limitations Glauber's 1951 paper explicitly acknowledged, after later literature settled the spinor-field extension and the soft-photon recoil problem. Within the exponential-coupling nucleon–meson model — the Hamiltonian $H(x)=\lambda\,f(x)\,e^{i\beta\varphi(x)}$ of the source paper, in both its scalar and pseudoscalar forms, treated to lowest order in $\lambda$ but non-perturbatively in $\beta$ , with the coupling-constant renormalization defined there — the following are open: (1) Derive the two-nucleon interaction generated by this Hamiltonian including the effects the source neglected — nucleon recoil, retardation, and relativistic kinematics — and determine whether that interaction admits two-nucleon bound states: in the scalar case, whose static potential is already known to be too singular at the origin to permit them, whether retaining recoil and retardation overturns that conclusion; and in the pseudoscalar case, which the source left undecided. (2) Retaining the momentum dependence of the interaction kernel $V(x-x')$ (rather than the four-dimensional delta-function approximation), compute the angular distribution of multiply produced mesons in nucleon–nucleon collisions, including the interference between amplitudes in which different numbers of mesons are emitted by each nucleon, and determine whether the model can reproduce the experimentally observed clustering of produced mesons around the collision axis, reporting the computed meson polar-angle spectrum about the collision axis and deciding whether it is forward-peaked — emission concentrated at small polar angles — as observed in the multiple-meson production reported by the source. (3) Establish the domain of validity, for physical meson–nucleon mass ratios $\mu<M$ , of the weak- $\beta$ (derivative-coupling) expansion, whose smallness the source's equivalence to pseudovector coupling controls only in the opposite regime $\mu\gg M$ . Questions (1)–(3) are independently posed.

References

R. J. Glauber, 'Some Notes on Multiple-Boson Processes', Physical Review 84(3), 395–400 (1951). doi:10.1103/PhysRev.84.395; https://doi.org/10.1103/PhysRev.84.395
F. Bloch and A. Nordsieck, 'Note on the Radiation Field of the Electron', Physical Review 52, 54–59 (1937). doi:10.1103/PhysRev.52.54; https://doi.org/10.1103/PhysRev.52.54
F. J. Dyson, 'The Interactions of Nucleons with Meson Fields', Physical Review 73, 929–930 (1948). doi:10.1103/PhysRev.73.929.2; https://doi.org/10.1103/PhysRev.73.929.2
H. W. Lewis, J. R. Oppenheimer, and S. A. Wouthuysen, 'The Multiple Production of Mesons', Physical Review 73, 127–140 (1948). doi:10.1103/PhysRev.73.127; https://doi.org/10.1103/PhysRev.73.127
D. R. Yennie, S. C. Frautschi, and H. Suura, 'The infrared divergence phenomena and high-energy processes', Annals of Physics 13(3), 379–452 (1961). doi:10.1016/0003-4916(61)90151-8; https://doi.org/10.1016/0003-4916(61)90151-8
G. Grammer, Jr. and D. R. Yennie, 'Improved Treatment for the Infrared-Divergence Problem in Quantum Electrodynamics', Physical Review D 8, 4332–4344 (1973). doi:10.1103/PhysRevD.8.4332; https://doi.org/10.1103/PhysRevD.8.4332
K. E. Cahill and R. J. Glauber, 'Density operators for fermions', Physical Review A 59, 1538–1555 (1999). doi:10.1103/PhysRevA.59.1538; https://doi.org/10.1103/PhysRevA.59.1538
S. Kamefuchi, L. O'Raifeartaigh, and A. Salam, 'Change of variables and equivalence theorems in quantum field theories', Nuclear Physics 28, 529–549 (1961). doi:10.1016/0029-5582(61)90056-6; https://doi.org/10.1016/0029-5582(61)90056-6
Z. Chyliński, 'Bremsstrahlung mechanism of the multiple production process in nucleon-nucleon collisions', Nuclear Physics 44, 58–77 (1963). doi:10.1016/0029-5582(63)90006-3; https://doi.org/10.1016/0029-5582(63)90006-3
A. Białas and T. Ruijgrok, 'On a bremsstrahlung theory for the multiple production of mesons', Il Nuovo Cimento 39, 1061–1081 (1965). doi:10.1007/BF02734541; https://doi.org/10.1007/BF02734541
R. J. Glauber, 'Coherent and incoherent states of the radiation field', Physical Review 131, 2766–2788 (1963). doi:10.1103/PhysRev.131.2766; https://doi.org/10.1103/PhysRev.131.2766
L. I. Schiff, 'Nonlinear Meson Theory of Nuclear Forces. I. Neutral Scalar Mesons with Point-Contact Repulsion', Physical Review 84, 1–9 (1951). doi:10.1103/PhysRev.84.1; https://doi.org/10.1103/PhysRev.84.1