The 1976 paper "Nonrelativistic dyon-dyon scattering" (Schwinger, Milton, Tsai, DeRaad, and Clark, Ann. Phys. 101, 451) solved the classical and quantum two-body scattering of dyons - particles carrying both electric and magnetic charge - in the nonrelativistic regime. Classically, the Lorentz force on an electric charge moving in a magnetic Coulomb field confines the relative trajectory to a cone, and the cross section exhibits rainbows and glories; quantum mechanically, the amplitude is a modified partial-wave series with non-integer effective angular momenta and Coulomb phase shifts, which the authors summed only numerically via Cesaro acceleration. The paper itself identified what remains open: (i) the analytic summation of these modified partial-wave expansions into closed-form amplitudes; (ii) the correct theory for gyromagnetic ratios γ\gamma far from 1 (in particular large γ\gamma), where the lower spin-1/2 angular eigenvalue falls below −1/4-1/4, the effective orbital angular momenta become complex, and an indefinite metric appears; (iii) a complete relativistic quantum scattering theory of magnetically charged particles beyond the eikonal (high-energy, distant-collision) approximation; and (iv) quantitative experimental criteria - e.g. δ\delta-ray distributions - discriminating dyons from monopoles. Only partial progress exists: exact analyticity results for the charge-monopole limit (Balachandran et al. 1978), the relativistic Dirac solution only for the physical γ=1\gamma=1 case (Kazama-Yang-Goldhaber 1977), string-independent relativistic cross sections only at zeroth-order eikonal (Urrutia 1978; Gamberg-Milton 2000), and modern on-shell pairwise-helicity S-matrix structures (Csáki et al. 2021) without strong-coupling two-body amplitudes. The δ\delta-ray discrimination program is dormant. The problem is open.

Background

A magnetic monopole is a hypothetical particle carrying magnetic charge gg; a dyon carries both electric charge ee and magnetic charge gg. Classical electrodynamics admits such objects symmetrically, but quantum mechanics restricts their charges severely: the electromagnetic field of two interacting charges carries angular momentum, and its quantization forces the combination m′=−(e1g2−e2g1)/(ℏc)m' = -(e_1 g_2 - e_2 g_1)/(\hbar c) to be an integer or half-integer (the Dirac-Schwinger-Zwanziger quantization condition). The source paper (Schwinger et al. 1976; see references) treats the nonrelativistic two-body scattering of dyons with momenta small compared with mcmc.

Classically, the magnetic Coulomb field of one dyon exerts a velocity-dependent Lorentz force on the other; the relative trajectory lies on the surface of a cone whose axis co-rotates with the system (the Poincaré cone), and the cross section exhibits semiclassical rainbows (angular maxima where two classical paths interfere constructively, in the sense of Ford-Wheeler semiclassical scattering analysis) and glories (enhancement near 180 degrees). Quantum mechanically, the conserved total angular momentum is J=L+m′r^\mathbf{J} = \mathbf{L} + m'\hat{\mathbf{r}}, where L\mathbf{L} is the mechanical orbital angular momentum, which fails to commute with the kinetic momentum when magnetic charge is present. Each partial wave of total angular momentum jj therefore carries an effective orbital angular momentum LL defined by L+1/2=(j+1/2)2−m′2L + 1/2 = \sqrt{(j+1/2)^2 - m'^2}, generally non-integer. At center-of-mass scattering angle θˉ\bar\theta the spinless amplitude takes the modified partial-wave form

2ikf(θˉ)=∑j≥∣m′∣(2j+1) Um′m′(j)(π−θˉ) e−i(πL−2δL),2ik f(\bar\theta) = \sum_{j \geq |m'|} (2j+1)\, U^{(j)}_{m'm'}(\pi - \bar\theta)\, e^{-i(\pi L - 2\delta_L)},

where Um′m′(j)U^{(j)}_{m'm'} is a Wigner rotation-matrix element, kk the relative momentum, and δL=arg⁡Γ(L+1+iη)\delta_L = \arg\Gamma(L+1+i\eta) the Coulomb phase shift, with η\eta the Coulomb parameter (the ratio of the electric Coulomb coupling to the relative momentum; η=0\eta = 0 for pure charge-monopole scattering). For spin-1/2 dyons with gyromagnetic ratio γ\gamma (the magnetic moment is μ=γeℏ/(2mc)\mu = \gamma e\hbar/(2mc) per unit spin), the eigenvalues of the angular operator are λ±=(J+1/2)2−m′2±(J+1/2)X\lambda_\pm = (J+1/2)^2 - m'^2 \pm (J+1/2)X with X=1+ν2(γ2−2γ)X = \sqrt{1 + \nu^2(\gamma^2 - 2\gamma)} and ν=m′/(J+1/2)\nu = m'/(J+1/2), and the effective orbital momenta satisfy λ±=L±(L±+1)\lambda_\pm = L_\pm(L_\pm + 1).

The series converges only conditionally (its terms decay like 1/j1/j), so the authors summed it numerically by Cesaro acceleration - replacing partial sums by their running averages σN=(1/N)∑n≤Nsn\sigma_N = (1/N)\sum_{n \leq N} s_n, which can converge when the original series does - reaching about 5% angular accuracy (about 1% at backward angles). Closed forms were obtained only in limits: the small-angle law dσ/dΩ≃(m′/2k)2sin⁡−4(θˉ/2)d\sigma/d\Omega \simeq (m'/2k)^2 \sin^{-4}(\bar\theta/2), and a semiclassical backward sum valid for η=0\eta = 0. For spin-1/2 dyons with γ\gamma far enough from 1 - in particular for large γ\gamma - the lower eigenvalue λ−\lambda_- of the pair can fall below −1/4-1/4, making the corresponding effective orbital angular momentum L−L_- complex; the quantum theory must then be written in a Hilbert space with an indefinite metric (a space whose inner product is not positive definite), which the authors regard as unphysical and expect to be cured only by a fully relativistic treatment.

The relativistic obstacle is well known: a local Lagrangian containing both electric and magnetic currents requires two potentials tied by a nonlocal constraint (Zwanziger's formalism) or a Dirac string - an unphysical line of singularities along which the vector potential is defined - whose choice must decouple from all observables. Perturbation theory is useless for magnetic charge because the expansion parameter is αg=g2/(4πℏc)≈34\alpha_g = g^2/(4\pi\hbar c) \approx 34 for the smallest magnetic charge, so only nonperturbative resummations in the eikonal approximation (the high-energy, small-angle limit in which the phase accumulates along nearly straight-line paths, a "distant-collision" approximation) have been possible. The paper closes by proposing δ\delta rays - energetic secondary electrons a fast projectile knocks out of detector material - as a way to discriminate a dyon (electric plus magnetic charge) from a pure monopole, motivated by the then-recent monopole-candidate event of Price et al. (1975).

Problem Statement

Complete the quantitative scattering theory of two particles carrying both electric and magnetic charge, as formulated and left open by Schwinger, Milton, Tsai, DeRaad, and Clark (Ann. Phys. 101, 451 (1976)). The problem has four linked components serving one objective - a complete, analytic where possible and relativistically well-founded quantum theory of dyon-dyon scattering with observable consequences - at the physically relevant parameter values (quantized m′=−(e1g2−e2g1)/(ℏc)m' = -(e_1 g_2 - e_2 g_1)/(\hbar c); the experimentally accessible regime is the subset ∣m′∣∼1|m'| \sim 1, ∣η∣≪1|\eta| \ll 1, γ≈1\gamma \approx 1, while components (1)-(2) require theoretical coverage for general and large γ\gamma):

(1) Closed-form amplitude. Sum the modified partial-wave series for the nonrelativistic spinless and spin-1/2 dyon-dyon amplitudes, 2ikf(θˉ)=∑j≥∣m′∣(2j+1)Um′m′(j)(π−θˉ)exp⁡[−i(πL−2δL)]2ik f(\bar\theta) = \sum_{j \geq |m'|} (2j+1) U^{(j)}_{m'm'}(\pi - \bar\theta) \exp[-i(\pi L - 2\delta_L)] with L+1/2=(j+1/2)2−m′2L + 1/2 = \sqrt{(j+1/2)^2 - m'^2} and δL=arg⁡Γ(L+1+iη)\delta_L = \arg\Gamma(L+1+i\eta), into closed analytic expressions (or an equivalent controlled analytic representation with proven convergence) valid for general quantized m′m', Coulomb parameter η\eta, gyromagnetic ratio γ\gamma, and all center-of-mass angles - replacing the Cesaro numerical summation - and thereby explain analytically the observed rainbow and glory angular structure.

(2) Gyromagnetic-ratio pathology. Determine the correct scattering theory for spin-1/2 dyons with gyromagnetic ratio γ\gamma outside the narrow window of width of order 1/(4∣m′∣)1/(4|m'|) around γ=1\gamma = 1 (in particular for large γ\gamma): there the lower angular eigenvalue λ−\lambda_- of the pair λ±=(J+1/2)2−m′2±(J+1/2)1+ν2(γ2−2γ)\lambda_\pm = (J+1/2)^2 - m'^2 \pm (J+1/2)\sqrt{1+\nu^2(\gamma^2-2\gamma)}, ν=m′/(J+1/2)\nu = m'/(J+1/2), falls below −1/4-1/4, so the effective orbital angular momentum L−L_- defined by λ−=L−(L−+1)\lambda_- = L_-(L_-+1) becomes complex (the eigenvalues λ±\lambda_\pm themselves remain real for real γ\gamma) and the nonrelativistic quantum theory must be written with an indefinite metric. Establish whether this complex-angular-momentum/indefinite-metric sector is an artifact cured by relativity, requires a deeper modification, or is simply unreachable physically, and give the resulting amplitudes or cross sections.

(3) Relativistic theory. Construct the complete relativistic quantum scattering theory of particles carrying magnetic charge - resolving the Dirac-string/gauge dependence of intermediate quantities and the quantization conditions, and going beyond the distant-collision (eikonal) approximation - yielding more accurate amplitudes and reducing, in the appropriate limit, to the nonrelativistic results of the source paper (including the fate of the sector identified in (2)).

(4) Experimental discrimination. Map the computed cross sections to measurable criteria that distinguish a dyon from a pure monopole (and from purely electric particles) in the accessible regime - for example, the yield, energy, and angular distributions of δ\delta rays (energetic secondary electrons) ejected in detector material - quantitatively enough that a search could quote expected signatures.

For (1), the general dyon-dyon amplitude is sought in closed analytic form, or in an equivalent controlled analytic representation with proven convergence and a bounded truncation error, valid for general quantized m′m', Coulomb parameter η\eta, and gyromagnetic ratio γ\gamma at all center-of-mass angles, with the rainbow/glory caustic structure derived analytically from that representation. For (2), the complex-L−L_-/indefinite-metric sector for γ\gamma outside the window around γ=1\gamma = 1 (in particular large γ\gamma) is to be settled as an artifact cured by relativity, as requiring a deeper modification, or as physically unreachable, with the corresponding amplitudes or cross sections. For (3), the relativistic construction is to be internally consistent (Lorentz invariant and free of residual Dirac-string/gauge dependence in observables), to go beyond the distant-collision (eikonal) approximation, and to reduce in the appropriate limit to the nonrelativistic results, including the sector of (2). For (4), the experimental discriminants are to be quantified as concrete measurable criteria (e.g. yield, energy, and angular distributions of δ\delta rays) separating a dyon from a pure monopole and from purely electric particles.

References

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