Completing the strong-coupling theory of the charged scalar mesotron field: intermediate coupling, recoil and gauge invariance, spin, and the unfinished canonical-transformation treatment (Schwinger, 1970)

Julian Schwinger's essay 'On the Charged Scalar Mesotron Field' (1970 Wentzel Festschrift; reprinted in A Quantum Legacy, 2000) gives a strong-coupling treatment of a charged scalar (spin-0, even-parity) mesotron field bound to an infinitely massive, spinless nucleon source: a point-source isobar spectrum $E_q = (g^2/\hbar c)^{-1}\mu c^2 (q-\tfrac{1}{2})^2$, elastic and charge-exchange cross sections of $\pi(\hbar/\mu c)^2(\mu c^2/E)^2$ each, and a bound charge density proportional to the squared Yukawa potential, plus a classical interpolation to arbitrary coupling in which charge quantization is imposed by hand. By its own account the essay leaves open: a quantum treatment at the physically relevant intermediate coupling; completion of the canonical-transformation derivation, whose steps are asserted but not performed; extension beyond the static source to a relativistic, gauge-invariant coupling with recoil; and inclusion of spin (the pseudoscalar theory of a sequel that was never published). None of these items has been addressed for the charged scalar model in the later literature.

Background

In the late 1930s Yukawa proposed that nuclear forces arise from the exchange of a heavy quantum, at first called the mesotron (the term then used for the meson, before the pion and the muon were distinguished in 1947). The quantum field theory of such particles was written down in analogy with quantum electrodynamics (QED). Two features made it far harder than QED. First, the coupling strength $g^2/(\hbar c)$ between nucleons (protons and neutrons) and the mesotron field is of order 10, so perturbation theory in powers of $g$ fails. Second, the mesotron is electrically charged, so its emission or absorption converts a proton into a neutron or vice versa (charge exchange), a discontinuous change of charge state with no classical analogue. In this theory the nucleon is described by isotopic spin operators $\tau$, whose two eigenstates represent the proton and the neutron.

To handle the large coupling, Gregor Wentzel - to whom the commemorative volume (a Festschrift: a book of essays honoring a scholar) containing this essay is dedicated - introduced around 1940 the strong-coupling expansion: instead of expanding in $g$, one expands in $1/g$, treating the coupling energy as the dominant term of the Hamiltonian. In this limit the mesotron cloud bound to the nucleon contains many quanta, the bound field becomes essentially classical, and the charge states of the nucleon appear as quantized excitations of a slowly rotating collective coordinate: the isobaric states, an infinite tower of states of charge $q = \dots, -1, 0, 1, 2, \dots$ with the proton and neutron as lowest members, which Bhabha and Heitler had postulated on phenomenological grounds. Pauli and Dancoff (1942) developed this program for the pseudoscalar coupling (the case in which the interaction involves the nucleon spin and the meson's spatial gradient, appropriate to a pion of odd parity), deriving the strong-coupling isobar spectrum. The static-source strong-coupling program remained active through the 1950s and 1960s: scattering theory (Feldman 1952; Wentzel 1952; Kaufman 1953), recoil corrections (Wentzel 1953), nucleon form factors (Morita 1963), and generalization to the pseudoscalar-meson octet - the group of eight mesons classified by the approximate flavor symmetry SU(3) of the strong interaction (Goebel 1966; Sakita 1968). Tomonaga-type intermediate-coupling treatments, which interpolate between perturbation theory and strong coupling by truncating the meson-cloud configuration space, were developed for the pseudoscalar theory in 1951-1954.

Schwinger's essay, written for that Festschrift volume (Quanta: Essays in Theoretical Physics Dedicated to G. Wentzel, 1970), treats the complementary, analytically simpler case: a charged scalar (spin-0, even-parity) complex mesotron field of mass $\mu$ coupled linearly, without spin, to a nucleon idealized as an infinitely massive source of density $K(\mathbf r)$ carrying isotopic spin (but no ordinary spin). Its Hamiltonian is

$H = \int \left\{ c^{2} \pi^{*}(\mathbf r) \pi(\mathbf r) + \nabla\phi^{*}(\mathbf r)\cdot\nabla\phi(\mathbf r) + \kappa^{2} \phi^{*}(\mathbf r) \phi(\mathbf r) \right\} d\mathbf r - (8\pi)^{1/2} g \int \phi(\mathbf r) K(\mathbf r) d\mathbf r\; \tfrac{1}{2}(\tau_{x} + i\tau_{y}) - (8\pi)^{1/2} g \int \phi^{*}(\mathbf r) K(\mathbf r) d\mathbf r\; \tfrac{1}{2}(\tau_{x} - i\tau_{y}),$

where $\kappa = \mu c/\hbar$, $\phi$ is the complex scalar field with conjugate momentum $\pi$, the source density is normalized to $\int K(\mathbf r) d\mathbf r = 1$, and $\tfrac{1}{2}(\tau_{x} \pm i\tau_{y})$ are the isotopic-spin raising and lowering operators acting on the nucleon, which absorb or emit the charged field quanta and so convert proton into neutron and vice versa. The essay performs a sequence of canonical transformations (changes of field variables that preserve the canonical commutation relations: passage to real field coordinates, elimination of the collective charge angle, and a final transformation separating free from bound-field variables) and expands the Hamiltonian in inverse powers of the coupling, within the stated domain of validity $g^{2}/\hbar c \gg 1$ for a point-like source ($\kappa a \ll 1$, where $a$ is the linear extension of the source) and $g^{2}/\hbar c \gg (\kappa a)^{3}$ for an extended source ($\kappa a \gg 1$). Its strong-coupling results include: the point-source isobar spectrum

$E_q = (g^2/\hbar c)^{-1}\mu c^2 \left(q - \tfrac{1}{2}\right)^2,$

elastic and charge-exchange scattering cross sections $\sigma_{M^+ \to M^+} = \sigma_{M^+ \to M^-} = \pi(\hbar/\mu c)^2(\mu c^2/E)^2$, equal to each other, so that the total cross section $2\pi(\hbar/\mu c)^2(\mu c^2/E)^2$ is half the naive value $4\pi(\hbar/\mu c)^2(\mu c^2/E)^2$ because only one of the two real components of the complex field is scattered; and a bound charge density proportional to the square of the Yukawa potential of the source (the static potential $e^{-\kappa r}/r$ of a massive-field source), decaying as $e^{-2\kappa r}/r^2$. A parallel classical treatment reproduces these results and yields an interpolation formula for the isobar energy valid at all couplings,

$E_q = \mu c^2\left\{\left[\left(q-\tfrac{1}{2}\right)^2 + \tfrac{1}{4}\left(g^2/\hbar c\right)^2\right]^{1/2} - \tfrac{1}{2}\left(g^2/\hbar c\right)\right\},$

but with charge quantization imposed by hand rather than derived. The essay itself flags what it does not do: the source is infinitely heavy, so recoil is absent and the finite-distance coupling to the extended source has no relativistic or gauge-invariant structure; spin is not treated (the physically relevant pseudoscalar case); only the strong-coupling limit is covered, while the physical coupling is intermediate; and the succession of canonical transformations is not carried out in detail - the transformed charge-density operator, for example, is asserted rather than derived. The follow-up treatment of the pseudoscalar theory, to which the essay defers the spin problem, never appeared in Schwinger's bibliography.

Problem Statement

Complete the quantum theory of the charged scalar mesotron field interacting with a nucleon source, removing the idealizations retained in Schwinger's 1970 treatment. For the model defined by the Hamiltonian of a complex (charged) scalar mesotron field of mass $\mu$ linearly coupled with strength $g$ to a static nucleon source $K(\mathbf r)$ carrying isotopic spin (the Hamiltonian written out in the Background above, with its strong-coupling domain of validity) - and, in the generalizations, spin and finite source mass - construct:

(i) a controlled non-perturbative quantum treatment valid at the physically relevant intermediate coupling ($g^2/\hbar c$ of order $1$ to $10$, between the perturbative and strong-coupling regimes) - controlled in the sense that the approximation scheme, the parameter governing its accuracy, and the sense in which its error is controlled are all stated - which interpolates between perturbation theory in the weak-coupling regime and the strong-coupling results, going beyond the heuristic classical interpolation in which charge quantization is imposed by hand;

(ii) the completion of the canonical-transformation diagonalization whose individual steps the 1970 treatment deferred - deriving rather than asserting the transformed operators (for example the charge-density operator) - together with a validation of the inequalities on $g$, $\mu$, and the source size $a$ under which the truncated expansion is internally consistent (namely $g^2/\hbar c \gg 1$ for $\kappa a \ll 1$ and $g^2/\hbar c \gg (\kappa a)^3$ for $\kappa a \gg 1$, with $\kappa = \mu c/\hbar$);

(iii) the extension beyond the infinitely heavy static source to include nucleon recoil within a coupling that is relativistically and gauge-invariantly structured, stating precisely which corrections survive, given that the strong-coupling expansion itself presupposes a static source;

(iv) the inclusion of spin - the pseudoscalar derivative coupling to a spin-$\tfrac{1}{2}$ source, which the essay deferred to a sequel that was never published - carried out as the spin generalization of the charged mesotron model defined above, within the same completion program as (i)-(iii) and beyond the static strong-coupling treatments of the symmetric pseudoscalar theory described in the Background above.

The completed theory must yield the isobar spectrum, the elastic and charge-exchange mesotron-nucleon scattering amplitudes, and the bound charge distribution, each as a function of the coupling strength, with Schwinger's strong-coupling formulas (the point-source spectrum $E_q = (g^2/\hbar c)^{-1}\mu c^2(q-\tfrac{1}{2})^2$, the total cross section $2\pi(\hbar/\mu c)^2(\mu c^2/E)^2$ split equally between the elastic and charge-exchange channels, and the charge density $\propto e^{-2\kappa r}/r^2$) recovered as limiting cases and perturbation theory recovered in the weak-coupling limit. These components are joined by one scientific objective - a non-perturbative theory of the mesotron-nucleon system at the physical coupling strength without the idealizations of the 1970 treatment - and all of (i)-(iv) are jointly required, so that no proper subset by itself resolves the task.

References

1. J. Schwinger, 'On the Charged Scalar Mesotron Field', book chapter in A Quantum Legacy: Seminal Papers of Julian Schwinger (ed. K. A. Milton, World Scientific, 2000), pp. 246-283; originally published in Quanta: Essays in Theoretical Physics Dedicated to G. Wentzel (1970), pp. 101-138. DOI: 10.1142/9789812795694_0013, https://doi.org/10.1142/9789812795694_0013 (INSPIRE records https://inspirehep.net/literature/63624 and https://inspirehep.net/literature/63468).
2. K. A. Milton (ed.), A Quantum Legacy: Seminal Papers of Julian Schwinger, World Scientific Series in 20th Century Physics, 2000. DOI: 10.1142/4168, https://doi.org/10.1142/4168
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