Schwinger's source-theoretic double spectral account of deep inelastic scattering: anomalous negative-$M_-^2$ spectral support, the $h_2$ cutoff ansatz, the $M_0$ mass-scale hypothesis, and the modified scaling variable $\omega_S$

In a 1975 PNAS paper, Julian Schwinger reanalyzed deep inelastic electron-nucleon scattering from the phenomenological standpoint of source theory, representing the forward virtual-photon-nucleon scattering amplitude by a double spectral representation in two spectral masses and constraining it with three empirical inputs: elastic form factors, the low-mass resonance spectrum, and high-energy diffractive photoproduction. The account reproduces Bjorken scaling and parametrizes its violations, and the paper itself acknowledges four unresolved issues: (i) matching the diffractive photoproduction behavior appears to require dominant spectral support at large negative crossed-channel mass $M_-^2$, whose analytic status and physical interpretation (including the danger of negative spectral weights $h_{1,2}$) was left open; (ii) the slow exponential cutoff ansatz for $h_2$ and the independence hypothesis from which scaling is derived were not obtained from dynamics; (iii) the analysis assumed no new large mass scale $M_0\gg m$; (iv) the modified scaling variable $\omega_S$ and its empirical $3/4$-shifted variant awaited systematic validation and microscopic derivation. The source-theoretic program was carried forward only briefly — through 1979, within Schwinger's own circle, with one partial result on anomalous spectral regions (Ivanetich et al. 1979) — and was then abandoned as quantum chromodynamics (QCD) became the accepted dynamical description. The four questions remain unresolved, and survive as a coherent open problem in the analyticity and spectral-representation tradition.

Background

Deep inelastic scattering (DIS) is the reaction $e+N\to e+X$ in which an electron exchanges a spacelike virtual photon of squared four-momentum transfer $q^2>0$ (Schwinger's sign convention) with a nucleon $N$ of mass $m$, and only the scattered electron is detected while the hadronic final state $X$ is summed over. Viewed as absorption of the virtual photon, DIS is governed by two invariant structure functions, $H_1(q^2,qp)$ and $H_2(q^2,qp)$, the imaginary parts of the forward Compton scattering amplitude of the photon on the nucleon, with $qp$ the photon energy in the nucleon rest frame (conventionally $m\nu$). The MIT-SLAC experiments of 1968-69 established that at large $q^2$ and large energy transfer the cross sections depend, to good approximation, only on the dimensionless Bjorken variable $\omega=2m\nu/q^2$ ("Bjorken scaling"), with a small ratio of longitudinal to total virtual-photon absorption cross section, $\sigma_L/\sigma_A$ with $\sigma_A=\sigma_L+\sigma_T$ and $\sigma_T$ the transverse piece.

Source theory was Schwinger's deliberately nonspeculative, phenomenological framework, which describes particle reactions through empirically constrained causal propagation of sources and refuses hypothetical constituents such as quarks or partons. In "Source Theory Viewpoints in Deep Inelastic Scattering" (1975), Schwinger wrote the forward amplitude as a double spectral representation, a dispersion-type representation in two mass variables $M_+^2$ and $M_-^2$ (the squared effective masses of the two causal propagation channels, in the lineage of the Mandelstam and Deser-Gilbert-Sudarshan representations), reducing the structure functions to Laplace-type integrals over a conjugate variable $\zeta$ of weight functions $h_{1,2}(\zeta, m^2/M_+^2)$. Three empirical inputs pin the weights: the elastic nucleon contribution, fixed by the dipole form factor $(1+q^2/m_0^2)^{-2}$ with $m_0=0.9\,m$; the discrete resonance spectrum, imposed through superconvergence relations (high-$q^2$ vanishing of spectral moments); and the high-energy real-photon ($q^2=0$) diffractive behavior, in which the photoproduction cross section tends to a nearly constant limit of similar size for proton and neutron.

From this representation the observed DIS characteristics emerge as an interpolation: scaling arises because $h_2$ is effectively independent of $m^2/M_+^2\ll 1$ for the finite $\zeta$ dominating the Laplace kernel; deviations from exact scaling are parametrized by a slow exponential decrease of $h_2$ in $\zeta$, of rate $m^2/(2M_+^2)$ (Eq. [36] of the paper), and by a modified ("improved") scaling variable $\omega_S$, printed in Eq. [52] of the paper as $\omega_S=[2m\nu+(3/2)M^2]/[q^2+(1/2)M^2]$, with $M^2$ the diffractive mass scale fixed by the high-energy diffractive condition $M_+^2+M_-^2-2m^2\simeq M^2$, a function of $2m\nu$ and $q^2$ that coincides with $\omega$ up to nucleon-mass terms and extends acceptable data fits down to $\omega_S>1.5$; an empirical variant replacing $\omega_S-1$ by $\omega_S-3/4$ fits data over $1.14<\omega<20$ and predicts the vanishing of $\sigma_L/\sigma_A$ as the scaling variable approaches unity. A footnote records the working hypothesis that the experiments do not reach excitation levels characterized by a new large mass $M_0\gg m$; such a scale would introduce the additional variable $M_+^2/M_0^2\sim(\omega-1)(q^2/M_0^2)$ and reveal itself as a deviation from scaling at large $q^2$.

The paper closes by flagging what remains unresolved: the dominant support of the double spectral weight appears to require large negative values of $M_-^2$ (since $M_+^2+M_-^2-2m^2\simeq M^2$, with $M^2$ the diffractive mass scale, under high-energy diffractive conditions), and the danger of serious conceptual problems posed by negative $h_{1,2}$ is explicitly left for further investigation. These questions stood at the moment the community was converging on quantum chromodynamics (QCD), whose asymptotic freedom and the Altarelli-Parisi evolution equations (1977) became the accepted dynamical account of scaling violations. The source-theoretic program was carried forward only briefly (through 1979, by Schwinger and his UCLA collaborators) and then abandoned; its specific questions remain unresolved.

Problem Statement

Determine whether Schwinger's source-theoretic double spectral account of deep inelastic scattering can be completed — or must be rejected — by answering the four interlocking questions arising from the issues his 1975 paper left open or acknowledged as unsettled, within the representation's own empirical constraints (elastic dipole form factor, low-mass resonance spectrum with superconvergence relations, and high-energy diffractive photoproduction):

(a) Anomalous spectral support. Matching the diffractive high-energy behavior forces the dominant support of the double spectral weight functions $h_{1,2}(M_+^2,M_-^2)$ onto large negative values of the crossed-channel spectral mass ($M_+^2+M_-^2-2m^2\simeq M^2$, the diffractive scale). Establish the analytic status and physical interpretation of this negative-$M_-^2$ region: can the representation be realized with non-negative, physically interpretable weights under all three constraints simultaneously, or does consistency force negative $h_{1,2}$ somewhere, contradicting the representation's physical foundation? The 1979 partial result (Ivanetich, Tsai, DeRaad, Milton, Urrutia) established kinematic consistency of anomalous spectral regions with single spectral representations in model cases; the full question under the DIS constraints stands.

(b) The $h_2$ cutoff ansatz. Derive from an underlying dynamics, or bound by data, the slow exponential decrease of $h_2(\zeta,m^2/M_+^2)$ (rate $m^2/(2M_+^2)$, Eq. [36]) together with the hypothesis that $h_2$ is independent of $m^2/M_+^2\ll 1$ for the finite $\zeta$ dominating the Laplace kernel — the assumption from which Bjorken scaling is derived in this framework.

(c) The new mass scale. Decide, within or against this framework, whether a new large mass scale $M_0\gg m$ is probed by these experiments, which would introduce the variable $M_+^2/M_0^2\sim(\omega-1)(q^2/M_0^2)$ and produce scaling deviations at large $q^2$ qualitatively beyond the parametrized ones. (QCD's logarithmic scaling violations and the observed quark thresholds address the phenomenon but not this formulation.)

(d) The scaling variable $\omega_S$. Systematically validate or refute the modified scaling variable $\omega_S$ and its empirical $3/4$-shifted variant as descriptions of deep inelastic data — including the predicted vanishing of $\sigma_L/\sigma_A$ as the scaling variable approaches unity — against modern high-precision structure-function data, in quantitative comparison with the accepted alternatives, or derive it (or its failure) from an underlying dynamics.

All four sub-questions (a)-(d) are jointly required: addressing some but not all of them does not complete the task. Answers to any component must respect the empirical constraints shared by the others, and a refutation of any one component is a result about the account as a whole; the objective is a single coherent verdict on whether the source-theoretic spectral account of deep inelastic scattering stands, and if so in what completed form.

References

1. Julian Schwinger, "Source Theory Viewpoints in Deep Inelastic Scattering", Proc. Natl. Acad. Sci. USA 72(1), 1-5 (1975). DOI: 10.1073/pnas.72.1.1, https://doi.org/10.1073/pnas.72.1.1 (open-access full text: https://europepmc.org/articles/pmc432227).
2. Julian Schwinger, "Source Theory Discussion of Deep Inelastic Scattering with Polarized Particles", Proc. Natl. Acad. Sci. USA 72(4), 1559-1563 (1975). DOI: 10.1073/pnas.72.4.1559, https://doi.org/10.1073/pnas.72.4.1559.
3. Julian Schwinger, "Deep inelastic scattering of leptons", Proc. Natl. Acad. Sci. USA 73(10), 3351-3354 (1976). DOI: 10.1073/pnas.73.10.3351, https://doi.org/10.1073/pnas.73.10.3351.
4. Lester L. DeRaad, Jr., Kimball A. Milton, Wu-yang Tsai, "Deep inelastic neutrino scattering: A double spectral form viewpoint", Phys. Rev. D 12, 3747-3757 (1975); Erratum: Phys. Rev. D 13, 3166 (1976). DOI: 10.1103/PhysRevD.12.3747, https://doi.org/10.1103/PhysRevD.12.3747.
5. R. J. Ivanetich, Wu-yang Tsai, Lester L. DeRaad, Jr., Kimball A. Milton, L. F. Urrutia, "Anomalous spectral regions in source theory", Physica A 96(1-2), 233-242 (1979), in: Themes in Contemporary Physics (Julian Schwinger Festschrift), North-Holland. DOI: 10.1016/0378-4371(79)90210-3, https://doi.org/10.1016/0378-4371(79)90210-3.
6. Kimball A. Milton, "My years with Julian Schwinger: From source theory through sonoluminescence", in: Julian Schwinger Centennial Conference (Singapore, 2018), World Scientific, pp. 221-235 (published 2019). DOI: 10.1142/9789811213144_0015, https://doi.org/10.1142/9789811213144_0015 (author preprint: arXiv:1808.01981, https://arxiv.org/abs/1808.01981).
7. Kimball A. Milton, "In Appreciation Julian Schwinger: From Nuclear Physics and Quantum Electrodynamics to Source Theory and Beyond", Physics in Perspective 9, 70-114 (2007). DOI: 10.1007/s00016-007-0326-6, https://doi.org/10.1007/s00016-007-0326-6.
8. Giulio Altarelli, Guido Parisi, "Asymptotic Freedom in Parton Language", Nuclear Physics B 126, 298-318 (1977). DOI: 10.1016/0550-3213(77)90384-4, https://doi.org/10.1016/0550-3213(77)90384-4.
9. Julian Schwinger, "Deep Inelastic Sum Rules in Source Theory", Nuclear Physics B 123, 223-239 (1977). DOI: 10.1016/0550-3213(77)90460-6, https://doi.org/10.1016/0550-3213(77)90460-6.
10. A. Bodek, M. Breidenbach, D. L. Dubin, J. E. Elias, J. I. Friedman, H. W. Kendall, J. S. Poucher, E. M. Riordan, M. R. Sogard, D. H. Coward, D. J. Sherden, "Experimental studies of the neutron and proton electromagnetic structure functions", Phys. Rev. D 20, 1471-1552 (1979). DOI: 10.1103/PhysRevD.20.1471, https://doi.org/10.1103/PhysRevD.20.1471.
11. Julian Schwinger, "Deep inelastic scattering of charged leptons", Proc. Natl. Acad. Sci. USA 73(11), 3816-3819 (1976). DOI: 10.1073/pnas.73.11.3816, https://doi.org/10.1073/pnas.73.11.3816.
12. Julian Schwinger, "Deep inelastic neutrino scattering and pion-nucleon cross sections", Phys. Lett. B 67(1), 89-90 (1977). DOI: 10.1016/0370-2693(77)90814-0, https://doi.org/10.1016/0370-2693(77)90814-0.
13. Wu-yang Tsai, Lester L. DeRaad, Jr., Kimball A. Milton, "Verification of virtual Compton-scattering sum rules in quantum electrodynamics", Phys. Rev. D 11, 3537-3540 (1975). DOI: 10.1103/PhysRevD.11.3537, https://doi.org/10.1103/PhysRevD.11.3537.
14. T. H. Bauer, R. D. Spital, D. R. Yennie, F. M. Pipkin, "The hadronic properties of the photon in high-energy interactions", Rev. Mod. Phys. 50(2), 261-436 (1978). DOI: 10.1103/RevModPhys.50.261, https://doi.org/10.1103/RevModPhys.50.261.